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Author: Charles A. Young
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Title: A Textbook of General Astronomy

For Colleges and Scientific Schools

Author: Charles A. Young

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FRONTISPIECE.

The Great Telescope of the Lick Observatory,

Mt. Hamilton, Cal.

Object-Glass made by A. Clark & Sons:

Aperture, 36 in.; Focal Length, 56 ft. 2 in.

Mounting by Warner & Swasey.

A T E X T - B O O K

OF

GENERAL ASTRONOMY

FOR

COLLEGES AND SCIENTIFIC SCHOOLS

BY

CHARLES A. YOUNG, Ph.D., LL.D.,

Professor of Astronomy in the College of New Jersey

(Princeton).

Boston, U.S.A., and London:

GINN & COMPANY, PUBLISHERS.

1889.

Entered at Stationers’ Hall.

Copyright, 1888, by

CHARLES A. YOUNG,

All Rights Reserved.

Typography by J. S. Cushing & Co., Boston, U.S.A.

Presswork by Ginn & Co., Boston, U.S.A.

PREFACE.

The present work is designed as a text-book of Astronomy
suited to the general course in our colleges and schools of science,
and is meant to supply that amount of information upon the subject
which may fairly be expected of every “liberally educated” person.

While it assumes the previous discipline and mental maturity usu-
ally corresponding to the latter years of the college course, it does
not demand the peculiar mathematical training and aptitude nec-
essary as the basis of a special course in the science—only the most
elementary knowledge of Algebra, Geometry, and Trigonometry is
required for its reading. Its aim is to give a clear, accurate, and
justly proportioned presentation of astronomical facts, principles,
and methods in such a form that they can be easily apprehended
by the average college student with a reasonable amount of effort.

The limitations of time are such in our college course that proba-
bly it will not be possible in most cases for a class to take thoroughly
everything in the book. The fine print is to be regarded rather as
collateral reading, important to anything like a complete view of
the subject, but not essential to the course. Some of the chapters
can even be omitted in cases where it is found necessary to abridge
the course as much as possible; e.g., the chapters on Instruments
and on Perturbations.

While the work is no mere compilation, it makes no claims to
special originality: information and help have been drawn from all
available sources.

The author is under great obligations to the
astronomical histories of Grant and Wolf, and especially to Miss

Clerke’s admirable “History of Astronomy in the Nineteenth Cen-
tury.” Many data also have been drawn from Houzeau’s valuable

“Vade Mecum de l’Astronomie.”

It has been intended to bring the book well down to date, and
to indicate to the student the sources of information on subjects
which are necessarily here treated inadequately on account of the

PREFACE.

v
limitations of time and space.

Special acknowledgments are due to Professor Langley and to
his publishers, Messrs. Ticknor & Co., for the use of a number of
illustrations from his beautiful book, “The New Astronomy”; and
also to D. Appleton & Co. for the use of several cuts from the au-
thor’s little book on the Sun. Professor Trowbridge of Cambridge
kindly provided the original negative from which was made the cut
illustrating the comparison of the spectrum of iron with that of the
sun. Warner & Swasey of Cleveland and Fauth & Co. of Washing-
ton have also furnished the engravings of a number of astronomical
instruments.

Professors Todd, Emerson, Upton, and McNeill have given most
valuable assistance and suggestions in the revision of the proof; as
indeed, in hardly a less degree, have several others.

The author will consider it a great favor if those who may use
the book will kindly communicate to him, either directly or through
the publishers, any errata, in order that they may be promptly
corrected.

Princeton, N. J., August, 1888.

Note. In this issue of the book a number of errors which appeared in the
first impression have been corrected.

April, 1889.

TABLE OF CONTENTS.

PAGES

INTRODUCTION .

.

.

.

.

.

.

.

.

.

.

.

1–4

CHAPTER I.—The Doctrine of the Sphere; Definitions
and General Considerations .

.

.

.

.

.

.

.

.

5–20

CHAPTER II.—Astronomical Instruments: the Telescope;

Time-Keepers and Chronograph; the Transit Instrument
and Accessories; the Meridian Circle and Reading Mi-
croscope;
the Altitude and Azimuth Instrument;
the

Equatorial Instrument and Micrometer; the Sextant .

.

21–58

CHAPTER III.—Corrections to Astronomical Observa-
tions: the Dip of the Horizon; Parallax; Semi-Diameter;

Refraction; and Twilight .

.

.

.

.

.

.

.

.

59–71

CHAPTER IV.—Problems of Practical Astronomy: the

Determination of Latitude, of Time, of Longitude, of a

Ship’s Place at Sea, of Azimuth, and of the Apparent

Right Ascension and Declination of a Heavenly Body; the

Time of Sunrise or Sunset .

.

.

.

.

.

.

.

.

72–93

CHAPTER V.—The Earth: the Approximate Determination
of its Dimensions and Form;

Proofs of its Rotation;

Accurate Determination of its Dimensions by Geodetic

Surveys and Pendulum Observations; Determination of its

Mass and Density .

.

.

.

.

.

.

.

.

.

. 94–120

CHAPTER VI.—The Earth’s Orbital Motion: the Motion
of the Sun among the Stars; the Equation of Time;

Precession; Nutation; Aberration; the Calendar .

.

. 121–148

CHAPTER VII.—The Moon: her Orbital Motion; Distance
and Dimensions; Mass, Density, and Superficial Gravity;

Rotation and Librations; Phases; Light and Heat; Physical

Condition;

Influence
exerted
on
the

Earth;

Surface

Structure; and Possible Changes .

.

.

.

.

.

. 149–176

TABLE OF CONTENTS.

vii

PAGES

CHAPTER VIII.—The Sun: Distance and Dimensions; Mass
and Density; Rotation; Solar Eye-Pieces, and Study of
the Sun’s Surface;

General Views as to Constitution;

Sun Spots: their Appearance, Nature, Distribution, and

Periodicity; the Spectroscope and the Solar Spectrum;

Chemical Composition of the Sun; the Chromosphere and

Prominences; the Corona .

.

.

.

.

.

.

.

. 177–216

CHAPTER IX.—The Sun’s Light and Heat: Comparison
of Sunlight with Artificial Lights; the Measurement of the

Sun’s Heat, and Determination of the Solar Constant;
the Pyrheliometer, Actinometer, and Bolometer; the Sun’s

Temperature; Maintenance of the Sun’s Radiation; and

Conclusions as to its Age and Future Endurance .

. 217–232

CHAPTER X.—Eclipses; Form and Dimensions of Shadows;

Lunar Eclipses; Solar Eclipses, Total, Annular, and Partial;

Ecliptic Limits, and Number of Eclipses in a Year; the

Saros; Occultations .

.

.

.

.

.

.

.

.

.

. 233–251

CHAPTER XI.—Central Forces; Equable Description of

Areas; Areal, Linear, and Angular Velocities; Kepler’s

Laws and Inferences from them; Gravitation demonstrated
by the Moon’s Motion; Conic Sections as Orbits; the

Problem of Two Bodies; the “Velocity from Infinity” and
its Relation to the Species of Orbit described by a Body
moving under Gravitation; Intensity of Gravitation .

. 252–273

CHAPTER XII.—The Problem of Three Bodies; Disturb-
ing Forces; Lunar Perturbations and the Tides .

.

. 274–299

CHAPTER XIII.—The Planets: their Motions, Appar-
ent and Real; the Ptolemaic, Tychonic, and Copernican

Systems;
the

Orbits
and
their

Elements;

Planetary

Perturbations .

.

.

.

.

.

.

.

.

.

.

. 300–322

CHAPTER XIV.—The Planets themselves; Methods of
determining their Diameters, Masses, Densities, Times of

Rotation, etc.; the “Terrestrial Planets,”—Mercury, Venus,
and Mars; the Asteroids; Intra-Mercurial Planets and the

Zodiacal Light .

.

.

.

.

.

.

.

.

.

.

. 323–355

CHAPTER XV.—The Major Planets,—Jupiter, Saturn,

Uranus, and Neptune .

.

.

.

.

.

.

.

.

. 356–383

TABLE OF CONTENTS.

viii

PAGES

CHAPTER XVI.—The Determination of the Sun’s Hor-
izontal Parallax and Distance; Oppositions of Mars
and Transits of Venus; Gravitational Methods; Determi-
nation by Means of the Velocity of Light .

.

.

. 384–402

CHAPTER XVII.—Comets: their Number, Motions, and

Orbits; their Constituent Parts and Appearance; their

Spectra, Physical Constitution, and Probable Origin .

. 403–438

CHAPTER XVIII.—Meteors:

Aerolites,
their Fall and

Physical Characteristics;

Shooting Stars and Meteoric

Showers; Connection between Meteors and Comets .

. 439–455

CHAPTER XIX.—The Stars: their Nature and Number;
the Constellations; Star-Catalogues; Designation of Stars;
their Proper Motions, and the Motion of the Sun in

Space; Stellar Parallax .

.

.

.

.

.

.

.

.

. 456–476

CHAPTER XX.—The Light of the Stars; Star Magni-
tudes and Photometry; Variable Stars; Stellar Spectra;

Scintillation of the Stars .

.

.

.

.

.

.

.

. 477–503

CHAPTER XXI.—Aggregations of Stars:

Double and

Multiple Stars; Clusters; Nebulæ; the Milky Way, and

Distribution of Stars in Space; Constitution of the Stellar

Universe; Cosmogony and the Nebular Hypothesis .

. 504–539

APPENDIX.—Tables of Astronomical Data .

.

.

. 540–547

INDEX .

.

.

.

.

.

.

.

.

.

.

.

.

.

. 548–567

INTRODUCTION.

1. Astronomy (ἄστρον ν´
ομος) is the science which treats of
the heavenly bodies. As such bodies we reckon the sun and moon,
the planets (of which the earth is one) and their satellites, comets
and meteors, and finally the stars and nebulæ.

We have to consider in Astronomy:—

(a) The motions of these bodies, both real and apparent, and
the laws which govern these motions.

(b) Their forms, dimensions, and masses.

(c) Their nature and constitution.

(d ) The effects they produce upon each other by their attrac-
tions, radiations, or by any other ascertainable influence.

It was an early, and has been a most persistent, belief that the
heavenly bodies have a powerful influence upon human affairs, so
that from a knowledge of their positions and “aspects” at critical
moments (as for instance at the time of a person’s birth) one could
draw up a “horoscope” which would indicate the probable future.

The pseudo-science which was founded on this belief was named

Astrology,—the elder sister of Alchemy,—and for centuries Astron-
omy was its handmaid; i.e., astronomical observations and calcula-
tions were made mainly in order to supply astrological data.

At present the end and object of astronomical study is chiefly
knowledge pure and simple; so far as now appears, its development
has less direct bearing upon the material interests of mankind than
that of any other of the natural sciences. It is not likely that great
inventions and new arts will grow out of its laws and principles,
such as are continually arising from physical, chemical, and biolog-
ical discoveries, though of course it would be rash to say that such
outgrowths are impossible. But the student of Astronomy must ex-
pect his chief profit to be intellectual, in the widening of the range
of thought and conception, in the pleasure attending the discovery

INTRODUCTION.

2
of simple law working out the most complicated results, in the de-
light over the beauty and order revealed by the telescope in systems
otherwise invisible, in the recognition of the essential unity of the
material universe, and of the kinship between his own mind and the
infinite Reason that formed all things and is immanent in them.

At the same time it should be said at once that, even from the
lowest point of view, Astronomy is far from a useless science. The
art of navigation depends for its very possibility upon astronomical
prediction. Take away from mankind their almanacs, sextants, and
chronometers, and commerce by sea would practically stop. The
science also has important applications in the survey of extended
regions of country, and the establishment of boundaries, to say noth-
ing of the accurate determination of time and the arrangement of
the calendar.

It need hardly be said that Astronomy is not separated from
kindred sciences by sharp boundaries. It would be impossible, for
instance, to draw a line between Astronomy on one side and Geology
and Physical Geography on the other. Many problems relating to
the formation and constitution of the earth belong alike to all three.

2. Astronomy is divided into many branches, some of which, as
ordinarily recognized, are the following:—

1. Descriptive Astronomy.—This, as its name implies, is
merely an orderly statement of astronomical facts and principles.

2. Practical Astronomy.—This is quite as much an art as a
science, and treats of the instruments, the methods of observation,
and the processes of calculation by which astronomical facts are
ascertained.

3. Theoretical Astronomy, which treats of the calculations
of orbits and ephemerides, including the effects of so-called “per-
turbations.”

4. Mechanical Astronomy, which is simply the application
of mechanical principles to explain astronomical facts (chiefly the
planetary and lunar motions). It is sometimes called Gravitational

Astronomy, because, with few exceptions, gravitation is the only
force sensibly concerned in the motions of the heavenly bodies. Un-
til within thirty years this branch of the science was generally des-
ignated as Physical Astronomy, but the term is now objectionable
because of late it has been used by many writers to denote a very

INTRODUCTION.

3
different and comparatively new branch of the science; viz.,—

5. Astronomical Physics, or Astro-physics.—This treats
of the physical characteristics of the heavenly bodies, their bright-
ness and spectroscopic peculiarities, their temperature and radia-
tion, the nature and condition of their atmospheres and surfaces,
and all phenomena which indicate or depend on their physical con-
dition.

6. Spherical Astronomy.—This, discarding all consideration
of absolute dimensions and distances, treats the heavenly bodies
simply as objects moving on the “surface of the celestial sphere”: it
has to do only with angles and directions, and, strictly regarded, is
in fact merely Spherical Trigonometry applied to Astronomy.

3. The above-named branches are not distinct and separate,
but they overlap in all directions. Spherical Astronomy, for instance,
finds the demonstration of many of its formulæ in Gravitational As-
tronomy, and their application appears in Theoretical and Practical

Astronomy. But valuable works exist bearing all the different titles
indicated above, and it is important for the student to know what
subjects he may expect to find discussed in each; for this reason it
has seemed worth while to name and define the several branches,
although they do not distribute the science between them in any
strictly logical and mutually exclusive manner.

In the present text-book little regard will be paid to these sub-
divisions, since the object of the work is not to present a complete
and profound discussion of the subject such as would be demanded
by a professional astronomer, but only to give so much knowledge
of the facts and such an understanding of the principles of the sci-
ence as may fairly be considered essential to a liberal education. If
this result is gained in the reader’s case, it may easily happen that
he will wish for more than he can find in these pages, and then he
must have recourse to works of a higher order and far more difficult,
dealing with the subject more in detail and more thoroughly.

To master the present book no further preparation is neces-
sary than a very elementary knowledge of Algebra, Geometry,
and Trigonometry, and a similar acquaintance with Mechanics and

Physics, especially Optics. While nothing short of high mathemati-
cal attainments will enable one to become eminent in the science, yet
a perfect comprehension of all its fundamental methods and princi-

INTRODUCTION.

4
ples, and a very satisfactory acquaintance with its main results, is
quite within the reach of every person of ordinary intelligence, with-
out any more extensive training than may be had in our common
schools. At the same time the necessary statements and demonstra-
tions are so much facilitated by the use of trigonometrical terms and
processes that it would be unwise to dispense with them entirely in
a work to be used by pupils who have already become acquainted
with them.

In discussing the different subjects which present themselves,
the writer will adopt whatever plan appears best fitted to convey to
the student clear and definite ideas, and to impress them upon the
mind. Usually it will be best to proceed in the Euclidean order, by
first stating the fact or principle in question, and then explaining its
demonstration. But in some cases the inverse process is preferable,
and the conclusion to be reached will appear gradually unfolding
itself as the result of the observations upon which it depends, just
as its discovery came about.

The frequent references to “Physics” refer to the “Elementary

Text-Book of Physics,” by Anthony & Brackett; 3d edition, 1887.

Wiley & Sons, N.Y.

THE DOCTRINE OF THE SPHERE.

5

CHAPTER I.

THE “DOCTRINE OF THE SPHERE,” DEFINITIONS, AND

GENERAL CONSIDERATIONS.

Astronomy, like all the other sciences, has a terminology of
its own, and uses technical terms in the description of its facts and
phenomena. In a popular essay it would of course be proper to avoid
such terms as far as possible, even at the expense of circumlocutions
and occasional ambiguity; but in a text-book it is desirable that the
reader should be introduced to the most important of them at the
very outset, and made sufficiently familiar with them to use them
intelligently and accurately.

4. The Celestial Sphere.—To an observer looking up to the
heavens at night it seems as if the stars were glittering points at-
tached to the inner surface of a dome; since we have no direct per-
ception of their distance there is no reason to imagine some nearer
than others, and so we involuntarily think of the surface as spher-
ical with ourselves in its centre. Or if we sometimes feel that the
stars and other objects in the sky really differ in distance, we still
instinctively imagine an immense sphere surrounding and enclosing
all. Upon this sphere we imagine lines and circles traced, resem-
bling more or less the meridians and parallels upon the surface of
the earth, and by reference to these circles we are able to describe
intelligently the apparent positions and motions of the heavenly
bodies.

This celestial sphere may be regarded in either of two different
ways, both of which are correct and lead to identical results.

(a) We may imagine it, in the first place, as transparent, and of
merely finite (though undetermined) dimensions, but in some way
so attached to, and connected with, the observer that his eye always
remains at its centre wherever he goes. Each observer, in this way
of viewing it, carries his own sky with him, and is the centre of his
own heavens.

(b) Or, in the second place,—and this is generally the more

THE DOCTRINE OF THE SPHERE.

6
convenient way of regarding the matter,—we may consider the ce-
lestial sphere as mathematically infinite in its dimensions; then, let
the observer go where he will, he cannot sensibly get away from
its centre. Its radius being “greater than any assignable quantity,”

the size of continents, the diameter of the earth, the distance of the
sun, the orbits of planets and comets, even the spaces between the
stars, are all insignificant, and the whole visible universe shrinks
relatively to a mere point at its centre. In what follows we shall use
this conception of the celestial sphere.1

The apparent place of any celestial
body will then be the point on the ce-
lestial sphere where the line drawn from
the eye of the observer in the direction in
which he sees the object, and produced
indefinitely, pierces the sphere. Thus, in

Figure 1, A, B, C are the apparent places
of a, b, and c, the observer being at O.

The apparent place of a heavenly body ev-
idently depends solely upon its direction,

Fig. 1.

and is wholly independent of its distance
from the observer.

5. Linear and Angular Dimensions.—Linear dimensions
are such as may be expressed in linear units; i.e., in miles, feet,
or inches; in metres or millimetres. Angular dimensions are ex-
pressed in angular units; i.e., in right angles, in radians,2 or (more

1To most persons the sky appears, not a true hemisphere, but a flattened
vault, as if the horizon were more remote than the zenith. This is a subjective
effect due mainly to the intervening objects between us and the horizon. The
sun and moon when rising or setting look much larger than when they are
higher up, for the same reason.

2A radian is the angle which is measured by an arc equal in length to radius.

Since a circle whose radius is unity has a circumference of 2π, and contains

360

360◦, or 21, 600 , or 1, 296, 000 , it follows that a radian contains

◦ or

2π

21600

1296000

, or

; i.e. (approximately), a radian = 57.3◦ = 3437.7 =

2π

2π

206264.8 . Hence, to reduce to seconds of arc an angle expressed in radians,
we must multiply it by the number 206264.8; a relation of which we shall have
to make frequent use.

See Halsted’s Mensuration, p. 25.

THE DOCTRINE OF THE SPHERE.

7
commonly in astronomy) in degrees, minutes, and seconds. Thus,
for instance, the linear semi-diameter of the sun is about 697,000
kilometers (433,000 miles), while its angular semi-diameter is about

16 , or a little more than a quarter of a degree. Obviously, angular
units alone can properly be used in describing apparent distances
and dimensions in the sky. For instance, one cannot say correctly
that the two stars which are known as “the pointers” are two or five
or ten feet apart: their distance is about five degrees.

It is sometimes convenient to speak of “angular area,” the unit
of which is a “square degree” or a “square minute”; i.e., a small
square in the sky of which each side is 1◦ or 1 . Thus we may
compare the angular area of the constellation Orion with that of

Taurus, in square degrees, just as we might compare Pennsylvania
and New Jersey in square miles.

6. Relation between the Distance and Apparent Size of

Object.—Suppose a globe having a radius BC equal to r. As seen
from the point A (Fig. 2) its apparent (i.e., angular ) semi-diameter
will be BAC or s, its distance being AC or R.

Fig. 2.

We have immediately from Trigonometry, since B is a right an-
gle,
r
sin s =

.

R

If, as is usual in Astronomy, the diameter of the object is small
as compared with its distance, we may write
r
s =

,

R
which gives s in radians (not in degrees or seconds). If we wish it
in the ordinary angular units,
r
r
s◦ = 57.3

,
or
s = 206264.8

.

R

R

In either form of the equation we see that the apparent diameter
varies directly as the linear diameter, and inversely as the distance.

THE DOCTRINE OF THE SPHERE.

8

In the case of the moon, R = about 239,000 miles; and r, 1081
miles. Hence s = 1081 = 1 of a radian, which is a little more

239000

221
than 1 of a degree.

4

It may be mentioned here as a rather curious fact that most persons
say that the moon appears about a foot in diameter ; at least, this seems
to be the average estimate. This implies that the surface of the sky
appears to them only about 110 feet away, since that is the distance at
which a disc one foot in diameter would have an angular diameter of 1

110
of a radian, or 1 ◦.

2

7. Vanishing Point.—Any system of parallel lines produced
in one direction will appear to pierce the celestial sphere at a sin-
gle point. They actually pierce it at different points, separated on
the surface of the sphere by linear distances, equal to the actual
distances between the lines, but on the infinitely distant surface
these linear distances, being only finite, become invisible, subtend-
ing at the centre angles less than anything assignable. The different
points, therefore, coalesce into a spot of apparently infinitesimal
size—the so-called “vanishing point” of perspective. Thus the axis
of the earth and all lines parallel to this axis point to the celestial
pole.

In order to describe intelligibly the apparent position of an ob-
ject in the sky, it is necessary to have certain points and lines from
which to reckon. We proceed to define some of these which are most
frequently used.

8. The Zenith.—The Zenith is the point vertically overhead,
i.e., the point where a plumb-line, produced upwards, would pierce
the sky: it is determined by the direction of gravity where the ob-
server stands.

If the earth were exactly spherical, the zenith might also be de-
fined as the point where a line drawn from the centre of the earth
upward through the observer meets the sky. But since, as we shall
see hereafter, the earth is not an exact globe, this second defini-
tion indicates a point known as the Geocentric Zenith, which is not
identical with the True or Astronomical Zenith, determined by the
direction of gravity.

9. The Nadir.—The Nadir is the point opposite the zenith—
under foot, of course.

THE DOCTRINE OF THE SPHERE.

9

Both zenith and nadir are derived from the Arabic, which lan-
guage has also given us many other astronomical terms.

10. Horizon.—The Hor¯ızon1 is a great circle of the celestial
sphere, having the zenith and nadir as its poles: it is therefore half-
way between them, and 90° from each.

A horizontal plane, or the plane of the horizon, is a plane per-
pendicular to the direction of gravity, and the horizon may also be
correctly defined as the intersection of the celestial sphere by this
plane.

Many writers make a distinction between the sensible and ratio-
nal horizons. The plane of the sensible horizon passes through the
observer; the plane of the rational horizon passes through the cen-
tre of the earth, parallel to the plane of the sensible horizon: these
two planes, parallel to each other, and everywhere about 4000 miles
apart, trace out on the sky the two horizons, the sensible and the
rational. It is evident, however, that on the infinitely distant sur-
face of the celestial sphere, the two traces sensibly coalesce into one
single great circle, which is the horizon as first defined. In strict-
ness, therefore, while we can distinguish between the two horizontal
planes, we get but one horizon circle in the sky.

11. The Visible Horizon is the line where sky and earth
meet. On land it is an irregular line, broken by hills and trees,
and of no astronomical value; but at sea it is a true circle, and of
great importance in observation. It is not, however, a great circle,
but, technically speaking, only a small circle; depressed below the
true horizon by an amount depending upon the observer’s elevation
above the water. This depression is called the Dip of the Horizon,
and will be discussed further on.

12. Vertical Circles.—These are great circles passing through
the zenith and nadir, and therefore necessarily perpendicular to the
horizon—secondaries to it, to use the technical term.

Parallels of Altitude, or Almucantars.—These are small cir-
cles parallel to the horizon: the term Almucantar is seldom used.

The points and circles thus far defined are determined entirely
by the direction of gravity at the station occupied by the observer.

1Beware of the common, but vulgar, pronunciation, Hór¯ızon.

THE DOCTRINE OF THE SPHERE.

10

13. The Diurnal Rotation of the Heavens.—If one watches
the sky for a few hours some night, he will find that, while certain
stars rise in the east, others set in the west, and nearly all the con-
stellations change their places. Watching longer and more closely, it
will appear that the stars move in circles, uniformly, in such a way
as not to disturb their relative configurations, but as if they were
attached to the inner surface of a revolving sphere, turning on its
axis once a day. The path thus daily described by a star is called
its “diurnal circle.”

It is soon evident that in our latitude the visible “pole” of this
sphere—the point about which it turns—is in the north, not quite
half-way up from the horizon to the zenith, for in that region the
stars hardly move at all, but keep their places all night long.

14. The Poles.—The Poles may be defined as the two points
in the sky, one in the northern hemisphere and one in the southern,
where a star’s diurnal circle reduces to zero; i.e., points where, if a
star were placed, it would suffer no apparent change of place during
the whole twenty-four hours. The line joining these poles is, of
course, the axis of the celestial sphere, about which it seems to
rotate daily.

The exact place of the pole may be found by observing some
star very near the pole at two times 12 hours apart, and taking the
middle point between the two observed places of the star.

The definition of the pole just given is independent of any theory
as to the cause of the apparent rotation of the heavens. If, however,
we admit that it is due to the earth’s rotation on its axis, then we
may define the poles as the points where the earth’s axis produced
pierces the celestial sphere.

15. The Pole-star (Polaris).—The place of the northern pole
is very conveniently marked by the Pole-star, a star of the second
magnitude, which is now only about 1 1 ◦ from the pole: we say

4
now, because on account of a slow change in the direction of the
earth’s axis, called “precession” (to be discussed later), the distance
between the pole-star and the pole is constantly changing, and has
been for several centuries gradually decreasing.

The pole-star stands comparatively solitary in the sky, and may
easily be recognized by means of the so-called “pointers,”—two stars
in the “dipper” (in the constellation of Ursa Major)—which point

THE DOCTRINE OF THE SPHERE.

11
very nearly to it, as shown in Fig. 3. The pole is very nearly on the
line joining Polaris with the star Mizar (ζ Urs. Maj., at the bend in
the handle of the dipper), and at a distance just about one-quarter
of the distance between the pointers, which are nearly 5◦ apart.

The southern pole, unfortunately, is not so marked by any con-
spicuous star.

Fig. 3.—The Pole Star and the Pointers.

16. The Celestial Equator, or Equinoctial Circle.—This
is a great circle midway between the two poles, and of course 90◦
from each. It may also be defined as the intersection of the plane
of the earth’s equator with the celestial sphere. It derives its name
from the fact that, at the two dates in the year when the sun crosses
this circle—about March 20 and Sept. 22—the day and night are
equal in length.

17. The Vernal Equinox, or First of Aries.—The Equinox,
strictly speaking, is the time when the sun crosses the equator, but
the term has come by accommodation to denote also the point where
it crosses, though in strictness it should be called the “Equinoctial

Point.” This crossing occurs twice a year, once in September and
once in March, and the Vernal Equinox is the point on the equator
where the sun crosses it in the spring. It is sometimes called the

Greenwich of the Celestial Sphere, because it is used as a reference

THE DOCTRINE OF THE SPHERE.

12
point in the sky, much as Greenwich is on the earth. Its position is
not marked by any conspicuous star.

Why this point is also called the “First of Aries” will appear
later, when we come to speak of the zodiac and its “signs.”

18. Hour-Circles.—Hour-circles are great circles of the celes-
tial sphere passing through its poles, and consequently perpendicu-
lar to the celestial equator. They correspond exactly to the meridi-
ans of the earth, and some writers call them “Celestial Meridians”;
but the term is objectionable, as likely to lead to confusion with the

Meridian, to be noted immediately.

19. The Meridian and Prime Vertical.—The Meridian is
the great circle passing through the pole and the zenith. Since it
is a great circle, it must necessarily pass through both poles, and
through the nadir as well as the zenith, and must be perpendicular
both to the equator and to the horizon.

It may also be correctly defined as the Vertical Circle which
passes through the pole; or, again, as the Hour-Circle which passes
through the zenith, since all vertical circles must pass through the
zenith, and all hour-circles through the pole.

The Prime Vertical is the Vertical Circle (passing through the
zenith) at right angles to the meridian; hence lying east and west
on the celestial sphere.

20. The Cardinal Points.—The North and South Points are
the points on the horizon where it is intersected by the meridian.

The East and West Points are where it is cut by the prime vertical,
and also by the equator. The North Point, which is on the horizon,
must not be confounded with the North Pole, which is not on the
horizon, but at an elevation equal (see Art. 30) to the latitude of
the observer.

With these circles and points of reference we have now the means
to describe intelligibly the position of a heavenly body, in several
different ways.

We may give its altitude and azimuth, or its declination and
hour-angle; or, if we know the time, its declination and right ascen-
sion. Either of these pairs of co-ordinates, as they are called, will

THE DOCTRINE OF THE SPHERE.

13
define its place in the sky.

21. Altitude and Zenith Distance (Fig. 4).—The Altitude
of a heavenly body is its angular elevation above the horizon, and is
measured by the arc of the vertical circle passing through the body,
and intercepted between it and the horizon.

The Zenith Distance of a body is simply its angular distance
from the zenith, and is the complement of the altitude. Altitude +

Zenith Distance = 90◦.

22. Azimuth and Amplitude (Fig. 4).—The Azimuth of a
body is the angle at the zenith, between the meridian and the vertical
circle, which passes through the body. It is measured also by the arc
of the horizon intercepted between the north or south point, and
the foot of this vertical. The word is of Arabic origin, and has the
same meaning as the true bearing in surveying and navigation.

Fig. 4.—The Horizon and Vertical Circles.

O, the place of the Observer. M , some Star.

OZ, the Observer’s Vertical. ZM H, arc of the Star’s Vertical Circle.

Z, the Zenith; P , the Pole.

T M R, the Star’s Almucantar.

SEN W , the Horizon.

Angle T ZM , or arc SW N EH, Star’s Azimuth.

SZP N , the Meridian.

Arc HM , Star’s Altitude.

EZW , the Prime Vertical.

Arc ZM , Star’s Zenith Distance.

The Amplitude of a body is the complement of the azimuth.

Azimuth + Amplitude = 90◦.

There are various ways of reckoning azimuth. Many writers express
it in the same manner as the bearing is expressed in surveying; i.e., so
many degrees east or west of north or south; N. 20◦ E., etc. The more
usual way at present is, however, to reckon it in degrees from the south
point clear round through the west to the point of beginning: thus an
object in the SW. would have an azimuth of 45◦; in the NW., 135◦; in
the N., 180◦; in the NE., 225◦; and in the SE., 315◦. For example, to
find a star whose azimuth is 260◦, and altitude 60◦, we must face N. 80◦

THE DOCTRINE OF THE SPHERE.

14

E., and then look up two-thirds of the way to the zenith. The object
in this case has an amplitude of 10◦ N. of W., and a zenith distance of

30◦. Evidently both the azimuth and altitude of a heavenly body are
continually changing, except in certain very special cases.

In Fig. 4, SEN W represents the horizon, S being the south
point, and Z the zenith. The angle SZM , which numerically equals
the arc SH, is the Azimuth of the star M ; while EZM , or EH is
its Amplitude. M H is its Altitude, and ZM its Zenith Distance.

23. Declination and Polar Distance (Fig. 5).—The Decli-
nation of a heavenly body is its angular distance north or south of
the celestial equator, and is measured by the arc of the hour-circle
passing through the object, intercepted between it and the equa-
tor. It is reckoned positive (+) north of the celestial equator, and
negative (−) south of it. Evidently it is precisely analogous to the
latitude of a place on the earth. The north-polar distance of a star
is its angular distance from the North Pole, and is simply the com-
plement of the declination. Declination + North-Polar Distance =

90◦.

The declination of a star remains always the same; at least,
the slow changes that it undergoes need not be considered for our
present purpose. “Parallels of Declination” are small circles parallel
to the celestial equator.

24. The Hour-Angle (Fig. 5).—The Hour-Angle of a star is
the angle at the pole between the meridian and the hour-circle pass-
ing through the star. It may be reckoned in degrees; but it also may
be, and most commonly is, reckoned in hours, minutes, and seconds
of time; the hour being equivalent to fifteen degrees, and the minute
and second of time being equal to fifteen minutes and seconds of
arc respectively.

Of course the hour-angle of an object is continually changing,
being zero when the object is on the meridian, one hour, or fifteen
degrees, when it has moved that amount westward, and so on.

25. Right Ascension (Fig. 5).—The Right Ascension of a star
is the angle at the pole between the star’s hour-circle and the hour-
circle (called the Equinoctial Colure), which passes through the ver-
nal equinox.

It may be defined also as the arc of the equator, intercepted
between the vernal equinox and the foot of the star’s hour-circle.

THE DOCTRINE OF THE SPHERE.

15

It is always reckoned from the equinox toward the east ; some-
times in degrees, but usually in hours, minutes, and seconds of time.

The right ascension, like the declination, remains unchanged by the
diurnal motion.

26. Sidereal Time (Fig. 5).—For many astronomical purposes
it is convenient to reckon time, not by the sun’s position in the sky,
but by that of the vernal equinox.

Fig. 5.—Hour-Circles, etc.

O, place of the Observer; Z, his

P XP the Equinoctial Colure, or Zero

Zenith.

Hour-Circle.

SEN W , the Horizon.

m, some Star.

P OP , line parallel to the Axis of the

Y m, the Star’s Declination; P m, its

Earth.

North-polar Distance.

P and P , the two Poles of the Heav-

Angle mP R = arc QY , the Star’s
ens.

(eastern) Hour-Angle; = 24h mi-

EQW T , the Celestial Equator, or
nus Star’s (western) Hour-Angle.

Equinoctial.

Angle XP m = arc XY , Star’s Right

X, the Vernal Equinox, or “First of

Ascension, Sidereal time at the

Aries.”

moment

=

24h
minus
angle

XP Q.

The Sidereal Time at any moment may be defined as the hour-
angle of the vernal equinox. It is sidereal noon, when the equinoctial
point is on the meridian; 1 o’clock (sidereal) when its hour-angle is

15◦; and 23 o’clock when its hour-angle is 345◦, i.e., when the vernal
equinox is an hour east of the meridian; the time being reckoned
round through the whole 24 hours. On account of the annual motion
of the sun among the stars, the Solar Day, by which time is reckoned
for ordinary purposes, is about 4 minutes longer than the sidereal
day. The exact difference is 3m 56s.394 (sidereal), or just one day in
a year; there being 366 1 sidereal days in the year, as against 365 1

4

4
solar days.

THE DOCTRINE OF THE SPHERE.

16

27. Observatory Definition of Right Ascension.—It is ev-
ident from the above definition of sidereal time, that we may also
define the Right Ascension of a star as the sidereal time taken the
star crosses the meridian. The Star and the Vernal Equinox are
both of them fixed points in the sky, and do not change their rel-
ative position during the sky’s apparent daily revolution; a given
star, therefore, always comes to the meridian of any observer the
same number of hours after the vernal equinox has passed; and this
number of hours is the sidereal time at the moment of the star’s
transit, and measures its right ascension. In the observatory, this
definition of right ascension is the most natural and convenient.

It is obvious that the right ascension of a star corresponds in the
sky exactly with the longitude of a place on the earth; terrestrial
longitude being reckoned from Greenwich, just as right ascension is
reckoned from the vernal equinox.

N.B.

We shall find hereafter that the stars have latitudes and
longitudes of their own; but unfortunately these celestial latitudes
and longitudes do not correspond to the terrestrial, and great care
is necessary to prevent confusion. (See Art. 179.)

28. An armillary sphere, or some equivalent apparatus, is al-
most essential to enable a beginner to get correct ideas of the points,
circles, and co-ordinates defined above, but the figures will perhaps
be of assistance.

The first of them (Fig. 4) represents the horizon, meridian, and
prime vertical, and shows how the position of a star is indicated
by its altitude and azimuth. This framework of circles, depending
upon the direction of gravity, of course always remains apparently
unchanged in position, as if attached directly to the earth, while
the sky apparently turns around outside it.

The other figure (Fig. 5) represents the system of points and cir-
cles which depend upon the earth’s rotation, and are independent
of the direction of gravity. The vernal equinox and the hour-circles
apparently revolve with the stars while the pole remains fixed upon
the meridian, and the equator and parallels of declination, revolving
truly in their own planes, also appear to be at rest in the sky. But
the whole system of lines and points represented in the figure (hori-
zon and meridian alone excepted) may be considered as attached
to, or marked out upon, the inner surface of the celestial vault and
whirling with it.

THE DOCTRINE OF THE SPHERE.

17

It need hardly be said that the “appearances are deceitful”—that
which is really carried around by the earth’s rotation is the observer,
with his plumb-line and zenith, his horizon and meridian; while the
stars stand still—at least, their motions in a day are insensible as
seen from the earth.

At the poles of the earth, which are, mathematically speaking, “sin-
gular” points, the definitions of the Meridian, of North and South, etc.,
break down.

There the pole (celestial) and zenith coincide, and any number of
circles may be drawn through the two points, which have now become
one. The horizon and equator coalesce, and the only direction on the
earth’s surface is due south (or north)—east and west have vanished.

A single step of the observer will, however, remedy the confusion:

zenith and pole will separate, and his meridian will again become deter-
minate.

29. To recapitulate: The direction of gravity at the point where
the observer stands determines the Zenith and Nadir, the Horizon,
and the Almucantars (parallel to the Horizon), and all the vertical
circles. One of the verticals, the Meridian, is singled out from the
rest by the circumstance that it passes through the pole of the sky,
marking the North and South Points where it cuts the horizon.

Altitude and Azimuth (or their complements, Zenith Distance
and Amplitude) are the co-ordinates which designate the position
of a body by reference to the Zenith and the Meridian.

Similarly, the direction of the earth’s axis (which is indepen-
dent of the observer’s place on the earth) determines the Poles, the

Equator, the Parallels of Declination, and the Hour-Circles. Two of
these Hour-Circles are singled out as reference lines; one of them, the

Meridian, which passes through the Zenith, and is a purely local ref-
erence line; the other, the Equinoctial Colure, which passes through
the Vernal Equinox, a point chosen from its relation to the sun’s
annual motion. Declination and Hour-Angle are the co-ordinates
which refer the place of a star to the Pole and the Meridian; while

Declination and Right Ascension refer it to the Pole and Equinoc-
tial Colure. The latter are the co-ordinates usually employed in
star-catalogues and ephemerides to define the positions of stars and
planets, and correspond exactly to Latitude and Longitude on the
earth, by means of which geographical positions are designated.

THE DOCTRINE OF THE SPHERE.

18

30. Relation of the Apparent Diurnal Motion of the Sky
to the Observer’s Latitude.—Evidently the apparent motions
of the stars will be considerably influenced by the station of the
observer, since the place of the pole in the sky will depend upon
it.

The Altitude of the pole, or its height in degrees above the
horizon, is always equal to the Latitude of the observer. Indeed, the

German word for latitude (astronomical) is Polhöhe; i.e., simply

“Pole-height.”

This will be clear from Fig. 6. The latitude of a place is the angle
between its plumb-line and the plane of the equator ; the angle ON Q
in the figure. [If the earth were truly spherical, N would coincide
with C, the centre of the earth. The ordinary definition of latitude
given in the geographies disregards the slight difference.]

Fig. 6.—Relation of Latitude to the Elevation of the Pole.

Now the angle H OP
is equal to ON Q, because their sides
are mutually perpendicular; and it is also the altitude of the pole,
because the line HH is horizontal at O, and OP is parallel to P P ,
the earth’s axis, and therefore points to the celestial pole.

This fundamental relation, that the altitude of the celestial pole
is the Latitude of the observer, cannot be too strongly impressed on
the student’s mind. The usual symbol for the latitude of a place is
φ.

31. The Right Sphere.—If the observer is situated at the
earth’s equator, i.e., in latitude zero (φ = 0), the pole will be in the
horizon, and the equator will pass vertically overhead through the
zenith.

The stars will rise and set vertically, and their diurnal circles will
all be bisected by the horizon, so that they will be 12 hours above it
and 12 below. This aspect of the heavens is called the Right Sphere.

THE DOCTRINE OF THE SPHERE.

19

32. The Parallel Sphere.—If the observer is at the pole of
the earth (φ = 90◦), then the celestial pole will be in the zenith,
and the equator will coincide with the horizon. If he is at the North

Pole, all stars north of the celestial equator will remain permanently
above the horizon, never rising or falling at all, but sailing around
on circles of altitude (or Almucantars) parallel to the horizon. Stars
in the Southern Hemisphere, on the other hand, would never rise to
view. As the sun and the moon move in such a way that during half
the time they are alternately north and south of the equator, they
will be half the time above the horizon and half the time below it.

The moon would be visible for about a fortnight at a time, and the
sun for six months.

Fig. 7.—The Oblique Sphere and Diurnal Circles.

33. The Oblique Sphere (Fig. 7).—At any station between
the pole and equator the stars will move in circles oblique to the
horizon, SEN W in the figure. Those whose distance from the ele-
vated pole is less than the latitude of the place will, of course, never
sink below the horizon,—the radius of the “Circle of Perpetual Ap-
parition,” as it is called (the shaded cap around P in the figure),
being just equal to the height of the pole, and becoming larger as
the latitude increases. On the other hand, stars within the same
distance of the depressed pole will lie within the “Circle of Perpetual

Occultation,” and will never rise above the horizon.

A star exactly on the celestial equator will have its diurnal circle

EQW Q bisected by the horizon, and will be above the horizon just
as long as below it. A star north of the equator (if the North Pole
is the elevated one) will have more than half of its diurnal circle
above the horizon, and will be visible more than half the time; as,

THE DOCTRINE OF THE SPHERE.

20
for instance, a star at A: and of course the reverse will be true of
stars on the other side of the equator.1 Whenever the sun is north
of the equator, the day will therefore be longer than the night for
all stations in northern latitude: how much longer will depend both
on the latitude of the place and the sun’s distance from the celestial
equator.

1A Celestial Globe will be of great assistance in studying these diurnal
circles. The north pole of the globe must be elevated to an angle equal to the
latitude of the observer, which can be done by means of the degrees marked
on the brass meridian. It will then at once be easily seen what stars never set,
which ones never rise, and during what part of the 24 hours any heavenly body
at a known distance from the equator is above or below the horizon.

ASTRONOMICAL INSTRUMENTS.

21

CHAPTER II.

ASTRONOMICAL INSTRUMENTS.

34. Astronomical observations are of various kinds: some-
times we desire to ascertain the apparent distance between two bod-
ies at a given time; sometimes the position which a body occupies
at a given time, or the moment it arrives at a given circle of the
sky, usually the meridian. Sometimes we wish merely to examine
its surface, to measure its light, or to investigate its spectrum; and
for all these purposes special instruments have been devised.

We propose in this chapter to describe very briefly a few of the
most important.

35. Telescopes in General.—Telescopes are of two kinds, re-
fracting and reflecting. The former were first invented, and are
much more used, but the largest instruments ever made are reflec-
tors. In both the fundamental principle is the same. The large lens,
or mirror, of the instrument forms at its focus a real image of the
object looked at, and this image is then examined and magnified by
the eye-piece, which in principle is only a magnifying-glass.

In the form of telescope, however, introduced by Galileo,1 and still
used as the “opera-glass,” the rays from the object-glass are intercepted
by a concave lens which performs the office of an eye-piece before they
meet at the focus to form the “real image.”

But on account of the
smallness of the field of view, and other objections, this form of telescope
is never used when any considerable power is needed.

1In strictness, Galileo did not invent the telescope. Its first invention seems
to have been in 1608, by Lipperhey, a spectacle-maker of Middleburg, in Hol-
land; though the honor has also been claimed for two or three other Dutch
opticians. Galileo, in his “Nuncius Sydereus,” published in March, 1610, him-
self says that he had heard of the Dutch instruments in 1609, and by so hearing
was led to construct his own, which, however, far excelled in power any that had
been made previously; and he was the first to apply the telescope to Astronomy.

See Grant’s “History of Astronomy,” pp. 514 and seqq.

ASTRONOMICAL INSTRUMENTS.

22

36. Simple Refracting Telescope.—This consists essentially
as shown in the figure (Fig. 8), of a tube containing two lenses:

a single convex lens, A, called the object-glass; and another, of
smaller size and short focus, B, called the eye-piece. Recalling the
principles of lenses the student will see that if the instrument be
directed at a distant object, the moon, for instance, all the rays,
a0a1a2, which fall upon the object-glass from a point at the top of
the moon, will be collected at a in the focal plane, at the bottom of
the image. Similarly rays from the bottom of the moon will go to b
at the top of the image; moreover, since the rays that pass through
the optical centre of the lens, o, are undeviated,1 the angle a0ob0
will equal boa; or, in other words, if the focal length of the lens
be five feet, for instance, then the image of the moon, seen from a
distance of five feet, will appear just as large as the moon itself does
in the sky,—it will subtend the same angle. If we look at it from
a smaller distance, say from a distance of one foot, the image will
look larger than the moon; and in fact, without using an eye-piece at
all, a person with normal eyes can obtain considerable magnifying
power from the object-glass of a large telescope. With a lens of ten
feet focal length, such as is ordinarily used in an 8-inch telescope,
one can easily see the mountains on the moon and the satellites of

Jupiter, by taking out the eye-piece, and putting the eye in the line
of vision some eight or ten inches back of the eye-piece hole.

Fig. 8.—Path of the Rays in the Astronomical Telescope.

The image is a real one; i.e., the rays that come from different
points in the object actually meet at corresponding points in the
image, so that if a photographic plate were inserted at ab, and
properly exposed, a picture would be obtained.

1In this explanation, we use the approximate theory of lenses (in which their
thickness is neglected), as given in the elementary text-books. The more exact
theory of Gauss and later writers would require some slight modifications in our
statements, but none of any material importance. For a thorough discussion,
see Jamin, “Traité de Physique,” or Encyc. Britannica,—Optics.

ASTRONOMICAL INSTRUMENTS.

23

If we look at the image with the naked eye, we cannot come
nearer to the image (unless near-sighted) than eight or ten inches,
and so cannot get any great magnifying power; but if we use a
magnifying-glass, we can approach much closer.

37. Magnifying Power.—If the eye-piece B is set at a dis-
tance from the image equal to its principal focal distance, then any
pencil of rays from any point of the image will, after passing the
lens, be converted into a parallel beam, and will appear to the eye
to come from a point at an infinite distance, as if from an object
in the sky. The rays which came from the top of the moon, for
instance, and are collected at a in the image, will reach the eye
as a beam parallel to the line ac, which connects a with the opti-
cal centre of the eye-piece. Similarly with the rays which meet at
b. The observer, therefore, will see the top of the moon’s disc in
the direction ck, and the bottom in the direction cl. It will appear
to him inverted, and greatly magnified; its apparent diameter, as
seen by the naked eye and measured by the angle aob (or its equal
b0oa0); having been increased to acb. Since both these angles are
subtended by the same line ab, and are small (the figure, of course,
is much out of proportion), they must be inversely proportional to
the distance ob and cb; i.e., boa : bca = cb : ob; or, putting this
into words: The ratio between the natural apparent diameter of the
object, and its diameter as seen through the telescope, is equal to
the ratio between the focal lengths of the eye-lens and object-glass.

This ratio is called the magnifying power of the telescope, and is

F
therefore given by the simple formula M =

, where F is the focal
f
length of the object-glass and f that of eye-piece,1 while M is the
magnifying power.

If, for example, the object-glass have a focal length of thirty feet,
and the eye-piece of one inch, the magnifying power will be 360; the
power may be changed at pleasure by substituting different eye-
pieces, of which every large telescope has an extensive stock.

1A magnifying power of 1 is no magnifying power at all. Object and image
subtend equal angles. A magnifying power denoted by a fraction, say 1 would

4
be a minifying power, making the object look smaller, as when we look at an
object through the wrong end of a spy-glass.

ASTRONOMICAL INSTRUMENTS.

24

38. Brightness of Image.—Since all the rays from a star
which fall upon the large object-glass are transmitted to the ob-
server’s eye (neglecting the losses by absorption and reflection), he
obviously receives a quantity of light much greater than he would
naturally get: as many times greater as the area of the object-lens
is greater than that of the pupil of the eye. If we estimate this latter
as having a diameter of one-fifth of an inch, then a 1-inch telescope
would increase the light twenty-five times, a 10-inch instrument 2500
times, and the great Lick telescope, of thirty-six inches’ aperture,

32,400 times, the amount being proportional to the square of the
diameter of the lens.

It must not be supposed, however, that the apparent brightness
of an object like the moon, or a planet which shows a disc, is in-
creased in any such ratio, since the eye-piece spreads out the light
to cover a vastly more extensive angular area, according to its mag-
nifying power; in fact, it can be shown that no optical arrangement
can show an extended surface brighter than it appears to the naked
eye. But the total quantity of light utilized is greatly increased by
the telescope, and in consequence, multitudes of stars, far too faint
to be visible to the unassisted eye, are revealed; and, what is practi-
cally very important, the brighter stars are easily seen by day with
the telescope.

39. Distinctness of Image.—This depends upon the exact-
ness with which the lens gathers to a single point in the focal image
all the rays which emanate from the corresponding point in the
object. A single lens, with spherical surfaces, cannot do this very
perfectly, the “aberrations” being of two kinds, the spherical aber-
ration and the chromatic. The former could be corrected, if it were
worth while, by slightly modifying the form of the lens-surfaces; but
the latter, which is far more troublesome, cannot be cured in any
such way. The violet rays are more refrangible than the red, and
come to a focus nearer the lens; so that the image of a star formed
by such a lens can never be a luminous point, but is a round patch
of light of different color at centre and edge.

40. Long Telescopes.—By making the diameter of the lens very
small as compared with its focal length, the aberration becomes less
conspicuous; and refractors were used, about 1660, having a length of
more than 100 feet and a diameter of five or six inches. The object-
glass was mounted at the top of a high pole and the eye-piece was on

ASTRONOMICAL INSTRUMENTS.

25
a separate stand below. With such an “aerial telescope,” of six inches
aperture and 120 feet focus, Huyghens discovered the rings of Saturn.

His object-glass still exists, and is preserved in the collection of the Royal

Society in London.

41. The Achromatic Telescope.—The chromatic aberration
of a lens, as has been said, cannot be cured by any modification of
the lens itself; but it was discovered in England about 1760 that it
can be nearly corrected by making the object-glass of two (or more)

lenses, of different kinds of glass, one of the lenses being convex and
the other concave. The convex lens is usually made of crown glass,
the concave of flint glass. At the same time, by properly choosing
the curves, the spherical aberration can also be destroyed, so that
such a compound object-glass comes reasonably near to fulfilling
the condition, that it should gather to a mathematical point in the
image all the rays that reach the object-glass from a single point in
the object.

These object-glasses admit of

1

2

3
considerable
variety
of
forms.

Formerly
they
were
generally
made, as in Fig. 9, No. 3, having

Clark

Gauss

Littrow
the two lenses close together, and

Fig. 9.—Different Forms of the

Achromatic Object-glass.

the adjacent surfaces of the same,
or nearly the same, curvature. In
small object-glasses the lenses are often cemented together with Canada
balsam or some other transparent medium. At present some of the best
makers separate the two lenses by a considerable distance, so as to ad-
mit a free circulation of air between them; in the Pulkowa and Princeton
object-glasses, constructed by Clark, the lenses are seven inches apart,
and in the Lick telescope six and a half inches; as in No. 1. In a form
devised by Gauss (No. 2), which has some advantages, but is difficult
of construction, the curves are very deep, and both the lenses are of
watch-glass form—concave on one side and convex on the other. In all
these forms the crown glass is outside; Steinheil, Hastings, and others
have constructed lenses with the flint-glass lens outside. Object-glasses
are sometimes made with three lenses instead of two; a slightly better
correction of aberrations can be obtained in this way, but the gain is too
small to pay for the extra expense and loss of light.

42. Secondary Spectrum.—It is not, however, possible with
the kinds of glass at present available to secure a perfect correction
of the color. Our best achromatic lenses bring the yellowish green

ASTRONOMICAL INSTRUMENTS.

26
rays to a focus nearer the lens than they do the red and violet. In
consequence, the image of a bright star is surrounded by a purple
halo, which is not very noticeable in a good telescope of small size,
but is very conspicuous and troublesome in a large instrument.

This imperfection of achromatism makes it unsatisfactory to use an
ordinary lens (visually corrected) for astronomical photography. To fit
it to make good photographs, it must either be specially corrected for
the rays that are most effective in photography, the blue and violet (in
which case it will be almost worthless visually), or else a subsidiary lens,
known as a “photographic corrector,” may be provided, which can be
put on in front of the object-glass when needed. A new form of object-
glass, devised independently by Pickering in this country and Stokes
in England, avoids the necessity of a third lens by making the crown-
glass lens of such a form that when put close to the flint lens, with the
flatter side out, it makes a perfect object-glass for visual purposes; but
by simply reversing the crown lens, with the more convex side outward,
and separating the lenses an inch or two, it becomes a photographic
object-glass. A 13-inch object-glass of this construction at Cambridge
performs admirably.

Much is hoped from the new kind of glass now being made at Jena.

In combination with crown glass it produces lenses almost free from
chromatic aberration, and if it can be produced in homogeneous pieces
of sufficient size, it will revolutionize the art of telescope making.

43. Diffraction and Spurious Disc.—Even if a lens were
perfect as regards the correction of aberrations, the “wave” nature
of light prevents the image of a luminous point from being also a
point; the image must necessarily consist of a central disc, bright-
est in the centre and fading to darkness at the edge, and this is
surrounded by a series of bright rings, of which, however, only the
smallest one is generally easily seen. The size of this disc-and-ring
system can be calculated from the known wave-lengths of light and
the dimensions of the lens, and the results agree very precisely with
observation. The diameter of the “spurious disc” varies inversely
with the aperture of the telescope. According to Dawes, it is about

4 .5 for a 1-inch telescope; and consequently 1 for a 4 1 -inch instru-

2
ment, 0 .5 for a 9-inch, and so on.

This circumstance has much to do with the superiority of large in-
struments in showing minute details. No increase of magnifying power
on a small telescope can exhibit things as sharply as the same power on
the larger one; provided, of course, that the larger object-glass is equally

ASTRONOMICAL INSTRUMENTS.

27
perfect in workmanship, and the air in good optical condition.

If the telescope is a good one, and if the air is perfectly steady,—
which unfortunately is seldom the case,—the apparent disc of a star
should be perfectly round and well defined, without wings or tails of
any kind, having around it from one to three bright rings, separated by
distances somewhat greater than the diameter of the disc. If, however,
the magnifying power is more than about 50 to the inch of aperture, the
edge of the disc will begin to appear hazy. There is seldom any advantage
in the use of a magnifying power exceeding 75 to the inch, and for most
purposes powers ranging from 20 to 40 to the inch are most satisfactory.

44. Eye-Pieces.—For many purposes, as for instance the ex-
amination of close double stars, there is no better eye-piece than
the simple convex lens; but it performs well only when the object
is exactly in the centre of the field. Usually it is best to employ for
the eye-piece a combination of two or more lenses which will give a
more extensive field of view.

Eye-pieces belong to two classes, the positive and the negative.

The former, which are much more generally useful, act as simple
magnifying-glasses, and can be used as hand magnifiers if desired.

The focal image formed by the object-glass lies outside of the eye-
piece.

In the negative eye-pieces, on the other hand, the rays from
the object-glass are intercepted before they come to the focus, and
the image is formed between the lenses of the eye-piece. Such an
eye-piece cannot be used as a hand magnifier.

45. The simplest and most common forms of these eye-pieces are
the Ramsden (positive) and Huyghenian (negative). Each is composed
of two plano-convex lenses, but the arrangement and curves differ, as
shown in Fig. 10. The former gives a very flat field of view, but is not
achromatic; the latter is more nearly achromatic, and possibly defines a
little better just at the centre of the field; but the fact that it is a negative
eye-piece greatly restricts its usefulness. In the Ramsden eye-piece the
focal lengths of the two component lenses, both of which have their flat
sides out, are about equal to each other, and their distance is about one-
third of the sum of the focal lengths. In the Huyghenian the curved sides
of the lenses are both turned towards the object-glass; the focal distance
of the field lens should be exactly three times that of the lens next the
eye, and the distance between the lenses one-half the sum of the focal
lengths. The peculiarity of the Steinheil “monocentric” eye-piece which
is a triple achromatic positive lens, consisting of a central convex lens of

ASTRONOMICAL INSTRUMENTS.

28
crown glass, with a concave meniscus of flint glass cemented to each side,
is that the curves are all struck from the same centre, the thickness of
the lenses being so computed as to produce the needed corrections. It is
free from all internal reflections, which in other eye-pieces often produce

“ghosts,” as they are called.

There are numerous

Steinheil
other forms of eye-piece,

‘Monocentric’

each with its own advan-

(Positive)

tages and disadvantages.

Ramsden

Huyghenian

The erecting
eye-piece,

(Positive)

(Negative)

used in spy-glasses, is es-
sentially a compound mi-
croscope, and gives erect
vision by again inverting
the already inverted im-

Fig. 10.—Various Forms of Telescope Eye-piece.

age formed by the object-
glass.

It is obvious that in a telescope of any size the object-glass is the most
important and expensive part of the instrument. Its cost varies from a
few hundred dollars to many thousands, while the eye-pieces generally
cost only from $5 to $20 apiece.

46. Reticle.—When a telescope is used for pointing, as in most
astronomical instruments, it must be provided with a reticle of some
sort. This is usually a metallic frame with spider lines stretched
across it, placed, not near the object-glass itself (as is often sup-
posed), but at the focus of the object-glass, where the image is
formed, as at a b in Fig. 8.

It is usually so arranged that it can be moved in or out a little to get
it exactly into the focal plane, and then, when the eye-piece (positive)

is adjusted for the observer’s eye to give distinct vision of the object,
the “wires,” as they are called, will also be equally distinct. As spider-
threads are very fragile, and likely to get broken or displaced, it is often
better to substitute a thin plate of glass with lines ruled upon it and
blackened. Of course, provision must be made for illuminating either
the field of view or the threads themselves, in order to make them visible
in darkness.

47. The Reflecting Telescope.—When the chromatic aber-
ration of lenses came to be understood through the optical discovery
of the dispersion of light by Newton, the reflecting telescope was in-
vented, and held its place as the instrument for star-gazing until

ASTRONOMICAL INSTRUMENTS.

29
well into the present century, when large achromatics began to be
made. There are several varieties of reflecting telescope, all agreeing
in the substitution of a large concave mirror in place of the object-
glass of the refractor, but differing in the way in which they get at
the image formed by this mirror at its focus in order to examine it
with the eye-piece.

48. In the Herschellian form, which is the simplest, but only suited
to very large instruments, the mirror is tipped a little, so as to throw
the image to the side of the tube, and the observer stands with his
back to the object and looks down into the tube. If the telescope is
as much as two or three feet in diameter, his head will not intercept
enough light to do much harm,—not nearly so much as would be lost by
the second reflection necessary in the other forms of the instrument. But
the inclination of the mirror, and the heat from the observer’s person, are
fatal to any very accurate definition, and unfit this form of instrument
for anything but the observation of nebulæ and objects which mainly
require light-gathering power.

In
the

Newtonian
telescope, a small plane
reflector standing at an

1
angle of 45° is placed in
the centre of the tube, so
as to intercept the rays

2
reflected
by
the
large
mirror a little before they
come to their focus, and

3
throw them to the side
of the tube, where the
eye-piece is placed.

Fig. 11.—Different Forms of Reflecting

In the Gregorian form

Telescope.

(which was the first in-

1. The Herschellian; 2. The Newtonian;
vented), the large mirror

3. The Gregorian.

is pierced through its cen-
tre, and the rays from it are reflected through the hole by a small concave
mirror, placed a little outside of the principal focus at the mouth of the
tube. With this instrument one looks directly at the stars as with a
refractor, and the image is erect.

The Cassegrainian form is very similar, except that the small concave
mirror of the Gregorian is replaced by a convex mirror, placed a little
inside the focus of the large mirror, which makes the instrument a little
shorter, and gives a flatter field of view.

ASTRONOMICAL INSTRUMENTS.

30

Formerly the great mirror was always made of a composition of cop-
per and tin (two parts of copper to one of tin) known as “speculum
metal.” At present it is usually made of glass silvered on the front sur-
face, by a chemical process which deposits the metal in a thin, brilliant
film. These silver-on-glass reflectors, when new, reflect much more light
than the old specula, but the film does not retain its polish so long.

It is, however, a comparatively simple matter to renew the film when
necessary.

The largest telescopes ever made have been reflectors. At the head
of the list stands the enormous instrument of Lord Rosse, constructed in

1842, with a mirror six feet in diameter and sixty feet focal length. Next
in order are a number of instruments of four feet aperture, first among
which is the great telescope of the elder Herschel, built in 1789, followed
by the telescope erected by Lassell at Malta in 1860, the Melbourne
reflector by Grubb in 1870, and the still more recent silver-on-glass re-
flector of the Paris observatory, which, however, has proved a failure,
owing to defective support of the mirror.

49. Relative Advantages of Refractors and Reflectors.

—There has been a good deal of discussion on this point, and each
construction has its partisans.

In favor of the reflectors we may mention,—

First. Ease of construction and consequent cheapness. The concave
mirror has but one surface to figure and polish, while an object-glass has
four. Moreover, as the light goes through an object-glass, it is evident
that the glass employed must be perfectly clear and of uniform density
through and through; while in the case of the mirror, the light does
not penetrate the material at all. This makes it vastly easier to get the
material for a large mirror than for a large lens.

Second (and immediately connected with the preceding). The possi-
bility of making reflectors much larger than refractors. Lord Rosse’s great
reflector is six feet in diameter, while the Lick telescope, the largest of
all refractors, is only three.

Third. Perfect achromatism. This is unquestionably a very great
advantage, especially in photographic and spectroscopic work.

But, on the whole, the advantages are generally considered to lie with
the refractors.

In their favor we mention:—

First. Great superiority in light. No mirror (unless, perhaps, a freshly
polished silver-on-glass film) reflects much more than three-quarters of
the incident light; while a good (single) lens transmits over 95 per cent.

In a good refractor about 82 per cent of the light reaches the eye, after

ASTRONOMICAL INSTRUMENTS.

31
passing through the four lenses of the object-glass and eye-piece. In a

Newtonian reflector, in average condition, the percentage seldom exceeds

50 per cent, and more frequently is lower than higher.

Second. Better definition.—Any slight error at a point in the surface
of a glass lens, whether caused by faulty workmanship or by distortion,
affects the direction of the ray passing through it only one-third as much
as the same error on the surface of a mirror would do.

If, for instance, in Fig. 12, an element of
the surface at P is turned out of its proper
direction, aa , by a small angle, so as to take
the direction bb , then the reflected ray will
be sent to f , and its deviation will be twice
the angle aP b. But since the index of re-
fraction of glass is about 1.5 the change in
the direction of the refracted ray from R to
r will only be about two-thirds of aP b.

Moreover, so far as distortions are con-

Fig. 12.—Effect of Surface
cerned, when a lens bends a little by its own

Errors in a Mirror and in a
weight, both sides are affected in a nearly

Lens.

compensatory manner, while in a mirror
there is no such compensation. As a consequence, mirrors very seldom
indeed give any such definition as lenses do. The least fault of workman-
ship, the least distortion by their own weight, the slightest difference
of temperature, between front and back, will absolutely ruin the image,
while a lens would be but slightly affected in its performance by the same
circumstances.

Third. Permanence. The lens, once made, and fairly taken care of,
suffers no deterioration from age; but the metallic speculum or the silver
film soon tarnishes, and must be repolished every few years. This alone
is decisive in most cases, and relegates the reflector mainly to the use of
these who are themselves able to construct their own instruments.

To these considerations we may add that a refractor, though more
expensive than a reflector of similar power, is not only more permanent,
and less likely to have its performance affected by accidental circum-
stances, but is lighter and more convenient to use.

50. Time-Keepers
and

Time-Recorders.—The Clock,

Chronometer, and Chronograph.—Modern practical astronomy
owes its development as much to the clock and chronometer as to the
telescope. The ancients possessed no accurate instruments for the
measurement of time, and until within 200 years, the only reason-
ably precise method of fixing the time of an important observation,

ASTRONOMICAL INSTRUMENTS.

32
as, for instance, of an eclipse, was by noting the altitude of the sun,
or of some known star at or very near the moment.

It is true that the Arabian astronomer Ibn Jounis had made
some use of the pendulum about the year 1000 a.d., more than

500 years before Galileo introduced it to Europeans. But it was
not until nearly a century after Galileo’s discovery that Huyghens
applied it to the construction of clocks (in 1657).

So far as the principles of construction are concerned, there is no
difference between an astronomical clock and any other. As a matter
of convenience, however, the astronomical clock is almost invariably
made to beat seconds (rarely half-seconds), and has a conspicuous
second-hand, while the hour-hand makes but one revolution a day,
instead of two, as usual, and the face is marked for twenty-four
hours instead of twelve. Of course it is constructed with extreme
care in all respects.

The Escapement, or “Scapement,” is often of the form known as
the “Graham Dead-beat”; but it is also frequently one of the numerous

“gravity” escapements which have been invented by ingenious mechani-
cians. The office of the escapement is to be “unlocked” by the pendulum
at each vibration, so as to permit the wheel-work to advance one step,
marking a second (or sometimes two seconds), upon the clock-face; while,
at the same time, the escapement gives the pendulum a slight impulse,
just equal to the resistance it has suffered in performing the unlocking.

The work done by the pendulum in “unlocking” the train, and the cor-
responding impulse, ought to be perfectly constant, in spite of all changes
in the condition of the train of wheels; and it is desirable, though not
essential, that this work should be as small as possible.

51. The pendulum itself is usually suspended by a flat spring,
and great pains should be taken to have the support extremely firm:

this is often neglected, and the clock then cannot perform well.

Compensation for Temperature.—In order to keep perfect time,
the pendulum must be a “compensation pendulum”; i.e., con-
structed in such a way that changes of temperature will not change
its length.

An uncompensated pendulum, with steel rod, changes its daily
rate about one-third of a second for each degree of temperature

(centigrade).

A wooden pendulum rod is much less affected by
temperature, but is very apt to be disturbed by changes of moisture.

ASTRONOMICAL INSTRUMENTS.

33

Graham’s mercurial pendulum (Fig. 13) is the
one most commonly used.

It consists simply of a
jar (usually steel), three or four inches in diame-
ter, and about eight inches high, containing forty or
fifty pounds of mercury, and suspended at the end
of a steel rod. When the temperature rises, the rod
lengthens (which would make the clock go slower);
but, at the same time, the mercury expands, from the
bottom upwards, just enough to compensate. This
pendulum will perform well only when not exposed to
rapid changes of temperature. Under rapid changes
the compensation lags. If, for instance, it grows warm
quickly, the rod will expand before the mercury does;
so that, while the mercury is growing warmer, the
clock will run slow, though after it has become warm
the rate may be all right.

A compensation pendulum, constructed on the
principle of the old gridiron pendulum of Harrison,
but of zinc and steel instead of brass and steel, is

Fig. 13.

Compensation
now much used. The compensation is not so eas-

Pendulums.

ily adjusted as in the mercurial pendulum, but when

1. Graham’s
properly made the mechanism acts well, and bears

Pendulum.

rapid alterations of temperature much better than

2. Zinc-Steel
the mercurial pendulum. The heavy pendulum-bob,

Pendulum.

a lead cylinder, is hung at the end of a steel rod,
which is suspended from the top of a zinc tube, and hangs through the
centre of it. This tube is itself supported at the bottom by three or four
steel rods which hang from a piece attached to the pendulum spring.

The standard clock at Greenwich has a pendulum of this kind.

52. Effect of Atmospheric Pressure.—In consequence of
the buoyancy of the air, and its resistance to motion, a pendulum
swings a little more slowly than it would in vacuo, and every change
in the density of the air affects its rate more or less. With mercurial
pendulums, of ordinary construction, the “barometric coefficient,”

as it is called, is about one-third of a second for an inch of the barom-
eter; i.e., an increase of atmospheric density which would raise the
barometer one inch would make the clock lose about one-third of a
second daily. It varies considerably, however, with different pendu-
lums.

It is not very usual to take any notice of this slight disturbance; but
when the extremest accuracy of time-keeping is aimed at, the clock is

ASTRONOMICAL INSTRUMENTS.

34
either sealed in an air-tight case from which the air is partially exhausted

(as at Berlin), or else some special mechanism, controlled by a barometer,
is devised to compensate for the barometric changes, as at Greenwich. In
the Greenwich clock a magnet is raised or lowered by the rise or fall of the
mercury in a barometer attached to the clock-case. When the magnet
rises, it approaches a bit of iron two or three inches above it, fixed to
the bottom of the pendulum, and the increase of attraction accelerates
the rate just enough to balance the retardation due to the air’s increased
density and viscosity. There are several other contrivances for the same
purpose.

53. Error and Rate.—The “error,” or “correction” of a clock
is the amount that must be added to the indication of the clock-face
at any moment in order to give the true time; it is, therefore, plus

(+) when the clock is slow, and minus (−) when it is fast. The rate
of a clock is the amount of its daily gain or loss; plus (+) when the
clock is losing. Sometimes the hourly rate is used, but “hourly” is
then always specified.

A perfect clock is one that has a constant rate, whether that
rate be large or small. It is desirable, for convenience’ sake, that
both error and rate should be small; but this is a mere matter of
adjustment by the user of the clock, who adjusts the error by setting
the hands, and the rate by raising or lowering the pendulum-bob.

The final adjustment of rate is often obtained by first setting the
pendulum-bob so that the clock will run slow a second or two daily, and
then putting on the top of the bob little weights of a gramme or two,
which will accelerate the motion. They can be dropped into place or
knocked off without stopping the clock or perceptibly disturbing it.

The very best clocks will run three or four years without being
stopped for cleaning, and will retain their rate without a change of more
than one-fifth of a second, one way or the other, during the whole time.

But this is exceptional performance. In a run as long as that, most clocks
would be liable to change their rate as much as half a second or more,
and to do it somewhat irregularly.

54. The Chronometer.—The pendulum-clock not being
portable, it is necessary to provide time-keepers that are.

The
chronometer is merely a carefully made watch, with a balance wheel
compensated to run, as nearly as possible, at the same rate in differ-
ent temperatures, and with a peculiar escapement, which, though
unsuited to watches exposed to ordinary rough usage, gives better

ASTRONOMICAL INSTRUMENTS.

35
results than any other when treated carefully.

The box-chronometer used on ship-board is usually about twice the
diameter of a common pocket watch, and is mounted on gimbals, so
as to keep horizontal at all times, notwithstanding the motion of the
vessel.

It usually beats half-seconds.

It is not possible to secure in
the chronometer-balance as perfect a temperature correction as in the
pendulum. For this and other reasons the best chronometers cannot
quite compete with the best clocks in precision of time-keeping; but
they are sufficiently accurate for most purposes, and of course are vastly
more convenient for field operations. They are simply indispensable at
sea. Never turn the hands of a chronometer backward.

Fig. 14.—A Chronograph by Warner and Swasey.

55. Before the invention of the telegraph it was customary to
note time merely “by eye and ear.” The observer, keeping his time-
piece near him, listened to the clock-beats, and estimated as closely
as he could, in seconds and tenths of seconds, the moment when the
phenomenon he was watching occurred—the moment, for instance,
when a star passed across a wire in the reticle of his telescope. At
present the record is usually made by simply pressing a “key” in the
hand of the observer, and this, by a telegraphic connection, makes

ASTRONOMICAL INSTRUMENTS.

36
a mark upon a strip or sheet of paper, which is moved at a uniform
rate by clock-work, and graduated by seconds-signals from the clock
or chronometer.

56. The Chronograph.—This is the instrument which carries
the marking-pen and moves the paper on which the time-record is
made. The paper is wrapped upon a cylinder, six or seven inches in
diameter, and fifteen or sixteen inches long. This cylinder is made
to revolve once a minute, by clock-work, while the pen rests lightly
upon the paper and is slowly drawn along by a screw-motion, so that
it marks a continuous spiral. The pen is carried on the armature of
an electro-magnet, which every other second (or sometimes every
second) receives a momentary current from the clock, causing it to
make a mark like those which break the lines in the figure annexed.

Fig. 15.—Part of a Chronograph Record.

The beginning of a new minute (the 60th sec.) is indicated either
by a double mark as shown, or by the omission of a mark. When
the observer touches his key he also sends a current through the
magnet, and thus interpolates a mark of his own on the record, as
at X in the figure: the beginning of the mark is the instant noted—
in this case 54.9s. Of course the minutes when the chronograph was
started and stopped are noted by the observer on the sheet, and
so enable him to identify the minutes and seconds all through the
record.

Many European observatories use chronographs in which the record
is made upon a long fillet of paper, instead of a sheet on a cylinder. The
instrument is lighter and cheaper than the American form, but much less
convenient.

The regulator of the clock-work must be a “continuous” regulator,
working continuously, and not by beats like a clock-escapement. There
are various forms, most of which are centrifugal governors, acting either
by friction (like the one in the figure) or by the resistance of the air; or
else “spring-governors,” in which the motion of a train, with a pretty
heavy fly-wheel, is slightly checked at regular intervals by a pendulum.

57. Clock-Breaks.—The arrangements by which the clock is
made to send regular electric signals are also various. One of the earliest

ASTRONOMICAL INSTRUMENTS.

37
and simplest is a fine platinum wire attached to the pendulum, which
swings through a drop of mercury at each vibration. All of the arrange-
ments, however, in which the pendulum itself has to make the electric
contact are objectionable, and for clocks using the Graham dead-beat
scapement no absolutely satisfactory means of giving the signals has yet
been devised. Clocks with the gravity escapements have a decided advan-
tage in this respect. Their wheel-work has no direct action in driving the
pendulum, and so may be made to do any reasonable amount of outside
work in the way of “key-manipulation” without affecting the clock-rate
in the least. Usually a wheel on the axis of the scape-wheel is made to
give the electric signals by touching a light spring with one of its teeth
every other second.

Chronometers are now also fitted up in the same way, to be used
with the chronograph.

The signals sent are sometimes “breaks” in a continuous current,
and sometimes “makes” in an open circuit. Usage varies in this respect,
and each method has its advantages. The break-circuit system is a little
simpler in its connections, and possibly the signals are a little more sharp,
but it involves a much greater consumption of battery material, as the
current is always circulating, except during the momentary breaks.

58. Meridian Observations.—A large proportion of all as-
tronomical observations are made at the time when the heavenly
body observed is crossing the meridian, or very near it. At that
moment the effects of refraction and parallax (to be discussed here-
after) are a minimum, and as they act only in a vertical plane, they
do not have any influence on the time at which the body crosses
the meridian.

59. The Transit Instrument is the instrument used, in con-
nection with a clock or chronometer, and often with a chronograph
also, to observe the time of a star’s “transit” across the meridian.

If the error of the (sidereal) clock is known at the moment, this
observation will determine the right ascension of the body, which, it
will be remembered, is simply the sidereal time at which it crosses
the meridian; i.e., the number of hours, minutes, and seconds by
which it follows the vernal equinox.

Vice versa, if the right ascension is known, the error or correction
of the clock will be determined.

The instrument (Fig. 16) consists essentially of a telescope
mounted upon a stiff axis perpendicular to the telescope tube. This

ASTRONOMICAL INSTRUMENTS.

38
axis is placed horizontal, east and west, and turns on pivots at its
extremities, in Y-bearings upon the top of two fixed piers or pil-
lars. A small graduated circle is attached, to facilitate “setting” the
telescope at any designated altitude or declination.

Fig. 16.—The Transit Instrument (Schematic).

The telescope carries at the eye-end, in the focal plane of the
object-glass, a reticle of some odd number of vertical wires,—five
or more,—one of which is always in the centre, and the others are
usually placed at equal distances on each side of it. One or two
wires also cross the field horizontally.

If the pivots are true, and the instrument accurately adjusted,
it is evident that the central vertical wire will always follow the
meridian as the instrument is turned ; and the instant when a star
crosses this wire will be the true moment of the star’s meridian
transit. The object in having a number of wires is, of course, simply
to gain accuracy by taking the mean of a number of observations
instead of depending upon a single one.

In order to “level ” the axis properly, a delicate spirit-level is
an essential adjunct; it is usual, also, (and important) to provide a
convenient “reversing apparatus,” by which the instrument can be
turned half round, making the eastern and western pivots change
places.

ASTRONOMICAL INSTRUMENTS.

39

The instrument must be thoroughly stiff
and rigid, without loose joints or shaky screws;
and the two pivots must be accurately round,
precisely in line with each other, free from ta-
per, and precisely of the same size; all of which
conditions may be summed up by saying that
they must be portions of one and the same ge-
ometrical cylinder.

Fig. 17.—Reticle of

The proper construction and grinding of these
the Transit
pivots, which are usually of hard bell metal (some-

Instrument.

times of steel), taxes the art of the most skilful
mechanician. The level, also, is a delicate instrument, and difficult to
construct.

Provision is made, of course, for illuminating the field of view at
night so as to make the reticle wires visible. Usually one (or both) of the
pivots is pierced, and a lamp throws light through the opening upon a
small mirror in the centre of the tube, which reflects it down upon the
reticle.

The Y’s are used instead of round bearings, in order to prevent any
rolling or shake of the pivots as the instrument turns.

Fig. 18 shows a modern transit instrument (portable) as actually
constructed by Fauth & Co.

Another form of the instrument is much used, which is often
designated as the “Broken Transit.” A reflector in the central cube
throws the rays coming from the object-glass, out at right angles
through one end of the axis, where the eye-piece is placed; so that
the observer does not have to change his position at all for different
stars, but simply looks straight forward horizontally.

It is very
convenient and rapid in actual work, but the observations require a
considerable correction for flexure of the axis.

60. Adjustments.—(1) Focus and verticality of wires. (2) Collima-
tion. (3) Level. (4) Azimuth.

First. The first thing to do after the instrument is set on its supports
and the axis roughly levelled, is to adjust the reticle. The eye-piece is
drawn out or pushed in until the wires appear perfectly sharp, and then
the instrument is directed to a star or to some distant object (not less
than a mile away), and without disturbing the eye-piece, the sliding-tube,
which carries the reticle, is drawn out or pushed in until the object is also
distinct at the same time with the wires. If this adjustment is correctly
made, motion of the eye in front of the eye-piece will not produce any

ASTRONOMICAL INSTRUMENTS.

40
apparent displacement of the object in the field, with reference to the
wires. To test the verticality of the wires, the telescope is moved up
and down a little, while looking at the object; if the axis is level and
the wires vertical, the wire will not move off from the object sideways.

There are screws provided to turn the reticle a little, so as to effect this
adjustment.

Fig. 18.—A 3-inch Transit, with reversing apparatus. Fauth & Co.

When the wires have been thus adjusted for focus and verticality,
the reticle-slide should be tightly clamped and never disturbed again.

The eye-piece can be moved in and out at pleasure, to secure distinct

ASTRONOMICAL INSTRUMENTS.

41
vision for different eyes, but it is essential that the distance between the
object-glass and the reticle remain constant.

Second.

Collimation.

The line joining the optical centre of the
object-glass with the middle wire of the reticle is called the “line of
collimation,” and this line must be made exactly perpendicular to the
axis of rotation by moving the reticle slightly to one side or the other by
means of the adjusting screws provided for the purpose. The simplest
way of effecting the adjustment is to point the instrument on some well-
defined distant object, like a nail-head or a joint in brickwork, and then
carefully to “reverse” the instrument without disturbing the stand. If
the middle wire, after reversal, points just as it did before, the “collima-
tion” is correct; if not, the middle wire must be moved half way towards
the object by the screws.

Collimator.—It is not always easy to find a distant object on which
to make this adjustment, and a “collimator ” may be substituted with
advantage. This is simply a telescope mounted horizontally on a pier
in front of the transit instrument, so that when the transit telescope is
horizontal, it can look straight into the collimator, which ought to be of
about the same size as the transit itself.

In the focus of the collimator object-glass are placed two wires form-
ing an X, and thus placed they can be seen by a telescope looking into
the collimator just as distinctly as if they were at an infinite distance and
really celestial objects. The instrument furnishes us a mark optically ce-
lestial, but mechanically within reach of our finger-ends for illumination,
adjustment, etc. If the pier on which it is mounted is firm, the collimator
cross is in all respects as good as a star, and much more convenient.

Third. Level. The adjustment for level is made by setting a striding
level on the pivots of the axis, reading the level, then reversing the level

(not the transit) and reading it again. If the pivots are round and of the
same size, the difference between the level-readings direct and reversed
will indicate the amount by which one pivot is higher than the other.

One of the Y’s is made so that it can be raised and lowered slightly by
means of a screw, and this gives the means of making the axis horizontal.

If the pivots are not of the same size (and they never are absolutely), the
astronomer must determine and allow for the difference.

Fourth. Azimuth. In order that the instrument may indicate the
meridian truly, its axis must lie exactly east and west; i.e., its azimuth
must be 90◦. This adjustment must be made by means of observations
upon the stars, and is an excellent example of the method of successive
approximations, which is so characteristic of astronomical investigation.

(a) After adjusting carefully the focus and collimation of the instrument,

ASTRONOMICAL INSTRUMENTS.

42
we set it north and south by guess, and level it as precisely as possible.

By looking at the pole star, and remembering how the pole itself lies
with reference to it, one can easily set the instrument pretty nearly; i.e.,
within half a degree or so. The middle wire will now describe in the sky
a vertical circle, which crosses the meridian at the zenith, and lies very
near the meridian for a considerable distance each side of the zenith.

(b) We must next get an “approximate” time; i.e., set our clock or
chronometer nearly right. To do this, we select from the list of several
hundred stars in the Nautical Almanac (which is to be regarded in about
the same light with the clock and the spirit level, as an indispensable
accessory to the transit) a star which is about to cross the meridian near
the zenith. The difference between the right ascension of the star as
given in the Almanac, and the time shown by the clock-face, will be very
nearly the error of the clock at the time of the observation: not exactly,
unless the declination of the star is such that it passes exactly through
the zenith, but very nearly, since the star crosses the meridian near the
zenith. We now have the time within a second or two.

(c) Next turn down the telescope upon some Almanac star, which
is soon to cross the meridian within 10◦ of the pole. It will appear to
move very slowly. A little before the time it should reach the meridian,
move the whole frame of the instrument until the middle wire points
upon it, and then, by means of the “Azimuth Screw,” which gives a
slight horizontal motion to one of the Y’s, follow the star until the in-
dicated moment of its transit ; i.e., until the clock (corrected for clock
error) shows on its face the star’s right ascension. If the clock correction
had been known with absolute exactness, the instrument would now be
truly in the meridian; as the clock error, however, is only approximate,
the instrument will only be approximately in the meridian; but—and
this is the essential point—it will be very much more nearly so than at
the beginning of the operation. The supposed incorrectness, amounting
perhaps to one or two seconds, in the time at which the instrument was
set on the circumpolar star will, on account of the slow motion of the
star, make almost no perceptible difference in the direction given to the
axis.

A repetition of the operation may possibly be needed to secure all the
desired precision. The accuracy of this azimuth adjustment can then be
verified by three successive “culminations” or transits of the pole star,
or any other circumpolar. The interval occupied in passing from the
upper to the lower culmination on the west side of the meridian ought,
of course, to be exactly equal to the time on the eastern side; i.e., twelve
sidereal hours.

ASTRONOMICAL INSTRUMENTS.

43

61. The final test of all the adjustments, and of the accurate
going of the clock, is obtained by observing a number of Almanac
stars of widely different declination. If they all indicate identically
the same clock correction, the instrument is in adjustment; if not,
and if the differences are not very great, it is possible to deduce from
the observations themselves the true clock error, and the adjustment
errors of the instrument.

It is to be added, in this connection,
that the astronomer can never assume
that adjustments are perfect : even if
once perfect, they would not stay so,
on account of changes of temperature
and other causes. Nor are observations
ever absolutely accurate. The problem
is, from observations more or less in-
accurate but honest, with instruments
more or less maladjusted but firm, to
find the result that would have been ob-
tained by a perfect observation with a
perfect and perfectly adjusted instru-
ment. It can be more nearly done than

Fig. 19.—The Meridian Circle
one might suppose. But the discussion

(Schematic).

of the subject belongs to Practical As-
tronomy, and cannot be entered into here.

62. Prime Vertical Instrument.—For certain purposes, a

Transit Instrument, provided with an apparatus for rapid reversal,
is turned quarter-way round and mounted with the axis north and
south, so that the plane of rotation lies east and west, instead of in
the meridian. It is then called a Prime Vertical Transit.

63. The Meridian Circle.—In order to determine the Decli-
nation or Polar Distance of an object, it is necessary to have some
instrument for measuring angles; mere time-observations will not
suffice. The instrument most used for this purpose is the Merid-
ian Circle, or Transit Circle, which is simply a transit instrument,
with a graduated circle attached to its axis, and revolving with the
telescope. Sometimes there are two circles, one at each end of the
axis.

Fig. 19 represents the instrument “schematically,” showing
merely the essential parts. Fig. 20 is a meridian circle, with a 4-inch

ASTRONOMICAL INSTRUMENTS.

44
telescope, constructed by Fauth & Co.

Fig. 20.—Meridian Circle.

A, B, C, D, the Reading Microscopes.

F , the Clamp. G, the Tangent Screw.

K, the Graduated Circle.

LL, the Level, only placed in position

H, the Roughly Graduated Setting
occasionally.

Circle.

M , the Right Ascension Micrometer.

I, the Index Microscope. This is usu-

W W , Counterpoises, which take part
ally, however, placed half way be-
of the weight of the instrument off
tween A and D.

from the Y’s.

In observatory instruments the circle is usually from two to four feet
in diameter; larger circles were once used, but it is found that their
weight, and the consequent strains and flexures, render them actually
less accurate than the smaller ones. The utmost resources of mechanical
art are exhausted in making the graduation as precise as possible and in

ASTRONOMICAL INSTRUMENTS.

45
providing for its accurate reading, as well as in securing the maximum
firmness and stability of every part of the instrument. The actual divi-
sions are usually 5 apart (in very large instruments sometimes only 2 ),
but the circle is “read” to seconds and tenths of seconds of arc by means
of reading microscopes, from two to six in number, fixed to the pier of
the instrument. In a circle of forty inches diameter, 1 is a little less than

1
of an inch, ( 20
inch), so that the necessity of fine workmanship

10000

206265
is obvious.

64. The Reading Microscope (Fig. 21).—This consists es-
sentially of a compound microscope, which forms a magnified image
of the graduation at the focus of its object-glass, where this image
is viewed by a positive eye-piece. At the place where the image is
formed a pair of parallel spider-lines or a cross is placed, movable in
the plane of the image by a “micrometer screw”; i.e., a fine screw
with a graduated head, usually divided into sixty parts. One revolu-
tion of the screw carries the wire 1 of arc, which makes one division
of the screw-head 1 , the tenths of seconds being estimated.

The adjustment of the microscope for

“runs,” as it is called (that is, to make one revo-
lution of the micrometer screw exactly equal to

1 ), is effected as follows. By setting the wires
first on one of the graduation marks visible in
the field of view, and then on the next mark, it
is immediately evident whether five revolutions
of the screw “run” over or fall short of 5 of
the graduation. If they overrun, it shows that
the image of the graduation formed by the mi-
croscope objective is too small to fit the screw,
and vice-versa. Now, by simply increasing or
decreasing the distance A B between the objec-
tive and the micrometer box, the size of the
image can be altered at will, and the objective

Fig. 21.—The Reading

Microscope.

is therefore so mounted that this can be done.

Of course, every change in the length of the mi-
croscope tube will also require a readjustment of the distance between
the “limb,” or graduated surface, of the circle and the microscope, in or-
der to secure distinct vision; but by a few trials the adjustment is easily
made sufficiently precise.

The reading of the circle is as follows: An extra index-microscope,
with low power and large field of view, shows by inspection the degrees
and minutes. The reading-microscopes are only used to give the odd

ASTRONOMICAL INSTRUMENTS.

46
seconds, which is done by turning the screw until the parallel spider-
lines are made to include one of the graduation lines half-way between
themselves; the head of the screw then shows directly the seconds and
tenths, to be added to the degrees and minutes shown by the index.

Thus in Fig. 22, the reading of the microscope is 3 22 .1, the 3 being
given by the scale in the field, the 22 .1 by the screw-head.

65. Method of observing
a Star.—A minute or two be-
fore the star reaches the meridian
the instrument is approximately
pointed, so that the star will
come into the field of view. As
soon as it makes its appearance,

Fig. 22.—Field of View of Reading

Microscope.

the instrument is moved by the
slow-motion tangent-screw until
the star is “bisected” by the fixed horizontal wire of the reticle, and
the star is kept bisected until it reaches the middle vertical wire
which marks the meridian. The microscopes are then read, and
their mean result is the star’s “circle-reading.”

Frequently the star is bisected, not by moving the whole instrument,
but by means of a “micrometer wire,” which moves up and down in the
field of view. The micrometer reading then has to be combined with the
reading of the microscope, to get the true circle-reading.

66. Zero Points.—In determining the declination or meridian
altitude of a star by means of its circle-reading, it is necessary to
know the “zero point ” of the circle. For declinations, the “zero
point” is either the polar or the equatorial reading of the circle; i.e.,
the reading of the circle when the telescope is pointed at the pole
or at the equator.

The “polar point” may be found by observing some circumpolar
star above the pole, and again, twelve hours later, below it. When
the two circle-readings have been duly corrected for refraction and
instrumental errors, their mean will be the polar point.

Suppose, for instance, that δ Ursæ Minoris, at the “upper culmina-
tion,” gives a corrected reading of 52◦ 18 25 .3, while at the lower culmi-
nation the reading is 45◦ 31 35 .7, then the mean of these, 48◦ 55 00 .5,
is the polar point, and of course the equatorial reading is 138◦ 55 00 .5,—
just 90◦ greater.

The polar distance of the star would be the half-
difference of the two readings, or 3◦ 23 24 .8.

ASTRONOMICAL INSTRUMENTS.

47

67. Nadir Point.—The determination of the polar point re-
quires two observations of the same star at an interval of twelve
hours. It is often difficult to obtain such a pair; moreover, the
refraction complicates the matter, and renders the result less trust-
worthy. Accordingly it is now usual to use the nadir or the horizon-
tal reading as the zero, rather than the polar point.

The nadir point is determined by pointing the
telescope downwards to a basin of mercury, mov-
ing the telescope until the image of the horizontal
wire of the reticle, as seen by reflection, coincides
with the wire itself.

Since the reticle is exactly
in the principal focus of the object-glass, rays of

Fig. 23.

light emitted by any point in the reticle will be-

The Collimating
come a parallel beam after passing the lens, and

Eye-Piece.

if this beam strikes a plane mirror perpendicularly
and is returned, the rays will come just as if from a real object in the
sky, and will form an image at the focal plane. When, therefore, the
image of the central wire of the reticle, seen in the mercury basin
by reflection, coincides with the wire itself, we know that the line
of collimation must be exactly perpendicular to the surface of the
mercury; i.e., vertical.

To make the image visible it is necessary to illuminate the reticle
by light thrown towards the object-glass from behind the wires, instead
of light coming from the object-glass towards the eye as usual. This
peculiar illumination is commonly effected by means of Bohnenberger’s

“collimating eye-piece,” shown in Fig. 23. In the simplest form it is
merely a common Ramsden eye-piece, with a hole in one side, and a thin
glass plate inserted at an angle of 45◦. A light from one side, entering
through the hole, will be (partially) reflected towards the wires, and will
illuminate them sufficiently.

The horizontal point of course differs just 90◦ from the nadir point.

It may also be found independently by noting the circle-readings of some
star observed one night directly, and the next night by reflection in mer-
cury; or, if the star is a close circumpolar, both observations may be
made the same evening, one a few minutes before its meridian passage,
the other just as long after. But the method of the collimating eye-piece
is fully as accurate and vastly more convenient.

ASTRONOMICAL INSTRUMENTS.

48

Fig. 24.—Altitude and Azimuth Instrument.

68. Differential Use of the Instrument.—We now know the
places of several hundred stars with so much precision that in many
cases it is quite sufficient to observe one or two of these “standard stars”

in connection with the bodies whose places we wish to determine. The
difference between the declination of the known star and that of any star
whose place is to be determined, will, of course, be simply the difference
of their circle-readings, corrected for refraction, etc. The meridian circle
is said to be used “differentially” when thus treated.

69. Errors of Graduation, etc.—If the circle is from a rep-
utable maker, and has four or six microscopes, and if the observa-
tions are carefully made and all the microscopes read each time,

ASTRONOMICAL INSTRUMENTS.

49
results of sufficient precision for most purposes may be obtained by
merely correcting the observations for “runs” and refraction. The
outstanding errors ought not to exceed a second or two. But when
the tenths of a second are in question, the case is different.

It
will not then do for the astronomer to assume the accuracy of the
graduation of his circle, but he must investigate the errors of its di-
visions, the errors of the micrometer screws in the microscopes, the
flexure of the telescope, and the effect of differences of temperature
in shifting the zero points of the circle, by slightly disturbing the
position or direction of the microscopes. Of course this is not the
place to enter into such details, but it is an opportunity to impress
again upon the student the fact that truth and accuracy are only
attainable by immense painstaking and labor.

70. Mural Circle.—This instrument is in principle the same as
the meridian circle, which has superseded it.

It consists of a circle,
carrying a telescope mounted on the face of a wall of masonry (as its
name implies) and free to revolve in the plane of the meridian. The wall
furnishes a convenient support for the microscopes.

71. Altitude and Azimuth Instru-
ment.—Since the transit instrument and
meridian circle are confined to the plane of
the meridian, their usefulness is obviously lim-
ited. Meridian observations, when they are to
be had, are better and more easily used than
any others, but are not always attainable. We
must therefore have instruments which will
follow an object to any part of the heavens.

The altitude and azimuth instrument is
simply a surveyor’s theodolite on a large scale.

It has a horizontal circle turning upon a verti-

Fig. 25.

cal axis, and read by verniers or microscopes.

The Equatorial

Upon this circle, and turning with it, are sup-

(Schematic).

ports which carry the horizontal axis of the
telescope with its vertical circle, also read by microscopes. Ob-
viously the readings of these two circles, when the instrument is
properly adjusted and the zero points determined, will give the al-
titude and azimuth of the body pointed on. Fig. 24 represents a
small instrument of this kind.

ASTRONOMICAL INSTRUMENTS.

50

Fig. 26.—The 23-inch Princeton Telescope.

72. The Equatorial.—The essential characteristic of this in-
strument is that its principal axis, i.e., the axis which rests in fixed
bearings, instead of being either horizontal or vertical, is inclined
at an angle equal to the latitude of the place, and directed towards
the pole, thus placing it parallel to the earth’s axis of rotation. This
axis of the instrument is called its polar axis; and the graduated cir-
cle which it carries, and which is parallel to the celestial equator, is
called the hour-circle, because its reading gives the hour-angle of the
body upon which the telescope happens to be pointed. Sometimes,
also, it is called the Right Ascension Circle. Upon this polar-axis
are secured the bearings of the declination axis, which is perpen-
dicular to the polar axis, and carries the telescope itself and the
declination circle.

In the instruments before described, the telescope is a mere
pointer, and wholly subsidiary to the circles; in the equatorial the

ASTRONOMICAL INSTRUMENTS.

51
telescope is usually the main thing, and the circles are subordinate,
serving only to aid the observer in finding or identifying the body
upon which the telescope is directed.

Fig. 25 exhibits schematically the ordinary form of equatorial
mounting, of which there are numerous modifications. Fig. 26 is
the 23-inch Clark telescope at Princeton, and Fig. 27 is the 4-foot

Melbourne reflector. The frontispiece is the great Lick telescope of
thirty-six inches diameter.

Fig. 27.—The Melbourne Reflector.

The advantages of the equatorial mounting for a large telescope
are very great as regards convenience. In the first place, when the
telescope is once pointed upon a star or planet, it is only necessary
to turn the polar axis with a uniform motion in order to “follow”

the star, which otherwise would be carried out of the field of view
in a few moments by the diurnal motion. This motion, since it
is uniform, can be, and in all large instruments usually is, given
by clock-work, with a continuous regulator of some kind, similar
to that used in the chronograph. The instrument once directed

ASTRONOMICAL INSTRUMENTS.

52
and clamped, and the clock-work started, the object will continue
apparently immovable in the field of view as long as may be desired.

In the next place, it is very easy to find an object, even if in-
visible to the naked eye, like a faint comet or nebula, or a star in
the daytime, provided we know its declination and right ascension,
and have the sidereal time; for which reason a sidereal clock or
chronometer is an indispensable adjunct of the equatorial.

To find an object, the telescope is turned in declination until the
reading of the declination circle corresponds to the declination of the
object, and then the polar axis is turned until the hour-circle of the
instrument (not to be confounded with an hour-circle in the sky) reads
the hour-angle of the object. This hour-angle, it will be remembered, is
simply the difference between the sidereal time and the right ascension
of the object. The hour-angle is east if the right ascension exceeds the
time; west, if it is less. When the telescope is thus set, the object will
be found (with a low magnifying power) in the field of view, unless it is
near the horizon, in which case refraction must be taken into account.

While the instrument cannot give very accurate determinations
of the positions of bodies by the direct readings of its circles, on
account of the irregular flexures of its axes, it may do so indirectly;
that is, it may be used to determine very accurately the difference
between the right ascension and declination of a comet or planet,
for instance, and that of some neighboring star, whose place has
been already determined by the meridian circle; and this is one of
the most important uses of the instrument.

73. The Micrometer.—Micro-
meters of various sorts are employed
for the purpose.

The most com-
mon and most generally useful is the
so-called “filar position-micrometer,”

Fig. 28, which is an indispensable
auxiliary of every good telescope.

It is a small instrument, much like
the upper part of the reading micro-
scope, but more complicated. It usu-
ally contains a reticle of fixed wires,
two or three parallel to each other,
and crossed at right angles by a sec-

Fig. 28.—The Filar
ond set. Then there are two or three

Position-Micrometer.

ASTRONOMICAL INSTRUMENTS.

53
wires parallel to the first set, and movable by an accurately made
screw with a graduated head and a counter, or scale, for indicating
the number of entire revolutions made by the screw. The box con-
taining these wires, and carrying the eye-piece and screw, can itself
be turned around in a plane perpendicular to the optical axis of
the telescope, and set in any desired position; for example, so that
the movable wires shall be parallel to the celestial equator, while
the other set run north and south. This “position angle” is read on
a graduated circle, which forms part of the instrument. Means of
illumination are provided, giving at pleasure either dark wires in a
bright field, or vice versa.

Fig. 29.—Construction of the Micrometer.

With this instrument one can measure the distance (in seconds of
arc), and the direction between any two stars which are near enough
to be seen at once in the same field of view. This range in small tele-
scopes may reach 30 of arc; while in the larger instruments, which,
with the same eye-pieces have much higher magnifying powers, it is
necessarily less,—not more than from 5 to 10 .

74. A new form of equatorial, known as the Equatorial Coudé, or

Elbowed Equatorial, has been recently introduced at the Paris Obser-
vatory. With large instruments of the ordinary form a great deal of
inconvenience is encountered by the observer, in moving about to follow
the eye-piece into the various positions into which it is forced by the
inconsiderateness of the heavenly bodies. Moreover, the revolving dome,
which is usually erected to shelter a great telescope, is an exceedingly
cumbrous and expensive affair.

In the Equatorial Coudé, Fig. 30, these difficulties are overcome by
the use of mirrors. The observer sits always in one fixed position, looking
obliquely down through the polar axis, which is also the telescope tube.

The Paris instrument has an object-glass about ten inches in diame-
ter, and performs very satisfactorily. The two reflections, however, cause
a considerable loss of light, and some injury to the definition. The mir-

ASTRONOMICAL INSTRUMENTS.

54
rors, and the consequent complications, also add heavily to the cost of
the instrument. Fig. 30 is from a photograph of this instrument.

Fig. 30.—The Equatorial Coudé.

75. All the instruments so far described, except the chronome-
ter, are fixed instruments; of use only when they can be set up
firmly and carefully adjusted to established positions. Not one of
them would be of the slightest use on ship-board.

We have now to describe the instrument which, with the help
of the chronometer, is the main dependence of the mariner. It is
an instrument with which the observer measures the angular dis-
tance between two objects; as, for instance, the sun and the visible
horizon, not by pointing first on one and then afterwards on the
other, but by sighting them both, simultaneously and in apparent
coincidence; which can be done even when he has no fixed position
or stable footing.

76. The Sextant.—The graduated limb of the sextant is car-
ried by a light framework, usually of metal, provided with a suitable
handle X. The arc is about one-sixth of a circle, as the name im-
plies, and is usually from five to eight inches radius. It bears a
graduation of half-degrees, numbered as whole degrees, so that it
can measure any angle less than 120◦.

An “index-arm,” M N in the figure, is pivoted at the centre
of the arc, and carries a vernier which slides along the limb, and

ASTRONOMICAL INSTRUMENTS.

55
can be fixed at any point by a clamp and delicately moved by the
attached tangent screw, T . The reading of this vernier gives the
angle measured by the instrument. The best instruments read to

10 .

Just over the centre of motion, the “index-mirror ” M , about
two inches by one and one-half in size, is fastened securely to the
index-arm, so as to be perpendicular to the plane of the limb. At

H, the “horizon-glass,” about an inch wide and of the same height
as the index-glass, is secured firmly to the frame of the instrument,
in such position that, when the vernier of the index-arm reads zero,
the index-mirror and horizon-glass will be parallel to each other.

Only half of the horizon-glass is silvered, the upper half being left
transparent. E is a small telescope.

Fig. 31.—The Sextant.

If the vernier stands near, but not at zero, the observer looking
into the telescope will see together in the field of view two separate
images of the object; and if, while still looking, he slides the vernier
a little, he will see that one of the images remains fixed, while the
other moves. The fixed image is due to the rays which reach the
object-glass of the telescope directly, coming through the unsilvered
half of the horizon-glass; the movable image, on the other hand, is
produced by rays which have suffered two reflections,—first, from
the index-mirror to the horizon-glass; and second, at the lower half
of the horizon-glass. When the two mirrors are parallel, and the

ASTRONOMICAL INSTRUMENTS.

56
vernier reads zero, the two images coincide, provided the object is
at a considerable distance.

If now the vernier does not stand at or near zero, the observer,
looking at any object directly through the horizon-glass, will see, not
only that object, but also whatever other object is so situated as to
send its rays to the telescope by reflection upon the mirrors; and the
reading of the vernier will give the angle at the instrument between
the two objects whose images thus coincide; the angle between the
planes of the two mirrors being just half that between the objects,
and the half-degrees on the limb being numbered as whole ones.

77. The principal use of the instrument is in measuring the
altitude of the sun. At sea the observer, holding the instrument with
his right hand and keeping the plane of the arc vertical, looks directly
towards the visible horizon at the point under the sun, through the
horizon-glass (whence its name); then by moving the vernier with
his left hand, he inclines the index-glass upwards until one edge of
the reflected image of the sun is brought just to touch the horizon-
line, noting the exact time by the chronometer, if necessary. The
reading of the vernier, after correcting for the semi-diameter of the
sun, the dip of the horizon, the refraction, and the parallax (and for
the “index-error” of the sextant, if the vernier does not read strictly
zero when the mirrors are parallel) gives the sun’s true altitude at
the moment.

78. On land the visible horizon is of no
use, and we have recourse to an “artificial
horizon,” as it is called.

This is merely a
shallow basin of mercury, covered, when nec-
essary to protect it from the wind, with a
roof made of glass plates having their sides
plane and parallel.

In this case we measure the angle be-
tween the sun’s image reflected in the mer-
cury and the sun itself. The reading of the
instrument, corrected for index-error, gives
twice the sun’s apparent altitude; which ap-

Fig. 32.—Principle of the

Sextant.

parent altitude, corrected as before for re-
fraction and parallax, but not for dip of the
horizon, gives the true altitude. The skilful use of the sextant requires
steadiness of hand and considerable dexterity, and from the small size of

ASTRONOMICAL INSTRUMENTS.

57
the telescope the angles measured are of course less precise than if deter-
mined by large fixed instruments. But its portability and applicability
at sea render it absolutely invaluable.

79. The principle that the true angle between the objects whose
images coincide is twice the angle between the mirrors (or between their
normals) is easily demonstrated as follows (Fig. 32):—

The ray SM coming from an object, after reflection first at M (the
index-mirror), and then at H (the horizon-glass), is made to coincide
with the ray OH coming from the horizon. We must prove that the
angle SEO, between the object and the horizon, as seen from the point

E in the instrument, is double the angle Q, between M Q and HQ, which
are normals to the mirrors, and therefore double Q , which is the angle
between the planes of the mirrors.

First, from the law of reflection, we have,

SM P = HM P, or SM H = 2 × P M H.

Similarly,

M HE = 2 × M HQ.

From the geometric principle that the exterior angle SM H of the
triangle HM E is equal to the sum of the opposite interior angles at H
and E, we get

HEM = SM H − M HE = 2 P M H − 2 M HQ = 2(P M H − M HQ).

Similarly, from the triangle HM Q, we have

HQM = P M H − M HQ,
which is half the value just found for HEM , and proves the proposition.

Of course with the sextant, as with all other instruments, it is
necessary for the observer who aims at the utmost precision to inves-
tigate, and take into account its errors of graduation, construction
and adjustment; but their discussion does not belong here.

80. Besides the instruments we have described, there are many
others designed for special work, some of which, as the zenith tele-
scope, and heliometer, will be mentioned hereafter as it becomes
necessary.

There is also a whole class of physical instruments,
photometers, spectroscopes, heat-measuring appliances, and pho-
tographic apparatus, which will have to be considered in due time.

But with clock, meridian circle, and equatorial and their usual
accessories, all the fundamental observations of theoretical and

ASTRONOMICAL INSTRUMENTS.

58
spherical astronomy can be supplied. The chronometer and sex-
tant are practically the only astronomical instruments of any use at
sea.

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

59

CHAPTER III.

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS, DIP OF

THE HORIZON, PARALLAX, SEMI-DIAMETER, REFRAC-

TION, AND TWILIGHT.

81. Dip of the Horizon.—In observations of the altitude of a
heavenly body at sea, where the measurement is made from the sea-
line, a correction is needed on account of the fact that this visible
horizon does not coincide with the true astronomical horizon (which
is 90◦ from the zenith), but falls sensibly below it by an amount
known as the Dip of the Horizon. The amount of this dip depends
upon the size of the earth and the height of the observer’s eye above
the sea-level.

In Fig. 33, C is the centre of the earth, AB a
portion of its level surface, and O the observer,
at an elevation h above A. The line OH is truly
horizontal, while the tangent line, OB, corre-
sponds to the line drawn from the eye to the vis-
ible horizon. The angle HOB is the dip. This is
obviously equal to the angle OCB at the centre
of the earth, if we regard the earth as spherical,
as we may do with quite sufficient accuracy for

Fig. 33.—Dip of
the purpose in hand.

the Horizon.

From the right-angled triangle OBC we have
directly

BC
cos OCB =

.

CO

Putting R for the radius of the earth, and ∆ for the dip, this be-
comes

R
cos ∆ =

.

R + h

This formula is exact, but inconvenient, because it gives the small
angle ∆ by means of its cosine. Since, however, 1 − cos ∆ = 2 sin2 1 ∆,

2

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

60
we easily obtain the following:—
h
sin 1 ∆ =

.

2

2(R + h)

This gives the true depression of the sea horizon, as it would be if
the line of sight, drawn from the eye to the horizon line, were straight.

On account of refraction it is not straight, however, and the amount of
this “terrestrial refraction” is very variable and uncertain. It is usual
to diminish the dip computed from the formula by one-eighth its whole
amount.

An approximate formula1 for the dip is

∆ (in minutes of arc) =
h (feet);

1

This approximate formula may be obtained thus:—
h
h
h

2 sin2( 1 ∆) =

=

+

1 +

.

2

R + h

R

R
h

But since
is a very small fraction, it may be neglected in the divisor

R
h

1 +

, and the expression becomes simply,

R
h
h

2 sin2 1 ∆ =

; whence sin 1 ∆ =

.

2

R

2

2R

Since ∆ is a very small angle,

∆ = sin ∆ = 2 sin 1 ∆,
so that

2
h
h

∆ (in radians) = 2

=

.

2R

1 R

2

To reduce radians to minutes, we must multiply by 3438, the number of
minutes in a radian. Accordingly,
h

∆ (in minutes of arc) = 3438

.

1 R

2

If we express h in feet, we must also use the same units for R. The mean
radius of the earth is about 20,884,000 feet, one-half of which is 10,442,000, and
the square root of this is 3231; so that the formula becomes

3438

∆ =
h (feet),

3231
which is near enough to that given in the text.

In fact, the refraction makes so much difference that after taking the nu-

3438
merical factor,

, as unity, the formula still gives ∆ about 1 part too

3231

20
large.

The formula ∆ =

3h (metres) is yet more nearly correct.

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

61
or, in words, the square root of the elevation of the eye (in feet) gives
the dip in minutes. This gives a value about 1 part too large.

20

Since the dip is applicable only to sextant observations made at
sea, where, from the nature of the instrument, and the rising and
falling of the observer with the vessel’s motion, it is not possible to
measure altitudes more closely than within about 15 , there is no
need of any extreme precision in its calculation.

82. Parallax.—In the most general sense, “parallax” is the
change of a body’s direction resulting from the observer’s displace-
ment.

In the restricted and technical sense in which we are to
employ it now, it may be defined as the difference between the di-
rection of a body as actually observed and the direction it would have
if seen from the earth’s centre. Thus in the figure, Fig. 34, where
the observer is supposed to be at O, the position of P in the sky (as
seen from O) would be marked by the point where OP produced
would pierce the celestial sphere. Its position as seen from C would
be determined in the same way by producing CP to which OX is
drawn parallel. The angle P OX, therefore, or its equal, OP C, is
the parallax of P for an observer at O.

Obviously, from the figure, we may
also give the following definition of the
parallax.

It is the angular distance

(number of seconds of arc) between the
observer’s station and the centre of the
earth’s disc, as seen from the body ob-
served. The moon’s parallax at any mo-
ment for me is my angular distance from
the earth’s centre, as seen by “the man
in the moon.”

Fig. 34.—Diurnal Parallax.

When a body is in the zenith its par-
allax is zero, and it is a maximum at the horizon. In all cases
it depresses a body, diminishing the altitude without changing the
azimuth.

The “law” of the parallax is, that it varies as the sine of the
zenith distance directly, and inversely as the linear distance (in
miles) of the body.

This follows easily from the triangle COP , where we have P C :

OC = sin COP : sin CP O.

Put D for P C, the distance of the body from the earth; R for the

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

62
earth’s radius, CO; CP O, the parallax; ζ for ZOP , the apparent
zenith distance, and remember that the sine of ζ is equal to the
sine of its supplement, COP : we then have as the translation of the
above proportion,

D : R = sin ζ : sin p.

R

This gives us
sin p =

· sin ζ;

D
or, since p is always a small angle,

R
p = 206265

· sin ζ.

D

83. Horizontal Parallax.—When a body is at the horizon (Ph
in the figure), then ζ becomes 90◦, and sin ζ = 1. In this case the
parallax reaches its maximum value, which is called the horizontal
parallax of the body. Taking ph as the symbol for this, we have

R

R
sin ph =

; or, nearly enough, ph = 206265

.

D

D

Comparing this with the formula above, we see that the parallax of a
body at any zenith distance equals the horizontal parallax multiplied
by the sine of the zenith distance; i.e., p = ph sin ζ.

N.B. A glance at the figure will show that we may define the
horizontal parallax, OP C, of any body, as the angular semi-diameter
of the earth seen from that body. To say, for instance, that the sun’s
horizontal parallax is 8 .8, amounts to saying that, seen from the
sun, the earth’s apparent diameter is twice 8 .8, or 17 .6.

84. Relation between Horizontal Parallax and Distance.

—Since we have

R
sin ph =

,

D
it follows of course that D = R ÷ sin ph;

206265
or, (nearly)

D =

× R.

ph

If the sun’s parallax equals 8 .8,

206265
its distance =

× R = 23439 R.

8.8

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

63

85. Equatorial Parallax.—Owing to the “ellipticity” or

“oblateness” of the earth the horizontal parallax of a body varies
slightly at different places, being a maximum at the equator, where
the distance of an observer from the earth’s centre is greatest. It is
agreed to take as the standard the equatorial horizontal parallax;
i.e., the earth’s equatorial semi-diameter as seen from the body.

86. Diurnal Parallax.—The parallax we have been discussing
is sometimes called the diurnal parallax, because it runs through all
its possible changes in one day.

When the sun, for instance, is rising, its parallax is a maximum,
and by throwing it down towards the east, increases its apparent right
ascension. At noon, when the sun is on the meridian, its parallax is
a minimum, and affects only the declination. At sunset it is again a
maximum, but now throws the sun’s apparent place down towards the
west. Although the sun is invisible while below the horizon, yet the par-
allax, geometrically considered, again becomes a minimum at midnight,
regaining its original value at the next sunrise.

The qualifier, “diurnal,” is seldom used except when it is nec-
essary to distinguish between this kind of parallax and the annual
parallax of the fixed stars, which is due to the earth’s orbital mo-
tion. The stars are so far away that they have no sensible diurnal
parallax (the earth is an infinitesimal point as seen from them); but
some of them do have a slight and measurable annual parallax, by
means of which we can roughly determine their distances.

(Chap.

XIX.)

87. Smallness of Parallax.—The horizontal parallax of even
the nearest of the heavenly bodies is always small. In the case of the
moon the average value is about 57 , varying with her continually
changing distance. Excepting now and then a stray comet, no other
heavenly body ever comes within a distance a hundred times as
great as hers. Venus and Mars approach nearest, but the parallax
of neither of them ever reaches 40 .

88. Semi-Diameter.—In order to obtain the true altitude of
an object it is necessary, if the edge, or “limb,” as it is called,
has been observed, to add or deduct the apparent semi-diameter
of the object. In most cases this will be sensibly the same in all
parts of the sky, but the moon is so near that there is quite a
perceptible difference between her diameter when in the zenith and

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

64
in the horizon.

A glance at Fig. 34 shows that in the zenith the moon’s distance is less
than at the horizon, by almost exactly the earth’s radius—the difference
between the lines OZ and OPh. Now this is very nearly one-sixtieth part
of the moon’s distance, and consequently the moon, on a night when its
apparent diameter at rising is 30 , will be 30
larger when near the
zenith. Since the semi-diameter given in the almanac is what would be
seen from the centre of the earth, every measure of the moon’s distance
from stars or from the horizon will require us to take into account this

“augmentation of the semi-diameter,” as it is technically called.

The formula, easily deduced from the figure by remembering that the
angle P CO = ζ − p (zenith distance − parallax), and that the appar-
ent and “almanac” diameters will be inversely proportional to the two
distances OP and CP , is
sin ζ
apparent semi-diameter = almanac s. d. ×

.

sin (ζ − p)

This measurable increase of the moon’s angular diameter at high
altitudes has nothing to do with the purely subjective illusion which
makes the disc look larger to us when near the horizon. That it is
a mere illusion may be made evident by simply looking through
a dark glass just dense enough to hide the horizon and intervening
landscape. The moon or sun then seems to shrink at once to normal
dimensions.

89. Refraction.—

Rays of light have their direc-
tion changed by refraction in
passing through the air, and as
the direction in which we see a
body is that in which its light
reaches the eye, it follows that
this refraction apparently dis-
places the stars and all bodies
seen through the atmosphere.

So far as the action is regular,
the effect is to bend the rays di-

Fig. 35.—Atmospheric Refraction.

rectly downwards, and thus to
make the objects appear higher in the sky. Refraction increases
the altitude of a celestial object without altering the azimuth. Like
parallax, it is zero at the zenith and a maximum at the horizon; but

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

65
it follows a different law. It is entirely independent of the distance
of the object, and its amount varies (nearly) as the tangent of the
zenith distance—not as the sine, as in the case of parallax.

90. This approximate law of the refraction is easily proved.

Suppose in Fig. 35 that the observer at O sees a star in the direction

OS, at the zenith distance ZOS or ζ. The light has reached him from

S by a path which was straight until the ray met the upper surface of
the air at A, but afterwards curved continually downwards as it passed
from rarer to denser regions.

We know that the atmosphere is very shallow as compared with the
size of the earth, and it is exceedingly rare in the upper portions, so that,
as far as concerns refraction, we may assume that the point A, where the
first perceptible bending of the ray occurs, is not more than fifty miles
high, and that the vertical AZ is sensibly parallel to OZ; consequently,
also, that all the successive “strata of equal density” are parallel to each
other and to the upper surface of the air.

[This amounts to neglecting the earth’s curvature between O and B.]

The true zenith distance (as it would be if there were no refraction) is

ZDS , which equals Z AS ; and since the retraction, r, may be defined
as the difference between the true and apparent zenith distances, this
true zenith distance will = ζ + r.

Now from optical principles, when a ray of light passes through a
medium composed of parallel strata, the final direction of the ray is the
same as if the medium had throughout the density of the last stratum,
and therefore the final direction, SO, will be the same as if all the air,
from A down, had the same density as at O, with the same index of
refraction, n. We may therefore apply the law of refraction directly at

A, and write sin Z AS = n sin BAC (= ZOS), or sin (ζ + r) = n sin ζ;

AC being drawn parallel to OS.

Developing the first member, we have
sin ζ cos r + cos ζ sin r = n sin ζ.

But r is always a small angle, never exceeding 40 ; we may therefore
take cos r = 1. Doing this and transposing the first term, we get
cos ζ sin r = n sin ζ − sin ζ = (n − 1) sin ζ.

Whence,
sin r = (n − 1) tan ζ;
or,
r = (n − 1)206265 tan ζ (nearly).

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

66

The index of refraction for air, at zero centigrade and a barometric
pressure of 760mm, is 1.000294; whence,
r = .000294 × 206265 × tan ζ = 60 .6 tan ζ.

This equation holds very nearly indeed down to a zenith distance
of 70◦, but fails as we approach the horizon. For rays coming nearly
horizontal, the points A and B are so far from O that the normal AZ is
no longer practically parallel to OZ; and many of the other fundamental
assumptions on which the formula is based also break down.

At the horizon, where ζ = 90◦ and tan ζ = infinity, the formula
would give sin r = infinity also; an absurdity, since no sine can exceed
unity. The refraction there is really about 37 , under the circumstances
of temperature and pressure above indicated.

91. Effect of Temperature and Barometric Pressure.—

The index of refraction of air depends of course upon its temper-
ature and pressure. As the air grows warmer, its refractive power
decreases; as it grows denser, the refraction increases. Hence, in all
precise observations of the altitude (or zenith distance), it is neces-
sary to note both the thermometer and the barometer, in order to
compute the refraction with accuracy. For rough work, like ordinary
sextant observations, it will answer to use the “mean refraction,”

corresponding to an average state of things.

Tables of Refraction.—The computation of the refraction is best
effected by special tables made for the purpose; of these, Bessel’s tables
are the most convenient, best known, and probably even yet the most
accurate. It must be always borne in mind, however, that from the action
of wind and other causes the condition of the air along the path of the
ray is seldom perfectly normal; in consequence, the actual refraction in
any given case is liable to differ from the computed by as much as one
or even two per cent. No amount of care in observation can evade this
difficulty; the only remedy is a sufficient repetition of observations under
varying atmospheric conditions. Observations at an altitude below 10◦
or 15◦ are never much to be trusted.

Lateral Refraction.—When the air is much disturbed, sometimes
objects are displaced horizontally as well as vertically.

Indeed, as a
general rule, when one looks at a star with a large telescope and high
power, it will seem to “dance” more or less—the effect of the varying
refraction which continually displaces the image.

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

67

92. Effect on the Time of Sunrise and Sunset.—The hor-
izontal refraction, ranging as it does from 34 to 39 , according to
temperature, is always somewhat greater than the diameter of ei-
ther the sun or the moon. At the moment, therefore, when the sun’s
lower limb appears to be just rising, the whole disc is really below
the plane of the horizon; and the time of sunrise in our latitudes is
thus accelerated from two to four minutes, according to the incli-
nation of the sun’s diurnal circle to the horizon, which inclination
varies with the time of the year. Of course, sunset is delayed by the
same amount, and thus the day is lengthened by refraction from
four to eight minutes, at the expense of the night.

93. Effect on the Form and Size of the Discs of the Sun
and Moon.—Near the horizon the refraction changes very rapidly.

While under ordinary summer temperature it is about 35 at the
horizon, it is only 29 at an elevation of half a degree; so that, as
the sun or moon rises, the bottom of the disc is lifted 6 more than
the top, and the vertical diameter is thus made apparently about
one-fifth part shorter than the horizontal. This distorts the disc into
the form of an oval, flattened on the under side. In cold weather
the effect is much more marked. As the horizontal diameter is not
at all increased by the refraction, the apparent area of the disc is
notably diminished by it; so that it is evident that refraction cannot
be held in any way responsible for the apparent enlargement of the
rising luminary.

94. Determination of the Refraction.—1. Physical Method.

Theory furnishes the law of astronomical refraction, though the
mathematical expression becomes rather complicated when we at-
tempt to make it exact. In order, therefore, to determine the as-
tronomical refraction under all possible circumstances, it is only
necessary to determine the index of refraction of air, and its vari-
ations with temperature and pressure, by laboratory experiments,
and to introduce the constants thus obtained into the formulæ. It is
difficult, however, to make these determinations with the necessary
precision. In fact, at present our knowledge of the constants of air
rests mainly on astronomical work.

2. By Observations of Circumpolar Stars. At an observatory
whose latitude exceeds 45◦ select some star which passes through
the zenith at the upper culmination. (Its declination must equal
the latitude of the observatory.) It will not be affected by refraction

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

68
at the zenith, while at the lower culmination, twelve hours later,
it will. With the meridian circle observe its polar distance in both
positions, determining the “polar point” of the circle as described on
pp. 46–47. If the polar point were not itself affected by refraction,
the simple difference between the two results for the star’s polar
distance, obtained from the upper and lower observations, would
be the refraction at the lower point.

As a first approximation, however, we may neglect the refraction
at the pole, and thus obtain a first approximate lower refraction. By
means of this we may compute an approximate polar refraction, and
so get a first “corrected polar point.” With this compute a second
approximate lower refraction, which will be much more nearly right
than the first; this will give a second “corrected polar point”; this
will in turn give us a third approximation to the refraction; and so
on. But it would never be necessary to go beyond the third, as the
approximation is very rapid. If the star does not go exactly through
the zenith, it is only necessary to compute each time approximate
refractions for its upper observation, as well as for the polar point.

At present, however, the refraction is so well known that the
method actually used is to form “equations of condition” from the
observations of the altitude of known stars under varying circum-
stances, and from these to deduce such corrections to the star places
and refraction constants as will best harmonize the whole mass of
material.

95. 3. By Observations of the Altitudes of Equatorial Stars made
at an Observatory near the Equator. For an observer so situated, stars
that are on the celestial equator (δ = 0) will come to the meridian at the
zenith, and will rise and fall vertically, with a motion strictly proportional
to the time; the true zenith distance of the star at any moment being
just equal to its hour-angle in degrees. We have only, then, to observe
the apparent zenith distance of a star with the corresponding time, and
the refraction comes out directly.

If the station is not exactly on the equator, and if the star’s decli-
nation is not exactly zero, it is only necessary to know the latitude and
declination approximately in order to get the refraction very accurately;
a considerable error in either latitude or declination will affect the result
but slightly.

4. The French astronomer Loewy has recently proposed a method
which promises well. He puts a pair of reflectors, inclined to each other
at a convenient angle of from 45◦ to 50◦ (a glass wedge with silvered

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

69
sides), in front of the object-glass of an equatorial. This will bring to the
eye two rays which make a strictly constant angle with each other, and
there is no difficulty in finding pairs of stars so situated that their images
will come into the field of view together. Now, were it not for refrac-
tion, these images would always keep their relative position unchanged,
notwithstanding the diurnal motion; but on account of the changes in
the refraction, as one star rises and the other falls, they will shift in the
field, and micrometric measures will determine the shifting, and so the
refraction, with great precision.

96. Twilight.—(Although this subject is outside the main purpose of
this chapter, which deals with corrections to be applied to astronomical ob-
servations, we treat it here because, like refraction, it is a purely atmospheric
phenomenon, and finds no other more convenient place.)

Twilight, the illumination of the sky which begins before sunrise,
and continues after sunset, is caused by the reflection of light to the
observer from the upper regions of the earth’s atmosphere. It is not
yet certain whether this is due to reflection from foreign matter in
the air, such as minute crystals of ice and salt, particles of dust of
various kinds, and infinitesimal drops of water, or whether the pure
gases themselves have some power of reflecting light. There is no
doubt, however, that air, under the ordinary conditions, possesses
considerable power of reflection; so that, as long as any air upon
which the sun is shining is visible to the observer, it will send him
more or less light, and appear illuminated.

Fig. 36.—Twilight.

Suppose the atmosphere to have the depth indicated in the fig-
ure. Then, if the sun is at S, Fig. 36, it will just have set to an observer at 1, but all the air within his range of vision will still
be illuminated. When, by the earth’s rotation, he has been trans-
ported to 2, he will see the “twilight bow” rising in the east, a faintly

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

70
reddish arc separating the illuminated part of the sky from the dark-
ened part below, which lies in the shadow of the earth. When he
reaches 3, the western half of the sky alone remains bright, but the
arc of separation between the light and darkness has become vague
and indefinite; when he reaches 4, only a glow remains in the west;
and when he comes to 5, night closes in on him. Nothing remains
in sight on which the sun is shining.

97. Duration of Twilight.—This depends upon the height of the
atmosphere, and the angle at which the sun’s diurnal circle cuts the
horizon. It is found as a matter of observation, not admitting, however,
of much precision, that twilight lasts until the sun has sunk about 18◦
below the horizon; that is to say, the angle 1 C 5 in the figure is about

18◦.

The time required to reach this point in latitude 40◦ varies from two
hours at the longest days in summer, to one hour thirty minutes about

Oct. 12 and March 1, when it is least. At the winter solstice it is about
one hour and thirty-five minutes.

In higher latitudes the twilight lasts longer, and the variation is more
considerable; the date of the minimum also shifts.

Near the equator the duration is shorter, hardly exceeding an hour
at the sea-level; while at high elevations (where the amount of air above
the observer’s level is less) it becomes very brief. At Quito and Lima it is
said not to last more than twenty minutes. Probably, also, in mountain
regions the clearness of the air, and its purity contribute to the effect.

98. Height of the Atmosphere.—It is evident from the fig-
ure that at the moment twilight ceases, the last visible portion of
illuminated air is at the top of the atmosphere, and just half-way
between the observer and the nearest point where the sun is set-
ting. If the whole arc 1, 5 is 18◦, 1, 3 is 9◦: then calling the height
of the atmosphere H and the earth’s radius R, and neglecting re-
fraction (i.e., supposing the lines 1 m and 5 m to be straight), we
have from the right-angled triangle 1 C m, C m = 1 C × sec 9◦, or

R+H = R×sec 9◦; whence H = R (sec 9◦−1) = 0.0125 R, or almost
exactly fifty miles. This must be diminished about one-fifth part
on account of the curvature of the lines 1 m and 5 m by refraction,
making the height of the atmosphere about forty miles.

The result must not, however, be accepted too confidently. It
only proves that we get no sensible twilight illumination from air at
a greater height: above that elevation the air is either too rare, or

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS.

71
too pure from foreign particles, to send us any perceptible reflection.

There is abundant evidence from the phenomena of meteors that the
atmosphere extends to a height of 100 miles at least, and it cannot
be asserted positively that it has any definite upper limit.

99. Aberration.—There is yet one more correction which has to
be applied in order to get the true direction of the line which at the
instant of observation joins the eye of the observer to the star he is
pointing at. The aberration of light is an apparent displacement of the
object observed, due to the combination of the earth’s orbital motion
with the progressive motion of light. It can be better discussed, however,
in a different connection (see Chap. VI.), and we content ourselves with
merely mentioning it here.

PROBLEMS OF PRACTICAL ASTRONOMY.

72

CHAPTER IV.

PROBLEMS OF PRACTICAL ASTRONOMY, LATITUDE, TIME,

LONGITUDE, AZIMUTH, AND THE RIGHT ASCENSION AND

DECLINATION OF A HEAVENLY BODY.

100. There are certain problems of Practical Astronomy
which have to be solved in obtaining the fundamental facts from
which we deduce our knowledge of the earth’s form and dimensions,
and other astronomical data.

The first of these problems is that of the

LATITUDE.

The latitude (astronomical ) of a place (Art. 30) is simply the
altitude of the celestial pole (Polhöhe), or, what comes to the same
thing, as is evident from Fig. 7 (Art. 33), it is the declination of
the zenith. It may also be defined, from the mechanical point of
view, as the angle between the plane of the earth’s equator and the
observer’s plumb-line or vertical.

Neither of these definitions assumes anything as to the form of the
earth, and we shall find farther on that this astronomical latitude is sel-
dom identical with the geocentric, nor even with the geodetic latitude
of a place. It is, however, the only kind of latitude which can be di-
rectly determined from astronomical observations, and its determination
is one of the most important operations of what may be called Economic

Astronomy.

101. Determination of Latitude.—First: By Circumpolars.

The most obvious method of determining the latitude is to observe,
with the meridian circle or some analogous instrument, the altitude
of a circumpolar star at its upper culmination, and again, twelve
hours later, at its lower. Each of the observations must be corrected
for refraction, and then the mean of the two corrected altitudes will
be the latitude.

This method has the advantage of being an independent one; i.e., it

PROBLEMS OF PRACTICAL ASTRONOMY.

73
does not require any data (such as the declination of the stars used) to
be accepted on the authority of previous observers. But to obtain much
accuracy it requires considerable time and a large fixed instrument. In
low latitudes the refraction is also very troublesome.

102. Second: By the meridian altitude or zenith distance of a
body of known declination.

In Fig. 37 the semi-circle AQP B is the meridian, Q and P being
respectively the equator and the pole, and Z the zenith. QZ is the
declination of the zenith, or the observer’s latitude (= P B = φ).

Suppose now that we observe Zs (= ζs), the zenith distance of a
star s (south of the zenith), as it crosses the meridian, and that its
declination Qs (= δs) is known; then evidently φ = δs + ζs.

In the same way, if the star were at n, between zenith and pole,
φ = δn − ζn.

If we use the meridian circle, we can always select stars that pass
near the zenith where the refraction will be small; moreover, we can
select them in such a way that some will be as much north of the zenith
as others are south, and thus eliminate the refraction errors. But we
have to get our star declinations out of catalogues made by previous
observers, and so the method is not an independent one.

103. At Sea the latitude is usually ob-
tained by observing with the sextant the
sun’s maximum altitude, which of course
occurs at noon. Since at sea it is seldom
that one knows beforehand precisely the
moment of local noon, the observer takes

Fig. 37.—Determination
care to begin to observe the sun’s altitude
of Latitude.

some ten or fifteen minutes earlier, repeat-
ing his observations every minute or two.

At first the altitude
will keep increasing, but immediately after noon it will begin to
decrease. The observer uses therefore the maximum1 altitude ob-
tained, which, corrected for refraction, parallax, semi-diameter, and

1On account of the sun’s motion in declination, and the northward or south-
ward motion of the ship itself, the sun’s maximum altitude is usually attained
not precisely on the meridian, but a few seconds earlier or later. This requires
a slight correction to the deduced latitude, explained in books on Navigation
or Practical Astronomy.

PROBLEMS OF PRACTICAL ASTRONOMY.

74
dip of the horizon, will give him the true latitude of his ship, by the
formula φ = δ ± ζ.

104. Third:

By Circum-meridian Altitudes.—If the observer
knows his time with reasonable accuracy, he can obtain his latitude from
observations made when the body is near the meridian, with practically
the same precision as at the moment of meridian passage. It would take
us a little too far to explain the method of reduction, which is given
with the necessary tables in all works on Practical Astronomy.

The
great advantage of this method is that the observer is not restricted to
a single observation at each meridian-passage of the sun or of the se-
lected star, but can utilize the half-hours preceding and following that
moment. The meridian-circle cannot be used, as the instrument must
be such as to make extra-meridian observations possible. Usually the
sextant or universal instrument is employed. This method is much used
in the French and German geodetic surveys.

105. Fourth: The Zenith Telescope Method.—(Sometimes known
as the American method, because first practically introduced by Captain Tal-
cott of the United States Engineers, in a boundary survey in 1845.)

The
essential
characteristic
of
the
method is the micrometric measurement
of the difference between the nearly equal
zenith distances of two stars which culmi-
nate within a few minutes of each other,
one north and the other south of the zenith,
and not very far from it: such pairs of stars
can always be found.

When the method
was first introduced, a special instrument,
known as the zenith telescope, was gener-
ally employed, but at present a simple tran-
sit instrument, with declination microme-
ter, and a delicate level attached to the tele-

Fig. 38.—Principle of
scope tube, is ordinarily used.

the Zenith Telescope.

The telescope is set at the proper alti-
tude for the star which first comes to the meridian, and the “latitude
level,” as it is called, is set horizontal; as the star passes through
the field of view its distance north or south of the central wire is
measured by the micrometer. The instrument is then reversed, and
so set by turning the telescope up or down (without, however, dis-
turbing the angle θ (Fig. 38) between the level and telescope), that

PROBLEMS OF PRACTICAL ASTRONOMY.

75
the level is again horizontal. After this reversal and adjustment, the
telescope tube is then evidently elevated at exactly the same angle,
ζ, as before, but on the opposite side of the zenith. As the second
star passes through the field, we measure with the micrometer its
distance north or south of the centre of the field; the comparison of
the two micrometer measures gives the difference of the two zenith
distances.

From Fig. 37 we have
for star south of zenith, φ = δs + ζs;
for star north of zenith, φ = δn − ζn.

Adding the two equations and dividing by 2, we have
δ
ζ
φ =
s + δn

+
s − ζn

.

2

2

The star catalogue gives us the declinations of the two stars

(δs + δn); and the difference of the zenith distances (ζs − ζn) is
determined with great accuracy by the micrometer measures.

The great advantage of the method consists
in its dispensing with a graduated circle, and
in avoiding almost wholly the errors due to re-
fraction; it virtually utilizes the circles of the
fixed observatories by which the star declina-
tions have been measured, without requiring
them to be brought into the field. Forty years
ago it was not always easy to find accurate de-

Fig. 39.—Latitude by
terminations of the declinations of the stars em-

Prime Vertical Transits.

ployed, but at present the star catalogues have
been so extended and improved that this difficulty has practically disap-
peared, so that this method of determining the latitude is now not only
the most convenient and rapid, but is quite as precise as any, if the level
is sufficiently sensitive. Evidently the limit of accuracy depends upon
the exactness with which the level measures the slight, but inevitable,
difference between the inclinations of the instrument when pointed on
the two stars.

106. Fifth: By the Prime Vertical Instrument (p. 44). —We ob-
serve simply the moment when a known star passes the prime vertical on
the eastern side, and again upon the western side. Half the interval will
give the hour-angle of the star when on the prime vertical; i.e., the angle

PROBLEMS OF PRACTICAL ASTRONOMY.

76

ZP S in Fig. 39, where Z is the zenith, P the pole, and SZS the prime
vertical. The distance P S of the star from the pole is the complement
of the star’s declination; and P Z is the complement of the observer’s
latitude. Since the prime vertical is perpendicular to the meridian at the
zenith, the triangle P ZS will be right-angled at Z, and from Napier’s
rule of circular parts (taking ZP S as the middle part) we shall have
cos ZP S = tan P Z cot P S,
or
cos t = cot φ tan δ;
whence
tan φ = tan δ sec t.

If δ nearly equals φ, t will be small, and a considerable error in the
observation of t will then produce very little change in its secant or in
the computed latitude.

The observations are not so convenient and easy as in the case of the
zenith telescope, and the number of stars available is less; but the method
presents the great advantage of requiring nothing but an ordinary transit
instrument, without any special outfit of micrometer and latitude level.

It also entirely evades the difficulties caused by refraction.

107. Sixth:

By
the

Gnomon.—The
ancients
had
no instruments such as we have
hitherto described, and of course
could not use any of the preceding
methods of finding the latitude.

They were, however, able to make
a very respectable approximation
by means of the simplest of all
astronomical
instruments,
the
gnomon. This is merely a vertical

Fig. 40.—Latitude by the
shaft or column of known height

Gnomon.

erected on a perfectly horizontal
plane; and the observation consists in noting the length of the
shadow cast at noon at certain times of the year.

Suppose, for instance, that on the day of the summer solstice,
at noon, the length of the shadow is AC, Fig. 40. The height AB
being given, we can easily compute in the right-angled triangle the
angle ABC, which equals SBZ, the sun’s zenith distance when far-
thest north. Again observe the length AD of the shadow at noon of
the shortest day in winter, and compute the angle ABD, which is

PROBLEMS OF PRACTICAL ASTRONOMY.

77
the sun’s corresponding zenith distance when farthest south. Now,
since the sun travels equal distances north and south of the ce-
lestial equator, the mean of the two results will give the angular
distance between the equator and the zenith; i.e., the declination of
the zenith, which (Art. 100) is the latitude of the place.

The method is an independent one, like that by the observation of
circumpolar stars, requiring no data except those which the observer
determines for himself. Evidently, however, it does not admit of much
accuracy, since the penumbra at the end of the shadow makes it impos-
sible to measure its length very precisely.

It should be noted that the ancients, instead of designating the po-
sition of a place by means of its latitude, used its climate instead; the
climate (from κλ´
ιµα) being the slope of the plane of the celestial equator,
the angle AEB, which is the complement of the latitude.

It is supposed, indeed known, that many of the Egyptian obelisks
were erected primarily to serve as gnomons, and were used for that pur-
pose.

108. Possible Variations of the Latitude.—It is an inter-
esting question whether the position of the earth’s axis is fixed with
reference to its mass and surface. Theoretically it is hardly possible that
it should be, because any change in the arrangement of the matter of the
earth, by denudation, subsidence, or elevation, would almost necessarily
disturb it. If so disturbed, the latitudes of places toward which the pole
approached would be increased, and those on the opposite side would
be decreased. At present we can only say that if such disturbance has
occurred, it must have been extremely slight for the last 200 years, not
exceeding 40 or 50 feet at most; but there are suspicions of a minute and
progressive change of the latitude of some of the observatories (notably

Pulkowa), which have drawn attention to the matter, and the subject is
under investigation.

TIME AND ITS DETERMINATION.

109. One of the most important problems presented to the as-
tronomer is the determination of Time. By universal consent the
apparent rotation of the heavens is made to furnish the standard,
and the determination of time is effected by ascertaining by obser-
vation the hour-angle of the object selected to mark the beginning of
the day by its transit across the meridian. In practice, three kinds
of time are now recognized, viz., sidereal time, apparent solar time,
and mean solar time.

PROBLEMS OF PRACTICAL ASTRONOMY.

78

110. Sidereal Time.—As has already been explained (Art. 26),
the sidereal time at any moment is the hour-angle of the vernal
equinox at that moment; or, what comes to the same thing, though
it sounds differently, it is the time marked by a clock which is so
set and adjusted as to show noon, or 0h 00m 00s, at each transit
of the vernal equinox. The sidereal day, thus defined, is the time
intervening between two successive transits of the same star; at
least, it is so within the hundredth part of a second, though on
account of the precession of the equinoxes (and the proper motions
of the stars) the agreement is not absolute, the difference amounting
to about one day in twenty-six thousand years.

111. Apparent Solar Time.—Just as sidereal time is the hour-
angle of the vernal equinox, so at any moment the apparent solar
time is the hour-angle of the sun. It is the time shown by the sun-
dial, and its noon is when the sun crosses the meridian. On account
of the annual eastward motion of the sun among the stars (due to
the earth’s orbital motion), this day is about four minutes longer
than the sidereal; and moreover, because the sun’s motion in right
ascension is not uniform, the apparent solar days are not all of
the same length, nor, consequently, its hours, minutes, or seconds.

December 23d is fifty-one seconds longer from (apparent) noon to
noon than Sept. 16th. For this reason, apparent solar time is not
satisfactory for scientific use, and has long been discarded in favor
of mean solar time. Until within about a hundred years, however,
it was the only kind of time commonly employed, and its use in the
city of Paris was not discontinued until the year 1816.

112. Mean Solar Time.—A “fictitious sun” is therefore imag-
ined, which moves uniformly and in the celestial equator, completing
its annual course in exactly the same time as that in which the ac-
tual sun makes the circuit of the ecliptic. It is mean noon when
this “fictitious sun” crosses the meridian, and at any moment the
hour-angle of this “fictitious sun” is the mean time for that moment.

Sidereal time will not answer for business purposes, because its noon

(the transit of the vernal equinox) occurs at all hours of night and day-
light in different seasons of the year. Apparent solar time is scientifically
unsatisfactory, because of the variation in the length of its days and
hours. And yet we have to live by the sun; its rising and setting, daylight
and night, control our actions. In mean solar time we find a satisfactory
compromise, an invariable time unit, and still an agreement with sun-

PROBLEMS OF PRACTICAL ASTRONOMY.

79
dial time close enough for convenience. It is the time now used for all
purposes except in certain astronomical work. The difference between
apparent time and mean time, never amounting to more than about a
quarter of an hour, is called the equation of time, and will be discussed
hereafter in connection with the earth’s orbital motion, Chap. VI.

The nautical almanac furnishes data by means of which the sid-
ereal time may be deduced from the corresponding solar, or vice
versa, by a very brief and simple calculation.

113. In practice the problem of determining the time always
takes the form of ascertaining the error of a time-piece; that is, the
amount by which a clock or watch is fast or slow of the time it
ought to show. The methods most in use by astronomers are the
following:—

First. By means of the transit instrument. Since the right ascen-
sion of a star is the sidereal time of its passage across the meridian

(Art. 26), it is obvious that the difference between the right ascen-
sion of a known star and the time shown by a sidereal clock at the
instant when the star crosses the middle wire of an accurately ad-
justed transit instrument, is the error of the clock at that moment.

Practically, it is usual to observe a number of stars (from eight to
ten), reversing the instrument once at least, so as to eliminate the
collimation error (Art. 60). With a good instrument a skilled ob-
server can determine this clock error or “correction” within about
one-thirtieth of a second of time, provided proper means are taken
to ascertain and allow for his “personal equation.”

114. Personal Equation.—It is found that every observer has
his own peculiarities of time observation with a transit, and his

“personal equation” is the amount to be added (algebraically) to
the time observed by him, in order to get the actual moment of
transit as it would be recorded by some supposable arrangement,
which should automatically register the moment when the star’s
image was bisected by the wire.

This personal equation differs for different observers, but is reason-
ably (though never strictly) constant for one who has had much practice.

In the case of observations with the chronograph, it is usually less than

±0s.2. It can be determined by an apparatus in which an artificial star,
resembling the real stars as much as possible in appearance, is made to
traverse the field of view and to telegraph its arrival at certain wires,
while the observer notes the moments for himself.

PROBLEMS OF PRACTICAL ASTRONOMY.

80

One of the most important problems of practical astronomy now
awaiting solution is the contrivance of some practical method of time
observation free from this annoying human element, the personal equa-
tion, which is always more or less uncertain and variable.

If mean time is wanted, it can be deduced from the sidereal time
by the data of the almanac.

The sun can also be observed instead of the stars, the moment
of the sun’s transit being that of apparent noon; but this obser-
vation, for many reasons, is far less accurate and satisfactory than
observations of the stars.

115. Second.

The method of equal altitudes.—If we observe
with a sextant in the forenoon the time shown by the chronome-
ter when the sun attains the height indicated by a certain reading
of the sextant, and then in the afternoon, the time when the sun
again reaches the same altitude, the moment of apparent noon will
be half-way between the two observed times; provided, of course,
that the chronometer runs uniformly during the interval, and also
provided that proper correction is made for the sun’s slight motion
in declination—a correction easily computed.

The advantage of this method is that the errors of graduation
of the sextant have no effect, nor is it necessary for the observer to
know his latitude except approximately.

Per contra, there is, of course,
danger that the afternoon obser-
vations may be interfered with by
clouds; and, moreover, both obser-
vations must be made at the same
place.

A modification of this method
is now coming into extensive use, in

Fig. 41.—Determination of Time
by a Single Altitude.

which two different stars of known
right ascension and of nearly the
same declination are used, at equal altitudes east and west of the
meridian.

116. Third. By a single altitude of the sun, the observer’s lat-
itude being known.—This is the method usual at sea. The altitude
of the sun having been measured with the sextant, and the corre-
sponding time shown by the chronometer having been accurately

PROBLEMS OF PRACTICAL ASTRONOMY.

81
noted, we compute the hour-angle of the sun, P , from the trian-
gle ZP S (Fig. 41), and this hour-angle corrected for the equation
of time, gives the true mean time at the observed moment. The
difference between this and that shown by the chronometer is the
error of the chronometer. In the triangle ZP S all three of the sides
are given, viz.: P Z is the complement of the latitude φ, which is
supposed to be known; P S is the complement of the declination δ,
which is found in the almanac, as is also the equation of time; while

ZS or ζ, is the complement of the sun’s altitude, as measured by the
sextant, and corrected for semi-diameter, refraction, and parallax.

The formula is

1
sin 1 [ζ + (φ − δ)] sin 1 [ζ − (φ − δ)]

2
sin 1 P =

2

2

.

2
cos φ cos δ

In order to accuracy, it is desirable that the sun should be on the
prime vertical, or as near it as practicable. It should not be near
the meridian. Any slight error in the assumed latitude produces no
sensible effect upon the result, if the sun is exactly east or west at
the time the observation is taken. The disadvantage of the method
is that any error of graduation of the sextant enters into the result
with its full effect.

In some cases a person is so situated that it is necessary to determine
his time roughly, without instruments; and this can be done within about
a half a minute by establishing a noon-mark, which is nothing but a line
drawn exactly north and south, with a plumb-line, or some vertical edge,
like the edge of a door-frame or window-sash, at its southern extremity.

The shadow will then always fall upon the meridian line at apparent
noon.

117. The Civil and the Astronomical Day.—The astro-
nomical day begins at mean noon. The civil day begins at midnight,
twelve hours earlier. Astronomical mean time is reckoned round
through the whole twenty-four hours, instead of being counted in
two series of twelve hours each. Thus, 10 a.m. of Wednesday, May 2,
civil reckoning, is Tuesday, May 1, 22h by astronomical reckoning.

Beginners need to bear this in mind in using the almanac.

LONGITUDE.

118. Having now methods of obtaining the true local time, we
can attack the problem of longitude, which is perhaps the most

PROBLEMS OF PRACTICAL ASTRONOMY.

82
important of all the economic problems of astronomy. The great
observatories at Greenwich and at Paris were established simply for
the purpose of furnishing the observations which could be made the
basis of the accurate determination of longitude at sea.

The longitude of a place on the earth is the angle at the pole be-
tween the meridian of Greenwich and the meridian passing through
the observer’s place; or it is the arc of the equator intercepted be-
tween these meridians; or, what comes to the same thing, since this
arc is measured by the time required for the earth to turn suffi-
ciently to bring the second meridian into the same position held by
the first, it is simply the difference of their local times,—the amount
by which the noon at Greenwich is earlier or later than at the ob-
server’s place. It is now usually reckoned in hours, minutes, and
seconds, instead of degrees.

Since it is easy for the observer to find his own local time by
the methods which have been given, the knot of the problem is
really this: being at any place, to find the corresponding local time
at Greenwich without going there.

The methods of finding the longitude may be classed under three
different heads:

First, By means of signals simultaneously observable at the
places between which the difference of longitude is to be found.

Second, By making use of the moon as a clock-hand in the sky.

Third, By purely mechanical means, such as chronometers and
the telegraph.

119. Under the first head we may make use of

[A] A Lunar Eclipse.—When the moon enters the shadow of
the earth, the phenomenon is seen at the same moment, no matter
where the observer may be. By noting, therefore, his own local
time at the moment, and afterwards comparing it with the time at
which the phenomenon was observed at Greenwich, he will obtain
his longitude from Greenwich. Unfortunately, the edge of the earth’s
shadow is so indistinct that the progress of events is very gradual,
so that sharp observations are impossible.

[B] Eclipses of the satellites of Jupiter may be used in the same
way, with the advantage that they occur very frequently,—almost
every night, in fact; but the objection to them is the same as to the
lunar eclipses,—they are not sudden.

PROBLEMS OF PRACTICAL ASTRONOMY.

83

[C] The appearance and disappearance of meteors may be and
has been used to determine the difference of longitude between
places not more than two or three hundred miles apart, and gives
very accurate results. (Now superseded by the telegraph.)

[D] Artificial signals, such as flashes of powder and rockets, can
be used between two stations not too far distant.

Early in the
century the difference of longitude between the Black Sea and the

Atlantic was determined by means of a chain of signal stations on
the mountain tops; so also, later, the difference of longitude between
the eastern and western extremities of the northern boundary of

Mexico. This method is now superseded by the telegraph.

120. Second, the moon regarded as a clock.

Since the moon revolves around the earth once a month, it is, of
course, continually changing its place among the stars; and as the
laws of its motion are now well known, and as the place which it
will occupy is predicted for every hour of every Greenwich day three
years in advance in the nautical almanac, it is possible to deduce
the corresponding Greenwich time by any observation which will
determine the place of the moon among the stars. The almanac
place, however, is the place at which the moon would be seen by
an observer at the centre of the earth, and consequently the actual
observations are in most cases complicated with very disagreeable
reductions for parallax before they can be made available.

The simplest lunar method is,

[A] That of Moon Culminations.—We merely observe with a
transit instrument the time when the moon’s bright limb crosses the
meridian of the place; and immediately after the moon we observe
one or more stars with the same instrument, to give us the error of
our clock. As the moon is observed on the meridian, its parallax
does not affect its right ascension, and accordingly, by a simple
reference to the almanac, we can ascertain the Greenwich time at
which the moon had the particular right ascension determined by
the observation. The method has been very extensively used, and
would be an admirable one were it not for the effects of personal
equation.

It seldom happens that the personal equation of an observer is the
same for such an object as the limb of the moon as it is for a star; and
since the moon’s motion among the stars is very slow, the effect of such
a difference is multiplied by about 30 (roughly the number of days in a

PROBLEMS OF PRACTICAL ASTRONOMY.

84
month) in its effect upon the longitude deduced.

[B] Lunar-Distances.—At sea it is, of course, impossible to ob-
serve the moon with a transit instrument, but we can observe its
distance from the stars near its path by means of a sextant. The
distance observed will not be the same that it would be if the ob-
server were at the centre of the earth, but by a mathematical process
called “clearing a lunar” the distance as seen from the centre of the
earth can be easily deduced, and compared with the distance given
in the almanac. From this the longitude can be determined. Any
error, however, in measuring a lunar-distance entails an error about
thirty times as great in the resulting longitude, and the method is
at present very little used, the moon having been superseded by the
chronometer for such purposes.

[C] Occultations.—Occasionally, in its passage through the sky,
the moon over-runs a star, or “occults” it. The star vanishes in-
stantaneously, and, of course, at the moment of its disappearance
the distance from the centre of the moon to the star is precisely
equal to the apparent semi-diameter of the moon; we thus have a

“lunar-distance” self-measured.

Observations of this kind furnish one of the most accurate meth-
ods of determining the difference of longitude between widely sepa-
rated places, the only difficulty arising from the fact that the edge
of the moon is not smooth, but more or less mountainous, so that
the distance of a star from the moon’s centre is not always the same
at the moment of its disappearance.

[D] In the same way a solar eclipse may be employed by observing
the moment when the moon’s limb touches that of the sun.

It will be noticed that these two last methods (the methods of occul-
tation and solar eclipse) do not belong in the same class with the method
of lunar eclipse, because the phenomena are not seen at the same instant
at different places, but the calculation of longitude depends upon the
determination of the moon’s place in the sky at the given time, as seen
from the earth’s centre.

There are still other methods, depending upon measurements of
the moon’s position by observations of its altitude or azimuth. In
all such cases, however, every error of observation entails a vastly
greater error in the final results. Lunar methods (excepting occul-
tations) are only used when better ones are unavailable.

PROBLEMS OF PRACTICAL ASTRONOMY.

85

121. Finally we have what may be called the mechanical meth-
ods of determining the longitude.

[A] By the chronometer ; which is simply an accurate watch that
has been set to indicate Greenwich time before the ship leaves port.

In order to find the longitude by the chronometer, the sailor has
to determine its “error” upon local time by an observation of the
altitude of the sun when near the prime vertical, as indicated on
page 80. If the chronometer indicates true Greenwich time, the
error deduced from the observation will be the longitude. Usually,
however, the indication of the chronometer face requires correction
for the rate and run of the chronometer since leaving port.

Chronometers are only imperfect instruments, and it is important,
therefore, that several of them should be used to check each other. It
requires three at least, because if only two chronometers are carried and
they disagree, there is nothing to indicate which one is the delinquent.

On very long voyages the errors of chronometers are cumulative, and
the error is found to accumulate, not merely in proportion to the time,
but more nearly in proportion to the square of the time; i.e., if the error
to be feared in the use of a chronometer in longitude determinations at
the end of a week is about two seconds of time, at the end of the month
it would be, not eight seconds, but about thirty-two seconds.

If, therefore, a ship is to be at sea, without making port, more than
three or four months at a time, the method becomes untrustworthy, and
it may be necessary to recur to lunar distances; for voyages of less than
a month the method is now, practically, all that could be desired.

[B] But the method which, wherever it is applicable, has super-
seded all others, is that of The Telegraph. When we wish to find the
longitude between two stations connected by telegraph, the process
is usually as follows: The observers at both stations, after ascer-
taining that they both have clear weather, proceed to determine
their own local time by extensive series of star observations with
the transit instrument. Then, at a agreed-upon time, the observer
at Station A “switches his clock” into the telegraphic circuit, so
that its beats are communicated along the line and received upon
the chronograph of the other, say the western station. After the
eastern clock has thus sent its signals, say for two minutes, it is
switched out of the circuit, and the western observer now switches
his clock into the circuit, and its beats are received upon the east-
ern chronograph. The operation is closed by another series of star
observations.

PROBLEMS OF PRACTICAL ASTRONOMY.

86

We have now upon each chronograph sheet an accurate compar-
ison of the two clocks, showing the amount by which the western
clock is slow of the eastern. If the transmission of electric signals
were instantaneous, the difference shown upon the two chronograph
sheets would agree precisely. Practically, however, there will always
be a small discrepancy amounting to twice the time occupied in the
transmission of the signals; but the mean of the two differences will
be the true difference of longitude of the places after the proper
corrections have been applied. Especial care must be taken to deter-
mine with accuracy, or to eliminate, the personal equations of the
observers.

It is customary to make observations of this kind on not less than five
or six evenings in cases where it is necessary to determine the difference
of longitude with the highest accuracy. The astronomical difference of
longitude between two places can thus be telegraphically determined
within about the one-hundredth part of a second of time; i.e., within
about ten feet or so, in the latitude of the United States.

It may be noted here that the time occupied by the transmission
of electric signals in longitude operations is not to be taken as the real
measure of “the velocity of the electric fluid” upon the wires, as was once
supposed. The time apparently consumed in the transmission is simply
the time required for the current at the receiving station (which current
probably begins at the very instant the key is touched at the other end
of the line) to become strong enough to do its work in making the signal;
and this time depends upon a multitude of circumstances.

122. Local and Standard Time.—In connection with time and
longitude determinations, a few words on this subject will be in place.

Until recently it has always been customary to use only local time, each
observer determining his own time by his own observations. Before the
days of the telegraph, and while travel was comparatively slow and in-
frequent, this was best; but the telegraph and railway have made such
changes that, for many reasons, it is better to give up the old system
of local times in favor of a system of standard time. It facilitates all
railway and telegraphic business in a remarkable degree, and makes it
practically easy for every one to keep accurate time, since it can be daily
wired from some observatory to every telegraph office.

According to the system that is now established in this country, there
are five such standard times in use,—the colonial, the eastern, the cen-
tral, the mountain, and the Pacific,—which differ from Greenwich time
by exactly four, five, six, seven, and eight hours respectively, the minutes
and seconds being identical everywhere. At most places only one of these

PROBLEMS OF PRACTICAL ASTRONOMY.

87
times is employed; but in cities where different systems join each other,
there are two standard times in use, differing from each other by exactly
one hour, and from the local time by about half an hour. In some such
places the local time also maintains its place.

In order to determine the standard time by observation, it is only
necessary to determine the local time by one of the methods given, and
correct it according to the observer’s longitude from Greenwich.

123. Where the Day Begins.—If we imagine a traveller start-
ing from Greenwich on Monday noon, and journeying westward as swiftly
as the earth turns to the east under his feet, he would, of course, keep
the sun exactly on the meridian all day long, and have continual noon.

But what noon? It was Monday when he started, and when he gets
back to London, twenty-four hours later, it is Tuesday noon there, and
there has been no intervening sunset. When does Monday noon become

Tuesday noon? The convention is that the change of date occurs at the

180th meridian from Greenwich. Ships crossing this line from the east
skip one day in so doing. If it is Monday forenoon when the ship reaches
the line, it becomes Tuesday forenoon the moment it passes it, the in-
tervening twenty-four hours being dropped from the reckoning on the
log-book. Vice versa, when a vessel crosses the line from the western
side, it counts the same day twice, passing from Tuesday forenoon back
to Monday, and having to do its Tuesday over again.

This 180th meridian passes mainly over the ocean, hardly touching
land anywhere. There is a little irregularity in the date upon the differ-
ent islands near this line. Those which received their earliest European
inhabitants via the Cape of Good Hope have, for the most part, the Asi-
atic date, belonging to the west side of the 180th meridian; while these
that were approached via Cape Horn have the American date.

When Alaska was transferred from Russia to the United States, it
was necessary to drop one day of the week from the official dates.

THE PLACE OF A SHIP AT SEA.

124. The determination of the place of a ship at sea is com-
mercially of such importance that, at the risk of a little repetition,
we collect together here the different methods available for its de-
termination. The methods employed are necessarily such that ob-
servations can be made with the sextant and chronometer, the only
instruments available under the circumstances.

The Latitude is usually obtained by observations of the sun’s
altitude at noon, according to the method explained in Art. 103.

PROBLEMS OF PRACTICAL ASTRONOMY.

88

The Longitude is usually found by determining the error upon
local time of the chronometer, which carries Greenwich time. The
necessary observations of the sun’s altitude should be made when
the sun is near the prime vertical, as explained in Art. 116.

In the case of long voyages, or when the chronometer has for
any reason failed, the longitude may also be obtained by measuring
a lunar-distance and comparing it with the data of the nautical
almanac.

By these methods separate observations are necessary for the
latitude and for the longitude.

125. Sumner’s Method.—Recently a new method, first pro-
posed by Captain Sumner, of Boston, in 1843, has been coming
largely into use. In this method, each observation of the sun’s alti-
tude, with the corresponding chronometer time, is made to define
the position of the ship upon a certain line, called the circle of po-
sition. Two such observations will, of course, determine the exact
place of the vessel at one of the intersections of the two circles.

At any moment the sun is vertically over some point upon the
earth’s surface, which may be called the sub-solar point. An ob-
server there would have the sun directly overhead. Moreover, if at
any point on the earth an observer measures the altitude of the
sun with his sextant, the zenith distance of the sun (which is the
complement of this altitude) will be his distance from the sub-solar
point at the moment of observation, reckoned in degrees of a great
circle.

If, then, I take a terrestrial globe, and, opening the dividers so as
to cover an arc equal to this observed zenith distance of the sun, put
one foot of the dividers upon the sub-solar point, and sweep a circle
on the surface of the globe around that point, the observer must
be somewhere on the circumference of that circle; and moreover, if
to the observer the sun is in the southwest, he himself must be in
the opposite direction from this sub-solar point; i.e., northeast of
it. In other words, the azimuth of the sun at the time of observation
informs him upon what part of the circle he is situated.

Suppose a similar observation made at the same place a few
hours later. The sub-solar point, and the zenith distance of the
sun, will have changed; and we shall obtain a new circle of position,
with its centre at the new sub-solar point. The observer must be

PROBLEMS OF PRACTICAL ASTRONOMY.

89
at one of its two intersections with the first circle—which of the
two intersections is easily determined from the roughly observed
azimuth of the sun.

If the ship moves between the two observations, the proper al-
lowance must be made for the motion. This is easily done by shift-
ing upon the chart that part of the first circle of position where the
ship was situated, carrying the line forward parallel to itself, by an
amount just equal to the ship’s run between the two observations,
as shown by the log. The intersection with the second circle then
gives the ship’s place at the time of the second observation.

The only problem remaining is to find the position of the “sub-
solar point” at any given moment. Now, the latitude of this point is
obviously the declination of the sun (which is found in the almanac).

If the sun’s declination is zero, the sun is vertically over some point
upon the equator. If its declination is +20◦, it is vertically over
some point on the twentieth parallel of north latitude, etc.

In the next place, its longitude is equal to the Greenwich appar-
ent solar time at the moment of observation; and this is given by the
chronometer (which keeps Greenwich mean solar time), by simply
adding or subtracting the equation of time; so that, by looking in
his almanac and at his chronometer, the observer has the position
of the sub-solar point immediately given him. (See note, page 93.)

Fig. 42.—Sumner’s Method.

Suppose, for example, that on May 20 (the sun’s declination being

+20◦), at 11 a.m., Greenwich apparent time (i.e., May 19, 23h by astro-
nomical reckoning), according to the chronometer, the sun is observed

PROBLEMS OF PRACTICAL ASTRONOMY.

90
to have an altitude of 40◦ by a ship in the North Atlantic. The sub-solar
point will then be (Fig. 42) at a point in Africa having a latitude of

+20◦, and an east longitude of 15◦—at A in the figure. And the radius
of the “circle of position,” i.e., the distance from A to C—will be 50◦.

Again, a second observation is made three hours later, when the
sun’s altitude is found to be 65◦. The sub-solar point will then be at B,
latitude 20◦, longitude 30◦ W., and the radius of the circle of position

BC be 25◦, C being the ship’s place.

Of course it would be impracticable to carry on a vessel a terrestrial
globe large enough for the accurate working out of the graphical opera-
tion indicated, but tables are provided, by which the necessary portions
of the position circles can be easily drawn upon the ordinary charts.

126. The peculiar advantage of the method is, that a single ob-
servation is used for all it is worth, giving accurately the position
of a line upon which the ship is somewhere situated, and approx-
imately (by the rough observation of the sun’s azimuth) the part
of that line upon which its place will be found. In approaching
the American coast, for instance, if an observation be taken in the
forenoon when the sub-solar point is over the continent of Africa,
the ship’s position circle will lie nearly parallel to the coast, and
then a single observation will give approximately the distance of
the ship from land, which may be all the sailor wishes to know.

The observations need not be taken at any particular time. We
are not limited to observations at noon, or to the time when the
sun is on the prime vertical. It is to be noted, however, that ev-
erything depends upon the chronometer, as much as in the ordinary
chronometric determination of longitude.

127. Determination of Azimuth.—A problem, important,
though not so often encountered as that of latitude and longitude
determinations, is that of determining the azimuth, or true bear-
ing, of a line upon the earth’s surface. The process is this: With
a theodolite having an accurately graduated horizontal circle the
observer points alternately upon the pole star and upon a distant
signal erected for the purpose; the signal being an artificial star con-
sisting of a small hole in a plate of metal, with a bull’s-eye lantern
or other light behind it. It is desirable that it should be at least
a mile away from the observer, so that any small displacement of
the instrument will be harmless. The theodolite must be carefully
adjusted for collimation, and especial pains must be taken to have

PROBLEMS OF PRACTICAL ASTRONOMY.

91
the telescope perfectly level.

The next morning by daylight the ob-
server measures the angle or angles between
the night-signal and the objects whose az-
imuth is required.

If the pole star were exactly at the pole,
the mere difference between the two readings
of the circle, obtained when the telescope is
pointed on the star and on the signal, would
directly give the azimuth of the signal, As this
is not the case, however, the time at which

Fig. 43.—
each observation of the pole star is made must

Determination of
be noted, and the azimuth of the star must be

Azimuth.

computed for that moment. This can easily
be done, as the right ascension and declination of this star are given
in the almanac for every day of the year.

Recurring to the Z.P.S. [zenith-pole-star] triangle, N (Fig. 43) being
the north point of the horizon, P the pole, and N Z the meridian, we at
once see that the side P S is the complement of the star’s declination;
the side P Z is the complement of the observer’s latitude (which must be
known); and the angle at P is the difference between the right ascension
of the pole star and the sidereal time of the observation; [(t − a) if the
star is west of the meridian at the time, and (a − t) if it is east.] This
will come out in hours, of course, and must be reduced to degrees before
making the computation. We thus have two sides of the triangle, viz.,

P S and P Z, with the included angle at P , from which to compute the
angle Z at the zenith. This is the star’s azimuth.

The pole star is used because, being so near the pole, any slight
error in the assumed latitude of the place or in the sidereal time of the
observation will hardly produce any effect upon the result, especially if
the star be caught between five and six hours before or after its upper
culmination, at a time when it changes its azimuth very slowly (near

S or S
in the figure). The sun, or any other heavenly body whose
position is given in the almanac, can also be used as a reference point in
the same way, provided sufficient pains are taken to secure an accurate
observation of the time at the instant when the pointing is made. The
altitude should not exceed thirty degrees or so.

But the results are
usually rough compared with these obtained by means of the pole star.

PROBLEMS OF PRACTICAL ASTRONOMY.

92

DETERMINATION OF THE POSITION OF A HEAVENLY

BODY.

128. The position of a heavenly body is defined by its right
ascension and declination. These quantities may be determined—

(1) By the meridian circle, provided the body is bright
enough to be seen by the instrument and comes to the meridian
in the night-time. If the instrument is in exact adjustment, the sid-
ereal time when the object crosses the middle wire of the reticle of
the instrument is directly (according to Art. 27) the right ascension
of the object.

The reading of the circle of the instrument, corrected for refrac-
tion and parallax if necessary, gives the polar distance of the object,
if the polar point of the circle has been determined (Art. 66); or it
gives the zenith distance of the object if the nadir point has been
determined (Art. 67). In either case the declination can be imme-
diately deduced, being the complement of the polar distance, and
equal to the latitude of the observer, minus the distance of the
star south of the zenith. One complete observation, then, with the
meridian circle, determines both the right ascension and declination
of the object.

If a body (a comet, for instance) is too faint to be observed by
the telescope of the meridian circle, which is seldom very powerful,
or if it does not come to the meridian during the night, we usually
accomplish our object—

129. (2) By the Equatorial, determining the position of the
body by measuring the difference of right ascension and declination
between it and some neighboring star, whose place is given in a star
catalogue, and of course has been determined by the meridian circle
of some observatory.

In measuring this difference of right ascension and declination, we
usually employ a filar micrometer fitted like the reticle of a meridian
circle. It carries a number of wires which lie north and south in the field
of view, and these are crossed at right angles by one or more wires which
can be moved by the micrometer screw. The difference of right ascension
between the star and the object to be determined is measured by simply
observing with the chronograph the transits of the two objects across
the north and south wires; the difference of declination, by bisecting
each object with one of the micrometer wires as it crosses the middle of

PROBLEMS OF PRACTICAL ASTRONOMY.

93
the field of view. The observed difference must be corrected for refraction
and for the motion of the body, if it is appreciable.

Other less complicated micrometers are also in use. One of them,
called the ring micrometer, consists merely of an opaque ring supported
in the field of view either by being cemented to a glass plate or by slender
arms of metal. The observations are made by noting the transits of the
comparison star and of the object to be determined across the outer and
inner edges of the ring. If the radius of the ring is known in seconds of
arc, we can from these observations deduce the differences both of right
ascension and declination. The results are less accurate than those given
by the wire micrometer, but the ring micrometer has the advantage that
it can be used with any telescope, whether equatorially mounted or not,
and requires no adjustment.

There are also many other methods of effecting the same object.

130. To Compute the Time of Sunrise or Sunset.—To
solve this problem, it is only necessary to work out the Z.P.S. triangle
and find the hour-angle P , having given precisely the same data as in
finding the time by a single altitude of the sun (Art. 116). P Z is the
observer’s co-latitude, P S is the complement of the sun’s declination

(given by the almanac); and the true distance from the zenith to the
centre of the sun at the moment when its upper edge is at the horizon
is 90◦ 50 , which is made up of 90◦, +16 (the mean semi-diameter of
the sun), plus 34 (the mean refraction at the horizon). The resulting
hour-angle P , corrected for the equation of time, gives the mean time

(local ) at which the sun’s upper limb touches the horizon, under the
average circumstances of temperature and barometric pressure. If it is
very cold, with the barometer standing high, sunrise will be accelerated,
or sunset retarded, by a considerable fraction of a minute. If the sun rises
or sets over the sea-horizon, and the observer’s eye is at any considerable
elevation above the sea-level, the dip of the horizon must also be added
to the 90◦ 50 before making the computation.

The beginning and end of twilight may be computed in the same way
by merely substituting 108◦ for 90◦ 50 .

131. Note to Art. 125. —In the explanation of Sumner’s method
it is assumed that the earth is a perfect sphere. In the actual application
of the method certain corrections are therefore necessary to take into
account the earth’s ellipticity.

THE EARTH.

94

CHAPTER V.

THE EARTH AS AN ASTRONOMICAL BODY.

Approximate

Dimensions—Proofs
of
its

Rotation—

Accurate Determination of its Form and Size by

Geodetic Operations and Pendulum Observations—

Astronomical, Geodetic and Geocentric Latitude—

Determination of the Earth’s Mass and Density.

132. Having discussed the methods of making astronomical
observations, we are now prepared to consider the earth in its as-
tronomical relations; i.e., those facts relating to the earth which are
ascertained by astronomical methods, and are similar to the facts
which we shall have to consider in the case of the other planets.

The facts are broadly these:—

1. The earth is a great ball, about 7918 miles in diameter.

2. It rotates on its axis once in twenty-four sidereal hours.

3. It is flattened at the poles, the polar diameter being nearly
twenty-seven miles, or one two hundred and ninety-fifth part less
than the equatorial.

4. It has a mean density of about five and six-tenths times that
of water, and a mass represented in tons by six with twenty-one
ciphers after it (or six sextillions of tons, according to the French
numeration).

5. It is flying through space in its orbital motion around the sun,
with a velocity of about nineteen miles a second ; i.e., about seventy-
five times as swiftly as any cannon-ball.

I.

133. The Earth’s Approximate Form and Size.—It is not
necessary to dwell upon the ordinary proofs of its globularity. We
merely mention them. 1. It can be circumnavigated. 2. The ap-
pearance of vessels coming in from sea indicates that the surface is

THE EARTH.

95
everywhere convex. 3. The fact that the sea-horizon, as seen from
an eminence, is everywhere depressed to the same extent below the
level line, shows that the surface is approximately spherical. 4. The
fact that as one goes from the equator toward the north, the ele-
vation of the pole increases proportionally to the distance from the
equator, proves the same thing. 5. The shadow of the earth, an seen
upon the moon at the time of a lunar eclipse, is that which only a
sphere could cast.

We may add as to the smoothness and globularity of the earth,
that if the earth be represented by an 18-inch globe, the difference
between the polar and equatorial diameter would only be about
one-sixteenth of an inch, the highest mountains upon the earth’s
surface would be represented by about one-eightieth of an inch, and
the average elevation of the continents would be hardly greater than
that of a film of varnish. The earth is really relatively smoother and
rounder than most of the balls in a bowling-alley.

134. An
approximate
measure of the diameter is
easily obtained.

Erect upon
a level plain three rods in
line, a mile apart, and cut
off their tops at the same
level,
carefully
determined

Fig. 44.—Curvature of the Earth’s

Surface.

with
a
surveyor’s
levelling
instrument.

It will then be
found that the line AC, Fig. 44, joining the extremities of the two
terminal rods, passes about eight inches below B, the top of the middle
rod.

Suppose the circle ABC completed, and that E is the point on the
circumference opposite B, so that BE equals the diameter of the earth

(= 2 R).

By geometry,

BD : BA = BA : BE,

BA2

BA2
whence

BE =

, or R =

.

BD

2BD

Now BA is one mile, and BD = 2 of a foot, or

1
of a mile.

3

7920

12

Hence 2R =

, or 7920 miles: a very fair approximation.

1

7920

On account of refraction, however, the result cannot be made exact
by any care in observation. The line of sight, AC, is not strictly straight,

THE EARTH.

96
but curves slightly towards the earth, and differently as the weather
changes.

135. The best method of ascertaining the size of the earth—
in fact the only one of real value—is by measuring the arcs of the
meridian in order to ascertain the number of miles or kilometers
in one degree, from which we immediately get the circumference of
the earth. This measure involves two distinct operations. One—the
measure of the number of miles—is purely geodetic; the other—
the determination of the number of degrees, minutes, and seconds
between the two stations—is purely astronomical.

We have to find by astronomical observation the angle between
two radii drawn from the centre of the earth to the two stations

(regarding the earth as spherical); or, what is the same thing, the
angular distance in the sky between their respective zeniths. The
two stations being on the same meridian, all that is necessary is
to measure their latitudes by any of the methods which have been
given in Chapter IV. and take the difference. This will be the angle
wanted.

If, for instance, the distance between the two stations
was found by measurement to be 120 miles, and the difference of
latitude was found by astronomical observations to be 1◦ 44 .2, we
should get 69.27 miles for one degree. Three hundred and sixty
times this would be the circumference of the earth, a little less than

25,000 miles, and the diameter would be found by dividing this by
π, which would give 7920 miles.

136. Eratosthenes of Alexandria seems to have understood the mat-
ter as early as 250 b.c. His two stations were Alexandria and Syene in

Upper Egypt. At Syene he observed that at noon of the longest day in
summer there was no shadow at the bottom of a well, the sun being then
vertically overhead. On the other hand, the gnomon at Alexandria, on
the same day, by the length of the shadow, gave him 1 of a circumfer-

50
ence, or 7◦ 12 as the distance of the sun from the zenith at that place,
which, therefore, is the difference of latitude between Alexandria and

Syene.

The weak place in his work was in the measurement of the distance
between the two places. He states it as 5000 stadia, thus making the
circumference of the earth 250,000 stadia; but we do not know the length
of his stadium, nor does he give any account of the means by which he
measured the distance, if he measured it at all. There seem to have been
as many different stadia among the ancient nations as there were kinds

THE EARTH.

97
of “feet” in Europe at the beginning of this century.

The first really valuable measure of the arc of a meridian was that
made by Picard in Northern France in 1671—the measure which served

Newton so well in his verification of the idea of gravitation.

II.

137. The Rotation of the Earth.—At the time of Coper-
nicus the only argument in favor of the earth’s rotation1 was that
the hypothesis was more probable than that the heavens themselves
revolved. All phenomena then known would be sensibly the same
on either supposition. A little later, analogy could be adduced, for
when the telescope was invented, we could see that the sun, moon,
and several of the planets are rotating globes.

At present we are able to adduce experimental proofs which
absolutely demonstrate the earth’s rotation, and some of them even
make it visible.

138. 1. The Eastward Deviation of Bodies
falling from a Great Height.—The idea that such
a deviation ought to occur was first suggested
by Newton. Evidently, since the top of a tower,
situated anywhere but at the pole of the earth,
describes every day a larger circle than its base,
it must move faster. A body which is dropped
from the top, retaining its excess of eastward mo-
tion as it descends, must therefore strike to the
east of the point which is vertically under its
starting-point, provided it is not deflected in its
fall by the resistance of the air or by air-currents.

Fig. 45 illustrates the principle. A body starting
from A, the top of the tower, reaches the earth at

D (BD being equal very approximately to AA ),

Fig. 45.

Eastward Deviation
while during its fall the bottom of the tower has
of a Falling Body.

only moved from B to B . The experiments are

1The word rotate denotes a spinning motion like that of a wheel on its
axis. The word revolve is more general in its application, and may be applied
either to describe such a spinning motion, or (and this is the more usual use in
astronomy) to describe the motion of one body around another, as that of the
earth around the sun.

THE EARTH.

98
delicate, since the deviation is very small, and it is not easy to avoid
the effect of air-currents. It is also extremely difficult to get balls
so perfectly spherical that they will not sheer off to one side or the
other in falling.

The best experiments of this kind so far have been these of Benzen-
berg, performed at Hamburg in 1802, and those of Reich, performed in

1831, in an abandoned mine shaft near Freiberg, in Saxony. The latter
obtained a free fall of 520 feet, and from the mean of 160 trials, the
eastern deviation observed was 1.12 inches, while theory would make it

1.08. The experiment also gave a southern deviation of 0.17 of an inch,
unexplained by theory. It seems to indicate the probable error of obser-
vation. The balls in falling sometimes deviated two or three inches one
side or the other from the average.

The formula given by Worms in his
treatise on “The Earth and its Mecha-
nism,” is

4πt(H − 1 ∆) cos φ
x =

2

;

3T
where x is the deviation, t is the number of
seconds occupied in falling, T the number
of seconds in a sidereal day, H the height
fallen through, and ∆ the difference be-
tween H and the height through which a
body would fall in t seconds if there were
no resistance (so that ∆ = 1 gt2 − H).

2

Finally, φ is the latitude of the place of
observation. In latitude 45◦ a fall of 576
feet should give, neglecting the resistance
of the air, a deviation of 1.47 inches. The

Fig. 46.—Foucault’s
resistance would increase it a little.

Pendulum Experiment.

It will be noted that at the pole, where
the cosine of the latitude equals zero, the experiment fails. The largest
deviation is obtained at the equator.

139. 2. Foucault’s Pendulum Experiment.—In 1851 Foucault,
that most ingenious of French physicists, devised and first executed
an experiment which actually shows the earth’s rotation to the eye.

From the dome of the Pantheon in Paris he suspended a heavy iron
ball about a foot in diameter by a wire more than 200 feet long

(Fig. 46). A circular rail some twelve feet across, with a little ridge
of sand built upon it, was placed under the pendulum in such a way

THE EARTH.

99
that a pin attached to the swinging ball would just scrape the sand
and leave a mark at each vibration. The ball was drawn aside by a
cotton cord and allowed to come absolutely to rest; then the cord
was burned, and the pendulum set to swinging in a true plane; but
this plane seemed to deviate slowly towards the right, cutting the
sand in a new place at each swing and shifting at a rate which would
carry it completely around in about thirty-two hours if the pendu-
lum did not first come to rest. In fact, the floor of the Pantheon was
soon turning under the plane of the pendulum’s vibration. The ex-
periment created great enthusiasm at the time, and has since been
very frequently performed and always with substantially the same
results.

140. The approximate theory of the experiment is very simple.

Such a pendulum, consisting of a round ball hung by a round wire
or else suspended on point, so as to be equally free to swing in any
plane (unlike the common clock pendulum in this freedom), being
set up at the pole of the earth, would appear to shift around in
twenty-four hours. Really, the plane of vibration remains invariable
and the earth turns under it, the plane of vibration in this case being
unaffected by the motion of the earth. This can be easily shown by
setting up a similar apparatus, consisting of a ball hung by a thread,
upon a table, and then turning the table around with as little jar
as possible. The plane of the swing will remain unchanged by the
motion of the table.

It is easy to see, further, that at the equa-
tor there would be no such tendency to shift.

In any other latitude the effect will be inter-
mediate, and the time required for the pendu-
lum to complete the revolution of its plane will
be twenty-four hours divided by the sine of the
latitude. The northern edge of the floor of a
room (in the northern hemisphere) is nearer the
axis of the earth than its southern edge, and
therefore is carried more slowly eastward by the
earth’s rotation.

Hence it must skew around

Fig. 47.

Explanation of the
continually, like a postage stamp gummed upon

Foucault Pendulum
a whirling globe anywhere except at the globe’s

Experiment.

equator. The southern extremity of every north
and south line on the floor continually works toward the east faster

THE EARTH.

100
than the northern extremity, causing the line itself to shift its direc-
tion accordingly, compared with the direction it had a few minutes
before. A free pendulum, set at first to swing along such a line,
must therefore apparently deviate continually at the same rate in
the opposite direction. In the northern hemisphere its plane moves
dextrorsum; i.e., with the hands of a watch: in the southern, its
motion is sinistrorsum.

141. Suppose a parallel of latitude drawn
through the place in question, and a series of tan-
gent lines drawn toward the north at points an
inch or so apart on this parallel. All these tan-
gents would meet at some point, V , Fig. 47, which
is on the earth’s axis produced; and taken together
these tangents would form a cone with its point at

V . Now if we suppose this cone cut down upon one
side and opened up (technically, “developed ”), it
would give us a sector of a circle, as in Fig. 48, and

Fig. 48.—Developed

Cone.

the angle of the sector would be the sum total of
the angles between all the adjacent meridians tan-
gent to the earth on that parallel. Now it is easy to prove that the angle
of this sector equals 360◦ × sin φ (φ being the latitude). (1) The circum-
ference of the parallel AB (Fig. 47) = 2π ×R cos φ, since R×cos φ = AD,
which is the radius of the parallel, R being AC, the radius of the globe,
and the angle ACD equal to 90◦ − φ. (2) The side AV of the cone (which
will be the radius of the sector when the cone is developed) = R cot φ;
so that the circumference of the circle which has V A for its radius would
be 2π × R cot φ.

Hence (ABA being the circumference of the parallel forming the edge
of the developed sector ABA V in Fig. 48), the angle of the sector AV A

(greater than 180◦ in the figure): 360◦ = arc ABA : whole circumference

ABA m; or angle V : 360◦ = R cos φ : R cot φ;
whence

V = 360◦ cos φ = 360◦ sin φ.

cot φ

V is the total angle described by the plane of the pendulum in a day.

At the pole the cone produced by the tangent lines becomes a little

“button,” a complete circle. At the equator it becomes a cylinder, and
the angle is zero.

In order to make the experiment successfully, many precautions must
be taken. It is specially important that the pendulum should vibrate in
a true plane, without any lateral motion. To secure this end, it must

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101
be carefully guarded against all jarring motion and air-currents.

To
diminish the effect of all such disturbances, which will always occur to a
certain extent, the pendulum should be very heavy and very long, and
of course the suspended ball must be truly round and smooth. Ordinary
clock-work cannot be used to keep the pendulum in vibration, since it
must be free to swing in every plane. Usually, the apparatus once started
is left to itself until the vibrations cease of their own accord; but Foucault
contrived a most ingenious electrical apparatus, which we have not space
to describe, by means of which the vibration could be kept up for days
at a time without producing any hurtful disturbance whatever.

It will be noticed that this experiment is most effective precisely
where the experiment of the falling bodies fails. This is best near the
pole, the other at the equator.

142. 3. By the Gyroscope, an ex-
periment also due to Foucault and pro-
posed and executed soon after the pen-
dulum experiment.

The instrument shown in Fig. 49
consists of a wheel so mounted in gim-
bals that it is free to turn in every
direction, and so delicately balanced
that it will stay in any position if
undisturbed.

If the wheel be set to
rotating rapidly, it will maintain the
direction of its axis invariable, unless
acted upon by extraneous force.

If,
then, we set the axis horizontal and
arrange a microscope to watch a mark
upon one of the gimbals, it will appear

Fig. 49.—Foucault’s
slowly to shift its position as the earth

Gyroscope.

revolves, in the same way as the plane
of the pendulum behaves.

143. 4. There are many other phenomena which depend upon and
really demonstrate the earth’s rotation. We merely mention them:—
a. The Deviation of Projectiles. In the northern hemisphere a pro-
jectile always deviates towards the right; in the southern hemisphere
toward the left.

b. The Trade Winds.

c. The Vorticose Revolution of the Wind in Cyclones. In the north-

THE EARTH.

102
ern hemisphere the wind in a cyclone moves spirally towards the centre
of the storm, whirling counter clock-wise, while in the southern, the spi-
ral motion is with the hands of a watch. The motion is explained in
either case by the fact that currents of air, setting out for the centre of
disturbance where the cyclone is formed, deviate like projectiles, to the
right in the northern hemisphere, and towards the left in the southern
hemisphere, so that they do not meet squarely in the centre of distur-
bance.

d. The Ordinary Law of Wind-change; that is, in the northern hemi-
sphere the north wind, under ordinary circumstances, changes to a north-
east, a northeast wind to an east, east to southeast, etc. When the wind
changes in the opposite direction, it is said to “back ” around. In the
southern hemisphere it of course usually backs around, much to the dis-
concertment of the early Australian settlers.

It might seem at first time the rotation of the earth, which occu-
pies twenty-four hours, is not a very rapid motion. A point on the
equator, however, has to move nearly one thousand miles an hour,
which is about fifteen hundred feet per second, and very nearly the
speed of a cannon-ball.

144. Invariability of the Earth’s Rotation.—It is a ques-
tion of great importance whether the day changes its length. Theo-
retically it must almost necessarily do so. The friction of the tides,
and the deposits of meteoric matter upon the earth both tend to
lengthen it; while on the other hand, the earth’s loss of heat by
radiation and consequent shrinkage must tend to shorten it. Then
geological changes, the elevation and subsidence of continents, and
the transportation of matter by rivers, act, some one way, some the
other. At present it can only be said that the change, if any has
occurred since astronomy became accurate, has been too small to
be detected. The day is certainly not longer or shorter by 1 of a

100
second than in the days of Ptolemy, and probably has not changed
by

1
of a second. The criterion is found in comparing the times

1000
at which celestial phenomena, such as eclipses, transits of Mercury,
etc., occur.

III.

145. The Earth’s Form, more accurately stated, is that of
a spheroid of revolution, having an equatorial radius of 6,377,377
metres, and a polar radius of 6,355,270 metres, according to Listing

THE EARTH.

103

(1873); or of 6,378,206.4 and 6,356,583.8 respectively, according to

Clarke.1 It must be understood, also, that this statement is only
a second approximation (the first being that the earth is a globe).

Owing to mountains and valleys, etc., the earth’s surface does not
strictly correspond to that of any geometrical solid whatever.

The flattening at the poles is the necessary consequence of the
earth’s rotation, and might have been cited in the preceding section
as proving it.

146. There are two ways of determining the
form of the earth: one, by measurement of dis-
tances upon its surface in connection with the lat-
itudes and longitudes of the points of observation.

This gives not only the form, but the dimen-
sions. The other method is by the observation
of the varying force of gravity at various points,—
observations which are made by means of a pendu-
lum apparatus of some kind, and determine only
the form, but not the size of the earth.

147. 1. Measurements of Arcs of Meridian in

Fig. 50.—A

Triangulation.

Different Latitudes.—To determine the size of the
earth regarded as a sphere, a single arc of meridian
in any latitude is sufficient. Assuming, however, that the earth is not
a sphere, but a spheroid with elliptical meridians, we must measure
at least two such arcs, one of which should be near the equator, the
other near the pole.

The astronomical work consists simply in finding with the great-
est possible accuracy the difference of latitude between the terminal
stations of the meridian arc. The geodetic work consists in measur-
ing their distance from each other in miles, feet, or metres, and it
is this part of the work which consumes the most time and labor.

The process is generally that known as triangulation.

Two stations are selected for the extremities of a base line six or seven
miles long, and the ground between them is levelled as if for a railroad.

The distance between these stations (A and B in Fig. 50) is then carefully
measured by an apparatus especially designed for the purpose and with
an error not to exceed half an inch or so in the whole distance. A third

1This is Clarke’s spheroid of 1866, and is adopted by the United States

Coast and Geodetic Survey. See Appendix for his spheroid of 1878.

THE EARTH.

104
station, 1, is then chosen, so situated that it will be visible from both

A and B, and all the angles of the triangle AB 1 are measured with
great care by a theodolite. A fourth station, 2, is then selected, such
that it will be visible from A and 1 (and if possible from B also), and
the angles of the triangle A 1 2 are measured in the same way. In this
manner the whole ground between the two terminal stations is covered
with a network of triangulation, the two terminal stations themselves
being made two of the triangulation points. Knowing one distance and
all the angles in this system, it is possible to compute with great accuracy
the exact length of the line 1 5 and its direction.

The sides of the triangles are usually from twenty-five to thirty
miles in length, though in a mountainous country not infrequently
much longer ones are available. Generally speaking, the fewer the
stations necessary to connect the extremities of the arc, and the
longer the lines, the greater will be the ultimate accuracy. In this
way it is possible to measure distances of 200 or 300 miles with a
probable error not exceeding two or three feet.

Many arcs of meridians have been measured in this way,—not less
than twenty or thirty in different parts of the earth, the most extensive
being the so-called Anglo-French arc, extending more than twelve degrees
in length; the Indian arc, nearly eighteen degrees long; and the great

Russo-Scandinavian arc, more than twenty-five degrees in length, and
reaching from Hammerfest to the mouth of the Danube. One short arc
has been measured in South America and one in South Africa.

In a general way, it appears that the higher the latitude the
longer the arc. Thus, near the equator the length of a degree has
been found to be 362,800 feet in round numbers, while in northern

Sweden, in latitude 66◦, it is 365,800 feet; in other words, the earth’s
surface is flatter near the poles. It is necessary to travel 3000 feet
further in Sweden than in India to increase the latitude one degree,
as measured by the elevation of the celestial pole.

148. The deduction of the exact form of the earth from such
measurements is an abstruse problem. Owing to errors of observa-
tion and local deviations in the direction of gravity, the different arcs
do not give strictly accordant results, and the best that can be done
is to find the result which most nearly satisfies all the observations.

If we assume that the form is that of an exact spheroid of rev-
olution, with all the meridians true ellipses and all exactly alike,
the problem is simplified somewhat, though still too complicated

THE EARTH.

105
for discussion here. Theory indicates that the form of a revolv-
ing mass, fluid enough to yield to the forces acting in such a case,
might, and probably would, be such a spheroid ; but other forms are
also theoretically possible, and some of the measurements rather in-
dicate that the equator of the earth is not a true circle, but an oval
flattened by nearly half a mile. On the whole, however, astronomers
are disposed to take the ground that since no regular geometrical
solid whatsoever can absolutely represent the form of the earth, we
may as well assume a regular spheroid for the standard surface, and
consider all variations from it as local phenomena, like hills and
valleys.

149. Each measurement of a degree
of latitude gives the “radius of curvature,”

as it is called, of the meridian at the de-
gree measured.

The length of a degree
from 44◦ 30 to 45◦ 30 , multiplied by 57.29

(the number of degrees in a radian), gives
the radius of the “osculatory circle,” which
would just fit the curve of the meridian at
that point, Having a table giving the actual
length of each degree of latitude, we could
construct the earth’s meridian graphically

Fig. 51.

Radii of Curvature of the
as follows:—

Meridian.

Draw the line AX, Fig. 51. On it lay off

Aa, equal to the radius of curvature of the first measured degree (that
is, 57.3 times the length of the degree), and with a as centre, describe an
arc AB, making the angle AaB just one degree. Next produce the line

Ba to b, making Bb the radius of curvature of the second degree, and
draw this second degree-arc; and so proceed until the whole ninety have
been drawn. This will give one quarter of the meridian, and of course
the three other quarters are all just like it. a, b, c, etc., are called the

“centres of curvature” of the different degrees.

If we assume the curve to be an ellipse, then the equatorial semi-
diameter AO, and the polar, P O, are given respectively by the two
formulas, AO = 3 qp2 and P O = 3 q2p, q and p being the radii of
curvature (Aa and P e in the figure) at the equator and pole.

150. The “ellipticity” or “oblateness” of an ellipse is the frac-
tion found by dividing the difference of the polar and equatorial

THE EARTH.

106
diameters by the equatorial, and is expressed by the equation

A − B
d =

.

A

In the case of the earth this is 1 , according to Clarke’s spheroid, of

295

1866. Until within the last few years Bessel’s smaller value, viz., 1 ,

299
was generally adopted. Listing’s larger value, 1 , is now preferred

288
by some.

The ellipticity of an ellipse must not be confounded with its
eccentricity. The latter is

√A2 − B2
e =

,

A
and is always a much larger numerical quantity than the ellipticity.

In the case of the earth’s meridian, it is

1
as against

1 . Its

12.1

295
symbol is usually e.

151. Arcs of longitude are also available for determining the earth’s
form and size. On a spherical earth a degree of longitude measured
along any parallel of latitude would be equal to one degree of the equator
multiplied by the cosine of the latitude. On an oblate or orange-shaped
spheroid (the surface of which lies wholly within the sphere having the
same equator) the degrees of longitude are evidently everywhere shorter
than on the sphere, the difference being greatest at a latitude of 45◦.

In fact, arcs in any direction between stations of which both the lati-
tude and longitude are known can be utilized for the purpose; and thus
the extensive surveys that have been made in different countries have
given us a pretty accurate knowledge of the earth’s dimensions. It is very
desirable, that in some way the chain of actual measurements should be
extended from the eastern continent to the western, but the immense
difficulties of so doing are obvious.

At present the distance from a point on the earth’s surface (say the
observatory at Washington) to any other point in the opposite hemi-
sphere (say the observatory at the Cape of Good Hope) is uncertain to
perhaps the extent of a quarter of a mile.

152. 2. Pendulum Experiments.—Since
l
π2l
t = π

(Physics, p. 72), g =

;
g
t2

THE EARTH.

107
we can therefore measure the variations of the force of gravity, g, at
different parts of the earth, either by taking a pendulum of invari-
able length and determining t, the time of its vibration; or by mea-
suring the length, l, of a pendulum which will vibrate seconds. Ex-
tensive surveys of this sort have been made, and are still in progress,
and it is found that the force of gravity at the pole exceeds that at
the equator by about 1 part. In other words, a person who weighs

190

190 pounds at the equator (by a spring balance) would, if carried to
the pole, show 191 pounds by the same balance.

The apparatus most used at present for the purpose of measuring
the force of gravity is a modification of the so-called Kater’s pendulum.

The pendulum itself usually consists of a brass tube about an inch in
diameter and about four feet long, carrying a ball three or four inches
in diameter at each end, both balls being exactly of the same size, but
one solid while the other is hollow. Two knife edges are inserted through
the rod at right angles, one near the heavy ball and the other at just
the same distance from the lighter one, and the weights and dimensions
of the apparatus are so adjusted that the time of vibration will be very
approximately the same whether the pendulum is swung heavy end up or
light end up, and will be not far from one second. The distance between
the knife edges will then, according to the theory of the pendulum, be
very nearly equal to the length of a simple pendulum vibrating in the
same time; and the small difference can be accurately calculated when
we know the exact time of vibration, each end up. The knife edges swing
on agate planes which are fastened upon a firm support; and great pains
must be taken to have the support really firm. Professor Peirce of our

Coast Survey a few years ago detected important errors in a majority of
the earlier pendulum observations, due to insufficient care in this respect.

153. The observations consist in comparing the pendulum with a
clock, either by noting the “coincidences,” or by an electrical record au-
tomatically made on a chronograph. A pin attached to the end of the
pendulum touches a globule of mercury (which is momentarily raised for
the purpose once in eight or ten minutes), and so records the swing upon
the chronograph sheet. The observations need to be carefully corrected
for temperature (which, of course, affects the distance between the knife
edges), for the length of arc through which the pendulum is swinging,
and for the resistance of the air. The observations determine the “force
of gravity” (French “pesanteur ”) at the station. This “force of gravity,”

however, thus determined, is not simply the earth’s attraction, but in-
cludes also the effects of the centrifugal force, due to the earth’s rotation,
which we must consider and allow for.

THE EARTH.

108

154. At the equator the centrifugal force acts vertically in direct
opposition to gravity, and is given by the well-known formula

V 2

C = R

(see Physics, p. 62), in which V is the velocity of the earth’s surface
at the equator, and R the earth’s radius. Since V is equal to the
earth’s circumference divided by the number of seconds in a sidereal
day, we have

2πR

4π2R

V =

, and C =

.

t
t2

Now R, the radius of the earth, equals

20,926,000 feet; and t equals 86,164 mean-
time seconds.

C, therefore, comes out to

0.111 feet, which is 1 of g, g being 32 1 feet.

289

6

We may remark in passing that if the rate of
rotation were seventeen times as great, C would
be 172, or 289 times greater than now, and would
equal gravity; so that on that supposition bodies

Fig. 52.

The Earth’s
at the equator would weigh absolutely nothing,

Centrifugal Force.

and any greater velocity of rotation would send
them flying.

At any other latitude, since M N = OQ cos M OQ,1 the centrifu-
gal force, c, equals C cos φ, acting at right angles to the axis of the
earth and parallel to the plane of the equator. Now, this centrifugal
force c is not wholly effective in diminishing the weight of a body,
but only that portion of it (M R in Fig. 52) which is directed ver-
tically. c is M T in the figure, and M R is equal to c multiplied by
the cosine of φ, which finally gives us C × cos2 φ for the amount
by which the centrifugal force diminishes gravity at a station whose
latitude is φ.

Every observation, therefore, of the “force of gravity,” obtained
by the pendulum, needs to be increased by the quantity
g × cos2 φ,

289
in order to get the real value of the earth’s gravitational attraction
at the point of observation.

1This is not exact, since M N in an oblate spheroid is less than OQ ×
cos M OQ; but the difference is unimportant in the case of the earth.

THE EARTH.

109

The other component of c (viz. M S) acts at right angles to gravity
and parallel to the earth’s surface, and is given by the formula

C cos φ sin φ = 1 C sin 2φ.

2

The direction of still water is determined by the resultant of the earth’s
attraction combined with this deflecting force acting towards the equator;
so that this surface is not perpendicular to a line drawn towards the
centre of the earth anywhere excepting at the equator and the poles.

155. Having a series of pendulum observations, we can then
form a table showing the force of gravity at each station; and cor-
recting this by adding the amount of the centrifugal force at each
place, we shall have the force of the earth’s attraction. This is
greater the nearer each station is to the centre of the earth; but
unfortunately there is no simple relation connecting the force with
the distance. The attraction depends not only on the distance from
the centre of the earth, but also upon the form of the earth and
the constitution of its interior, and the arrangement of its strata of
different density. We may safely assume, however, that the earth
is made up concentrically, so to speak; the strata of equal den-
sity being arranged like the coats of an onion. On this hypothesis

Clairaut, in 1742, demonstrated the relation given below, which is
always referred to as Clairaut’s equation.

Let w be the loss of weight between the equator and the pole,
and C the centrifugal force at the planet’s equator, both being ex-
pressed as fractions of the equatorial force of gravity, and let d be
the ellipticity of the planet.

Then, as Clairaut proved,
d + w = 2 1 × C;

2
whence
d = 2 1 C − w.

2

In the case of the earth we have

1

1
d = 2 1 ×

−

,

2

289

190

1
which gives
d =

.

292.8

Considering all the data, the most that can safely be said as to
d is that it lies between the fractions 1 and 1 . (Clarke’s later

290

295
values for d are larger than that adopted by the Coast Survey.)

THE EARTH.

110

156. Astronomical, Geographical, and Geocentric Lati-
tudes.—The astronomical latitude of a place has been defined as
the elevation of the pole, or, what comes to the same thing, it is the
angle between the plane of the equator and the direction of gravity at
that place, however that direction may be affected by local causes.

The geocentric latitude, on the other hand, is the angle made at
the centre of the earth (as the word implies) between the plane of
the equator and a line drawn from the observer to the centre of the
earth, which line of course does not coincide with the direction of
gravity, since the earth is not spherical.

The geographical or geodetic latitude of a station is the angle
formed with the plane of the equator by a line drawn from the
station perpendicular to the surface of the standard spheroid.

If the earth’s surface were strictly spheroidal,
and there were no local variations of gravity, the
astronomical latitude and the geographical lati-
tude would coincide—and they never differ greatly;
but the geocentric latitude differs from them by a
very considerable quantity—as much as 11 in lati-
tude 45◦. The geocentric latitude is but little used
except in certain astronomical calculations where

Fig. 53.

parallax is involved.

Astronomical and

In Fig. 53, the angle M OQ is the geocentric

Geocentric Latitude.

latitude of M , while M N Q is the geographical lat-
itude. M N Q is also the astronomical latitude, unless there is some local
disturbance of the direction of gravity. The angle OM N , which is the
difference between the geocentric and astronomical latitudes, is called

“the angle of the vertical.”

157. It will be noticed that the astronomical latitude of a place
is the only one of these three latitudes which is determined directly
by observation. In order to know the geocentric and geographical
latitudes of a place, we must know the form and dimensions of the
earth, which are ascertained only by the help of observations made
elsewhere.

The geocentric degrees are longer near the equator than near
the poles, and it is worth noticing that if we form a table giving
the length of each degree of geographical latitude from the equator
to the pole, the same table, read backwards, gives the length of
geocentric degrees.

THE EARTH.

111

Since the earth is ellipsoidal instead of spherical, it is evident
that lines of “level” on the earth’s surface are affected by the earth’s
rotation. If this rotation were to cease, the direction of gravity
would be so much changed that the Gulf of Mexico would run up
the Mississippi River, because the distance from the centre of the
earth to the head of the river is less by some thousands of feet than
the distance from the mouth of the river to the centre of the earth.

158. Station Errors.—The irregularities in the direction of grav-
ity are by no means insensible as compared with the accuracy of modern
astronomical observation, and the difference between the astronomical
latitude and longitude of a place and the geographical latitude and lon-
gitude of the same place constitute what is called the “station error.” In
the eastern part of the United States these station errors, according to
the Coast Survey observations, average about 1 1 . Errors of from 4 to

2

6 are not uncommon, and in mountainous countries, as for instance in
the Caucasus and in Northern India, these errors occasionally amount
to 30 or 40 . They are not “errors” in the sense that the astronomical
latitude of the place has not been determined correctly, but are merely
the effects of the irregular distribution of matter in the crust of the earth
in altering the direction of gravity. Pendulum observations show local
variations in the force of gravity quite proportional to the deviations
which the station-errors show in its direction.

IV.

159. The Earth’s Mass and Density.—The ‘mass’ of a body
is the quantity of matter that it contains, the unit of mass being
the quantity of matter contained in a certain arbitrary body which
is taken as a standard. For instance, a “kilogram” is the quantity
of matter contained in the block of platinum preserved at Paris as
the standard of mass.1 A pound is similarly defined by reference to
the prototypes at Washington and London.

Two masses of matter are defined as equal which require the
same expenditure of energy to give them the same velocity; or vice
versa, those are equal which, when they have the same velocity, pos-
sess the same energy, and, in giving up their motion and coming to
rest, do the same amount of work.

1This was meant to be just equal to the mass contained in a cubic decimeter
of water at its maximum density, and is so very nearly indeed.

THE EARTH.

112

Masses can therefore be compared by placing them in the same field
of force and comparing the energies developed in them when they have
moved equal distances under the action of the force. This method, how-
ever, is seldom convenient.

160. Proportionality of Mass to Weight.—Newton showed
by his experiments with pendulums of different substances, that at
any given point the attraction of the earth for a body of any kind
of matter is proportional to the mass of that body; the attraction
being measured as a pull or “stress” in this case, and called “the
weight ” of the body. In other and more common language, the
mass of a body is proportional to its weight (we must not say it is its
weight), provided the weighing of the bodies thus compared is done,
in cases where scientific accuracy is essential, at the same place
on the earth’s surface. Practically, therefore, we usually measure
the masses of bodies by simply weighing them. It is to be carefully
observed, however, that the words “kilogram,” “pound,” “ton,” etc.,
have also a secondary meaning, as denoting units of pull and push,—
of “stress” speaking strictly and technically,—or of “force,” as that
much abused word is very generally used.

It is, from a literary point of view, just as proper to speak of a stress
or a pull of a hundred pounds1 as of a mass of a hundred pounds, but
the word “pound” means an entirely different thing in the two cases.

At the surface of the earth the relation between the ideas, however, is
so close that the way in which the ambiguity came about is perfectly
obvious, and it is hardly probable that language will ever change so as
to remove it. To a certain extent it is admittedly unfortunate, and the
student must always be on his guard against it. At the earth’s surface
a mass of 100 pounds always “weighs” very nearly 100 pounds; but, to
anticipate slightly, at an elevation of 4000 miles above the surface, the
same mass would “weigh” only 25 pounds; at the distance of the moon
about half an ounce; while on the surface of the sun it would “weigh”

nearly 2800 pounds.

161. Gravity.—The law of gravitation discovered by Newton
declares that any particle of matter attracts any other particle with
a force (“stress,” if the bodies are prevented from moving) pro-
portional inversely to the square of the distance between them, and

1Of course the student will remember that we have a unit of stress,—the
dyne,—which is wholly free from this objection of ambiguity.

THE EARTH.

113
directly to the product of their masses; or, as a formula, we may
write,

M

F = k

1 × M2 ,
d2
in which M1 and M2 are the two masses, and d the distance between
them, while k is a constant numerical factor depending upon the
units employed.1

We must not imagine the word “attract ” to mean too much.

It
merely states the fact that there is a tendency for the bodies to move
toward each other, without including or implying any explanation of
the fact. So far, no explanation has appeared which is less difficult to
comprehend than the fact itself. Whether bodies are drawn together by
some outside action, or pushed together; or whether they themselves can
act across space with mathematical intelligence,—in what way it is that

“attraction” comes about, is still unknown,—apparently as inscrutable
as the very nature and constitution of an atom of matter itself; it is
simply a fundamental fact.

162. When the distance between attracting bodies is large as
compared with their own magnitude, then reckoning the distance
between their centres of mass as their true distance, the formula is
sensibly true for them as it would be for mere particles. When, how-
ever, the distance is not thus great, the calculation of the attraction
becomes a very serious problem, involving what is known as a “dou-
ble integration.” We must find the attraction of each particle of the
first body upon each particle of the other body, and take the sum
of all these infinitesimal stresses. Newton, however, showed that if
the bodies are spheres, either homogeneous or of concentric struc-
ture, then they attract and are attracted precisely as if the matter in
them were wholly collected at their centres. The earth, for instance,
attracts a body at its surface very nearly as if it were all collected

1It will not do to write the formula

M1 × M2

F =
d2

(omitting the k), unless the units are so chosen that the unit of force shall be
equal to the attraction between two masses each of one unit, at a distance of
one unit. It is not true that the attraction between two particles, each having
a mass of one pound, at a distance of one foot, is equal to a stress of one pound

(of force), as would rather naturally be inferred if we should write the equation
without the constant factor.

THE EARTH.

114
at its own centre, 4000 miles distant; not exactly so, because the
earth is not strictly spherical; but in what follows we shall neglect
this slight inaccuracy.

163. In order, then, to find the mass of the earth in kilograms,
pounds, or tons, we must find some means of accurately comparing
its attraction for some object on its own surface with the attraction
of the same object by some body of known mass, at a measured
distance. The difficulty lies in the fact that the attraction produced
by any body, not too large to be handled conveniently, is so exces-
sively small that only the most delicate operations serve to detect
and measure it.

The first successful attack upon the problem was made in 1774
by Maskelyne, the Astronomer Royal, by means of what is now
usually referred to as,—

164. 1. “The Mountain Method,” because, in fact, the
earth in this operation is weighed against a mountain.

Two stations were chosen on the same
meridian, one north and one south of the
mountain Schehallien, in Scotland.

In
the first place, a careful topographical
survey was made of the whole region, giv-

Fig. 54.

ing the precise distance between the sta-

The Mountain Method of
tions, as well as the exact dimensions of

Determining the Earth’s

Density.

the mountain, which is a “hog-back” of
very regular contour. From the known dimensions of the earth and
the measured distance, the difference of the geographical latitudes
of the two places M and N (Fig. 54) can be accurately computed;
i.e., the angle which the plumb lines at M and N would have made
if there were no mountain there.

In this case it was 41 . The next operation was to
observe the astronomical latitude at each station. This
astronomical difference of latitude, i.e., the angle which
the plumb lines actually do make, was found to be 53 ,
the plumb lines at M and N being drawn inward out of
their normal position by the attraction of the mountain
to the extent of 6 on each side; so that the astronom-
ical difference of latitude was increased by 12 over the Fig. 55.

geographical.

THE EARTH.

115

Now, in such a case the ratio of gravity to the deflecting force, ac-
cording to the laws of the composition of forces, is that of aM to aA in
the figure (Fig. 55), or the ratio of 1 to the tangent of the deflection, δ;
g
that is, calling the deflecting force f , we have

= cot δ, = cot 6 in this
f
case.

By the law of gravitation, the earth’s attracting force at its surface
is given by the formula

E
g = k

,

R2
where E is the mass of the earth (the unknown quantity of our problem),
and R its radius, 4000 miles. Similarly, if C in the figure is the centre of
attraction of the mountain, we have
m
f = k

,
d2
m being the mass of the mountain, and d the distance from C to the
station. Combining this with the preceding, we get

E
g

R 2

=

,
m
f
d
or, in this case,

E

R 2

= cot 6

.

m
d

We thus get the ratio of the earth’s mass to that of the mountain;
and provided we can find the mass of the mountain in tons or any
other known unit of mass, the problem will be completely solved. By
a careful geological survey of the mountain, with deep borings into
its strata, the mass of the mountain was determined as accurately
as it could be (though here is the weakest point of the method), and
thus the mass of the earth was finally computed.

Now, knowing the diameter of the earth, its volume in cubic
feet is easily found, and from the volume and the known number
of mass-pounds, (62 1 nearly) in a cubic foot of water, the weight

2
the earth would have, if composed of water, follows. Comparing
this with the mass actually found, we get the density, which in this
experiment came out 4.71.

A repetition of the work in 1832 at Arthur’s Seat, near Edin-
burgh, gave 5.32.

THE EARTH.

116

165. 2. Much more trustworthy re-
sults, however, are obtained by the method
of the torsion balance, first devised by

Michell, but first employed by Cavendish
in 1798. A light rod, carrying two small
balls at its extremities, is suspended hori-
zontally at its centre by a long fine metal-
lic wire. If it be allowed to come to rest,
and then a very slight deflecting force be
applied, the rod will be pulled out of posi-
tion by an amount depending on the stiff-
ness and length of the wire, as well as the
force itself. When the deflecting force is re-

Fig. 56.—Plan of the
moved, the rod will vibrate back and forth

Torsion Balance.

until brought to rest by the resistance of
the air. The “torsional coefficient,” as it is called (i.e., the stress
corresponding to a torsion of one revolution), can be accurately de-
termined by observing the time of vibration when the dimensions
and weight of the rod and balls are known. If, now, two large balls

A and B are brought near the smaller ones, as in Fig. 56, a deflec-
tion will be produced by their attraction, and the small balls will
move from a and b to a and b . By shifting the large balls to the
other side at A and B , we get an equal deflection in the opposite
direction, i.e., to a and b , and the difference between the two po-
sitions assumed by the small balls, i.e., a a and b b , will be twice
the deflection.

It is not necessary, nor even best, to wait for the balls to come to
rest. We note the extremities of their swing. The middle point of the
swing gives the point of rest, and the time occupied by the swing is the
time of vibration, which we need in determining the coefficient of torsion.

We must also measure accurately the distance, Aa and Bb between the
centre of each of the large balls and the point of rest of the small ball
when deflected.

The earth’s attraction on each of the small balls of course equals
the ball’s weight. The attractive force of the large ball on the small
one near it is found directly from the experiment. If the deflection,
for instance, is 1◦ and the coefficient of torsion is such that it takes
one grain to twist the wire around one whole revolution, then the
deflecting force, which we will call f as before, will be 1 of a grain.

360

Call the mass of the large ball B, and let d be the measured distance

THE EARTH.

117
from its centre to that of the deflected ball. We shall then have

B
f = k

,
d2
also, w being the weight of the small ball,

E
w = k

,

R2
whence we get, very much as in the preceding case,

E
w

R 2

=

,

B
f
d
which gives the mass of the earth in terms of B.

The method differs from the preceding in that we use a large ball
of metal instead of a mountain, and measure its deflecting force by
a laboratory experiment instead of comparing astronomical obser-
vations with geodetic measurements.

166. In the earlier experiments by this method the small balls were
of lead, about two inches in diameter, at the extremities of a light wooden
rod, five or six feet long, enclosed in a case with glass ends, and their
position and vibration was observed by a telescope looking directly at
them from a distance of several feet. The attracting masses, B, were balls
also of lead, about one foot in diameter, mounted on a frame pivoted in
such a way that they could be easily brought to the required positions.

Great difficulty was caused by air currents in the case, and it was
necessary to enclose the whole apparatus in a small room of its own
which was covered with tin-foil on the outside, and to avoid going near
the room or allowing any radiant heat to strike it for hours before the
observations. Baily, in England, and Reich, in Germany, between 1838
and 1842, made very extensive series of observations of this kind. Baily
obtained 5.66 for the earth’s density, and Reich 5.48.

The experiment was repeated in 1872 by Cornu, in Paris, with a
modified apparatus.

The horizontal bar was in this case only half a metre long, of alu-
minium, with small platinum balls at the end. For the large balls, glass
globes were used, which could be pumped full of mercury or emptied at
pleasure. The whole was enclosed in an air-tight case, and the air ex-
hausted by an air-pump. The deflections and vibrations were observed
by means of a telescope watching the image of a scale reflected in a

THE EARTH.

118
small mirror attached to the aluminium beam near its centre, accord-
ing to the method first devised by Gauss and now so generally used in
galvanometers and similar apparatus. Cornu obtained 5.56 as the re-
sult, and showed that Baily’s figure required a correction which, when
applied, would reduce it to 5.55.

167. 3. Potsdam Observations.—During 1886 and 1887 an-
other series of observations was made by Wilsing, at Potsdam, with
apparatus similar in principle to the torsion balance, except that
the bar carrying the balls to be attracted was vertical, and turned
on knife edges very near its centre of gravity. The knife edges, like
those of an ordinary balance, rested upon agate planes, and the
centre of gravity of the apparatus was so adjusted that one vibra-
tion of the pendulum, under the influence of gravity alone, would
occupy from two to four minutes. The deflecting weights in this
case were large cylinders of cast iron, suspended in such a way that
they could be brought opposite the small balls, first on one side
and then on the other. The whole was set up in a basement, and
carefully and very effectually guarded against all changes of tem-
perature, the arrangements being such that all manipulations and
observations could be effected from the outside without entering the
room. The deflections and vibrations were observed by a reflected
scale, as in Cornu’s observations. The result obtained was 5.59.

Several other methods have been used; of less scientific value,
however.

168. a. The mass of the earth can he deduced by ascertaining the
force of gravity at the top of a mountain and at its base, by means of
pendulum experiments. The mass of the mountain must be determined
by a survey, just as in the Schehallien method, which makes the method
unsatisfactory. At the top of a mountain the height of which is h, and
the distance of its centre of attraction from the top is d, gravity will be
made up of two parts, one the attraction of the earth at a distance from
its centre equal to R + h, and the other the attraction of the mountain
alone considered. Calling the mass of the mountain m, and gravity at its
summit g (g being the force of gravity at the earth’s surface), we shall
have the proportion

E

E
m
g : g =

:

+

,

R2

(R + h)2
d2
the second fraction in the last term of the proportion being the attrac-
tion of the mountain. When g and g are ascertained by the pendulum

THE EARTH.

119
experiments, E remains as the only unknown quantity, and can be read-
ily found. Observations of this kind were made by Carlini, in 1821, at

Mt. Cenis, and the result was 4.95.

169. b. By means of pendulum observations at the earth’s surface
compared with those at the bottom of a mine of known depth.

This
method was employed by Airy in 1843, at Harton Colliery, 1200 feet
deep; result, 6.56. In this case the principle involved is somewhat differ-
ent. At any point within a hollow, homogeneous, spherical shell, gravity
is zero, as Newton has shown. The attraction balances in all directions.

If, then, we go down into a mine, the effect on gravity is the same as if
a shell composed of all that part of the earth above our level had been
removed. At the same time our distance from the earth’s centre has been
decreased by d, the depth of the mine.

E

At the surface
g = k

, as before.

R2

E − “shell”

At the bottom of the mine
g = k

.

(R − d)2

Comparing the two equations, we find E in the terms of the shell,
since the ratio of g to g is given by pendulum observations. Obviously,
however, the mass of the “shell” is difficult to determine with accuracy.

And it is by no means homogeneous, so that there is no great reason
for surprise at the discordant result. g was found to be actually greater
than g, showing that although at the centre of the earth the attraction
necessarily becomes zero, yet as we descend below the surface, gravity
increases for a time down to some unknown but probably not very great
depth, where it becomes a maximum.

170. c. By experiments with a common balance. If a body be hung
from one of the scale-pans of a balance, its apparent weight will obviously
be increased when a large body is brought very near it underneath; and
this increase can be measured. Poynting in England and Jolly in Ger-
many have recently used this method, and have obtained results agree-
ing very fairly with these got from the torsion balance. The experiment,
with some modifications, is soon to be tried again on a very large scale
in Germany.

171. Constitution of the Earth’s Interior.—Since the av-
erage density of the earth’s crust does not exceed three times that
of water, while the mean density of the whole earth is about 5.58

(taking the average of all the most trustworthy results), it is obvious
that at the centre the density must be very much greater than at

THE EARTH.

120
the surface,—very likely as high as eight or ten times that of water,
and equal to the density of the heavier metals. There is nothing
in this that might not have been expected. If the earth were ever
fluid, it is natural to suppose that in the solidification the densest
materials would settle towards the interior.

Whether the interior of the earth is solid or fluid it is difficult to
say with certainty. Certain tidal phenomena, to be discussed hereafter,
have led Sir William Thomson and the younger Darwin to conclude that
the earth as a whole is solid throughout, and “more rigid than glass,”

volcanic centres being mere pustules in the general mass. To this many
geologists demur.

As regards the temperature at the earth’s centre, it is hardly an
astronomical question, though it has very important astronomical re-
lations. We can only take space to say that the temperature appears
to increase from the surface downward at the rate of about one degree

Fahrenheit for every fifty or sixty feet, so that at the depth of a few miles
the temperature must be very high.

THE EARTH’S ORBITAL MOTION.

121

CHAPTER VI.

THE APPARENT MOTION OF THE SUN AMONG THE STARS,

AND THE EARTH’S ORBITAL MOTION.—THE EQUATION

OF TIME, PRECESSION, NUTATION, AND ABERRATION.—

VARIOUS KINDS OF “YEAR.”—THE CALENDAR.

172. The Annual Motion of the Sun.—The apparent an-
nual motion of the sun must have been one of the earliest noticed
of all astronomical phenomena. Its discovery antedates history.

As seen by the people in Europe and Asia, the sun, starting
in the spring, mounts higher in the sky each day at noon for three
months, appears to stand still for a few days at the summer solstice,
and then descends towards the south, reaching in the autumn the
same noonday elevation it had in the spring. It keeps on its south-
ward course to a winter solstice in December, and then returns to
its original height at the end of the year, marking and causing the
seasons by its course. A year, the interval between the successive
returns of the sun to the same position, was very early found to
consist of a little more than three hundred and sixty days.

Nor is this all. The sun’s motion is not merely a north-and-south
motion, but it also moves eastward among the stars; for in the spring
the stars which are rising in the eastern horizon at sunset are dif-
ferent from these which are found there in the summer or winter.

In the spring, the most conspicuous of the eastern constellations at
sunset are Leo and Boötes; a little later, Virgo appears; in the sum-
mer, Ophiuchus and Libra; still later, Scorpio; and in mid-winter,

Orion and Taurus are in the eastern sky.

173. So far as mere appearances go, everything would be ex-
plained by assuming that the earth is at rest and the sun moving
around it; but equally by the converse supposition,—for if the earth
as seen from the sun appears at any point in the heavens, the sun as
seen from the earth must appear in exactly the opposite point, and
must keep opposite, moving through the same path in the sky (but

THE EARTH’S ORBITAL MOTION.

122
six months behind), and always in the same “angular direction,” if
we may use the expression. (Just as two opposite teeth on a gear-
wheel move in the same angular direction, though at any moment
they are moving in opposite linear directions.)

174. That it is really the earth which moves, and not the sun,
is absolutely demonstrated by two phenomena, too minute and del-
icate for pre-telescopic observations, but accessible enough to mod-
ern methods. We can only mention them here, leaving their fuller
discussion for the present. One of them is the aberration of light, the
other the annual parallax of the fixed stars. These can be explained
only by the actual motion of the earth.

175. The Ecliptic.—By observing with a meridian circle daily
the declination of the sun, and the difference between its right as-
cension and that of some star (Flamsteed used α Aquilæ for the
purpose), we shall obtain a series of positions of the sun’s centre
which can be plotted on a celestial globe; and we can thus make
out the path of the sun among the stars, and find the place where
it cuts the celestial equator, and the angle it makes. This path
turns out to be a great circle, as is shown by its cutting the equa-
tor at two points just 180◦ apart (the so-called equinoctial points
or equinoxes), and makes an angle with it of approximately 23 1 ◦.

2

This great circle is called the Ecliptic, because, as was early dis-
covered, eclipses happen only when the moon is crossing it. It may
be defined as the trace of the plane of the earth’s orbit upon the
celestial sphere, just as the celestial equator is the trace of the plane
of the terrestrial equator on the same sphere.

176. Definitions.—The angle which the ecliptic makes with
the equator is called the Obliquity of the ecliptic, and the points
midway between the equinoxes are called the Solstices (sol-stitium),
because at these points the sun “stands” or stops moving in decli-
nation for a short time.

Two circles parallel to the equator, drawn through the solstices,
are called the Tropics (Greek τ ρ´πω), or “turning-lines,” because
there the sun turns from its northward motion to a southward, or
vice versa. The obliquity is, of course, simply equal to the sun’s
maximum declination, or greatest distance from the equator, which
is reached in June and December.

The ancients were accustomed to determine it by means of the gno-

THE EARTH’S ORBITAL MOTION.

123
mon1 (Art. 107). The length of the shadow at noon on the solstitial
days determines the zenith distance of the sun on these days, and the
difference of the zenith distances at the two solstices is twice the angle
desired. The gnomon also determined for the ancients the length of the
year, it being only necessary to observe the interval between days in the
spring or autumn, when the shadow had the same length at noon.

177. The Zodiac and its Signs.—A belt 16◦ wide, 8◦ on
each side of the ecliptic, is called the Zodiac. The name is said to
be derived from ζ ´
ωoν, a living creature, because the constellations
in it (except Libra) are all figures of animals. It was taken of that
particular width by the ancients simply because the moon and the
then known planets never go further than 8◦ from the ecliptic.

This belt is divided into the so-called signs, each 30◦ in length,
having the following names and symbols:—



Aries,



Libra,





Spring

Taurus,

Autumn

Scorpio,



Gemini,



Sagittarius,



Cancer,



Capricornus,





Summer

Leo,

Winter

Aquarius,



Virgo,



Pisces,

The symbols are for the most part conventionalized pictures of the
objects. The symbol for Aquarius is the Egyptian character for water.

The origin of the signs for Leo, Virgo, and Capricornus is not quite clear.

It has been suggested that
is simply a “cursive” form for Λ, the initial
of Λ´ων;
for Παρ (ΠαρΘ´νoς), and
for Tρ (Tρ ´
αγoς).

CELESTIAL LATITUDE AND LONGITUDE.

178. Since the moon and all the principal planets always keep
within the zodiac, the ecliptic is a very convenient circle of reference,
and was used us such by the ancients. Indeed, until the invention
of pendulum clocks, it was on the whole more convenient than the
equator, and more used.

The two points in the heavens 90◦ distant from the ecliptic are
called the Poles of the ecliptic. The northern one is in the constel-

1The Chinese claim to have made an observation of this kind about

4000 b.c., and the result given is very nearly what it should have been at
that time. (The obliquity changes slightly in centuries.) If their observation is
genuine, it is by far the oldest of all astronomical records.

THE EARTH’S ORBITAL MOTION.

124
lation of Draco, about half-way between the stars δ and ζ Draconis.

Now, suppose a set of great circles drawn, like meridians, through
these poles of the ecliptic, and hence perpendicular to that cir-
cle; these are Circles of latitude or secondaries to the ecliptic. The

Longitude of a star or any other heavenly body is, then, the angle
made at the pole of the ecliptic, between the circle of latitude, which
passes through the vernal equinox, and the circle of latitude passing
through the body; or, what comes to the same thing, it is the arc
of the ecliptic included between the vernal equinox and the foot of
the circle of latitude passing through the body. Celestial longitude
is always reckoned eastward from the vernal equinox, completely
around the ecliptic, so that the longitude of the sun when 10◦ west
of the vernal equinox would be written as 350◦, and not as −10◦.

The Latitude of a star is simply its distance north or south of
the ecliptic measured on the star’s circle of latitude.

179. It will be seen that longitude differs from right ascension
in being reckoned on the ecliptic instead of on the equator, nor can it
be reckoned in time, but only in degrees, minutes, and seconds. Lat-
itude differs from declination in that it is reckoned from the ecliptic
instead of from the equator.

The relation between right ascension
and declination on the one hand, and lon-
gitude and latitude on the other, may be
made clearer by the accompanying dia-
gram (Fig. 57), in which EC is the ecliptic
and EQ the equator, E being the vernal
equinox. S being a star, its right ascension

(α) is ER and its declination (δ) is SR; its
longitude (λ) is EL, and its latitude (β) is

Fig. 57.

Relation between Celestial

SL. P and K are the poles of the equa-

Latitude and Longitude,
tor and ecliptic respectively, and the circle
and Right Ascension and

KP CQ is the Solstitial Colure, so called.

Declination.

The student can hardly take too great care
to avoid confusion of celestial latitude and longitude with right ascen-
sion and declination or with terrestrial latitude and longitude. It is, of
course, unfortunate that latitude in the sky should not be analogous to
latitude upon the earth, or celestial longitude to terrestrial. The terms
right ascension and declination are, however, of comparatively recent
introduction, and found the ground preoccupied, celestial latitude and

THE EARTH’S ORBITAL MOTION.

125
longitude being much older.

180. Conversion of λ and β into α and δ, or Vice Versa.

—Right ascension and declination can, of course, always be converted
into longitude and latitude by a trigonometrical calculation. We proceed
as follows: In the triangle ERS, right-angled at R, we have given ER
and RS (α and δ), from which we find the hypothenuse ES and the
angle RES. Next in the triangle ELS, right-angled at L, we have the
hypothenuse ES and the angle LES, which is equal to RES − LEQ

(LEQ being ω, the obliquity of the ecliptic). Hence we easily find EL
and LS.

181. The Earth’s Orbit in Space.—The ecliptic is not the
earth’s orbit, and must not be confounded with it. It is a great circle
of the infinite celestial sphere, the trace made upon the sphere by
the plane of the earth’s orbit, as was stated in its definition. The
fact that it is a great circle gives us no information about the earth’s
orbit, except that the orbit all lies in one plane passing through the
sun. It tells us nothing as to its real form and size.

By reducing the observations of the sun’s right ascension and
declination through the year to longitude and latitude (the latitude
will always be zero, of course, except for some slight perturbations)

and combining them with observations of the sun’s apparent diame-
ter, we can, however, ascertain the real form of the earth’s orbit and
the law of its motion in this orbit. But the size of the orbit—the
scale of miles—cannot be fixed until we can find the sun’s distance.

182. To find the Form of the

Orbit, we may proceed thus: Take a
point S for the sun and draw from it
a line SO, Fig. 58, directed towards
the vernal equinox as the origin of
longitudes. Lay off from S indefinite
lines, making angles with SO equal
to the earth’s longitude on each of
the days observed through the year;
i.e., the angle OS 10, is the longitude
at the time of the 10th observation;

Fig. 58.

Determination of the Form of the
and so on. We shall thus get a sort

Earth’s Orbit.

of “spider,” showing the directions as
seen from the earth on these days.

Next, as to relative distances. While the apparent diameter of

THE EARTH’S ORBITAL MOTION.

126
the sun does not tell us its real distance from the earth, unless
we first know the sun’s real diameter in miles, the changes in the
apparent diameter do inform us as to the relative distance of the
earth at different times, since the nearer we are, the larger the sun
appears,—the distance being inversely proportional to the apparent
diameter (Art. 6). If, then, we lay off on the arms of our “spider”

distances inversely proportional to the number of seconds of arc in
the sun’s measured diameter at each date, these distances will be
proportional to the true distance of the earth from the sun, and the
curve joining the points thus obtained will be a true map of the
earth’s orbit, though without any scale of miles upon it.

When the operation is performed, we
find that the orbit is an ellipse of small
eccentricity (about one-sixtieth), with
the sun, not in the centre, but at one
focus.

183. For the benefit of any who may
not have studied conic sections we define
the ellipse. It is a curve such that the sum

Fig. 59.—The Ellipse.

of the two distances from any point on its
circumference to two points within, called the foci, is always constant,
and equal to what is called the major-axis of the ellipse, SP +P F = AA ,
in Fig. 59. AC is called the semi-major-axis, and is usually denoted by

A or a. BC is the semi-minor-axis, denoted by B or b. The eccentricity,

SC
denoted by e, is the fraction

.

AC

Since BS is equal to A,

SC =

A2 − B2;

√A2 − B2
and
e =

.

A

The points where the earth is nearest to and most remote from
the sun are called respectively perihelion and aphelion and the line
that joins them is, of course, the major axis of the orbit. This line,
considered as indefinitely produced in both directions, is called the
line of apsides,—the major-axis being a limited piece or “sect” of
the line of apsides.

184. The variations of the sun’s diameter are too small to be de-
tected without a telescope (amounting to only about three per cent), so
that the ancients were unable to perceive them. Hipparchus, however,

THE EARTH’S ORBITAL MOTION.

127
about 150 b.c., discovered that the earth is not in the centre of the cir-
cular orbit which he supposed the sun to describe around it. Everybody
assumed, on à priori grounds, never disputed until the time of Kepler,
that the sun’s orbit must be a circle and described with a uniform mo-
tion, because a circle is the only “perfect” curve, and uniform motion
the only perfect motion. Obviously, however, the sun’s apparent motion
is not uniform, because it takes 186 days for the sun to pass from the
vernal equinox to the autumnal through the summer months, and only

179 days to return during the winter. Hipparchus explained this differ-
ence by the hypothesis that the earth is out of the centre of the sun’s
path.

185. To find the Eccentricity of the Orbit.—Having the
greatest and least apparent diameters of the sun, the eccentricity,
e, is easily found. In Fig. 59, since, by definition, e = CS ÷ CA, we
have CS = CA × e, or Ae. The perihelion distance AS is therefore
equal to A × (1 − e), and the aphelion distance SA to A(1 + e).

Suppose now that the greatest and least measured diameters of the
sun are p and q. This gives us the proportion p : q = A(1+e) : A(1−
e), since the diameters are inversely proportional to the distances.

From this we get
p − q
e =

.

p + q

The actual values of p and q are 32 36 .4 and 31 31 .8, which give
e = 0.01678 : this is about 1 , as has been stated.

60

186. To
find
the

Law
of
the

Earth’s

Motion.—By comparing the
measured apparent diameter with the dif-
ferences of longitude from day to day, we
can also deduce the law of the earth’s mo-
tion. On making a table of daily motions
and apparent diameters, we find that these
daily motions vary directly as the squares

Fig. 60.

of the diameters; from which it directly fol-

Equable Description of
lows that the earth moves in such a way

Areas.

that its radius-vector describes areas pro-
portional to the times (a law which Kepler first brought to light in

1609). The radius-vector is the line which joins the earth to the sun
at any moment.

THE EARTH’S ORBITAL MOTION.

128

187. Consider a small elliptical sector, dSc (Fig. 60), described by
the earth in a unit of time. Regarding it as a triangle, its area is given
by the formula 1 SC × SD sin cSd; and calling this angle θ (which will

2
be very small), and considering that in so short a time Sd and Sc would
remain sensibly equal, each being equal to R (the radius-vector at the
middle point of the arc), this formula becomes,

Area of sector = 1 R2θ.

2

Now, calling the sun’s apparent diameter D, we have:

k

R =

,

D

(k being a constant, and depending on the sun’s diameter in miles);
k2
whence

R2 =

.

D2

But our measurements show that θ = k1D2, k1 being another constant.

Substitute these values of R2 and θ in the formula for the area, and we have
k2

Area of sector = 1

× k
k2k

2

1D2 = 1

1,

D2

2
a constant; that is, the area described by the radius-vector in a unit
of time is always the same. The planet near perihelion moves so much
faster, that the areas aSb, cSd, and eSf are all equal to each other, if
the arcs are described in the same time.

188. Kepler’s Problem.—As the
case stands so far, this is a mere fact of
observation; but as we shall see hereafter,
and as was demonstrated by Newton, the
fact shows that the earth moves under the
action of a force always directed in line
with the sun. In such a case the “equable
description of areas” is a necessary me-

Fig. 61.—Kepler’s
chanical consequence. It is true in every

Problem
case of elliptical motion, and enables us to
find the position of the earth or any planet in its orbit at any time,
when we once know the time of its orbital revolution (technically
the period), and the time when it was at perihelion. Thus, the angle

ASP (Fig. 61), which is called the Anomaly of the planet, must be
such that the area of the elliptical sector ASP will be that portion
t
of the whole ellipse which is represented by the fraction

, t be-

T
ing the number of days since the planet last passed the perihelion,

THE EARTH’S ORBITAL MOTION.

129
and T the number of days in the whole period. For instance, if
the earth last passed perihelion on Dec. 31 (which it did), its place
on May 1 must be such that the sector ASP will be 121 of the

365 1

4
whole of the earth’s orbit; since from Dec. 31 to May 1 is 121 days.

The solution of this problem, known as “Kepler’s problem,” leads
to transcendental equations, and lies beyond our scope.

See Watson’s “Theoretical Astronomy,” pp. 53 and 54, or any other
similar work.

189. Anomaly and Equation of the Centre.—The angle

ASP , which has been termed simply the “Anomaly,” is strictly
the true Anomaly, as distinguished from the mean Anomaly. The
former may be defined as the angle actually made at any time by
the radius-vector of a planet with the line of apsides, the angle be-
ing reckoned from the perihelion point completely around in the
direction of the planet’s motion. The mean Anomaly is what the

Anomaly would be at the given moment if the planet had moved with
uniform angular velocity, completing the orbit in the same period,
and passing perihelion at the same time, as it actually does. The
difference between the two anomalies is called the Equation of the

Centre. This is zero at perihelion and aphelion, and a maximum
midway between them. In the case of the sun, its greatest value
is nearly 2◦, the sun getting alternately that amount ahead of, and
behind, the position it would occupy if its apparent daily motion
were uniform.

190. The Seasons.—The earth in its motion around the sun
always keeps its axis parallel to itself, for the mechanical reason
that a revolving body necessarily maintains the direction of its axis
invariable, unless disturbed by extraneous force, as is very prettily
illustrated by the gyroscope. About March 20 the earth is so situ-
ated that the plane of its equator passes through the sun, the sun’s
declination being zero on that day.

At that time, the line which separates the illuminated portions
of the earth passes through the two poles, and day and night are
everywhere equal. The same is again true of the 22d of September,
when the sun is at the autumnal equinox on the opposite side of the
orbit.

About the 21st of June the earth is so situated that its north
pole is inclined towards the sun by about 23 1 ◦, which is the sun’s

2

THE EARTH’S ORBITAL MOTION.

130
northern declination on that date. The south pole is then in the
obscure half of the earth’s globe, while the north pole receives sun-
light all day long; and in all portions of the northern hemisphere
the day is longer than the night, the difference between the day and
night depending upon the latitude of the place, while in the south-
ern hemisphere the days are shorter than the nights. At the time of
the winter solstice these conditions are reversed. At the equator (of
the earth) the day and night are equal at all times of the year. The
sun when in northern declination of course always rises at a point
on the horizon north of east, and sets at a point north of west, so
that for a portion of the time each day it shines on the north side
of a house.

191. Diurnal Phenomena near the Pole.—At the north pole,
where the celestial pole is in the zenith, and the diurnal circles are par-
allel with the horizon, the sun will maintain the same elevation all day
long, except for the slight change caused by the variation of its decli-
nation in twenty-four hours. The sun will appear on the horizon at the
date of the vernal equinox (in fact, about three days before, on account
of refraction), and slowly wind upward in the sky until it reaches its
maximum elevation of 23 1 ◦ on June 21. Then it will retrace its course

2
until a day or two after the autumnal equinox, when it sinks out of sight.

At points between the north pole and the polar circle the sun will
appear above the horizon earlier in the year than March 20, and will rise
and set daily until its declination becomes equal to the observer’s dis-
tance from the pole, when it will make a complete circuit of the heavens,
touching the horizon at midnight at the northern point; and after that
never setting again until it reaches the same declination in its southward
course after passing the solstice. From that time it will again rise and set
daily until it reaches a southern declination just equal to the observer’s
polar distance, when the long night begins; to continue until the sun,
having passed the southern solstice, returns again to the same declina-
tion at which it made its appearance in the spring. At the polar circle
itself (or, more strictly speaking, owing to refraction, about one-half a
degree south of it) the “midnight sun” will be seen on just one day in
the year, the day of the summer solstice; and there will also be one abso-
lutely sunless day, viz., the day of the winter solstice. The same remarks
apply in the southern hemisphere, by making the obvious changes.

192. Effects on Temperature.—The changes in the dura-
tion of “insolation” (exposure to sunshine) at any place involve
changes of temperature and other climatic conditions, thus produc-

THE EARTH’S ORBITAL MOTION.

131
ing the seasons. Taking as a standard the amount of heat received
in twenty-four hours on the day of the equinox, it is clear that the
surface of the soil at any place in the northern hemisphere will re-
ceive more than this average amount of heat whenever the sun is
north of the celestial equator, for two reasons.

1. Sunshine lasts more than half the
day.

2. The mean elevation of the sun dur-
ing the day is greater than when it is at
the equinoxes, since it is higher at noon,
and in any case reaches the horizon at ris-
ing and setting. Now, the more obliquely

Fig. 62.

the rays strike, the less heat they bring to

Effect of Sun’s Elevation
each square inch of surface, as is obvious
on Amount of Heat
from Fig. 62. A beam of sunshine having

Imparted to the Soil.

the cross-section ABCD, when it strikes
the surface at an angle h (equal to the sun’s altitude) is spread over
a much larger surface, Ac, and of course the amount of heat per
square inch is proportionately reduced. If Q is the amount of heat
per square inch brought by the ray when falling perpendicularly, as
on the surface AC, then on Ac the amount per square inch will be

Q × sin h, since AB = Ab × sin h. This difference in favor of the
more nearly vertical rays is exaggerated by the absorption of heat
in the atmosphere, because rays that are nearly horizontal have to
traverse a much greater thickness of air before reaching the ground.

For these two reasons, at a place in the northern hemisphere,
the temperature rises rapidly as the sun comes north of the equator,
thus giving us our summer.

193. Time of Highest Temperature.—We, of course, re-
ceive the most heat per diem at the time of the summer solstice;
but this is not the hottest time of the summer, for the obvious rea-
son that the weather is then all the time getting hotter, and the
maximum will not be reached until the increase ceases; that is, not
until the amount of heat lost in the night equals that stored up by
day.

If the earth’s surface threw off the same amount of heat hourly
whether it were hot or cold, then this maximum would not come un-
til the autumnal equinox. This, however, is not the case. The soil loses
heat faster when warm than it does when cold, the loss being nearly pro-

THE EARTH’S ORBITAL MOTION.

132
portional to the difference between the temperature of the soil and that
of surrounding space; (Newton’s law of cooling); and so the time of the
maximum is made to come not far from the end of July, or the first of

August, in our latitude. For similar reasons the minimum temperature of
winter occurs about Feb. 1, about half-way between the solstice and the
vernal equinox. Since, however, our weather is not entirely “made on the
spot where it is used,” but is affected by winds and currents that come
from great distances, the actual time of the maximum temperature can-
not be determined by any mere astronomical considerations, but varies
considerably from year to year.

194. Difference
between

Seasons
in

Northern
and

Southern Hemispheres.—Since in December the distance of the
earth from the sun is about three per cent less than it is in June, the
earth (as a whole) receives hourly about six per cent more heat in

December than in June, the heat varying inversely as the square of
the distance. For this reason the southern summer, which occurs in

December and January, is hotter than the northern. It is, however,
seven days shorter, because the earth moves more rapidly in that
part of its orbit. The total amount of heat per acre, therefore,
received during the summer is sensibly the same in each hemisphere,
the shortness of the southern summer making up for its increased
warmth.

195. The southern winter, however, is both longer and colder than
the northern; and it is maintained by certain geologists, Mr. Croll es-
pecially, that, on the whole, the mean annual temperature of the hemi-
sphere which has its winter at the time when the earth is in aphelion is
lower than that of the opposite one. It has been attempted to account
for the glacial epochs in this way. It is certain that at present, at any
place in the southern hemisphere, the difference between the maximum
temperature of summer and the minimum of winter must be greater than
in the case of a station in the northern hemisphere, similarly situated
as to elevation, etc. We say “at present” because, on account of cer-
tain slow changes in the earth’s orbit, to be spoken of immediately, the
state of things will be reversed in about ten thousand years, the northern
summer being then the hotter and shorter one.

196. Secular Changes in the Orbit of the Earth.—The
orbit of the earth is not absolutely unchangeable in form or position,
though it is so in the long run as regards the length of its major axis
and the duration of the year.

THE EARTH’S ORBITAL MOTION.

133

197. 1. Change in Obliquity of the Ecliptic.—The ecliptic
slightly and very slowly shifts its position among the stars, thus
altering the latitudes of the stars and the angle between the eclip-
tic and equator, i.e., the obliquity of the ecliptic. This obliquity
is at present about 24 less than it was 2000 years ago, and is still
decreasing about half a second a year. It is computed that this
diminution will continue for about 15,000 years, reducing the obliq-
uity to 22 1 ◦, when it will begin to increase. The whole change,

4
according to J. Herschel, can never exceed about 1◦ 20 on each side
of the mean.

198. 2. Change of Eccentricity.—At present the eccentricity of
the earth’s orbit (which is now 0.0168) is also slowly diminishing.

According to Leverrier, it will continue to decrease for about 24,000
years, until it becomes 0.003, and the orbit will be almost circular.

Then it will increase again for 40,000 years, until it becomes 0.02.

In this way the eccentricity will oscillate backwards and forwards, al-
ways, however, remaining between zero and 0.07; but the oscillations are
not equal either in amount or time, and so cannot properly be compared
to the “vibrations of a mighty pendulum,” which is rather a favorite
figure of speech.

199. 3. Revolution of the Apsides of the Earth’s Orbit.—The
line of apsides of the orbit, which now stretches in both directions
towards the constellations of Sagittarius and Gemini, is also slowly
and steadily moving eastward, and at a rate which will carry it
around the circle in about 108,000 years.

200. These so-called “secular ” changes are due to the action of
the other planets upon the earth. Were it not for their attraction,
the earth would keep her orbit with reference to the sun strictly un-
altered from age to age, except that possibly in the course of millions
of years the effects of falling meteoric matter and the attraction of
the nearer fixed stars might make themselves felt.

Besides these secular perturbations of the earth’s orbit, the earth
itself is continually being slightly disturbed in its orbit. On account of
its connection with the moon, it oscillates each month a few hundred
miles above and below the true plane of the ecliptic, and by the action
of the other planets it is sometimes set forwards or backwards to the
extent of a few thousand miles. Of course every such change produces a
corresponding slight change in the apparent position of the sun.

THE EARTH’S ORBITAL MOTION.

134

201. Equation of Time.—We have stated a few pages back

(Art. 111), that the interval between the successive passages of the
sun across the meridian is somewhat variable, and that for this rea-
son apparent solar, or sun-dial, days are unequal. On this account
mean time has been adopted, which is kept by a “fictitious” or

“mean” sun moving uniformly in the equator at the same average
rate as that of the real sun in the ecliptic. The hour-angle of this
mean sun is, as has been already explained, the local mean time (or
clock time); while the hour-angle of the real sun is the apparent or
sun-dial time. The Equation of Time is the difference between these
two times, reckoned as plus when the sun-dial is slower than the
clock, and minus when it is faster. It is the correction which must
be added (algebraically) to apparent time in order to get mean time.

As it is the difference between the two hour-angles, it may also be
defined as the difference between the right ascensions of the mean
sun and the true sun; or as a formula we may write: E = αt − αm,
in which αm is the right ascension of the mean sun, and αt, of the
true sun.

The principal causes of this difference are two:—

202. 1. The Variable Motion of the

Sun in the Ecliptic, due to the Eccen-
tricity of the Earth’s Orbit.—Near per-
ihelion, which occurs about Dec. 31,
the sun’s motion in longitude is most
rapid. Accordingly, at this time the ap-
parent solar days exceed the sidereal by
more than the average amount, mak-
ing the sun-dial days longer than the Fig. 63. Effect of Obliquity of
mean. (The average solar day, it will Ecliptic in producing Equation
be remembered, is 3m 56s longer than
of Time.

the sidereal.) The sun-dial will there-
fore lose time at this season, and will continue to do so for about
three months, until the angular motion of the sun falls to its mean
value. Then it will gain until aphelion, when, if the clock and the
sun were started together at perihelion, they will once more be to-
gether. During the next half of the year the action will be reversed.

Thus, twice a year, so far as the eccentricity of the earth’s orbit is
concerned, the clock and sun would be together at perihelion and
aphelion, while half-way between they would differ by about eight

THE EARTH’S ORBITAL MOTION.

135
minutes; the equation of time (so far as due to this cause only)

being +8 minutes in the spring, and −8 minutes in the autumn.

203. 2. The Inclination of the Ecliptic to the Equator.—Even
if the sun’s (apparent ) motion in longitude (i.e., along the ecliptic)

were uniform, its motion in right ascension would be variable. If
the true and fictitious suns started together at the equinox, they
would indeed be together at the solstices and at the other equinox,
because it is just 180◦ from equinox to equinox, and the solstices
are exactly half-way between them. But at intermediate points,
between the equinoxes and solstices, they would not be together on
the same hour-circle. This is best seen by taking a celestial globe
and marking on the ecliptic a point, m, half-way between the vernal
equinox and the solstice, and also marking a point n on the equator,

45◦ from the equinox. It will at once be seen that the former point,
m in Fig. 63, 1 is west of n, so that m in the daily westward motion
of the sky will come to the meridian first; in other words, when the
sun is half-way between the vernal equinox and the summer solstice,
the sun-dial is faster than the clock, and the equation of time is
minus. The difference, measured by the arc m n, amounts to nearly
ten minutes; and of course the same thing holds, mutatis mutandis,
for the other quadrants.

Fig. 64.—The Equation of Time.

204. Combination of the Effects of the Two Causes.—

We can represent graphically these two components of the equation
of time and the result of their combination as follows (Fig. 64):—

1Fig. 63 represents a celestial globe viewed from the west side, the axis
being vertical, and K, the pole of the ecliptic, on the meridian, while E is the
vernal equinox.

THE EARTH’S ORBITAL MOTION.

136

The central horizontal line is a scale of dates one year long, the
letters denoting the beginning of each month. The dotted curve
shows the equation of time due to the eccentricity of the earth’s
orbit, above considered. Starting at perihelion on Dec. 31, this
component is then zero, rising from there to a value of about +8m
on April 2, falling to zero on June 30, and reaching a second max-
imum of −8m on Oct. 1. In the same way the broken-line curve
denotes the effect of the obliquity of the ecliptic, which, by itself
alone considered, would produce an equation of time having four
maxima of, approximately, 10m each, about the 6th of February,

May, August, and November (alternately + and −), and reducing
to zero at the equinoxes and solstices.

The full-lined curve represents their combined effect, and is con-
structed by making its ordinate at each point equal to the sum

(algebraic) of the ordinates of the two other curves. At the 1st of

February, for instance, the distance F , 3, in the figure = F, 1 + F, 2.

So, also, M, 6 = M, 4+M, 5; the components, however, in this case
have opposite signs, so that the difference is actually taken.

The equation of time is zero four times a year, viz.: on April 15,

June 14, Sept. 1, and Dec. 24.

The maxima are February 11,

+14m 32s; May 14, −3m 55s; July 26, +6m 12s, and Nov. 2, −16m 18s.

But the dates and amounts vary slightly from year to year.

The two causes above discussed are only the principal ones effective
in producing the equation of time. Every perturbation suffered by the
earth comes in with its own effect; but all other causes combined never
alter the equation by more than a few seconds.

205. Precession of the Equinoxes.—The length of year was
found in two ways by the ancients:—

1. By the gnomon, which gives the time of the equinox and
solstice; and

2. By observing the position of the sun with reference to the
stars,—their rising and setting at sunrise or sunset.

Comparing the results of observations made by these two meth-
ods at long intervals, Hipparchus (120 b.c.) found that the two do
not agree; the former year (from equinox to equinox) being 20m 23s
shorter than the other (according to modern data). The equinox is
plainly moving westward on the ecliptic, as if it advanced to meet
the sun on each annual return. He therefore called the motion the

“precession” of the equinoxes.

THE EARTH’S ORBITAL MOTION.

137

On comparing the latitudes of the stars in the time of the an-
cient astronomers with the present latitudes, we find that they have
changed very slightly indeed; and we know therefore that the eclip-
tic and the plane of the earth’s orbit maintains its position sensibly
unaltered. On the other hand, the longitudes of the stars have been
found to increase regularly at the rate of about 50 .2 annually,—
fully 30◦ in the last 2000 years. Since longitudes are reckoned from
the equinox (the intersection between the ecliptic and equator), and
since the ecliptic does not move, it is evident that the motion must
be in the celestial equator ; and accordingly we find that both the
right ascension and the declination of the stars are constantly chang-
ing.

206. Motion of the Pole of the Equator around the Pole
of the Ecliptic.—The obliquity of the ecliptic, which equals the
distance in the sky between the pole of the equator and the pole
of the ecliptic (Art. 178), has remained nearly constant. Hence the
pole of the equator must be describing a circle around the pole of the
ecliptic in a period of about 25,800 years (360◦ divided by 50 .2).

The pole of the ecliptic has remained practically fixed among the
stars, but the other pole has changed its position materially. At
present the pole star is about 1 1 ◦ from the pole. At the time of the

4
star catalogue of Hipparchus it was 12◦ distant from it, and during
the next century it will approach to within about 30 , after which
it will recede.

207. If upon a celestial globe we take the pole of the ecliptic as a
centre, and describe about it a circle with a radius of 23 1 ◦, we shall

2
get the track of the celestial pole among the stars, and shall find that
the circle passes very near the star α Lyræ, on the opposite side of
the pole of the ecliptic from the present pole star. About 12,000 years
hence α Lyræ will be the pole star. Reckoning backwards, we find that
some 3000 years ago γ Draconis was the pole star; and it is a curious
circumstance that certain of the tunnels in the pyramids of Egypt face
exactly to the north, and slope at such an inclination that this star at its
lower culmination would have been visible from their lower end at the
date when the pyramids are supposed to have been built. It is probable
that these passages were arranged to be used for the purpose of observing
the transits of their then pole star.

208. Effect of Precession upon the Signs of the Zodiac.

—Another effect of precession is that the signs of the zodiac do not now

THE EARTH’S ORBITAL MOTION.

138
agree with the constellations which bear the same name. The sign of

Aries is now in the constellation of Pisces; and so on, each sign having

“backed,” so to speak, into the constellation west of it.

209. Physical Cause of Precession.—The
physical cause of this slow conical rotation of the
earth’s axis around the pole of the ecliptic lies in the
two facts that the earth is not exactly spherical, and
that the attractions of the sun and moon1 act upon
the equatorial ring of matter which projects above the
true sphere, tending to draw the plane of the equator
into coincidence with the plane of the ecliptic by their
greater attraction on the nearer portions of the ring.

The action is just what it would be if a spheroidal ball
of iron of the shape of the earth had a magnet brought
near it. The magnet, as illustrated in Fig. 65, would
tend to draw the plane of the equator into the line

Fig. 65.

CM joining its pole with the centre of the globe, be-

Effect of
cause it attracts the nearer portion of the equatorial

Attraction on
protuberance at E more strongly than the remoter at
a Spheroid.

Q. If it were not for the earth’s rotation, this attrac-
tion would bring the two planes of the ecliptic and equator together;
but since the earth is spinning on its axis, we get the same result
that we do with the whirling wheel of a gyroscope by hanging a
weight at one end of the axis. We then have the result of the com-
bination of two rotations at right angles with each other, one the
whirl of the wheel, the other the “tip” which the weight tends to
give the axis. (See Brackett’s Physics, pp. 53–56.)

210. In this case, if the wheel of the gyroscope is turning swiftly
clock-wise, as seen from above (Fig. 66), the weight at the (lower)

end of the axis will make the axis move slowly around, counter-
clockwise, without at all changing its inclination. If we regard the
horizontal plane passing through the gyroscope as representing the
ecliptic, and the point in the ceiling vertically above the gyroscope
as the pole of the ecliptic, the line of the axis of the wheel produced
upward would describe on the ceiling a circle around this imaginary

1The planets, by their action upon the plane of the earth’s orbit (Art. 197),
slightly disturb the equinox in the opposite direction. This effect amounts to
about 0 .16 annually.

THE EARTH’S ORBITAL MOTION.

139
ecliptic pole, acting precisely as does the pole of the earth’s axis in
the sky. The swifter the wheel’s rotation, the slower would be this

“precessional” motion of its axis; and of course, the rate of motion
also depends upon the magnitude of the suspended weight.

Fig. 66.

Precession Illustrated by the Gyroscope.

211. A full treatment of the subject would be too complicated for
our pages. An elementary notion of the way the action takes place,
correct as far as it goes, is easily obtained by reference to Fig. 67. Let

XY be the axis of the gyroscope, the wheel being seen in section edge-
wise, and the eye being on the same level as the centre of the wheel;
the wheel turning so that the point B is coming towards the observer.

Now, suppose a weight hung on the lower end of the axis. If the wheel
were not turning, the point B would come to some point F in the same
time it now takes to reach C (that is, after a quarter of a revolution).

By combination of the two motions it will come to a point K at the
end of the same time, having crossed the horizontal plane AD at L; and
this can be effected only by a backward “skewing around” of the whole
wheel, axis and all. This does not, of course, explain why the inclination
of the axis does not change under the action of the weight, but is only
a very partial illustration, showing merely why the plane of the wheel
regresses. A complete discussion would require the consideration of the
motion of every point on the wheel by a thorough and difficult analytical
treatment, in order to give the complete explanation of the reason why
the depressing weight, however heavy, does not cause the end of the axis
to fall perceptibly. (See article, “Gyroscope,” in Johnson’s Cyclopædia.)

212. Why Precession is so Slow.—The slowness of the pre-
cession depends on three things: (a) the enormous “moment of ro-
tation” of the earth—a point on the equator moves with the speed
of a cannon ball; (b) the smallness of the mass (compared with that

THE EARTH’S ORBITAL MOTION.

140
of the whole earth) of the protuberant ring to which precession is
due; and (c) the minuteness of the force which tends to bring this
ring into coincidence with the ecliptic, a force which is not constant
and persistent, like the weight hung on the gyroscope axis, but very
variable.

213. The Equation of the Equinox.—Whenever the sun is
in the plane of the equator (which is twice a year, at the time of the
equinoxes), the sun’s precessional force disappears entirely, its attraction
then having no tendency to draw the equator out of its position. The
moon’s action, on account of her proximity, is still more powerful than
that of the sun; on the average two and a half times as great. Now, the
moon crosses the celestial equator twice every month, and at these times
her action ceases.

There is still another cause for variation
in the effectiveness of the moon’s attraction.

As we shall see hereafter, she does not move
in the ecliptic, but in a path which cuts the
ecliptic at an angle of about 5◦, at two points
called the Nodes; the ascending node being
the point where she crosses the ecliptic from
south to north. These nodes move westward
on the ecliptic (Art. 455), making the circuit

Fig. 67.

Regression of the Gyroscope
once in about nineteen years. Now, when

Wheel.

the ascending node of the moon’s orbit is at

B (Fig. 68), near the autumnal equinox F , its inclination to the equator
will be, as the figure shows, less than the obliquity of the ecliptic by
about 5◦; i.e., it will be only about 18◦. On the other hand, nine and a
half years later, when the node has backed around to a point A, near the
vernal equinox, the inclination of the moon’s orbit to the equator will
be nearly 28◦. When the node is in this position, the moon will produce
nearly twice as much precessional movement each month as when the
node was at B.

The precession, therefore, is not
uniform, but variable, almost ceas-
ing at some times and at others be-
coming rapid. The average amount,

Fig. 68.

as has been stated, is 50 .2 a year;

Variation in the Inclination of
and the variation is taken account

Moon’s Orbit to Equator.

of in what is called the equation of
the equinox, which is the difference between the actual position of the
equinox at any time and the position it would have at that moment if

THE EARTH’S ORBITAL MOTION.

141
the precession had been all the time going on uniformly.

214. Nutation.—Not only does the precessional force vary in
amount at different times, but in most positions of the disturbing
body with respect to the earth’s equator there is a slight thwartwise
component of the force, tending directly to accelerate or retard the
precessional movement of the pole—just as if one should gently
draw the weight W (Fig. 63) horizontally. The consequence is what
is called Nutation or “nodding.” The axis of the earth, instead of
moving smoothly in a circle, nods in and out a little with respect
to the pole of the ecliptic, describing a wavy curve resembling that
shown in Fig. 69, but with nearly 1400 indentations in the entire
circumference traversed in 26,000 years.

215. We distinguish three of these nutations,

(a) The Lunar Nutation, depending upon the mo-
tion of the moon’s nodes. This has a period of a
little less than nineteen years, and amounts to 9 .2,

(b) The Solar Nutation, due to the changing dec-
lination of the sun. Its period is a year, and its
amount 1 .2. (c) The Monthly Nutation, precisely
like the solar nutation, except that it is due to the
moon’s changes of declination during the month. It

Fig. 69.—

Nutation.

is, however, too small to be certainly measured, not
exceeding one-tenth of a second.

Nutation was detected by Bradley in 1728, but not fully explained
until 1748.

Neither precession nor nutation affects the latitudes of the stars,
since they are not due to any change in the position of the ecliptic,
but only to displacements of the earth’s axis. The longitudes alone are
changed by them.

The right ascension and declination of a star are both affected.

216. The Three Kinds of Year.—In consequence of the mo-
tion of the equinoxes caused by precession, the sidereal year and
the equinoctial or “tropical ” year do not agree in length. Although
the sidereal year is the one which represents the earth’s true orbital
revolution around the sun, it is not used as the year of chronology
and the calendar, because the seasons depend on the sun’s place
in relation to the equinoxes. The tropical year is the year usually
employed, unless it is expressly stated to the contrary. The length
of the Sidereal year is 365d 6h 9m 9s; that of the Tropical year is

THE EARTH’S ORBITAL MOTION.

142
about 20m less, 365d 5h 48m 46s.

The third kind of year is the anomalistic year, which is the time
from perihelion to perihelion again. As the line of apsides of the
earth’s orbit moves always slowly towards the east, this year is a
little longer than the sidereal. Its length is 365d 6h 13m 48s.

217. The Calendar.—The natural units of time are the day,
the month, and the year. The day, however, is too short for conve-
nient use in designating extended periods of time, as for instance
in expressing the age of a man. The month meets with the same
objection, and for all chronological purposes, therefore, the year is
the unit practically employed. In ancient times, however, so much
regard was paid to the month, and so many of the religious beliefs
and observances connected themselves with the times of the new and
full moon, that the early history of the calendar is largely made up
of attempts to fit the month to the year in some convenient way.

Since the two are incommensurable, the problem is a very difficult,
and indeed strictly speaking, an impossible, one.

In the earliest times matters seem to have been wholly in the
hands of the priesthood, and the calendar then was predominantly
lunar, with months and days intercalated from time to time to keep
the seasons in place. The Mohammedans still use a purely lunar
calendar, having a year of twelve lunar months, and containing al-
ternately 354 and 355 days. In their reckoning the seasons fall,
of course, continually in different months, and their calendar gains
about one year in thirty-three upon the reckoning of Christian na-
tions.

218. The Metonic Cycle.—Among the Greeks the discovery
of the so-called lunar or Metonic cycle by Meton, about 433 b.c.,
considerably simplified matters. This cycle consists of 235 synodic
months (from new moon to new again), which is very approximately
equal to 19 common years of 365 1 days.

4

235 months equal 6939d

16h

31m;

19 tropical years equal

6939d 14h 27m; so that at the end of the 19 years, the new and full
moon recur again on the same days of the year, and at the same time of
day within about two hours. The calendar of the phases of the moon,
for instance, for 1889 is the same as for 1870 and 1908 (except that
intervening leap-years may change the dates by one day).

The “Golden number ” of a year is its number in this Metonic cycle,
and is found by adding 1 to the “date-number” of the year and dividing

THE EARTH’S ORBITAL MOTION.

143
by 19. The remainder, unless zero, is the “golden number” (if it comes
out zero, 19 is taken instead). Thus the golden number for 1888 is found
by dividing 1889 by 19, and the remainder 8, is the golden number of
the year.

This cycle is still employed in the ecclesiastical calendar in finding
the time of Easter.

For further information on the subject, consult Johnson’s Encyclopædia, or

Sir Edmund Becket’s “Astronomy without Mathematics.”

219. Julian Calendar.—Until the time of Julius Cæsar the

Roman calendar seems to have been based upon the lunar year of
twelve months, or 355 days, and was substantially like the modern

Mohammedan calendar, with arbitrary intercalations of months and
days made by the priesthood and magistrates from time to time in
order to bring it into accordance with the seasons. In the later
days of the Republic, the confusion had become intolerable. Cæsar,
with the help of the astronomer Sosigenes, whom he called from

Alexandria for the purpose, reformed the system in the year 45 b.c.,
introducing the so-called “Julian calendar,” which is still used either
in its original shape or with a very slight modification. He gave
up entirely the attempt to coordinate the month with the year,
and adopting 365 1 days as the true length of the tropical year, he

4
ordained that every fourth year should contain an extra day, the
sixth day before the Kalends of March on that year being counted
twice, whence the year was called “bissextile” Before his time the
year had begun in March (as indicated by the Roman names of
the months,—September, seventh month; October, eighth month,
etc.), but he ordered it to begin on the 1st of January, which in that
year (45 b.c.) was on the day of the new moon next following the
winter solstice. In introducing the change it was necessary to make
the preceding year 445 days long, and it is still known in the annals
as “the year of confusion.” He also altered the name of the month

Quintilis, calling it “July” after himself.

There was some irregularity in the bissextile years for a few years
after Cæsar’s death, from a misunderstanding of his rule for the inter-
calary day; but his successor Augustus remedied that, and to put himself
on the same level with his predecessor, he took possession of the month

Sextilis, calling it “August”; and to make its length as great as that of

July, he robbed February of a day.

From that time on, the Julian calendar continued unbrokenly in use

THE EARTH’S ORBITAL MOTION.

144
until 1582; and it is still the calendar of Russia and of the Greek Church.

220. The Gregorian Calendar.—The Julian calendar is not
quite correct. The true length of the tropical year is 365 days 5 hours

48 minutes and 46 seconds, and this leaves a difference of 11 minutes
and 14 seconds by which the Julian calendar year is the longer,
being exactly 365 1 days. As a consequence, the date of the equinox

4
comes gradually earlier and earlier by about three days in 400 years.

m

(400 × 11 1

= 4467 minutes = 3d 2h 27m.) In the year 1582, the

6
date of the vernal equinox had fallen back 10 days to the 11th of

March, instead of occurring on the 21st of March, as at the time of
the Council at Nice, 325 a.d. Pope Gregory, therefore, acting under
the advice of the Jesuit astronomer, Clavius, ordered that the day
following Oct. 4 in the year 1582 should be called not the 5th, but
the 15th, and that the rule for leap-year should be slightly changed
so as to prevent any such future displacement of the equinox. The
rule now stands: All years whose date-number is divisible by four
without a remainder are leap-years, unless they are century years

(1700, 1800, etc.). The century years are not leap-years unless their
date-number is divisible by 400, in which case they are: that is,

1700, 1800, and 1900 are not leap years; but 1600, 2000, and 2400
are.

221. Adoption of the New Calendar.—The change was im-
mediately adopted by all Catholic nations; but the Greek Church
and most of the Protestant nations, rejecting the Pope’s authority,
declined to accept the correction. In England it was at last adopted
in the year 1752, at which time there was a difference of eleven
days between the two calendars. (The year 1600 was a leap-year
according to the Gregorian system as well as the Julian, but 1700
was not.) Parliament in 1751 enacted that the day following the 2d
of September, in the year 1752, should be called the 14th instead
of the 3d; and also that this year (1752), and all subsequent years,
should begin on the first of January.

The change was made under very great opposition, and there were
violent riots in consequence in different parts of the country, especially at

Bristol, where several persons were killed. The cry of the populace was,

“Give us back our fortnight,” for they supposed they had been robbed
of eleven days, although the act of Parliament was carefully framed to
prevent any injustice in the collection of interest, payment of rents, etc.

At present, since the year 1800 was not a leap-year according to the

THE EARTH’S ORBITAL MOTION.

145

Gregorian calendar, while it was so according to the Julian, the difference
between the two calendars amounts to twelve days; thus in Russia the

19th of August would be reckoned as the 7th. In Russia, however, for
scientific and commercial purposes both dates are very generally used, so
that the date mentioned would be written Aug. 7 . When Alaska was

19
annexed to the United States, its calendar had to be altered by eleven
days. (See Art. 123.)

222. The Beginning of the Year.—The beginning of the year
has been at several different dates in the different countries of Europe.

Some have regarded it as beginning at Christmas, the 25th of December;
others, on the 1st of January; others still, on the 1st of March; others, on
the 25th; and others still, at Easter, which may fall on any day between
the 22d of March and the 25th of April.

In England previous to the year 1752 the legal year commenced on
the 25th of March, so that when the change was made, the year 1751
necessarily lost its months of January and February, and the first twenty-
four days of March. Many were slow to adopt this change, and it becomes
necessary, therefore, to use considerable care with respect to English
dates which occur in the months of January, February, or March about
that period. The month of February, 1755, for instance, would by some
writers be reckoned as occurring in 1754. Confusion is best avoided by
writing, Feb. 1754 .

1755

223. First and Last Days of the Year.—Since the ordinary
civil year consists of 365 days, which is 52 weeks and one day, the last
day of each common year falls on the same day as the first; so that any
given date will fall one day later in the week than it did on the preceding
year, unless a 29th of February has intervened, in which case it will be
two days later; that is, if the 3d of January, 1889, falls on Thursday, the
same date in 1890 will fall on Friday.

224. Aberration.—Although in strictness the discussion of aberra-
tion does not belong to a chapter describing the earth and its motions, yet
since it is a phenomenon due to the earth’s motion, and affects the right ascen-
sion and declination of the stars in much the same ways as do precession and
nutation, it may properly enough be considered here.

Aberration is the apparent displacement of a star, due to the
combination of the motion of light with the motion of the observer.

The direction in which we have to point our telescope in ob-
serving a star is not the same that it would be if the earth were at
rest. It lies beyond our scope to show that according to the wave
theory of light the apparent direction of a ray will be affected by

THE EARTH’S ORBITAL MOTION.

146
the observer’s motion precisely in the same way (within very narrow
limits) as it would be if light consisted of corpuscles shot off from a
luminous body, as Newton supposed. This is the case, however, as

Doppler and others have shown; and assuming it, the explanation
of aberration is easy:—

Suppose an observer standing at
rest with a tube in his hand in a shower
of rain where the drops are falling verti-
cally. If he wishes to have the drops de-
scend axially through the tube without
touching the sides, he must of course
keep it vertical; but if he advances in
any direction, he must draw back the
bottom of the tube by an amount which
equals the advance he makes in the

Fig. 70.—Aberration of a

Raindrop.

time while the drop is falling through
the tube, so that when the drop falling from B reaches A , the bot-
tom of the tube will be there also; i.e., he must incline the tube
forward by an angle α, such that tan α = u ÷ V , where V is the
velocity of the raindrop and u that of his own motion. In Fig. 70

BA = V and AA = u.

225. Now take the more general case.

Suppose a star sending us light with a ve-
locity V in the direction SP , Fig. 71, which
makes the angle θ with the line of the ob-
server’s motion. He himself is carried by the
earth’s orbital velocity in the direction QP .

In pointing the telescope so that the light
may pass exactly along its optical axis, he
will have to draw back the eye-end by an
amount QP which just equals the distance Fig. 71.—Aberration of

Light.

he is carried, by the earth’s motion during
the time that the light moves from O to P .

The star will thus apparently be displaced towards the point to-
wards which he is moving, the angle of displacement P OQ, or α,
being determined by the relative length and direction of the two
sides OP and QP of the triangle OP Q. These sides are respectively
proportional to the velocity of light, V , and the orbital velocity of
the earth, u.

THE EARTH’S ORBITAL MOTION.

147

The angle at P being θ, the angle OQP will be (θ − α), and we shall
have from trigonometry the proportion sin α : sin (θ − α) = u : V .

To find α from this, develop the second term of the proportion and
divide the first two terms by sin α, which gives us

1 : sin θ cot α − cos θ = u : V,
whence
u sin θ cot α = V + u cos θ,

V + u cos θ
and
cot α =
u sin θ

Taking the reciprocal of this we have
u
tan α =
sin θ.

V + u cos θ

The second term in the denominator is insensible, since u is only about
one ten-thousandth of V , so that we may neglect it.1 This gives the
formula in the shape in which it ordinarily appears, viz.,
u
tan α =
sin θ.

V

The value of α (denoted by α0) which obtains when θ = 90◦ and
sin θ = unity, is called the Constant of Aberration.

The latest, and probably the most accu-
rate, determination of this constant (derived
from the Pulkowa Observations by Nyrén in

1882) is 20 .492. Aberration was discovered
and explained by Bradley, the English As-
tronomer Royal, in 1726.

226. The Effect of Aberration upon
the Apparent Places of the Stars.—As
the earth moves in an orbit nearly circular,

Fig. 72.

and with a velocity so nearly uniform that we

Aberrational Orbit of
may for our present purpose disregard its vari-
a Star.

ations, it is clear that a star at the pole of the

1The velocity of light, according to the latest determinations of Newcomb
and Michelson, is 299860 kilometers ± 30 kilometers (which equals 186,330
miles ± 20 miles). The mean velocity of the earth in its orbit, if we assume
the solar parallax to be 8 .8, is 29.77 kilometers, or 18.50 miles; this makes the
constant of aberration 20 .478, a little smaller than that given in the text.

THE EARTH’S ORBITAL MOTION.

148
ecliptic will be always displaced by the same amount of 20 .5, but
in a direction continually changing. It must, therefore, appear to
describe a little circle 41 in diameter during the year, as shown in

Fig. 72. Now the direction of the earth’s orbital motion is always
in the plane of the ecliptic, and towards the right hand as we stand
facing the sun. At the vernal equinox, therefore, we are moving
toward the point of the ecliptic, which is 90◦ west of the sun, i.e.,
towards the winter solstitial point, and the star is then displaced
in that direction. Three months later the star will be displaced in
a line directed towards the vernal equinox, and so on. The earth,
therefore, so to speak, drives the star before it in the aberrational
orbit, keeping it just a quarter of a revolution ahead of itself.

A star on the ecliptic simply appears to oscillate back and forth
in a straight line 41 long.

Generally, in any latitude whatever, the aberrational orbit is an
ellipse, having its major axis parallel to the ecliptic, and always 41
long, while its minor axis is 41 × sin β, β being the star’s latitude,
or distance from the ecliptic.

226*. Diurnal Aberration.—The motion of an observer due to
the earth’s rotation also produces a slight effect known as the diurnal
aberration. Its “constant” is only 0 .31 for an observer situated at the
equator; anywhere else it is 0 .31 cos φ, φ being the latitude of the ob-
server.

For any given star it is a maximum when the star is crossing the
meridian, and then its whole effect is slightly to increase the right ascen-
sion by an amount given by the formula

∆α = 0 .31 cos φ sec δ,
δ being the star’s declination.

See Chauvenet, “Practical Astronomy,” I. p. 638.

THE MOON.

149

CHAPTER VII.

THE MOON: HER ORBITAL MOTION AND VARIOUS KINDS

OF MONTH.—DISTANCE AND DIMENSIONS, MASS, DEN-

SITY AND GRAVITY.—ROTATION AND LIBRATIONS.—

PHASES.—LIGHT

AND

HEAT.—PHYSICAL

CONDITION

AND INFLUENCES EXERTED ON THE EARTH.—TELE-

SCOPIC ASPECT.—SURFACE AND POSSIBLE CHANGES

UPON IT.

227. We pass next to a consideration of our nearest neighbor
in the celestial spaces, the moon, which is a satellite of the earth
and accompanies us in our annual motion around the sun. She is
much smaller than the earth, and compared with most of the other
heavenly bodies, a very insignificant affair; but her proximity makes
her far more important to us than any of them except the sun. The
very beginnings of Astronomy seem to have originated in the study
of her motions and in the different phenomena which she causes,
such as the eclipses and tides; and in the development of modern
theoretical astronomy the lunar theory with the problems it raises
has been perhaps the most fertile field of invention and discovery.

228. Apparent Motion of the Moon.—Even superficial ob-
servation shows that the moon moves eastward among the stars
every night, completing her revolution from star to star again in
about 27 1 days. In other words, she revolves around the earth in

4
that time; or, more strictly speaking, they both revolve about their
common centre of gravity. But the moon is so much smaller than
the earth that this centre of gravity is situated within the ball of
the earth on the line joining the centres of the two bodies at a point
about 1100 miles below its surface.

As the moon moves eastward so much faster than the sun, which
takes a year to complete its circuit, she every now and then, at the
time of the new moon, overtakes and passes the sun; and as the
phases of the moon depend upon her position with reference to the

THE MOON.

150
sun, this interval from new moon to new moon is what we ordinarily
understand as the month.

229. Sidereal and Synodic Revolutions.—The Sidereal
revolution of the moon is the time occupied in passing from a
star to the same star again, as the name implies. It is equal to

27d 7h 43m 11s.545 ± 0s.01, or 27d.32166. The moon’s mean daily
motion among the stars equals 360◦ divided by this, which is 13◦ 11

(nearly).

The Synodic revolution is the interval from new moon to new
moon again, or from full to full. It varies somewhat on account of the
eccentricity of the moon’s orbit and of that of the earth around the
sun, but its mean value is 29d 12h 44m 2s.684 ± 0s.01, or 29d.53059;
and this is the ordinary month. (The word synodic is derived from
the Greek σύν and ὁδός, and has nothing to do with the nodes of
the moon’s orbit. The word is syn-odic, not sy-nodic).

A synodical revolution is longer than the sidereal, because during
each sidereal month of 27.3 days the sun has advanced among the
stars, and must be caught up with.

230. Elongation, Syzygy, etc.—The angular distance of the
moon from the sun is called its Elongation. At new moon it is zero,
and the moon is then said to be in “Conjunction.” At full moon it
is 180◦, and the moon is then in “Opposition.” In either case the
moon is said to be in “Syzygy” (σύν ζυγόν). When the elongation
is 90◦, as at the half-moon, the moon is in “Quadrature.”

231. Determination of the Moon’s Sidereal Period.—

This is effected directly by observations of the moon’s right ascen-
sion and declination (with the meridian circle), kept up systemati-
cally for a sufficient time.

If it were not for the so-called “secular acceleration” of the
moon’s motion (Arts. 459–461), an exceedingly accurate determina-
tion of the moon’s synodic period could be obtained by comparing
ancient eclipses with modern.

The earliest authentically recorded eclipse is one that was ob-
served at Nineveh in the year 763 b.c. between 9 and 10 o’clock on
the morning of June 15th.

By comparing this eclipse with (say) the eclipse of August, 1887,
we have an interval of more than 35,000 months, and so an error of
ten hours even, in the observed time of the Nineveh eclipse, would

THE MOON.

151
make only about one second in the length of the month. But the
month is a little shorter now than it was 2000 years ago.

232. Relation of Sidereal and Synodic Periods.—The
fraction of a revolution described by the moon in one day equals

1 , M being the length of the sidereal month. In the same way 1

M

E
represents the earth’s daily motion in its orbit, E being the length
of the year. The difference of these two equals the fraction of a
revolution which the moon gains on the sun during one day. In a
synodic month, S, it gains one whole revolution, and therefore must
gain each day 1 of a revolution; so that we have the equation

S

1

1

1

−

=

;

M

E

S
or, substituting the numerical values of E and S,

1

1

1

−

=

,

M

365.25635

29.53059
whence we derive the value of M .

Another way of looking at it is this: In a year there must be exactly
one more sidereal revolution than there are synodic revolutions, because
the sun completes one entire circuit in that time. Now the number of
synodic revolutions in a year is given by the fraction

365 14 = 12.369+.

S

There will therefore be 13.369 sidereal revolutions in the year, and the
length of one sidereal revolution equals 365 1 days divided by this number

4

13.369, which will be found to give the length of the sidereal revolution
as before.

233. Moon’s Path among the Stars.—By observing with
the meridian circle the right ascension and declination of the moon
daily during the month, just as in the case of the sun, we obtain
the position of the moon for each day, and joining the points thus
found, we can draw the path of the moon in the sky. It is found
to be a great circle inclined at a mean angle of 5◦ 8 to the ecliptic,
which it cuts in two points called the nodes (from nodus, a “knot”).

We say the path is found to be a great circle. This must be taken
with some reservation, since at the end of the month the moon never
returns precisely to the position it occupied at the beginning, owing
to the regression of the nodes and other so-called “perturbations,”

which will be discussed hereafter.

THE MOON.

152

234. Moon’s Meridian Altitude.—Since the moon’s orbit is
inclined to the ecliptic 5◦ 8 , its inclination to the equator varies from

28◦ 36 (23◦ 28 + 5◦ 8 ), when the moon’s ascending node is the vernal
equinox, to 18◦ 20 , when, 9 1 years later, the same node is at the autum-

2
nal equinox. In the first case the moon’s declination will change during
the month by 57◦ 12 , from −28◦ 36 to +28◦ 36 . In the other case it will
change only by 36◦ 40 , so that at different times the difference in the
behavior of the moon in this respect is very striking.

235. Interval between Moon’s Transits.—On the average
the moon gains 12◦ 11 .4 on the sun daily, so that it comes to the
meridian about 51 minutes of solar time later each day.

To find the mean interval between the successive transits of the
moon we may use the proportion

(360◦ − 12◦ 11 .4) : 360◦ = 24h : x;
whence x = 24h 50m.6.

The variations of the moon’s motion in right ascension, which are
very considerable (much greater than in the case of the sun), cause
this interval to vary from 24h 38m to 25h 06m.

236. The Daily Retardation of the Moon’s Rising and

Setting.—The average daily retardation of the moon’s rising and
setting is, of course, the same as that of her passage across the
meridian, viz., 51m; but the actual retardation of rising is subject
to very much greater variations than these of the meridian passage,
being affected by the moon’s changes in declination as well as by
the inequalities of her motion in right ascension. When the moon is
very far north, having her maximum declination of 28◦ 36 , she will
rise in our latitudes much earlier than when she is farther south.

In the latitude of New York the least possible daily retardation of
moon-rise is 23 minutes, and the greatest is 1 hour and 17 minutes.

In higher latitudes the variation is greater yet.

237. Harvest and Hunter’s Moons.—The variations in the
retardation of the moon’s rising attract most attention when they occur
at the time of the full moon. When the retardation is at its minimum, the
moon rises soon after sunset at nearly the same time for several successive
evenings; whereas, when the retardation is greatest, the moon appears
to plunge nearly vertically below the horizon by her daily motion. When
the full moon occurs at the time of the autumnal equinox, the moon
itself will be near the first of Aries.

THE MOON.

153

Now, as will be seen by reference to Fig. 73, the portion of the ecliptic
near the first of Aries makes a much smaller angle with the eastern
horizon than the equator.

[The line HN is the horizon, E being the east point—the figure being
drawn to represent a celestial globe, as if the observer were looking at
the eastern side of the celestial sphere from the outside.]

EQ is the equator. Now, when the autumnal equinoctial point or
first of Libra is on the horizon at E, the position of the ecliptic will be
that represented by ED; more steeply inclined to the horizon than EQ
is, by the angle QED, 23 1 ◦. But when the first of Aries is at E, the

2
ecliptic will be in the position J J . And if the ascending node of the
moon’s orbit happens then to be near the first of Aries, the moon’s path
will be M M .

Fig. 73.—Explanation of the Harvest Moon.

Accordingly, when the moon is in Aries, it, so to speak, coasts along
the eastern horizon from night to night, its time of rising not varying
very much; and this, when it occurs near the full of the moon, gives
rise to the phenomenon known as the harvest moon, the harvest moon
being the full moon nearest to the autumnal equinox. The full moon next
following is called the hunter’s moon.

In Norway and Sweden, under these circumstances, the moon’s orbit
may actually coincide with the horizon, so that she will rise at absolutely
the same time for a considerable number of successive evenings.

238. The Moon’s Orbit.—As in the case of the sun, the ob-
servation of the moon’s path in the sky gives no information as to
the real size of its orbit; but its form may be found by measuring
the apparent diameter of the moon, which ranges from 33 30 to

29 21 at different points. The orbit turns out to be an ellipse
like the orbit of the earth, but with an eccentricity more than three
times as great—about 1 on the average, but varying from 1 to 1

18

14

22
on account of perturbations.

THE MOON.

154

The extremities of the major axis of the moon’s orbit are called
the perigee and apogee (from περ`ι γη and ἀπ`
ο γη).

The line of apsides, which passes through these two points,
moves around towards the east once in about nine years, also on
account of perturbations.

239. Parallax and Distance of the Moon.—These can be
found in several ways, of which the simplest is the following: At
two observatories B and C (Fig. 74) on, or very nearly on, the
same meridian and very far apart (in the northern and southern
hemispheres if possible; Greenwich and the Cape of Good Hope,
for instance) let the moon’s zenith distance ZBM and Z CM be
observed simultaneously with the meridian circle. This gives in the
quadrilateral BOCM the two angles at B and C, each of which
is the supplement of the geocentric zenith distance. The angle at
the centre of the earth, BOC, is the difference of the geocentric
latitudes and is known from the geographical positions of the two
observatories. Knowing the three angles in the quadrilateral, the
fourth at M is of course known, since the sum of the four must he
four right angles. The sides BO and CO are known, being radii of
the earth; so that we can solve the whole quadrilateral by a simple
trigonometrical process.

First find from the triangle BOC
the partial angles OCB and OBC, and
the side BC. Then in the triangle BCM
we have BC and the two angles CBM
and M CB, from which we can find the
two sides BM and CM .

Finally, in
the triangle OBM , we now know the
sides OB and BM and the included an-
gle OBM , so that the side OM can be
computed, which is the distance of the

Fig. 74.—Determination of the
moon from the earth’s centre. Knowing

Moon’s Parallax.

this, the horizontal parallax KM O, or
the semi-diameter of the earth as seen from the moon, follows at once.

The moon’s parallax can also be deduced from observations at a sin-
gle station on the earth, but not so simply. If she did not move among
the stars, it would be very easy, as all we should have to do would be to
compare her apparent right ascension and declination at different points
in her diurnal circle. Near the eastern horizon the parallax (always de-
pressing an object) increases her right ascension; at setting, vice versa.

THE MOON.

155

On the meridian the declination only is affected. But the motion of the
moon must be allowed for, as the observations to be compared are nec-
essarily separated by considerable intervals of time, and this complicates
the calculation.

A third, and a very accurate, method is by means of occultations of
stars, observed at widely separated points on the earth. These occul-
tations furnish the moon’s place with great accuracy, and so determine
the parallax very precisely; but the calculation is not very simple, as
the moon’s motion in this case also enters into it, since the observations
cannot be simultaneous.

240. The Distance of the Moon is continually changing
on account of the eccentricity of its orbit, varying all the way, ac-
cording to Neison, between 252,972 and 221,614 miles; the mean
distance being 238,840 miles, or 60.27 times the equatorial radius
of the earth. The mean parallax of the moon is 57 2 , subject to
a similar percentage of change. This value of the parallax, it will
be noted, indicates that the earth, as seen from the moon, has a
diameter of nearly 2◦.

Knowing the size of the moon’s orbit and the length of the
month, the velocity of her motion around the earth is easily cal-
culated. It comes out 2288 miles per hour, or about 3350 feet a
second.

Fig. 75.—Moon’s Path with Reference to the Sun.

241. Form of the Moon’s Orbit with Reference to the

Sun.—While the moon moves in a small elliptical orbit around the
earth, it also moves around the sun in company with the earth.

This common motion of the moon and earth, of course, does not
affect their relative motion; but to an observer outside the system
the moon’s motion around the earth would only be a very small
component of the moon’s movement as seen by him.

The distance of the moon from the earth, 239,000 miles, is very
small compared with that of the earth from the sun, 93,000,000
miles—being only about

1
part. The speed of the earth in its

400

THE MOON.

156
orbit around the sun is also more than thirty times faster than that
of the moon in its orbit around the earth, so that for the moon the
resulting path in space is one which is always concave towards the
sun, as shown in Fig. 75. It is not like Figs. 76 and 77, as often
represented. If we represent the orbit of the earth by a circle with a
radius of 100 inches (8 feet 4 inches), the moon would only move out
and in a quarter of an inch, crossing the circumference twenty-five
times in going once around it.

Fig. 76.

Fig. 77.

False Representations of Moon’s Motions.

242. Diameter of the Moon.—The mean apparent diameter
of the moon is 31 7 . This gives it a real diameter of 2163 miles

(plus or minus one mile), which equals 0.273 of the earth’s diameter.

Since the surfaces of globes are as the squares of their diameters,
and their volumes as their cubes, this makes the surface of the moon

0.0747 of the earth’s (between 1 and 1 ); and the volume 0.0204

13

14
of the earth’s volume (almost exactly 1 ); that is, it would take 49

49
balls each as large as the moon in bulk to make a ball of the size of
the earth.

243. Mass of the Moon.—This is about 1 of the earth’s

80
mass, different authorities giving the value from 1 to 1 . It is not

75

85
easy to determine it with accuracy. In fact, though the moon is the
nearest of all the heavenly bodies, it is more difficult to “weigh”

her than to weigh Neptune, although he is the most remote of the
planets.

There are four ways of approaching the problem: (1) (perhaps
easiest to understand) by finding the position of the common centre
of gravity of the earth and moon with reference to the centre of the
earth. Since it is this common centre of gravity of the two bodies
which describes around the sun the ellipse which we have called
the earth’s orbit, and since the earth and moon revolve around this
common centre of gravity once a month, it follows that this monthly
motion of the earth causes an alternate eastward and westward dis-
placement of the sun in the sky, which can be measured. At the

THE MOON.

157
time of the new and full moon this displacement is zero, the centre
of gravity being on the line which joins the earth and sun; but when
the moon is at quadrature (that is, 90◦ from the sun, as at the time
of half-moon), the sun is apparently displaced in the sky towards
the moon, as is evident from Fig. 78. It will be about 6 .3 east of
its mean place at the first quarter of the moon, Fig. 78 (B), and as
much west at the time of the last quarter, Fig. 78 (A); (i.e., when
the angle M GS is 90◦, the angle M CS is always less than 90◦ by

6 .3, which is therefore the value of the angle CSG). Now since the
parallax of the sun (which is the earth’s semi-diameter seen from
the sun—the angle CSK) is about 8 .8, it follows that the distance
of the centre of gravity of the earth and moon from the centre of the
earth is the fraction 6.3 of the earth radius, or, about 2830 miles.

8.8

This is just about 1 of the distance from the earth to the moon,

80
whence we conclude that the mass of the earth is 80 times that of
the moon.

244. (2) A second method is by
comparing the moon’s actual period
with the computed period to which a sin-
gle particle at the moon’s distance from
the earth ought to have, according to the
known force of gravity of the earth, as
determined by pendulum experiments.

The explanation of this method cannot
be given until we have further studied
the motion of bodies under the law of
gravitation. It is not susceptible of great
accuracy.

(3) Still another method is by com-
paring the tides produced by the moon
with these produced by the sun.

This
gives us the mass of the moon as com-
pared with that of the sun; and the

Fig. 78.

Apparent Displacement of Sun
mass of the sun compared with that of
at First and Third Quarters of
the earth being known, it gives us ulti-
the Month.

mately the mass of the moon compared
with that of the earth.

(4) The ratio of the moon’s mass to the sun’s can also be computed
from the nutation of the earth’s axis, (See Chap. XIII.)

THE MOON.

158

245. No other satellite is nearly as large as the moon, in comparison
with its primary planet. The earth and moon together, as seen from a
distant star, are really in many respects more like a double planet than
like a planet and satellite, as ordinarily proportioned to each other. At
a time, for instance, when Venus happens to be near the earth, at a
distance of about twenty-five millions of miles, the earth to her would
appear just about as bright as Venus at her best does to us; and the
moon would be about as bright as Sirius, at a distance of about half a
degree from the earth.

246. Density and Superficial Gravity of the Moon.—
mass

Since the density of a body is equal to

, the density of the
volume
moon as compared with the earth is found from the fraction

1

0.125

80 , or

.

1

0.0204

49

This makes the moon’s density 0.613 of the earth’s density, or
about 3 4 the density of water—somewhat above the average den-

10
sity of the rocks which compose the crust of the earth.

This small density of the moon is not surprising, nor at all inconsis-
tent with the belief that it once formed part of the same mass with the
earth, since if such were the case, the moon was probably formed by the
separation of the outer portions of that mass, which would be likely to
have a smaller specific gravity than the rest.

247. The superficial gravity, or the attraction of the moon
for bodies at its own surface, may be found by the equation
m
g = g ×

,
r2
in which g signifies the superficial gravity of the moon, g is the
force of gravity on the earth, while m and r are the mass and radius
of the moon as compared with those of the earth. This gives us

0.0125
g = g ×

,

0.0747
or (very approximately) g equals one-sixth of g; that is, a body
which weighs six pounds on the earth’s surface would at the surface
of the moon weigh only one (in a spring balance). A man on the
moon could jump six times as high as he could on earth and could

THE MOON.

159
throw a stone six times as far. This is a fact to be remembered in
connection with the enormous scale of the surface-structure of the
moon. Volcanic forces, for instance, upon the moon would throw
the rejected materials to a vastly greater distance there than on the
earth.

248. Rotation of the Moon.—The moon ro-
tates on its axis once a month, in precisely the same
time as that occupied by its revolution around the
earth. In the long run it therefore keeps the same
side always towards the earth: we see to-day pre-
cisely the same face and aspect of the moon as

Galileo did when he first looked at it with his tele-
scope, and the same will continue to be the case for

Fig. 79.

thousands of years more, if not forever.

It is difficult for some to see why a motion of this sort should be
considered a rotation of the moon, since it is essentially like the motion
of a ball carried on a revolving crank. See Fig. 79. Such a ball, they
say, “revolves around the shaft, but does not rotate on its own axis.” It
does rotate, however. The shaft being vertical and the crank horizontal,
suppose that a compass needle be substituted for the ball, as in Fig. 80.

The pivot turns underneath it as the crank whirls, but the compass
needle does not rotate, maintaining always its own direction with the
marked end north. On the other hand, if we mark one side of the ball (in
the preceding figure), we shall find the marked side presented successively
to every point of the compass as the crank revolves, so that the ball as
really turns on its own axis as if it were whirling upon a pin fastened to
a table. The ball has two distinct motions by virtue of its connection
with the crank: first, the motion of translation, which carries its centre
of gravity, like that of the compass needle, in a circle around the axis of
the shaft; secondly, an additional motion of rotation around a line drawn
through its centre of gravity parallel to the shaft.

248*. Definition
of

Rotation.—A
body “rotates” whenever a line drawn from its
centre of gravity outward, through any point
selected at random in its mass, describes a circle
in the heavens. In every rotating body, one such
line can be so drawn that the circle described
by it in the sky becomes infinitely small, This

Fig. 80.

is the axis of the body. Another set of points
can be found such that lines drawn from the centre of gravity outward

THE MOON.

160
through them describe a great circle in the sky 90◦ distant from the
point pierced by the axis, and these points constitute the equator of the
body.

249. Librations of the Moon.—1. Libration in Latitude. The
axis of revolution of the moon is not perpendicular to its orbit. It
makes a constant angle of about 88 1 ◦ with the ecliptic, and the

2
moon’s equator is so placed that it is always edge-wise to the earth
when the moon is at her node, being maintained in that position
by an action of the earth, which produces a precessional motion of
the moon’s axis. The angle between the moon’s equator and the
plane of her orbit, therefore, is 1 1 ◦+ the inclination of the moon’s

2
orbit, which together make up an angle of a little more than 6 1 ◦;

2
but, as the inclination of the moon’s orbit to the ecliptic is con-
stantly varying slightly, this inclination of the moon’s axis to her
orbit also changes correspondingly. This inclination of the moon’s
axis produces changes in the aspect of the moon towards the earth
similar to these produced by the inclination of the earth’s axis to-
wards the ecliptic. At one time, just as the north pole of the earth
is turned towards the sun, so also the north pole of the moon is
tipped towards the earth at an angle of 6 1 ◦, and in the opposite

2
half of the moon’s orbit the south pole is similarly presented to us.

In consequence we alternately look over the northern and southern
portions of the moon’s disc.

The period of this libration is the time of the moon’s revolution from
node to node, called a nodical revolution. This is 27.21 days—about

2 hours and 38 minutes shorter than the sidereal revolution of the moon,
since the nodes always move westward, completing the circuit in about

19 years.

250. 2. Libration in Longitude.

The moon’s orbit being ec-
centric, she moves faster when near perigee, and slower when near
apogee; half-way between perigee and apogee she is more than 6◦
ahead of the position she would have if she had moved with the mean
angular velocity. Now the rotation is uniform. A point, therefore,
on the moon’s surface which is directed toward the earth at perigee
will not have revolved far enough to keep it directed toward the
earth when she is half-way (in time) between perigee and apogee,
as is evident from Fig. 81. For in the quarter-month next follow-
ing the perigee, the moon will travel to a point M , considerably
more than half-way to apogee. But the point a will have made only

THE MOON.

161
one quarter-turn, which is not enough to bring it to the line M E.

We shall therefore see a little around the western edge. Similarly on
the other side of the orbit, half-way between apogee and perigee, we
shall look around the eastern edge to the same extent. At perigee
and apogee both, the libration is, of course, zero. The amount of
this libration is evidently at any moment just the same as that of
the so-called “equation of the centre,” which, it will be remembered,
is the difference between the mean and true anomalies of the moon
at any moment. Its maximum possible value is 7◦ 45 .

The period of this libration is
the time it takes the moon to go
around from perigee to perigee—
the so-called anomalistic revolution,
which is 27.555 days, about 5 hours
and 36 minutes longer than the sid-
ereal month, and 8 hours 14 minutes
longer than the moon’s nodical rev-
olution, which determines the libra-
tion in latitude.

The cause of the increased length
of the anomalistic revolution is of
course the fact that the line of apsides

Fig. 81.—The Libration in
continually advances eastward, mak-

Longitude.

ing one revolution every nine years.

251. 3. Diurnal Libration. This is strictly a libration not of
the moon, but of the observer; still, as far as the aspect of the
moon goes, the effect is precisely the same as if it were a true lunar
libration. The moon’s motions have reference to the earth’s centre.

We, on the surface of the earth, look down over the western edge of
the moon when it is rising, and over the eastern when it is setting, by
an amount which is equal to the semi-diameter of the earth as seen
from the moon; that is, about one degree (the moon’s parallax).

On the whole, taking all three librations into account, we see
considerably more than half the moon, the portion which never dis-
appears being about forty-one per cent of the moon’s surface, that
never visible also forty-one per cent, while that which is alternately
visible and invisible is eighteen per cent.

252. The agreement between the moon’s time of rotation and
of her orbital revolution cannot be accidental. It is probably due to

THE MOON.

162
the action of the earth on some slight protuberance on the moon’s
surface, analogous to a tidal wave. If the moon were ever plastic,
such a bulge must have been formed on the side of the moon next
the earth, and would serve as the handle by which the earth always
keeps the same face of the moon towards herself. This subject will
be resumed later.

Fig. 82.—Explanation of the Phases of the Moon.

253. The Phases of the Moon.—Since the moon is an
opaque globe, shining entirely by reflected light, we can see only
that hemisphere of her surface which happens to be illuminated,
and of course only that part of the illuminated hemisphere which
is at the time turned towards the earth. At new moon, when the
moon is between the earth and the sun, the dark side is towards
us. A week later, at the end of the first quarter, half of the illumi-
nated hemisphere is seen, and we have the half moon, just as we do
a week after the full. Between the new moon and the half moon,
during the first and last quarters of the lunation, we see less than
half of the illuminated portion, and then have the “crescent” phase.

See Fig. 82 (in which the light is supposed to come from a point
far above the moon’s orbit). Between the half moon and the full,
during the second and third quarters of the lunation, we see more

THE MOON.

163
than half of the moon’s illuminated side, and have what is called
the “gibbous” phase.

Since the terminator or line which separates the dark portion of
the disc from the bright is always a semi-ellipse (being a semi-circle
viewed obliquely), the illuminated surface is always a figure made
up of a semi-circle plus or minus a semi-ellipse, as shown in Fig. 83,

A.

It is sometimes incorrectly attempted to repre-
sent the crescent form by a construction like Fig. 83,

B (where a smaller circle is cut by a larger one). It
is to be noticed that ab, the line which joins the
cusps, is always perpendicular to the line directed
to the sun, and the horns are always turned away

Fig. 83.

from the sun; so that the precise position in which
they will stand at any time is always predictable, and has nothing what-
soever to do with the weather. Artists are sometimes careless in the
manner in which they introduce the moon into landscapes. One occa-
sionally sees the moon near the horizon with the horns turned downwards,
a piece of drawing fit to go with Hogarth’s barrel which shows both its
heads at once.

254. Earth-Shine on the Moon.—Near the time of new moon
the whole disc of the satellite is easily visible, the portion on which
sunlight does not fall being illuminated by a pale ruddy light. This light
is earth-shine, the earth as seen from the moon being then nearly full; for
seen from the moon the earth shows all the phases that the moon does,
the earth’s phase in every case being exactly supplementary to that of
the moon as seen by us.

As the earth has a diameter nearly four times that of the moon,
the earth-shine at any phase would be about thirteen times as strong as
moonlight, if the reflective power of the earth’s surface were the same.

Probably, taking the clouds and snow into account, the earth’s surface on
the whole is rather more brilliant than the moon’s, so that near new moon
the earth-shine, by which the dark side of the moon is then illuminated,
is from fifteen to twenty times as strong as full moonlight. The ruddy
color is due to the fact that light sent to the moon from the earth has
twice penetrated our atmosphere and so has acquired the sunset tinge.

255. Physical Characteristics of the Moon.—1. Its Atmo-
sphere. The moon’s atmosphere, if it has any at all, is extremely
rare, probably not producing a barometric pressure to exceed 1 of

25
an inch of mercury, or 1 of the pressure at the earth’s surface.

750

THE MOON.

164

The evidence on this point is twofold.

(a) The telescopic appearance. The parts of the moon near the
edge of the disc, which, if there were any atmosphere, would be
seen through its greatest possible depth, are seen without the least
distortion: there is no haze, and all shadows are perfectly black.

There is no sensible twilight at the cusps of the moon; no evidences
of clouds or storms, or anything like atmospheric phenomena.

(b) The absence of refraction when the moon intervenes between
us and any more distant object. For instance, at an eclipse of the
sun there is no distortion of the sun’s limb where the moon cuts
it, nor any ring of light running out on the edge of the moon like
that which encircles the disc of Venus at the time of a transit. The
most striking evidence of this sort comes, however, from occulta-
tions of the stars. When the moon hides a star from sight, the
phenomenon, if it occurs at the moon’s dark edge, is an exceedingly
striking one. The star retains its full brightness in the field of the
telescope until all at once, without the least warning, it simply is
not there, the disappearance generally being absolutely instanta-
neous. Its reappearance is of the same sort, and still more startling.

Now if the moon had any perceptible atmosphere (or the star any
sensible diameter) the disappearance would be gradual. The star
would change color, become distorted, and fade away more or less
gradually.

The spectroscope adds its evidence in the same direction. There
is no modification of the spectrum of the star in any respect at the
time of its disappearance; and we may add that the spectrum of
moonlight is identical with that of sunlight pure and simple, there
being no traces of any effect whatever produced upon the sunlight
by its reflection from the moon, nor any signs of its having passed
through an atmosphere.

256. The time during which a star would be hidden behind
the moon would also be decreased by the refraction of any sensible
atmosphere, making the observed duration of an occultation less
than that computed from the known diameter of the moon and its
rate of motion. Certain Greenwich observations apparently show
a difference, amounting to about two seconds of time. This may
possibly be due in some part to the action of a real, but exceed-
ingly rare, lunar atmosphere; for if the whole phenomenon were due
simply to atmospheric action, it would indicate an atmosphere hav-

THE MOON.

165
ing a density about

1
of our own,—far within the limits which

2000
were stated above. But the difference may be, and very probably
is, attributable, in part at least, to a slight error in the measured
diameter of the moon, due to irradiation: the diameter of a bright
object always appears a little larger than it really is. An error of
about 2 of this sort would explain the whole discrepancy, without
any need of help from an atmosphere.

257. What has become of the Moon’s Atmosphere.—if
the moon ever formed a part of the same mass as the earth, she must
once have had an atmosphere. There are a number of possible and more
or less probable hypotheses to account for its disappearance. It has been
surmised (1) that there may be great cavities left within the moon’s
mass by volcanic eruptions, and that the rocks themselves have been
transformed into a sort of pumice-stone structure, and that the air has
retired into these internal cavities.

(2) That the air has been absorbed by the inner lunar rocks in cooling.

A heated rock expels any gases that it may have absorbed; but if it
afterwards cools slowly, it reabsorbs them, and can take up a very great
quantity. The earth’s core is supposed to be now too intensely heated to
absorb much gas; but if it goes on cooling, it will absorb more and more,
and in time it may rob the surface of the earth of all its air. There are
still other hypotheses, which we can not take space even to mention.

258. Water on the Moon’s Surface.—Of course without an
atmosphere there can be no water, since the water would immedi-
ately evaporate and form an atmosphere of water vapor if there were
no air present. It is not impossible, however, or even improbable,
that solid water, that is, ice and snow, may exist on the moon’s sur-
face at a temperature too low for any sensible evaporation. There
are many things in the moon’s appearance that seem to indicate the
former existence of seas and oceans on her surface, and the same
hypotheses have been suggested to account for their disappearance
that were suggested in the case of the moon’s atmosphere. It may
be added also that many kinds of molten rock in crystallizing would
take up large quantities of water of crystallization, not merely ab-
sorbed as a sponge absorbs water, but chemically united with the
other constituents of the rock. In whatever way, however, it may
have come about, it is certain that now no substances that are
gaseous, or that can be evaporated at low temperatures, exist in
any quantity on the moon’s surface—at least, not on our side of

THE MOON.

166
the moon.

There have been speculations that on the other side—that celestial
country so near us and so absolutely concealed from us—there may be
air and water and abundant life; the idea being that our side of the
moon is a great table land many miles in elevation, while the other side
is a corresponding depression, like the valley of the Caspian Sea, only
vastly deeper. An insufficiently grounded conclusion of Hansen’s, that
the centre of gravity of the moon is some thirty miles farther from us
than its centre of figure, for a time gave color to the idea, but it is now
practically abandoned, Hansen’s conclusion having been shown to be
unwarranted by the facts.

259. The Moon’s Light.—As to quality it is simple sunlight,
showing a spectrum which, as has been said, is identical in every
detail with that of light coming directly from the sun. Its bright-
ness as compared with that of sunlight is difficult to measure accu-
rately, and different experimenters have found results for the ratio
between full moonlight and sunlight ranging all the way from

1

300000

(Bouguer) to

1

(Wollaston). The value now usually accepted

800000
is that determined by Zöllner, viz.,

1

. According to this, if the

618000
whole visible hemisphere were packed with full moons, we should
receive from it about one-eighth part of the light of the sun.

It is found, also, that the half moon does not give even nearly half
as much light as the full moon. The law which connects the phase of
the moon with the amount of light given at the time, is rather compli-
cated, but the gist of the matter is that at any time, except at the full,
the visible surface is more or less darkened by the shadows cast by the
irregularities of the surface. Zöllner has calculated that an average angle
of 52◦ for these elevations and depressions would account for the law of
illuminations actually observed.

The average “albedo,” or reflecting power of the moon’s surface,

Zöllner states as 0.174; that is, the moon’s surface reflects a little
more than one-sixth part of the light that falls upon it. This is about
the albedo of a rather light-colored sandstone, and agrees well with
the estimate of Sir John Herschel, who found the moon to be very
exactly of the same brightness as the rock of Table Mountain when
it was setting behind it, illuminated as were the rocks themselves
by the light of the rising sun. There are, however, great variations
in the brightness of different portions of the moon’s surface. Some
spots are nearly as white as snow or salt, and others as dark as

THE MOON.

167
slate.

260. Heat of the Moon.—For a long time it was impossible
to detect the moon’s heat. It is too feeble to be detected by the most
delicate mercurial thermometer even when concentrated by a large
lens. The first sensible effect was obtained by Melloni, in 1846, with
the then newly invented thermopile, by a series of observations from
the summit of Vesuvius. Since then several physicists have worked
upon the subject with more or less success, but the most recent
and reliable investigations are those of Lord Rosse and Professor

Langley. With modern apparatus there is no difficulty in detecting
the heat in the lunar radiations, but measurements are extremely
difficult and liable to error. A considerable percentage of the lunar
heat seems to be heat simply reflected (like light), while the rest,
perhaps three-fourths of the whole, is “obscure heat ”; that is, heat
which has been first absorbed by the moon’s surface and then ra-
diated, like the heat from a brick surface that has been warmed by
sunshine. This is shown by the fact that a comparatively thin plate
of glass cuts off some 86 per cent of the heat received from the moon
in the same way that it does the heat of a stove, while the heat of
direct sunlight, or of an electric arc, would pass through the same
plate with very little diminution. The same thing appears also from
direct measurements upon the heat-spectrum of the moon made by

Langley with his bolometer, described further on. (Art. 343.)

261. As to the temperature of the moon’s surface, it is difficult
to affirm much with certainty. On one hand, the lunar rocks are
exposed to the sun’s rays in a cloudless sky for fourteen days at
a time, so that if they were blanketed by air like our own rocks
they would certainly become intensely heated. A few years ago,

Lord Rosse inferred from his observations that the temperature of
the lunar surface rose at its maximum (about three days after full
moon) far above that of boiling water.

But his own later investigations and these of Langley throw great
doubt on this conclusion. There is no air-blanket at the moon’s
surface to prevent it from losing heat as fast as it receives it; and it
now seems rather more probable that the temperature never rises
above the freezing-point of water, as is the case on the highest of
our mountains, where there is perpetual ice, and the temperature
is always low even at noon. So far as we can judge, the condition of
things on the moon’s surface must correspond to an elevation many

THE MOON.

168
times higher than any mountain on the earth; for no terrestrial
mountain is so high that the density of the air at its summit is even
nearly as low as that of the densest supposable lunar atmosphere.

This idea, that the temperature is low, is borne out, also, by the
fact that the bolometer shows the presence, in the lunar radiations,
of a considerable quantity of heat having a wave-length greater than
that of the heat radiated from a block of ice.

At the end of the long lunar night of fourteen days the temper-
ature must fall appallingly low, certainly 200◦ below zero.

The whole amount of heat sent by the full moon to the earth is
estimated by Rosse as about one eighty-thousandth part of that sent
by the sun.

262. Lunar Influences on the Earth.—The moon’s attrac-
tion cooperates with that of the sun in producing tides, of which we
shall speak hereafter. There are also certain distinctly ascertained
disturbances of terrestrial magnetism connected with the approach
and recession of the moon at perigee and apogee; and this ends the
chapter of ascertained lunar influences.

The multitude of current beliefs as to the controlling influence of
the moon’s phases and changes over the weather and the various condi-
tions of life are mostly unfounded, and in the strict sense of the word

“superstitions,”—mere survivors from a past credulity.

It is quite certain that if there is any influence at all of the sort
it is extremely slight—so slight that it cannot be demonstrated with
certainty, although numerous investigations have been made expressly
for the purpose of detecting it. We have never been able to ascertain,
for instance, with certainty, whether it is warmer or not, or less cloudy
or not, at the time of the full moon. Different investigations have led to
contradictory results.

The frequency of the moon’s changes is so great that it is always easy
to find instances by which to verify a belief that changes of the moon
control conditions on the earth. A change of the moon necessarily occurs
about once a week, the interval from quarter to quarter being between
seven and eight days. All changes, of the weather for instance, must
therefore occur within three and three-fourths days of a change of the
moon, and fifty per cent of them ought to occur within forty-six hours
of a change, even if there were no causal connection whatever.

Now it requires only a very slight prepossession in favor of a belief in
the effectiveness of the moon’s changes to make one forget a few of the
weather changes that occur too far from the proper time. Coincidences

THE MOON.

169
enough can easily be found to justify a preëxisting belief.

THE MOON’S SURFACE.

263. Even to the naked eye the moon is a beautiful object,
diversified with darker and lighter markings which have given rise
to numerous popular superstitions. With a powerful telescope these
naked-eye markings mostly vanish, and are replaced by a countless
multitude of smaller details, which are interesting in the highest
degree. The moon on the whole, on account of this diversity of
detail, is the finest of all telescopic objects; especially to moderate-
sized instruments, say from six to ten inches in diameter, which
generally give a more pleasing view of our satellite than instruments
either larger or smaller.

264. How near the Tele-
scope brings the Moon.—An
instrument of this size, with mag-
nifying powers between 250 and

500, brings up the moon virtually
to a distance ranging from 1000
miles to 500; and since an object
a mile in diameter on the moon
subtends an angle of about 0 .86,

Fig. 84.—A Normal Lunar Crater

(Nasmyth.)

with the higher powers of such an
instrument objects less than a mile in diameter become visible un-
der favorable atmospheric conditions. A long line or streak, even
less than a quarter of a mile across, could probably be seen. With
larger telescopes the power can now and then be carried at least
twice as high, and correspondingly smaller details made out. When
everything is at its best, the great Lick telescope of 30 inches aper-
ture, with a power of 2500 or so, may possibly reduce the virtual
distance of our satellite to about 100 miles for visual purposes. It is
evident that while with our telescopes we should be able to see such
objects as lakes, rivers, forests, and great cities, if they exist on the
moon, it will be hopeless to expect to distinguish single buildings, or
any of the ordinary operations and indications of life, if such there
are.

There are a few mountains on the earth from which a range of 100
miles is obtained in the landscape. Those who have seen such a landscape
know how little is to be made out with the naked eye at that distance.

THE MOON.

170

Still, the comparison is not quite fair, because in looking at a terrestrial
object a hundred miles away the line of vision passes through a dense
atmosphere, while in looking upward towards the moon it penetrates a
much less thickness of air.

265. The Moon’s Surface Structure.—The moon’s surface
for the most part is extremely uneven and broken, far more so than
that of the earth. The structure, however, is not like that of the
earth’s surface. On the earth the mountains are mostly in long
ranges, such as the Alps, the Andes, and Himalayas. On the moon
such mountain ranges are few in number, though they exist; but the
surface is pitted all over with great craters, resembling very closely
the volcanic craters on the earth’s surface, though on an immensely
greater scale. One of the largest craters upon the earth, if not the
largest, is the Aso San in Japan, about seven miles across. Many of
those on the moon are fifty and sixty miles in diameter, and some
are over 100 miles across, while smaller ones from a half-mile to
eight or ten miles in diameter are counted by the thousand.

Fig. 85.—Map of the Moon.

THE MOON.

171

The normal lunar crater is nearly circular, surrounded by an
elevated ring of mountains which rise anywhere from 1000 to 20,000
feet above the surrounding country. Within the floor of the crater
the surface may be either above or below the outside level. Some
craters are deep, some filled nearly to the brim. In some cases the
surrounding mountain ring is entirely absent, and the crater is a
mere hole in the plain. In the centre of the crater there usually
rises a group of peaks, which attain about the same elevation as
the encircling ring, and these central peaks often show little holes
or craterlets in their summits.

In most cases the resemblance of these formations to terrestrial vol-
canic structures, like those exemplified by Vesuvius and others in the
surrounding region, makes it natural to assume that they had a similar
origin. This, however, is not absolutely certain, for there are consider-
able difficulties in the way, especially in the case of the great “Bulwark

Plains,” so called, which are so extensive that a person standing in the
centre could not see the summit of the surrounding ring at any point; and
yet no line of demarcation can be drawn between them and the smaller
craters. The series is continuous. Moreover, on the earth, volcanoes
necessarily require the action of air and water, which do not now exist
on the moon. It is obvious, therefore, that if these lunar craters are the
result of true volcanic eruptions, they must be fossil formations; for it is
quite certain that no evidence of existing volcanic activity has ever been
found. The moon’s surface appears to be absolutely quiescent—still in
death.

On some portions of the moon these craters stand very thickly.

Older craters have been encroached upon, or more or less completely
obliterated by the newer, and the whole surface is a chaos, of which
the counterpart is hardly to be found on the earth, even in the
roughest portions of the Alps. This is especially the case near the
moon’s south pole. It is noticeable that, as on the earth the newest
mountains are generally the highest, so on the moon the more newly
formed craters are generally deeper and more precipitous than the
older ones.

266. Lunar Nomenclature.—The great plains were called
by Galileo oceans or seas (Maria), and some of the smaller ones
marshes (Paludes) and lakes, for he supposed that the grayish sur-
faces visible to the naked eye, and conspicuous in a small telescope,
were covered with water. Thus we have the “Oceanus Procellarum,”

the “Mare Imbrium,” and a number of other “seas,” of which “Mare

THE MOON.

172

Fecunditatis,” “Mare Serenitatis,” and “Mare Tranquilitatis,” are
the most conspicuous. There are twelve of them in all, and eight or
nine Paludes, Lacus, and Sinus.

The ten mountain ranges on the moon are mostly named after
terrestrial mountains, as Caucasus, Alps, Apennines, though two or
three bear the names of astronomers, like Leibnitz, Dörfel, etc.

The conspicuous craters bear the names of the more eminent ancient
and mediæval astronomers and philosophers, as Plato, Archimedes,

Tycho, Copernicus, Kepler, and Gassendi; while hundreds of smaller
and less conspicuous formations bear the names of more modern or
less noted astronomers.

The system seems to have originated with Riccioli in 1650, but most
of the names have been more recently assigned by the later map-makers,
the most eminent of whom have been the German astronomers Beer and

Maedler (who published their map in 1837), and Schmidt of Athens,
whose great map of the moon, on a scale seven feet in diameter, was
published by the Prussian government a few years ago. It is not at all
too much to say that our maps of the earth’s surface do not, on the
whole, compare in fulness and accuracy with our maps of the moon. Of
course this is not true of such countries as France and England, or others
that have been trigonometrically surveyed; but there are no such lucunæ
in our maps of the moon as exist in our maps of Asia and Africa, for
instance.

267. Other Lunar Forma-
tions.—The craters and moun-
tains are not the only interest-
ing formations on the moon’s sur-
face.

There are many deep, nar-
row, crooked valleys that go by the
name of “rills” (German Rillen),
some of which may once have been
watercourses. Then there are nu-
merous “clefts,” half a mile or so
wide and of unknown depth, run-
ning in some cases several hundred
miles, straight through mountain
and valley, without any apparent
regard for the accidents of the sur-

Fig. 86.—Archimedes and the
face. They seem to be deep cracks

Apennines (Nasmyth).

THE MOON.

173
in the crust of our satellite. Several of them are shown in Fig. 86.

Most curious and interesting of all are the light-colored streaks or

“rays” which radiate from certain of the craters, extending in some
cases a distance of several hundred miles. They are usually from five
to ten miles wide, and neither elevated nor depressed to any extent
with reference to the general surface. They pass across mountain
and valley, and sometimes through craters without any change in
width or color. We do not know whether they are like the so-called

“trap-dykes” on the earth,—fissures which have been filled up from
below with some light-colored material,—or whether they are mere
surface markings. No satisfactory explanation has ever been given.

The most remarkable system of “rays” of this kind is the one
connected with the great crater Tycho, not very far from the moon’s
south pole. They are not very conspicuous until within a few days
of full moon, but at that time they, and the crater from which they
radiate, constitute by far the most striking feature of the whole
lunar landscape.

268. Changes on the Moon.

—It is certain that there are no
conspicuous changes.

The ob-
server has before him no such ever-
varying vision as he would have in
looking toward the earth,—no fly-
ing clouds, no alternations of sea-
sons with the transformation of the
snowy wastes to green fields, nor
any considerable apparent move-
ment of objects on the disc. The
sun rises on them slowly as they
come one after the other to the ter-
minator, and sets as slowly. At the
same time it is confidently main-

Fig. 87.—Gassendi (Nasmyth).

tained by many observers that here
and there changes are still going on in the details of the surface.

Others as stoutly dispute it.

269. Probably the most notable and best advocated instance of
such a change is that of the little crater Linné, in the Mare Serenitatis.

It was observed by Schroeter very early in the century, and is figured
and described by Beer and Maedler as being about five and a half or

THE MOON.

174
six miles in diameter, quite deep and very bright. In 1866 Schmidt, who
had several times observed it before, announced that it had disappeared.

A few months later it was visible again, and there were many reported
changes in its appearance during the next year or two. There is no
question that it does not now at all agree in conspicuousness and size
with the representation of Beer and Maedler, for it is at present, and
has been for several years, only a minute dark spot, with a whitish spot
surrounding it. Astronomers would feel more confident that this was a
case of real change were it not that Schroeter’s earlier picture much more
resembles the present appearance than does that of Beer and Maedler.

As the latter observers worked with rather a small telescope, and had no
reason for taking any special pains in the delineation of this particular
object, the evidence is less conclusive than it might seem at first. The
change, however, if real, was certainly as great as in the instance of

Krakat˜
ao, the great volcano whose eruption in 1883 filled the earth’s
atmosphere with smoke and vapor for more than two years, and caused
the “twilight conflagrations” of the sky. The phenomenon in the case
of Linné, if real, was probably a falling in of the walls of the crater,
exposing fresh unweathered surfaces.

The reason why it is so difficult to be sure of changes lies in the great
variations in the appearance of a lunar object under the varying illumi-
nation. To insure certainty in such delicate observations, comparisons
must be made between the appearance seen at precisely the same phase
of the moon, with telescopes (and eyes too) of equal power; and under
substantially the same conditions otherwise, such as the height of the
moon above the horizon, the clearness and steadiness of the air, etc. It
is of course very difficult to secure such identity of conditions.

270. Measurements of Heights of

Lunar Mountains.—When the termina-
tor approaches a lunar mountain, the top of
the mountain catches the sunlight first, and
appears as a star entirely detached from the
rest of the illuminated portion, like the little
bright spots opposite a and b in Fig. 88. As
time passes, the bright spot becomes larger
as the light extends lower down the moun-
tain side, until the terminator reaches and

Fig. 88.

passes it.

If now we measure the apparent distance, AD or a, Fig. 89, from
the peak to the terminator at the moment when it first appears like

THE MOON.

175
a star, it is easy to compute AB, and from this, the height of the
mountain.

In Fig. 89 the angle S CN very approximately the moon’s “elonga-
tion” at the time of observation, since the line from the earth to the sun
is nearly parallel to S C (the moon’s distance being only about

1
of

400
the sun’s). Now the angle BAD = GBE = 90◦ − S CN , so that AB,
or b, = AD (or a)÷ sin S CN . Knowing b, and the radius of the moon
r, we get

(r + h)2 = r2 + b2,
in which h is the height of the mountain, and b is the distance AB. From
this equation we find
h =
r2 + b2 − r,
b2
or (very nearly)

h = 1

.

2 r

If, for instance, b were 60 miles,

3600
h(in miles) =
or 1.664 miles = 8845 feet.

2 × 1081

Since the terminator is very
ragged, it is sometimes best to
measure from the mountain top
clear across to the edge of the
moon, as indicated by the lit-
tle arrows in Fig. 88.

The
position of the theoretical ter-
minator (the terminator as it
would be if the moon were a
smooth sphere) is known from
the moon’s age, so that AB can
be deduced by measuring from

Fig. 89.

Measurement of the Height of a Lunar
the limb as well as from the ter-

Mountain.

minator.

The height of a mountain can also be ascertained by measur-
ing the length of its shadow in cases where the shadow falls on a
reasonably level surface. A few of the lunar mountains reach the
height of 22,000 or 23,000 feet, but there are none which attain the
elevation of the very highest terrestrial mountains. Heights ranging
from 10,000 to 20,000 feet are common.

THE MOON.

176

271. The Best Time to look at the Moon with a Tele-
scope.—The moon when full is not so satisfactory an object as when
near the half, because at the full moon there are no shadows, so that at
that time the “relief” of the surface structure is entirely lost. Certain
features, however, as has been before mentioned, are then best seen, as,
for instance, the streaks or rays. Generally, any particular mountain,
crater, rill, or cleft, is best studied when it is just on or very near the
terminator, that is, at the time when the sun is rising or setting near
it, because then the shadows are longest. The best general view of the
moon is that obtained a few days after the half moon, when Copernicus
and Tycho are both near the terminator, and Plato is still near enough
to it to show very well.

272. Photographs of the Moon.—A great deal of attention
has been paid to this subject, and some fine results have been reached.

The earliest success was that of Bond in 1850, with the old daguerreo-
type process; then followed the work of De la Rue in England, and of

Dr. Henry Draper, and especially of Mr. Rutherfurd in this country.

Rutherfurd’s pictures have remained absolutely unrivalled until very re-
cently.

To these older experimenters the moon’s motion offered a great dif-
ficulty, but now that with the sensitive plates at present used, a fraction
of a second is a sufficient exposure, that difficulty has disappeared, and
the plates which have recently been taken at Cambridge, Mass., are far
in advance even of Rutherfurd’s, showing such craters as Copernicus or

Ptolemy with a diameter of two inches, on a scale larger than that of

Schmidt’s map. The Lick Observatory has also taken up the work, and
is making admirable pictures.

THE SUN.

177

CHAPTER VIII.

THE SUN: DISTANCE AND DIMENSIONS.—MASS AND DEN-

SITY.—ROTATION.—STUDY OF THE SURFACE: GENERAL

VIEWS AS TO THE SUN’S CONSTITUTION.—SUN SPOTS:

THEIR

APPEARANCE,

NATURE,

DISTRIBUTION,

AND

PERIODICITY.—THE SPECTROSCOPE AND THE SOLAR

SPECTRUM.—CHEMICAL

ELEMENTS

RECOGNIZED

IN

THE SUN.—THE CHROMOSPHERE AND PROMINENCES.—

THE CORONA.

273. The SUN is simply a star ; a hot, self-luminous globe of
enormous magnitude as compared with the earth and the moon,
though probably only of medium size among its stellar compeers.

But to the earth and the other planets which circle around it, it
is the grandest of all physical objects. Its attraction confines its
planets to their orbits and controls their motions, and its rays sup-
ply the energy which maintains every form of activity upon their
surfaces and makes them habitable.

274. Its Distance and Dimensions.—Its distance is deter-
mined by finding its horizontal parallax; that is, the semi-diameter
of the earth as seen from the sun. The mean value of this parallax
is probably very near 8 .8,1 plus or minus 0 .03.

We reserve to a separate chapter the discussion of the methods
by which this most fundamental and important of all astronomical
data has been ascertained, merely remarking here that the problem
is one of extreme practical difficulty, though the principles involved
are simple enough.

1In the American Ephemeris the value deduced by Newcomb in 1867 is
used, viz., 8 .85. The British “Nautical Almanack” uses the same value, and
the French the value deduced by Leverrier a little earlier, 8 .86; but more
recent observations seem to show that this value is a little too large, and that
the number stated, 8 .8, is more probably correct. The difference is of no
importance for almanac purposes.

THE SUN.

178

Assuming the parallax at 8 .8, the mean distance of the sun

(putting r for the earth’s radius) equals
r ÷ sin 8 .8 = 23,439 × r.

With Clarke’s value of r (Art. 145), this gives 149,500000 kilome-
ters, or 92,897,000 miles; which, however, is uncertain by at least

200,000 miles, and is variable, also, to the extent of about three
million miles on account of the eccentricity of the earth’s orbit, the
earth being nearer the sun in December than in June.

275. This distance is so much greater than any with which we
have to do on the earth that it is possible to reach a conception of
it only by illustrations of some sort. Perhaps the simplest is that
drawn from the motion of a railway train. Such a train going 1000
miles a day (nearly forty-two miles an hour, and faster than the

Chicago Limited on the Pennsylvania Railroad) would take 254 13
years to make the journey.

If sound were transmitted through interplanetary space, and at
the same rate as through our own atmosphere, it would make the
passage in about fourteen years; i.e., an explosion on the sun would
be heard by us fourteen years after it actually occurred. A cannon-
ball moving unretarded, at the rate of 1700 feet per second, would
travel the distance in nine years. Light does it in 499 seconds.

276. Diameter.—The sun’s mean apparent diameter is

32 04 ± 2 . Since at the sun one second equals 450.36 miles, its
diameter equals 866,500 miles, or 109 1 times the diameter of the

2
earth. It is quite possible that this diameter is variable to the extent
of a few hundred miles, since, as will appear hereafter, the sun (at
least the surface which we see) is not solid.

Representing the sun by a globe two feet in diameter, the earth
would be 22 of an inch in diameter,—the size of a very small pea,

100
or a “22-calibre” round pellet. Its distance from the sun on that
scale would be just about 220 feet, and the nearest star (still on the
same scale) would be eight thousand miles away, at the antipodes.

If we were to place the earth in the centre of the sun, supposing
it to be hollowed out, the sun’s surface would be 433,000 miles away
from us. Since the distance of the moon is only about 239,000 miles,
it would be only a little more than half-way out from the earth to
the inner surface of the hollow globe, which would thus form a very
good background for the study of the lunar motions.

THE SUN.

179

It is perhaps worth noticing, as a help to memory, that the sun’s
diameter exceeds the earth’s just about as many times as it is itself
exceeded by the radius of the earth’s orbit; or, in other words, the sun’s
diameter is nearly a mean proportional between the earth’s distance from
the sun and the earth’s diameter, 110 being the common ratio.

Fig. 90.—Dimensions of the Sun compared with the Moon’s Orbit.

277. Surface and Volume.—Since the surfaces of globes are
proportional to the squares of their radii, the surface of the sun
exceeds that of the earth in the ratio of (109.5)2 to 1; that is, its
surface is about 12,000 times the surface of the earth.

The volumes of spheres are proportional to the cubes of their
radii; hence the sun’s volume is (109.5)3, or 1,300000 times that of
the earth.

278. The Sun’s Mass.—The mass of the sun is very nearly
three hundred and thirty-two thousand times that of the earth, sub-
ject to a probable error of at least one per cent. There are various
ways of getting at this result. For our purpose here, perhaps the
most convenient is by comparing the earth’s attraction for bodies
at her surface (as determined by pendulum experiments) with the
attraction of the sun for the earth,—the central force which keeps
her in her orbit. Put f for this force (measured, like gravity, by the
velocity it generates in one second), g for the force of gravity (32 feet

2 inches per second), r the earth’s radius, R the sun’s distance, and
let E and S be the masses of the earth and sun respectively. Then,

THE SUN.

180
by the law of gravitation, we have the proportion

S

E
f

R 2
f : g ::

:

, or S = E

.

(a)

R2
r2
g
r

R

Now,

= 23, 440 (nearly).

r

Its square equals 549,433,600. g = 386 inches. To find f we have
from Mechanics (Physics, p. 62),

V 2
f =

,

(b)

R
this being the expression for the “central force” in the case of a
body revolving in a circle. (We may neglect the eccentricity of the
earth’s orbit in a merely approximate treatment of the problem.)

V is the orbital velocity of the earth, which is found by dividing
the circumference of the orbit, 2πR, by T , the number of seconds
in a sidereal year. This velocity comes out 18.495 miles per second.

Putting this into formula (b), we get f = 0.2333 inches,
f

1
so that

= 0.0006044 =

(nearly);
g

1654

1
whence

S = E ×

× 519,433,600; or S equals 332,000.

1654

We may note in passing that half of f expresses the distance by
which the earth falls towards the sun every second, just as half g
is the distance a body at the earth’s surface falls in a second. This
quantity (0.116 inch), a trifle more than a ninth of an inch, is the
amount by which the earth’s orbit deviates from a straight line in
a second. In travelling eighteen and one-half miles the deflection is
only one-ninth of an inch.

2πR

278*. By substituting
for V in equation (b), we get

T

4π2R
f =

;

T 2
and putting this value f into equation (a) and reducing, we obtain

4π2
r

R 3

S = E

,

(c)

T 2
g
r

R

1
or, since

=
r
sin p

THE SUN.

181

(p being the sun’s horizontal parallax), we have finally

4π2
r

1

S = E

.

(d)

T 2
g
sin3 p

It will be noticed that in this expression the cube of the parallax
appears, and this is the reason why an uncertainty of one per cent in p
involves an uncertainty of three per cent in S.

In obtaining the mass of the sun it will be seen that we require as
data, T , the length of the sidereal year in seconds; the value of gravity, g

(which is derived from pendulum experiments); the radius of the earth, r

(deduced from geodetic surveys); and finally (and most difficult to get),
the sun’s parallax, p, or else, what comes to the same thing, the ratio
r .

R

279. The Sun’s Density.—This density1 as compared with
that of the earth is found by simply dividing its mass by its volume

(both as compared with the earth); that is, it equals the fraction

332 000 = 0.255,

1300 000
a little more than a quarter of the earth’s density. To get its “specific
gravity” (i.e., density as compared with water ), we must multiply
this by 5.58, the earth’s mean specific gravity. This gives 1.41; that
is, the sun’s mean density is not 1 1 times that of water,—a most

2
significant result as bearing on its physical condition.

280. Superficial Gravity.—This is found by dividing its mass
by the square of its radius; that is,

332 000 ,

(109 1 )2

2
which equals 27.6. A body weighing one pound on the earth’s sur-
face would there weigh 27.6 lbs. A body would fall 444 feet in a
second, instead of sixteen feet, as here.

1The determination of the sun’s density does not necessarily involve its
parallax. Put ρ for the sun’s radius, and Ds for its density: also let De be
the earth’s mean density. Substitute in equation (c), and we have 4 πρ3Ds =

3

3

3
r

R
r

R
ρ

4 πr3De

, whence Ds = De

. But

= sin Σ, Σ

3
g
r
g
ρ

R
r

1
being the sun’s (angular) semi-diameter. Hence, finally, Ds = De

.

g
sin3 Σ

THE SUN.

182

281. The Sun’s Rotation.—The sun’s surface often shows
spots upon it, which pass across the disc from east to west. These
are evidently attached to its surface, and not bodies circling around
the sun at a distance above it, as was imagined by some early as-
tronomers, because, as Galileo early demonstrated, they continue in
sight just as long as the time during which they are invisible; which
would not be the case if they were at any considerable elevation.

Period of Rotation.—The average time occupied by a spot in
passing around the sun and returning to the same position again
is 27.25 days,—average because different spots show considerable
differences in this respect. This interval, however, is not the true
time of solar rotation, but the synodic, since the earth advances in
the interval of a revolution so that the sun has to turn on its axis
a little farther each time to bring the spot again into conjunction
with the earth. The equation by which the true period is deduced
from the synodic is the same as in the case of the moon (Art. 232),
viz.:

1

1

1

−

=

,

T

E

S

T being the true period of the sun’s rotation, E the length of the
year, and S the observed synodic rotation;

1

1

1
whence

=

+

,

T

27.25

365.25
which gives T = 25d.35. Different observers get slightly different
results. Carrington finds 25d.38; Spoerer, 25d.23.

282. Position of the Sun’s Axis.—On watching the spots
with care as they cross the disc, it appears that they usually describe
paths more or less oval, showing that the sun’s axis is inclined to
the ecliptic. Twice a year, however, the paths become straight, at
the times when the earth is in the plane of the sun’s rotation. These
dates are about June 3 and Dec. 5.

The ascending node of the sun’s equator is in celestial longitude

73◦ 40 (Carrington), and the inclination of its equator to the plane of
the ecliptic is 7◦ 15 . Its inclination to the plane of the terrestrial equator
is 26◦ 25 . The position of the point in the sky towards which the sun’s
pole is directed is in right ascension 18h 44m, declination +64◦, almost
exactly half-way between the bright star α Lyræ and the Pole Star.

THE SUN.

183

283. Peculiar Law of the Sun’s Rotation.—Equatorial Ac-
celeration. The earth rotates as a whole, every point on its surface
making its diurnal revolution in the same time; so also with the
moon and with the planet Mars. Of course it is necessarily so with
any solid globe. But this is not the case with the sun. It was no-
ticed quite early that the different spots give different results for the
rotation period, but the researches of Carrington about thirty years
ago first brought out the fact that the differences follow a regular
law, showing that at the solar equator the time of rotation is less
than on either side of it. Thus spots near the sun’s equator give

T = 25 days; at solar latitude 20◦, T = 25.75 days; at solar latitude

30◦, T = 26.5 days; at solar latitude 40◦, T = 27 days. The time
of rotation in latitude 40◦ is fully two days longer than at the solar
equator; but we are unable to follow the law further towards the
poles, because the spots are rarely found beyond the parallels of

45◦ on each side of the equator, and there are no well-defined mark-
ings between this point and the poles by which we can accurately
determine the motion.

284. Various formulæ have been proposed to represent this law of
rotation. Carrington gives for the daily motion of a spot X = 865 −

7

165 × sin 4 l, l being the solar latitude of the spot. Faye, from the same
observations, considering that the exponent 7 could have no physical

4
justification, deduced X = 862 − 186 × sin2 l, which agrees almost as
well with the observations. Still other formulæ have been deduced by

Spoerer, Zöllner, and Tisserand, all giving substantially the same results.

The law, in any case, is simply empirical ; that is, it is deduced
from the observations, without being based upon any satisfactory
physical explanation, for no such explanation of this strange equa-
torial acceleration has yet been found. Probably it has its origin
somehow in the effects produced by the outpour of heat from the
sun’s surface; still, just how such a result should follow in the case
of a cooling globe, of which the particles are free to move among
each other, is not yet evident.

285. It is possible that the spots move on the surface of the sun,
changing their places just as do clouds or railroad trains upon the sur-
face of the earth, so that their motion does not represent the sun’s true
rotation. This however, as we shall see later, is hardly probable, and
if it were the case, it would still be no less difficult to account for this
systematic difference in their behavior at the solar equator and in the

THE SUN.

184
higher latitudes.

It has been suggested that the spots may be
due to the fall of matter from a considerable el-
evation above the sun’s surface, matter which
has remained at that elevation for some time,
and acquired a corresponding velocity of rota-
tion due to that elevation. It can be shown that
if the matter forming the spots had thus fallen
from an elevation of about 20,000 miles, it would

Fig. 91.—Telescope
account for their apparent acceleration. Matter
and Screen.

so falling would have an apparent eastward mo-
tion, just as do bodies on the earth when falling from the summit of
a tower (Art. 138). From this point of view it is very interesting to
inquire whether the minuter markings upon the sun’s surface, such as
the “granules,” to be spoken of very soon, do, or do not, possess the
same rate of motion as the spots. There is no decisive observational1
evidence at present that they do not. But the subject is an extremely
difficult one; and yet important, because the solution of the problem of
the sun’s equatorial acceleration will probably throw much light upon its
real constitution.

286. The Phenomena of the Sun’s Surface.—In order to
study the sun with the telescope it is necessary to be provided with
some special forms of apparatus. Its heat and light are so intense
that it is impossible to look directly at it, as we do at the moon.

A very convenient method of exhibiting the sun to a number of
persons at once is simply to attach to the telescope a frame carrying
a screen of white paper at a distance of a foot or more from the eye-
piece, as shown in Fig. 91. On pointing the instrument to the sun
and properly adjusting the focus, a distinct image is formed on the
screen, which shows the main features very fairly. It is, however,
much more satisfactory to look at it directly, with a proper eye-

1Mr. Crew has recently made at Baltimore, under the direction of Professor

Rowland, an extensive series of observations upon the displacement of the lines
of the spectrum at the eastern and western limbs of the sun. This displacement,
which is very slight, is due, according to Doppler’s principle (Art. 321), to the rotation of the sun; and Mr. Crew’s results, so far as they can be considered
decisive, go to show that the absorbing layer of gases by which the Fraunhofer
lines are formed does not behave like the sun spots, but is slightly retarded at the
sun’s equator. The observations are so delicate, however, that the conclusion,
though made very probable, can hardly be considered to be absolutely proved
beyond question.

THE SUN.

185
piece. With a small telescope, not more than two and a half or
three inches in diameter, a mere dark glass between the eye-piece
and the eye can be used, but this dark glass soon becomes very hot,
and is apt to crack. With larger instruments, it is necessary to use
eye-pieces especially designed for the purpose and known as solar
eye-pieces or helioscopes.

The simplest of them, and a very good one
for ordinary purposes, is one known as Her-
schel’s, in which the sun’s rays are reflected
at right angles by a plane of unsilvered glass

(Fig. 92).

This reflector is made either of a
prismatic form or concave, in order that the re-
flection from the back surface may not interfere
with that from the front. About nine-tenths of
the light passes through this reflector, and is
allowed to pass out uselessly through the open
end of the tube. The remaining tenth is sent

Fig. 92.—Herschel

Eye-piece.

through the eye-piece, and though still too in-
tense for the eye to endure, it requires only a
comparatively thin shade of neutral-tinted glass to reduce it sufficiently,
and in this case the shade does not become uncomfortably heated. It
is well to have the shade-glass made wedge-shaped,—thinner at one end
than at the other—so that one can choose the particular thickness which
is best adapted to the magnifying power employed.

287. The polarizing eye-pieces are still better when well made. In
these the light is reflected twice at plane surfaces of glass at the “angle of
polarization” (Physics, p. 480), and is then received on a second pair of
reflectors of black glass. When the upper pair of reflectors is in either of
the two positions shown in Fig. 93, a strong beam of light is received at

C,—too strong for the eye to bear, although more than ninety per cent
of it has already been rejected; but by simply turning the box which
carries the upper reflectors one-quarter of a revolution around the line

BB as an axis, the light may be wholly extinguished; and any desired
gradation may be obtained by setting it at the proper angle, without the
use of a shade-glass.

288. It may be asked why it will not answer merely to “cap”

the object-glass, and so cut off part of the light, instead of rejecting
it after it has once been allowed to enter the telescope. It is because
of the fact, mentioned in Article 43, that the smaller the object-lens
of the telescope, the larger the image it makes of a luminous point,

THE SUN.

186
or the wider its image of a sharp line. To cut down the aperture,
therefore, is to sacrifice the definition of delicate details. With a low
power there is no objection to reducing the amount of heat admitted
into the telescope tube in that way, but with the higher powers the
whole aperture should always be used.

289. Photography.—In the study
of the sun’s surface photography is for
some purposes very advantageous and
much used. The instrument must have
a special object-glass (Article 42), with
an apparatus for the quick exposure
of plates. Such instruments are called
photo-heliographs, and with them pho-
tographs of the sun are made daily at
numerous observatories.

The neces-
sary exposure varies from 1 to 1 of

500

10
a second, in different cases. The pic-
tures made by these instruments are
usually from two inches up to eight or
ten inches in diameter, and some of

Fig. 93.—Polarizing

Janssen’s, made at Meudon, bear en-

Helioscope.

larging up to forty inches in diameter.

Photographs have the advantage of freedom from prejudice and pre-
possession on the part of the observer; but they take no advantage
of the instants of fine seeing. They represent the surface as it hap-
pened to be at the moment when the plate was uncovered.

290. The study of the sun has become so important from a scientific
point of view that several observatories have recently been established
mainly for that purpose, though most of them connect with it that of
other topics in astronomical physics. The two most important of these
solar or astro-physical observatories, are the observatory at Meudon and
the so-called “Sonnenwarte” at Potsdam. There ought to be one in this
country.

291. General Views.—Before passing to a discussion of the
details of the different solar phenomena, it will be well to give a
very brief summary of the objects and topics to be considered.

1. The photosphere; i.e., the luminous surface of the sun directly
visible to our telescopes. It is probably a sheet of luminous clouds

THE SUN.

187
formed by condensation into little drops and crystals (like the water-
drops and ice-crystals in our terrestrial clouds) of certain substances
which within the central mass of the sun exist in a gaseous form,
but are cooled at its surface below the temperature necessary for
their condensation; perhaps such substances as carbon, boron, and
silicon. The granules, faculæ, and spots are all phenomena in this
photosphere.

2. The so-called “reversing layer ” is a stratum of unknown thick-
ness, but probably shallow, just above the photosphere, containing
the vapors of many of the familiar terrestrial elements; of which
the presence, and to some extent their physical condition, can be
investigated by means of the spectroscope.

3. Above the photosphere, in-
terpenetrating the atmosphere of
vapors just spoken of, and per-
haps indistinguishable from it, is
an envelope of permanent gases;
that is, gases which, under the
solar conditions, cannot be con-
densed into clouds of solid or liq-
uid particles.

Among them hy-
drogen is most conspicuous. This
envelope is the so-called Chromo-
sphere; and from it the promi-
nences of various kinds rise, some-
times to the height of hundreds of
thousands of miles. These beau-
tiful objects are best seen at total
eclipses of the sun, but to a certain
extent they can also be studied at
any time by the help of a spectro-
scope.

Fig. 94.

Constitution of the Sun. From “The

4. Higher yet rises the myste-

Sun,” by permission of the
rious Corona, of material still less

Publishers.

dense, and so far observable only
during total eclipses of the sun.

Fig. 94 shows the relative positions of these different elements
of the solar constitution.

5. A fifth subject deals with the measurement of the sun’s light

THE SUN.

188
and the relative brightness of different parts of the solar surface.

6. Another most interesting and important topic relates to the
amount of heat radiated by the sun,—the sun’s probable temperature
and the mechanism by which its heat-supply is maintained.

Fig. 95.

The Great Sun Spot of September, 1870, and the Structure of the

Photosphere. From a Drawing by Professor Langley. From “The New

Astronomy,” by permission of the Publishers.

292. The Photosphere.—The sun’s visible surface is called
the photosphere, and when studied under favorable atmospheric
conditions, with rather a low magnifying power, it looks like rough
drawing-paper. With higher powers it is seen to be, as shown in

Fig. 95, made up of a comparatively darkish background sprinkled
over with grains, or “nodules,” as Herschel called them, of some-
thing much more brilliant,—like snow-flakes on gray cloth, accord-
ing to Langley. These are from 400 to 600 miles across, and in the
finest seeing are themselves resolved into more minute “granules.”

For the most part, these nodules are about as broad as they are
long, though of irregular form; but here and there, especially in the
neighborhood of the spots, they are drawn out into long streaks.

Nasmyth seems first to have observed this structure, and called the
filaments “willow leaves.” Secchi called them “rice grains.” Accord-
ing to Huggins they were “dots”; and there was for a long time a

THE SUN.

189
pretty lively controversy as to their true form. Their shape, how-
ever, unquestionably varies very much in different parts of the sur-
face and under different circumstances. They are probably luminous
clouds floating in a less luminous atmosphere.

Near the edge the photosphere appears generally much less bril-
liant; but certain bright streaks called “faculæ” (from fax, a torch),
which though visible are not very obvious at points further from
the limb, become there conspicuous. These faculæ are elevations,—
masses of the same material as the rest of the photosphere but el-
evated above the general level and intensified in brightness. When
one of them passes off the edge of the sun, it is sometimes seen as a
little projection. They are most abundant near the sun-spots, and
they are more conspicuous near the edge of the disc, as shown in

Fig. 96, because the sun’s surface is overlaid by a gaseous atmo-
sphere which absorbs more of the light there than it does near the
centre, and these faculæ push up through it like mountains.

Fig. 96.—Faculæ at Edge of the Sun. (De La Rue.)

293. The Sun Spots.—Whenever these are present upon the
sun’s surface, they are the most conspicuous objects to be seen upon
it. The appearance of a normal sun spot, Fig. 97, fully formed and
not yet beginning to break up, is that of a dark central “umbra,”

more or less nearly circular, with a fringing “penumbra,” composed
of filaments directed radially. The umbra itself is not uniformly
dark throughout, but is overlaid with filmy clouds which require
a good telescope and helioscope to make them visible. Usually,
also, in the umbra there are several round and very black spots,
which are sometimes called “nucleoli,” but are often referred to us

“Dawes’ holes,” after the name of their first discoverer. But while

THE SUN.

190
this is the appearance of what may be taken as a normal spot, very
few are strictly normal. Most of them are more or less irregular in
form. They are often gathered in groups with a common penumbra,
and partly covered by brilliant “bridges” extending across from the
outside photosphere. Often the umbra is out of the centre of the
penumbra, or has a penumbra only on one side, and the penumbral
filaments, instead of being strictly radial, are frequently distorted in
every conceivable way. In fact, the normal spots form a very small
proportion of the whole number.

Fig. 97.—A Normal Sun Spot. (Secchi; modified.)

The darkest portions of the umbra are dark only by contrast.

Photometric observations (by Langley) show that even the nucleus
gives at least one per cent as much light as a corresponding area of
the photosphere; that is to say, as we shall see hereafter, the darkest
portion of a sun spot is brighter than a calcium light.

294. The spots are unquestionably cavities or depressions in
the photosphere, filled with gases and vapors which are cooler than
the surrounding portions, and therefore absorb a considerable pro-
portion of light. The fact that they are cavities is shown by the
change in the appearance of a spot as it approaches the edge of the
disc. When a normal spot is near the centre of the disc, the nucleus
is nearly central. As it approaches the edge, the penumbra becomes
wider on the outer edge and narrower on the inner, and just before
the spot disappears around the limb of the sun, the penumbra on
the inner edge entirely disappears,—the appearance being precisely
such as would be shown by a saucer-shaped cavity in the surface of
a globe, the bottom of the cavity being painted black to represent
the umbra, and the sloping sides gray for the penumbra. Fig. 98

THE SUN.

191
represents the phenomena in a schematic way. Observations upon a
single spot would hardly be sufficient to substantiate this, because
the spots are so irregular in their form; but by observing the be-
havior of several hundred of them the truth comes out quite clearly.

Occasionally, when a very large spot passes off the sun’s limb, the
depression can be seen with the telescope.

Fig. 98.—Sun Spots as Cavities.

The fact was first discovered by Wilson of Glasgow something
more than a hundred years ago. Previously it had very commonly
been supposed that the spots were elevated above the general sur-
face of the sun, and the idea still survives in certain quarters, though
certainly incorrect.

295. The penumbra is usually composed of “thatch-straws,” or
long drawn-out granules of photospheric matter, which, as has been
said, converge in a general way towards the centre of the spot. At
the inner edge the penumbra, from the convergence of these fila-
ments, is usually brighter than the outer. The inner ends of the
filaments are generally club-formed; but sometimes they are drawn
out into fine points, which seem to curve downward into the umbra
like the rushes over a pool of water. The outer edge of the penumbra
is usually pretty definite, and the penumbra there is darker. Around
the spot the photosphere is much disturbed and elevated into fac-
ulæ, which sometimes radiate outward from the spot like streams
of lava from a crater, though, of course, they are really nothing of
the sort.

296. Dimensions of Sun Spots.—The diameter of the umbra
of a sun spot ranges all the way from 500 to 1000 miles in the
case of a very small one, to 50,000 or 60,000 miles in the case of
the larger ones. The penumbra surrounding a group of spots is
sometimes 150,000 miles across, though that would be rather an
exceptional size. Not infrequently sun spots are large enough to be
seen by the naked eye, and they have been often so seen at sunset

THE SUN.

192
or through a fog. The depth by which the umbra is depressed below
the general surface of the photosphere is very difficult to determine,
but according to Faye, Carrington, and others, it seldom exceeds

2500 miles, and more often is between 500 and 1500.

297. Development and Changes of Form.—Generally the
origin of a sun spot fails to be observed. It begins from an insensible
point, and rapidly grows larger, the penumbra usually appearing
only after the nucleus is fairly developed.

If the disturbance which causes the spot is violent, the spot usu-
ally breaks up into several fragments, and these again into others
which tend to separate from each other. At each new disturbance
the forward portions of the group show a tendency to advance east-
ward on the sun’s surface, leaving behind them a trail of smaller
spots.

298. The “segmentation” of a spot, as Faye calls it, is usually ef-
fected by the formation of a “bridge,” or streak of brilliant light, which
projects itself across the penumbra and umbra from the outside photo-
sphere. These bridges are mere extensions of the surrounding faculæ,
and are often intensely bright.

Occasionally a spot shows a distinct cyclonic motion, the filaments
being drawn inward spirally; and in different members of the same group
of spots the cyclonic motions are not seldom in opposite directions.

When a spot at last vanishes it is usually by the rapid encroachment
of the photospheric matter, which, as Secchi expresses it, appears to

“fall pell-mell into the cavity,” completely burying it and leaving its
place covered by a group of faculæ. Figs. 99–104 (see page 193) show
the changes which took place in the great spot of September, 1870. They
are from photographs by Mr. Rutherfurd of New York, and are borrowed
from “The New Astronomy” of Professor Langley, through the courtesy
of his publishers.

299. Spots within 15◦ or 20◦ of the sun’s equator generally, on
the whole, drift a little towards it, while these in higher latitudes
drift away from it; but the motion is slight, and exceptions are
frequent.

In and around the spot itself the motion is usually inward to-
wards the centre, and downward at the centre. Not infrequently the
fragments at the inner end of the penumbral filaments appear to
draw off, move towards the centre of the spot, and then descend.

Occasionally, though seldom, the motion is vigorous enough to be
detected by the displacement of lines in the spectrum.

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193

Fig. 99.—Sept. 19.

Fig. 100.—Sept. 20.

Fig. 101.—Sept. 21.

Fig. 102.—Sept. 22.

Fig. 103.—Sept. 23.

Fig. 104.—Sept. 26.

The Great Sun Spot of 1870.

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194

300. Duration.—The duration of the spots is very various,
but, astronomically speaking, they are always short-lived phenom-
ena, sometimes lasting for only a few days, more frequently, perhaps,
for a month or two. In a single instance, a spot has been observed
through as many as eighteen successive revolutions of the sun.

301. Distribution.—It is a significant fact that the spots are
confined mostly to two zones of the sun’s surface between 5◦ and

40◦ of latitude north and south. A few are found near the equator,
none beyond the latitude of 45◦. Fig. 105 shows the distribution of
several thousand spots as observed by Carrington and Spörer.

Fig. 105.—Distribution of Sun Spots in Latitude.

Occasionally, what Trouvelot calls “veiled spots” are seen be-
yond the 45◦ limits—grayish patches surrounded by faculæ, which
look as if a dark mass were submerged below the surface and dimly
seen through a semi-transparent medium.

302. Theories as to the Nature of the Spots.—We first
mention (a) the theory of Sir William Herschel, because it still
finds place in certain text-books, though certainly incorrect. His
belief was that the spots were openings through two luminous strata,
which he supposed to surround the central globe of the sun. This
globe he supposed to be dark (and even habitable! ). The outer
stratum, the photosphere, was the brighter of the two, and the
opening in it the larger, while the inner shell between it and the solid
globe was of less luminous substance, and formed the penumbra. He
thought the opening through these might be caused by volcanoes
on the globe beneath.

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195

303. (b) Another theory, now abandoned, was proposed inde-
pendently both by Secchi and Faye about 1868. They supposed that
the spots were openings in the photosphere caused by the bursting
outward of the imprisoned gases underneath it; the photosphere at
that time being supposed to be liquid.

They explained the darkness of the centre of the spot by the fact
that a heated gas at a given temperature has a lower radiating power
and sends out much less light than a liquid surface, or than clouds formed
by the condensation of the same material at even a lower temperature.

This is true of gases at low pressure, but not of gases under great com-
pression, such as must be the case within the body of the sun. Besides,
if the gases possessed the small radiating power necessary to this theory,
they would also possess small absorbing power, and therefore would be
transparent; the inner side of the photosphere on the opposite side of the
sun would therefore be visible through the opening, so that the centre of
such an eruption would not be dark, but, if anything, brighter than the
general solar surface. Moreover, as we now know from the spectroscopic
evidence, the motion at the centre of a spot is inward, not outward.

304. (c) Faye more recently has proposed and now maintains
a theory which has numerous good points about it, and is accepted
by many, viz.: that the spots are analogous to storms on the earth,
being cyclones, due to the fact that the portions of the sun’s surface
near the equator make their revolution in a shorter time than these
in higher latitudes. This causes a relative drift in adjacent portions
of the photosphere, and according to him gives rise to vortices or
whirlpools like these in swiftly running water. The theory explains
the distribution of the spots (which abound precisely in the regions
where this relative drift is at the maximum) and many other facts,
such as their “segmentation.” According to it, however, all spots
should be cyclonic, and the spiral motion of all the spots in the
northern hemisphere should be clock-wise, while in the southern
hemisphere they should be counter-clock-wise. Now, as a matter
of fact, only a very few of the spots show such spiral motions, and
there is no such agreement in the general direction of the motion as
the theory requires.

Faye attempts to account for this by saying that we do not see the
vortex itself, but only the cloud of cooler materials which is drawn to-
gether by the down-rushing vortex, itself hidden beneath this cloud. Still,
it would seem that in such a case the cloud itself should gyrate. More-
over, the relative drift of the adjacent portions of the photosphere is too

THE SUN.

196
small to account for the phenomena satisfactorily. In the solar latitude
of 20◦ two points separated by 1 of the sun’s surface (123 miles) have a
relative daily drift of only about four and one-sixth miles, insufficient to
produce any sensible whirling.

305. (d ) Secchi’s later theory. He supposed the spots to be due
to eruptions from the inner portions of the sun’s surface, not in the
spot, however, but only near it; the spot itself being formed by the
settling down upon the photosphere of materials thrown out by the
eruption and cooled by their expansion and their motion through
the upper regions. We have, however, in fact, as a usual thing, not a
single eruption, but a ring of eruptions all around every large spot,
all of them converging their bombardment, so to speak, upon the
same centre,—a fact very difficult to explain if the spot originates
in the eruption, but not difficult to understand if the eruptions are
the result of the spot.

Perhaps the true explanation may be that when an eruption oc-
curs at any spot, the photosphere somewhere in the neighborhood
settles down in consequence of the diminution of the pressure be-
neath, thus forming a “sink,” so to speak, which is of course covered
by a greater depth of cooler vapors above, and so looks dark.

306. (e) Mr. Lockyer, in his recent work on the chemistry of
the sun, revives an old theory, first suggested by Sir John Herschel
and accepted by the late Professor Peirce, that the spots are not
formed by any action from within, but by cool matter descending
from above,—matter very likely of meteoric origin; but it is difficult
to see how the distribution of the spots with reference to the sun’s
equator can be accounted for in this way.

On the whole it is impossible to say that the problem of the
origin of sun spots is yet satisfactorily solved. There is no question
that sun spots are closely associated with eruptions from beneath;
but which is cause and which effect, or whether both are due to
some external action, remains undetermined.

307. Periodicity of Sun Spots.—In 1851 Schwabe of Dessau,
by the comparison of an extensive series of observations running over
nearly thirty years, showed that the sun spots are periodic, being at
times vastly more numerous than at others, with a roughly regular
recurrence every ten or eleven years. This had been surmised by

Horrebow more than a century before, though not proved.

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197

Subsequent study fully confirms this remarkable result of

Schwabe. Wolf of Zurich has collected all the observations discov-
erable and finds a pretty complete record back to 1610. From these
records is constructed the annexed diagram, Fig. 106. The ordinates
of the curve represent what Wolf calls his “relative numbers,”1 which
he has adopted as representing the spottedness.

Fig. 106.—Wolf’s Sun-Spot Numbers.

The average period is eleven and one-tenth years, but, as the
figure shows, the spot maxima are quite irregular, both in time and
as to the extent of spottedness. The last spot maximum occurred
in 1883–84 (a year or two behind time), and we are now (1888)

approaching a minimum. During a maximum the surface of the sun
is never free from spots, from twenty-five to fifty being frequently
visible at once. During a minimum, on the other hand, weeks often
pass without the appearance of a single one.

308. Possible Cause of the Periodicity.—The cause of this
periodicity is not known.

It has been attempted to connect it with

1This “relative number” is formed in rather an arbitrary manner from the
observations which Wolf hunted up as the basis of his investigation. The formula
is, r (the relative number) − k(10g + f ), in which g is the number of groups
and isolated spots observed, f the total number of spots which can be counted
in these groups and singly, while k is a coefficient which depends upon the
observer and the size of his telescope: it is large for a small telescope and not
very persistent observer, and approaches unity the more likely the observer
may be supposed to have noted every sun spot that appeared during the time
covered by his observations.

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198
planetary action. Some things in the Kew statistics of the sun spots
look as if Venus, Mercury, and the Earth had something to do with it,
the sun’s surface being more spotted when these planets approach nearer;
but the evidence is insufficient, or at least needs to be supplemented by
further comparisons. Jupiter also has been suspected. His period is

11.86 years, which is not very different from the mean sun-spot period;
but an examination of the different spot maxima show that some of them
have occurred when he was near perihelion, and others when he was near
aphelion; and on the whole there is very little reason for supposing that
he has any considerable influence in the matter.

Sir John Herschel suggested that it might be due to streams of me-
teors moving in an oval orbit with a period of about eleven years, and
approaching so near at perihelion that numerous members of the mete-
oric group actually fall into the sun; but, as has been said before, the
distribution of the spots would seem to contradict the idea.

309. Terrestrial Influence of the Sun Spots.—One influ-
ence of the sun spots upon the earth is perfectly demonstrated.

When the spots are numerous, magnetic disturbances (the so-called
magnetic storms) are most numerous and violent upon the earth, a
fact not to be wondered at since violent disturbances upon the sun’s
surface have been in many individual cases immediately followed by
magnetic storms, with a brilliant exhibition of the Aurora Borealis.

The nature and mechanism of the connection is as yet unknown,
but of the fact there can be no question. The dotted lines in the
figure of the sun-spot periodicity (Fig. 106) represent the magnetic
storminess of the earth at the indicated dates; and the correspon-
dence between these curves and the curve of spottedness makes it
impossible to doubt the connection.

310. It has been attempted, also, to show that greater or less
disturbance of the sun’s surface, as indicated by the greater fre-
quency of the sun’s spots, is accompanied by effects upon the me-
teorology of the earth, upon its temperature, barometric pressure,
storminess, and the amount of rain-fall. The researches of Mr. Mel-
drum of Mauritius with respect to the cyclones in the Indian Ocean
appear to bear out the conclusion that there may be some such
connection in that case, but the general results are by no means
decisive. In some parts of the earth the rain-fall seems to be greater
during a spot maximum; in others, less.

As to the temperature, it is still uncertain whether it is higher

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199
or lower at the time of a spot maximum. The spots themselves are
cooler (as Henry, Secchi, and Langley have shown) than the general
surface of the photosphere; but their extent is never sufficient to
reduce the amount of heat radiated from the sun by us much as

1

1000
part. On the other hand, when the spots are most numerous, the
generally disturbed condition of the photosphere would, as Langley
has shown, necessarily be accompanied by an increased radiation.

Dr. Gould considers that the meteorological records in the Ar-
gentine Republic between 1875 and 1885 show an indubitable con-
nection between the wind currents and the number of sun spots.

But the demonstration of such a relation really requires observa-
tions running through several spot periods. On the whole, it is
now quite certain that whatever influence the sun spots exert upon
terrestrial meteorology is very slight, if it exists at all.

THE SOLAR SPECTRUM AND ITS REVELATIONS.

311. About 1860 the spectroscope appeared in the field as a
new and powerful instrument of astronomical research, at once re-
solving many problems as to the nature and constitution of the
heavenly bodies which before had not seemed to be even open to
investigation.

The essential part of the apparatus is either a prism or train of
prisms, or else a diffraction grating,1 which is capable of perform-
ing the same office of dispersing—that is, of spreading and sending
in different directions—the light rays of different wave-lengths. If,
with such a “dispersion piece,” as it may be called (either prism
or grating), one looks at a distant point of light, as a star, he will
see instead of a point a long streak of light, red at one end and
violet at the other. If the object looked at be not a point, but a
line of light parallel to the edge of the prism or to the lines of the
grating, then, instead of a mere colored streak without width, one
gets a spectrum, a colored band of light, which may show markings
that will give the observer most valuable information. (Physics,
pp. 458–460.) For convenience’ sake it is usual to form this line of
light by admitting the light through a narrow “slit,” which is at one

1The grating is merely a piece of glass or speculum metal, ruled with many
thousand straight, equidistant lines, from 5000 to 20,000 in the inch. Usually
the surface before ruling is accurately plane, but for some purposes the concave
gratings, originated by Professor Rowland, are preferable.

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200
end of a tube having at the other end an achromatic object-glass
at such a distance that the slit is in its principal focus. This tube
with slit and lens constitutes the “collimator,” so called because it
is precisely the same as the instrument used in connection with the
transit instrument to adjust its line of collimation (Article 60).

Instead of looking at the spectrum with the naked eye, however,
it is better in most cases to use a small telescope; called the “view-
telescope,” to distinguish it from the large telescope, to which the
spectroscope is often attached.

312. Construction of the Spectroscope.—The instrument,
therefore, as usually constructed, and shown in Fig. 107, consists of
three parts,—collimator, dispersion-piece, and view-telescope; but
in the direct-vision spectroscope, shown in the figure, the view-
telescope is omitted. If the slit, S, be illuminated by strictly homo-
geneous light all of one wave-length, say yellow, the “real image”

of the slit will be found at Y . If at the same time light of a differ-
ent wave-length be also admitted, say red, a second image will be
formed at R, and the observer will see a spectrum with two “bright
lines,” the lines being really nothing more than images of the slit. If
light from a candle be admitted, there will be an infinite number of
these slit-images, close packed, like the pickets in a fence, without
interval or break, and we then get a continuous spectrum; but if we
look at sunlight or moonlight, we shall find a spectrum continuous
in the main, but crossed by numerous dark lines, as if some of the

“pickets” had been knocked off, leaving gaps.

Fig. 107.—Different Forms of Spectroscope.

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201

313. Integrating and Analyzing Spectroscope.—If we
simply direct the collimator of a spectroscope towards a distant
luminous object, every part of the slit receives light from every part
of the object, so that in this case every elementary streak of the
spectrum is a spectrum of the entire body, without distinction of
parts. A spectroscope used in this way is said to be an integrating
instrument.

If, however, we interpose a lens (the object-glass of a telescope)

between the luminous object and the slit, so as to have in the plane
of the slit a distinct, real image of the object, then the top of the
slit, for instance, will be illuminated wholly by light from one part
of the object, the middle of it by light from another point, and
the bottom by light from still a third. The spectrum formed by
the top of the slit belongs, then, to the light from that particular
point of the object whose image falls upon that part of the slit;
and so of the rest. We thus separate the spectra of the different
parts of the object, and so optically analyze it. An instrument thus
used is spoken of as an “analyzing spectroscope.” The combined
instrument formed by attaching a spectroscope to a large telescope
for the spectroscopic observation of the heavenly bodies has been
called by Mr. Lockyer a “telespectroscope.” Fig. 108 shows the
apparatus used by the writer for some years at Dartmouth College.

For solar purposes a grating spectroscope is generally better than
a prismatic, being less complicated and more compact for a given
power.

Fig. 108.—The Telespectroscope.

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202

314. Principles upon which Spectrum Analysis depends.

—These, substantially as announced by Kirchoff in 1858, are the
three following:—

1st, A continuous spectrum is given by every incandescent body,
the molecules of which so interfere with each other as to prevent
their free, independent, luminous vibration; that is, by bodies which
are either solid or liquid, or, if gaseous, are under high pressure.

2d, The spectrum of a gaseous element, under low pressure, is
discontinuous, made up of bright lines, and these lines are charac-
teristic; that is, the same substance under similar conditions always
gives the same set of lines, and generally does so even under widely
different conditions.

Fig. 109.—Reversal of the Spectrum.

3d, A gaseous substance absorbs from white light passing
through it precisely these rays of which its own spectrum consists.

The spectrum of white light which has been transmitted through it
then exhibits a “reversed ” spectrum of the gas; that is, one which
shows dark lines instead of the characteristic bright lines.

Fig. 109 illustrates this principle. Suppose that in front of the
slit of the spectroscope we place a spirit lamp with a little carbonate
of soda and some salt of thallium upon the wick. We shall then get
a spectrum showing the two yellow lines of sodium and the green
line of thallium, bright. If now the lime-light be started right behind
the lamp flame, we shall at once get the effect shown in the lower
figure,—a continuous spectrum crossed by black lines just where the

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203
bright lines were before. Insert a screen between the lamp flame and
the lime, and the dark lines instantly show bright again.

315. Chemical Constituents of
the Sun.—By taking advantage of these
principles we can detect the presence of a
large number of well-known terrestrial el-
ements in the sun. The solar spectrum is
crossed by dark lines, which, with an in-
strument of high dispersion, number sev- Fig. 110.—The Comparison

Prism.

eral thousand, and by proper arrange-
ments it is possible to identify among
these lines many which are due to the presence in the sun’s lower
atmosphere of known terrestrial elements in the state of vapor. To
effect the comparison necessary for this purpose, the spectroscope
must be so arranged that the observer can have before him, side
by side, the spectrum of sunlight and that of the substance to be
tested. In order to do this, half of the slit is fitted with a little

“comparison prism,” so-called (Fig. 110), which reflects into it the
light from the sun, while the other half of the slit receives directly
the light of some flame or electric spark. On looking into the eye-
piece of the spectroscope, the observer will then see a spectrum,
the lower half of which, for instance, is made by sunlight, while the
upper half is made by light coming from an electric spark between
two metal points, say of iron.

Fig. 111.

Comparison of the Solar Spectrum with that of Iron. From a Negative by

Prof. Trowbridge.

Photography may also be most effectively used in these com-
parisons instead of the eye.

Fig. 111 is a rather unsatisfactory
reproduction, on a reduced scale, of a negative recently made by

Professor Trowbridge at Cambridge. The lower half is the violet
portion of the spectrum of the sun, and the upper half that of the
vapor of iron in an electric arc. The reader can see for himself with
what absolute certainty such a photograph indicates the presence of

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204
iron in the solar atmosphere. A few of the lines in the photograph
which do not show corresponding lines in the solar spectrum are
due to impurities in the carbon, and not to iron.

316. As the result of such comparisons we have the following
list of twelve elements, which are certainly known to exist in the
sun, viz.:—
hydrogen,
cobalt,
iron,
chromium,
titanium,
barium,
calcium,
sodium,
manganese,
magnesium,
nickel,
platinum.

There are evidences, perhaps not quite conclusive, of the presence
of nearly as many more, viz.:—
copper,
uranium,
palladium,
aluminium,
vanadium,
cadmium,
molybdenum,
carbon,
lead.

As to carbon, however, the spectrum is so peculiar, consisting of
bands rather than lines, that it is very difficult to be sure, but the
tendency of the latest investigations (of Rowland and Hutchins) is
to establish its right to a place on the list.

The more recent researches have thrown much doubt on the presence
of several substances which, a few years ago, were usually included in the
list, as, for instance, strontium and cerium. It has been generally admit-
ted also that the photographs of Dr. Henry Draper had demonstrated
the presence of oxygen in the sun, represented in the solar spectrum, not
by dark lines like other elements, but by certain wide, bright bands. The
latest work, while it does not absolutely refute Dr. Draper’s conclusion,
appears however to turn the balance of evidence the other way.

317. It will be noticed that all the bodies named in the list,
carbon alone excepted, are metals (chemically hydrogen is a metal),
and that a multitude of the most important terrestrial elements fail
to appear; oxygen (?), nitrogen, chlorine, bromine, iodine, sulphur,
phosphorus, silicon, and boron are all missing. We must be cautious,
however, as to negative conclusions. It is quite conceivable that the
spectra of these bodies under solar conditions may be so different
from their spectra as presented in our laboratories that we cannot

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205
recognize them; for it is now quite certain that some substances,
nitrogen, for instance, under different conditions, give two or more
widely different spectra.

Among the many thousand lines of the solar spectrum only a
few hundred are so far identified.

318. Mr. Lockyer’s Views.—Mr. Lockyer thinks it more
probable that the missing substances are not truly elementary, but
are decomposed or “dissociated” on the sun by the intense heat,
and so do not exist there, but are replaced by their components;
he believes, in fact, that none of our so-called elements are really
elementary, but that all are decomposable, and, to some extent ac-
tually decomposed in the sun and stars, and some of them by the
electric spark in our own laboratories. Granting this, a crowd of
interesting and remarkable spectroscopic facts find easy explana-
tion. At the same time the hypothesis is encumbered with great
difficulties and has not yet been finally accepted by physicists and
chemists. For a full statement of his views the reader is referred to
his “Chemistry of the Sun.”

319. The Reversing Layer.—According to Kirchoff’s theory
the dark lines are formed by the passing of light from the minute
solid and liquid particles of which the photospheric clouds are sup-
posed to be formed, through vapors containing the substances which
we recognize in the solar spectrum. If this be so, the spectrum of
the gaseous envelope, which by its absorption forms the dark lines,
should by itself show a spectrum of corresponding bright lines. The
opportunities are of course rare when it is possible to obtain the
spectrum of this gas-stratum alone by itself; but at the time of a
total eclipse, at the moment when the sun’s disc has just been ob-
scured by the moon, and the sun’s atmosphere is still visible beyond
the moon’s limb, if the slit of the spectroscope be carefully adjusted
to the proper point, the observer ought to see this bright-line spec-
trum. The author succeeded in making this very observation at
the Spanish eclipse of 1870. The lines of the solar spectrum, which
up to the final obscuration of the sun had remained dark as usual

(with the exception of a few belonging to the spectrum of the chro-
mosphere), were suddenly “reversed,” and the whole length of the
spectrum was filled with brilliant-colored lines, which flashed out
quickly and then gradually faded away, disappearing in about two
seconds,—a most beautiful thing to see. Substantially the same

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206
thing has since then been several times observed.

320. The natural interpretation of this phenomenon is, that the
formation of the dark lines in the solar spectrum is mainly, at least,
produced by a very thin layer close down to the photosphere, since the
moon’s motion in two seconds would cover a thickness of only about 500
miles. It was not possible, however, to be certain that all the dark lines
were reversed, and in this uncertainty lies the possibility of a different
interpretation. Mr. Lockyer doubts the existence of any such thin stra-
tum. According to his views the solar atmosphere is very extensive, and
these lines of iron, which correspond to the more complex combinations
of its constituents, are formed only in the regions of lower temperature,
high up in the sun’s atmosphere. They should appear early at the time
of an eclipse and last long, but not be very bright. Those due to the
constituents of iron which are found only close down to the solar sur-
face should be short and bright; and he thinks that the numerous bright
lines observed under the conditions stated are due to such substances
only. Observation needs to be directed to the special point to determine
whether all of the dark lines are reversed at the edge of the sun, or only
a few ; and if so, what ones.

321.

Sun-Spot

Spectrum.—

This is like the general solar spec-
trum, except that certain lines are
much widened, while certain others are
thinned, and sometimes the lines of hy-
drogen become bright. It is to be no-
ticed that by far the larger proportion

Fig. 112.

The C line in the Spectrum of
of the dark lines of any given substance
a Sun Spot, Sept. 22, 1870.

are not affected at all in the spot spec-
trum, but only a certain few of them, a point which Mr. Lockyer
considers very important. Not infrequently it happens that certain
lines of the spectrum are crooked and broken in connection with sun
spots, as shown by Fig. 112. Such phenomena are caused, according
to Doppler’s principle,1 by the swift motion of matter towards or

1Doppler’s principle is this: that when we are approaching, or approached
by, a body which is emitting regular vibrations, then the number of waves
received by us in a second is increased, and their wave-length correspondingly
diminished ; and vice versa when the distance of the vibrating body is increasing.

Thus the pitch of a musical tone rises while we are approaching the sounding
body, and falls as we recede; in just the same way the “refrangibility” of the

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207
from the observer. In the particular case shown in the figure, hydro-
gen is the substance, and the greatest motion indicated was towards
the observer at the rate of about 300 miles a second—an unusual
velocity. These effects are most noticeable, not in the spots, but
near them, usually just at the outer edge of the penumbra.

The dark and apparently continuous spectrum which is due to the
nucleus of a sun spot is not truly continuous, but under high dispersion is
resolved into a range of extremely fine, close-packed, dark lines, separated
by narrow spaces. At least this is so in the green and blue portions of
the spectrum; it is more difficult to make out this structure in the yellow
and red. It appears to indicate that the absorbing medium which fills
the hollow of a sun spot is gaseous, and not composed of precipitated
particles like smoke, as has been suggested.

322. The Chromosphere.—The chromosphere is a region of
the sun’s gaseous envelope which lies close above the photosphere,
the “reversing layer,” if it exists at all, being only the most dense
and hottest part of it. The chromosphere is so called, because as
seen for an instant, during a total solar eclipse, it is of a bright
scarlet color, the color being due to the hydrogen which is its main
constituent. It is from 5000 to 10,000 miles in thickness, and in
structure is very like a sheet of scarlet flame, not being composed
of horizontal sheets, but of (approximately) upright filaments. Its
appearance has been compared very accurately to that of “a prairie
on fire”; but the student must carefully guard against the idea that
there is any real “burning” in the case; i.e., any process of combina-
tion between hydrogen and some other substance. The temperature
rays, say of hydrogen, emanating from the sun is increased (the wave-length
being shortened) whenever we are approaching it with a speed which bears a
sensible ratio to the velocity of light. Calling λ the wave-length of the ray
when the observer and the luminous object are relatively at rest, and λ the
wave-length as affected by their relative motion; putting V also for the velocity
of light (about 186,330 miles per second), and s for the speed with which the
observer and source of light approach each other, we have

V
λ = λ

.

V + s

[If the distance is increasing instead of diminishing, the denominator will be

(V − s).] With the most powerful spectroscopes motions of from one to two
miles per second along the line of sight can thus be detected.

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208
is altogether too high for any formation of hydrogen compounds at
the sun’s surface.

323. The Prominences.—At a total eclipse, after the total-
ity has fairly set in, there are usually to be seen at the edge of
the moon’s disc a number of scarlet, star-like objects, which in the
telescope appear as beautiful, fiery clouds of various form and size.

These are the so-called “prominences,” which very non-committal
name was given while it was still doubtful whether they were so-
lar or lunar. Photography, in 1860, proved that they really belong
to the sun, for the photographs taken during the totality showed
that the moon obviously moves over them, covering those upon the
eastern limb, and uncovering those upon the western.

In 1868, during the observation of the eclipse which occurred
that year in India, the spectroscope showed them to be gaseous, and
that their main constituent is hydrogen. Their spectrum contains
also a conspicuous yellow line known as the D3 line, because it is
very near the two “D” lines of sodium. This line is not due to
hydrogen, and is not identified with any known element; but an
element has been assumed for it and called provisionally “helium.”

In connection with this eclipse, Janssen, who observed it in In-
dia, found that the lines of the prominence spectrum were so bright
that he was able to observe them the next day after the eclipse in
full sunlight; and he also found that by a proper management of
his instrument he could study the form and behavior of the promi-
nences nearly as well without an eclipse as during one. Lockyer,
in England, some time earlier had come to similar conclusions from
theoretical grounds, and he practically perfected his discovery a few
weeks later than Janssen, although without knowledge of what he
had done. By a remarkable but accidental coincidence their discov-
eries were communicated to the French Academy on the same day;
and in their honor the French have struck a medal bearing their
united effigies.

324. How the Spectroscope makes the Prominences

Visible.—The only reason we cannot see the prominences at any
time is on account of the bright illumination of our own atmosphere.

We can screen off the direct light of the sun; but we cannot screen
off the reflected sunlight coming from the air which is directly be-
tween us and the prominences themselves; a light so brilliant that
the prominences cannot be seen through it without some kind of

THE SUN.

209
aid.

The spectrum of this air-light is, of course,
just the same as that of the sun—a continu-
ous spectrum with the same dark lines upon it.

When, therefore, we arrange the apparatus as
indicated in Fig. 113, pointing the telescope so
that the image of the sun’s limb just touches
the slit of the spectroscope, then, if there is

Fig. 113.

a prominence at that point, we shall have in

Spectroscope Slit
adjusted for
our spectroscope two spectra superposed upon

Observation of the
each other; namely, the spectrum of the air-

Prominences.

illumination and that of the prominence. The
latter is a spectrum of bright lines, or, if the slit is opened a little, of
bright images of whatever part of the prominence may fall within
the edges of the slit. Now, the brightness of these images is not
affected by any increase of dispersion in the spectroscope. Increase1
of dispersion merely sets these images farther apart, without mak-
ing them fainter. The spectrum of the aerial illumination, on the
other hand, is made very faint by its extension; and, moreover, it
presents dark lines (or spaces when the slit is opened) precisely at
the points where the bright images of the prominences fall.

A spectroscope of dispersive power
sufficient to divide the two E lines,
attached to a telescope of four or five
inches aperture, gives a very satisfac-
tory view of these beautiful objects;
the red image corresponds to the C
line, and is by far the best for such ob-
servations, though the D3 line or the

F line can also be used. When the
instrument is properly adjusted, the
slit opened a little, and the image of
the sun’s limb brought exactly to the

Fig. 114.

The Chromosphere and
edge of the slit, the observer at the

Prominences seen in the
eye-piece of the spectroscope will see

Spectroscope.

things about as we have attempted to represent them in Fig. 114; as

1Too high dispersion injures the definition, however, because the lines in
the spectrum of hydrogen are rather broad and hazy.

THE SUN.

210
if he were looking at the clouds in an evening sky through a slightly
opened window-blind.

325. Different Kinds of Prominences; Their Forms and

Motions.—The prominences may be broadly divided into two
classes,—the quiescent or diffused, and the eruptive or “metallic,”

as Secchi calls them, because they show in their spectrum the lines of
many metals besides hydrogen. The former, illustrated by Fig. 115

(see p. 211), are immense clouds, often 60,000 miles in height, and of
corresponding horizontal dimensions, either resting upon the chro-
mosphere or connected with it by slender stems like great banyan-
trees. They are not very brilliant, and are composed almost entirely
of hydrogen and “helium.” They often remain nearly unchanged for
days together as they pass over the sun’s limb. They are found on
all portions of the disc, at the poles and equator as well as in the
spot zones. Some of them are clouds floating entirely detached from
the sun’s surface.

Usually these clouds are simply the remnants of prominences
which appear to have been thrown up from below, but in some
cases they actually form and grow larger without any visible con-
nection with the chromosphere—a fact of considerable importance,
as showing in these regions the presence of hydrogen, invisible to
our spectroscopes until somehow or other it is made to give out
the rays of its familiar spectrum. All the forms and motions of
the prominences, it may be said further, seem to indicate the same
thing—that they exist and move, not in a vacuum, but in a medium
of density comparable with their own, as clouds do in our own at-
mosphere.

326. The eruptive prominences, on the other hand, are bril-
liant and active, not usually so large as the quiescent, but at times
enormous, reaching elevations of 100,000, 200,000, or even 400,000
miles. They are illustrated by Fig. 116. Most frequently they are in
the form of spikes or flames; but they present also a great variety
of other fantastic shapes, and are sometimes so brilliant as to be
visible with the spectroscope on the surface of the sun itself, and
not merely at the limb. Generally prominences of this class are
associated with active sun spots, while both classes appear to be
connected with the faculæ. The figures given are from drawings of
individual prominences that have been observed by the author at
different times.

THE SUN.

211

Clouds.

Diffuse.

Filamentary.

Stemmed.

Plumes.

Horns.

Fig. 115.

Quiescent Prominences. Scale 75,000 Miles to the Inch. From “The Sun,”

by Permission of D. Appleton & Co.

THE SUN.

212

Vertical Filaments.

Prominences Sept. 7, 1871, 12.30 p.m.

Cyclone.

Same at 1.15 p.m.

Flame.

Jets near Sun’s Limb, Oct. 5, 1871.

Fig. 116.

Eruptive Prominences. From “The Sun.” By Permission of D. Appleton & Co.

THE SUN.

213

These solar clouds are most fascinating objects to watch, on ac-
count of the beauty of their forms, and the rapidity of their changes.

In the case of the eruptive prominences the swiftness of the changes
is sometimes wonderful—portions can be actually seen to move, and
this implies a real velocity of at least 250 miles a second, so that it
is no exaggeration to speak of such phenomena as veritable “explo-
sions”: of course, in such cases the lines in the spectrum are greatly
broken and distorted, and frequently a “magnetic storm” follows
upon the earth, with a brilliant Aurora Borealis.

The number visible at a single time is variable, but it is not very
unusual to find as many as twenty on the sun’s limb at once.

327. The Corona.—This is a halo, or “glory,” of light which
surrounds the sun at the time of the total eclipse. From the remotest
times it has been well known, and described with enthusiasm, as
being certainly one of the most beautiful of natural phenomena.

The portion of the corona nearest the sun is almost dazzlingly
bright, with a greenish, pearly tinge which contrasts finely with
the scarlet blaze of the prominences. It is made up of streaks and
filaments which on the whole radiate outwards from the sun’s disc,
though they are in many places strangely curved and intertwined.

Usually these filaments are longest in the sun-spot zones, thus giving
the corona a more or less quadrangular figure. At the very poles of
the sun, however, there are often tufts of sharply defined threads.

For the most part the streamers have a length not much exceed-
ing the sun’s radius, but some of them at almost every eclipse go far
beyond this limit. In the clear air of Colorado during the eclipse of

1878, two of them could be traced for five or six degrees,—a distance
of at least 9,000000 miles from the sun. A most striking feature of
the corona usually consists of certain dark rifts which reach straight
out from the moon’s limb, clear to the extremest limit of the corona.

The corona varies much in brightness at different eclipses, and
of course the details are never twice the same. Its total light under
ordinary circumstances is at least two or three times as great as
that of the full moon.

328. Photographs of the Corona.—While the eye can per-
haps grasp some of its details more satisfactorily than the photo-
graphic plate can do, it is found that drawings of the corona are
hardly to be trusted. At any rate, it seldom happens that the rep-

THE SUN.

214
resentations of two artists agree sufficiently to justify any confidence
in their scientific accuracy. Photographs, on the other hand, may
be trusted as far as they go, though they may fail to bring out some
things which are conspicuous to the eye. Fig. 117 is from the pho-
tograph of the Egyptian eclipse of 1882, when a little comet was
found close to the sun.

Fig. 117.—Corona of the Egyptian Eclipse, 1882.

Of course, as in the case of the prominences, the only reason we
cannot see the corona without an eclipsed sun is the illumination of
the earth’s atmosphere. If we could ascend above our atmosphere, and
manage to exist and to observe there, we could see it by simply screening
off the sun’s disc. So long, however, as the brightness of the illuminated
air is more than about sixty times that of the corona, it must remain
invisible to the eye. Dr. Huggins has thought that it might be possible
by means of photographs to detect differences of illumination less than

1

(the limit of the eye’s perception), and so to obtain pictures of the

60
corona at any time; especially as it appears that the coronal light is
far richer in ultra-violet rays (the photographic rays) than the general
sunlight with which the air is illuminated. His attempts so far, however,
have yielded only doubtful success.

329. Spectrum of the Corona.—This was first definitely ob-
served in 1869 during the eclipse which passed over the western part
of the United States in that year. It was then found that its most
remarkable characteristic is a bright line in the green, which the
writer identified as coinciding with the dark line at 1474 on the
scale of Kirchoff’s map (λ = 5316). This line was also observed by

Harkness.

THE SUN.

215

This result was for a time very puzzling, since the dark line in ques-
tion is given by ˚

Angstrom and other authorities as due to the spectrum
of iron. The mystery has since been removed, however, by the discovery
that under high dispersion the line is double, and that the corona line
coincides with the more refrangible of the two components, while the
other one is the line due to iron. We have as yet been unable to identify
with any terrestrial element the substance to which this line is due, but
the provisional name “coronium” has been proposed for it. The recent
researches of Grünewald make it somewhat probable that both coronium
and helium are components of hydrogen, which (in line with Mr. Lock-
yer’s speculations) is supposed to be partially decomposed under solar
conditions.

Besides this conspicuous green line, the hydrogen lines are also
faintly visible in the spectrum of the corona; and by means of a pho-
tographic camera used during the Egyptian eclipse of 1882, it was
found that the upper or violet portion of the spectrum is very rich
in lines, among which H and K are specially conspicuous. There is
also, through the whole spectrum, a faint continuous background,
which, however, according to Mr. Lockyer’s statements, is not of
uniform brightness, but “banded.” In it some observers have re-
ported the presence of a few of the more conspicuous dark lines of
the ordinary solar spectrum, but the evidence on this point is rather
conflicting.

If during the totality we look at the eclipsed sun with a diffrac-
tion grating, or through a prism of high dispersive power, we see
three rings which are really images of the corona. One of them, the
brightest and the largest, is the green ring due to the 1474 line; the
others are a red ring due to C, and a blue one due to the F line of
hydrogen.

330. Nature of the Corona.—It is evident that the corona is
a truly solar and not merely an optical or atmospheric phenomenon,
from two facts: first, the identity of detail in photographs made at
widely separate stations. In 1871, for instance, photographs were
obtained at the Indian station of Bekul, in Ceylon, and in Java,
three stations separated by many hundreds of miles; but, excepting
minute differences of detail, such as might be expected to have re-
sulted from the changes that would naturally go on in the corona,
during the half-hour while the moon’s shadow was travelling from

Bekul to Java, all the photographs agree exactly, which of course
would not be the case if the corona depended in any way upon the

THE SUN.

216
atmospheric conditions at the observer’s station.

Second (but first historically), the presence of bright lines in
the spectrum of the corona proves that it cannot be a terrestrial or
lunar phenomenon, by demonstrating the presence in the corona of
a self-luminous gas, which observation fails to find either near to
the moon or in our own atmosphere. It must, therefore, be at the
sun.

But while it is thus certain that the corona contains luminous
gas, it also is very likely that finely divided solid or liquid matter
may be present in the corona; that is, fog or dust of some kind.

331. The corona cannot be a true “solar atmosphere” in any
strict sense of the word. No gaseous envelope in any way analogous
to the earth’s atmosphere could possibly exist there in gravitational
equilibrium under the solar conditions of pressure and temperature.

The corona is probably a phenomenon due somehow to the intense
activity of the forces there at work; meteoric matter, cometic mat-
ter, matter ejected from within the sun, are all concerned.

That this matter is inconceivably rare is evident from the fact
that in several cases comets have passed directly through the corona
without experiencing the least perceptible disturbance of their mo-
tions. It is altogether probable that at a very few thousand miles
above the sun’s surface, its density becomes far less than that of
the best vacuum we can make in an electric lamp.

THE SUN’S LIGHT AND HEAT.

217

CHAPTER IX.

THE SUN’S LIGHT AND HEAT: COMPARISON OF SUNLIGHT

WITH

ARTIFICIAL

LIGHTS.—MEASUREMENT

OF

THE

SUN’S HEAT, AND DETERMINATION OF THE “SOLAR

CONSTANT.”—PYRHELIOMETER,

ACTINOMETER,

AND

BOLOMETER.—THE SUN’S TEMPERATURE.—THEORIES

AS TO THE MAINTENANCE OF THE SUN’S RADIATION,

AND CONCLUSIONS AS TO THE SUN’S POSSIBLE AGE

AND FUTURE DURATION.

332. The Sun’s Light.—The Quantity of Sunlight. It is very
easy to compare (approximately) sunlight with the light of a stan-
dard1 candle; and the result is, that when the sun is in the zenith,
it illuminates a white surface about 60,000 times as strongly as a
standard candle at a distance of one metre. If we allow for the
atmospheric absorption, the number would he fully 70,000. If we
then multiply 70,000 by the square of 150,000 million (roughly the
number of metres in the sun’s distance from the earth), we shall
get what a gas engineer would call the sun’s “candle power.” The
number comes out 1575 billions of billions (English); i.e., 1575 with
twenty-four ciphers following.

333. One way of making the comparison is the following: Arrange
matters as in Fig. 118. The sunlight is brought into a darkened room by
a mirror M , which reflects the rays through a lens L of perhaps half an
inch in diameter. After the rays pass the focus they diverge and form on
the screen S a disc of light, the size of which may be varied by changing
the distance of the screen. Suppose it so placed that the illuminated
circle is just ten feet in diameter; that is, 240 times the diameter of the

1A standard candle is a sperm candle weighing one-sixth of a pound and
burning 120 grains an hour. The French “Carcel burner,” used as a standard
in their photometry, gives just ten times the quantity of light given by this
standard candle. An ordinary gas-burner consuming five feet of gas hourly
gives a light equivalent to from twelve to fifteen standard candles.

THE SUN’S LIGHT AND HEAT.

218
lens. The illumination of the disc will then be less than that of direct
sunlight in the ratio of 2402 (or 57,600) to 1 (neglecting the loss of light
produced by the mirror and the lens, a loss which of course must he
allowed for). Now place a little rod like a pencil near the screen, as at

P , light a standard candle, and move the candle back and forth until the
two shadows of the pencil, one formed by the candle, and the other by
the light from the lens, are equally dark. It will be found that the candle
has to be put at a distance of about one metre from the screen; though
the results would vary a good deal from day to day with the clearness of
the air.

Fig. 118.—Comparison of Sunlight with a Standard Candle.

334. When the sun’s light is compared with that of the full
moon and of various stars, we find, as stated (Art. 259), that it is
about 600,000 times that of the full moon. It is 7,000,000000 times
as great as the light received from Sirius, and about 40,000,000000
times that from Vega or Arcturus.

335. The Intensity of the Sun’s Luminosity.—This is a
very different thing from the total quantity of its light, as expressed
by its “candle power” (a surface of comparatively feeble luminos-
ity can give a great quantity of light if large enough). It is the
amount of light per square inch of luminous surface which deter-
mines the intensity. Making the necessary computations from the
best data obtainable (only roughish approximations being possible),
it appears that the sun’s surface is about 190,000 times as bright as
that of a candle flame, and about 150 times as bright as the lime
of the calcium light. Even the darkest part of a solar spot outshines

THE SUN’S LIGHT AND HEAT.

219
the lime. The intensely brilliant spot in the so-called “crater” of an
electric arc comes nearer sunlight than anything else known, being
from one-half to one-fourth as bright as the surface of the sun itself.

But either the electric arc or the calcium light, when interposed
between the eye and the sun looks like a dark spot on the disc.

336. Comparative Brightness of Different Portions of
the Sun’s Surface.—By forming a large image of the sun, say a
foot in diameter, upon a screen, we can compare with each other the
rays coming from different parts of the sun’s disc. It thus appears
that there is a great diminution of light at the edge, the light there,
according to Professor Pickering’s experiments, being just about
one-third as strong as at the centre. There is also an obvious differ-
ence of color, the light from the edge of the disc being brownish red
as compared with that from the centre. The reason is, that the red
and yellow rays of the spectrum lose much less of their brightness
at the limb than do the blue and violet. According to Vogel, the
latter rays are affected nearly twice as much as the former. For this
reason, photographs of the sun exhibit the darkening of the limb
much more strongly than one usually sees it in the telescope.

337. Cause of the Darkening of
the Limb.—It is due unquestionably to
the general absorption of the sun’s rays
by the lower portion of the overlying at-
mosphere. The reason is obvious from
the figure (Fig. 119). The thinner this
atmosphere, other things being equal,
the greater the ratio between the percent-

Fig. 119.

Cause of the Darkening of
age of absorption at the centre and edge
the Sun’s Limb.

of the disc, and the more obvious the
darkening of the limb.

Attempts have been made to determine from the observed differ-
ences between the brightness of centre and limb the total percentage
of the sun’s light thus absorbed. Unfortunately we have to supple-
ment the observed data with some very uncertain assumptions in
order to solve the problem; and it can only be said that it is probable
that the amount of light, absorbed by the sun’s atmosphere lies be-
tween fifty and eighty per cent; i.e., the sun deprived of its gaseous
envelope would probably shine from two to five times as brightly as
now. It is noticeable also, as Langley long ago pointed out, that thus

THE SUN’S LIGHT AND HEAT.

220
stripped, the “complexion” of the sun would be markedly changed
from yellowish white to a good full blue, since the blue and violet
rays are much more powerfully absorbed than these at the lower
end of the spectrum.

THE SUN’S HEAT.

338. Its Quantity; the “Solar Constant.” By the “quantity
of heat ” received by the earth from the sun we mean the number of
heat-units received in each unit of time by a square unit of surface
when the sun is in the zenith. The heat-unit most employed by en-
gineers is the calorie, which is the quantity of heat required to raise
the temperature of one kilogram of water one degree centigrade. It
is found by observation that each square metre of surface exposed
perpendicularly to the sun’s rays receives from the sun each minute
from twenty-five to thirty of these calories; or rather it would do
so if a considerable portion of the sun’s heat were not stopped by
the earth’s atmosphere, which absorbs some thirty per cent of the
whole, even when the sun is vertical, and a much larger proportion
when the sun is near the horizon. This quantity, twenty-five calo-
ries1 per square metre per minute (using the smaller of the values
mentioned, which certainly is not too large), is known as the “Solar

Constant.”

339. Method of determining the “Solar Constant.”—

The method by which the solar constant is determined is simple
enough in principle, though complicated with serious practical dif-
ficulties which affect its accuracy. It is done by allowing a beam
of sunlight of known cross-section to shine upon a known weight
of water (or other substance of known specific heat ) for a known
length of time, and measuring the rise of temperature. It is neces-
sary, however, to determine and allow for the heat received from

1For many scientific purposes the engineering calorie is inconveniently large,
and a smaller one is employed, which replaces the kilogram of water by the gram
heated one degree—the smaller calorie being thus only

1
of the engineering

1000
unit. As stated by many writers (Langley, for instance), the solar constant is
the number of these small calories received per square centimetre of surface in
a minute. This would make the number 2.5 instead of 25. It would perhaps
be better to bring the whole down to the “c.g.s. system” by substituting the
second for the minute; and this would give us for the solar constant, on the

“c.g.s. system,” 0.0417 (small ) calories per square centimetre per second.

THE SUN’S LIGHT AND HEAT.

221
other sources during the experiment, and for that lost by radiation.

Above all, the absorbing effect of our own atmosphere is to be taken
into account, and this is the most difficult and uncertain part of the
work, since the atmospheric absorption is continually changing with
every change of the transparency of the air, or of the sun’s altitude.

340. Pyrheliometers
and

Acti-
nometers.—The
instruments
with
which
these measurements are made, are known as

“pyrheliometers” and “actinometers.” Fig. 120
represents the pyrheliometer of Pouillet, with
which in 1838 he made his determination of the
solar constant, at the same time that Sir John

Herschel was experimenting at the Cape of

Good Hope in practically the same way. They
were the first apparently to understand and
attack the problem in a reasonable manner.

The pyrheliometer consists essentially of a
little cylindrical box ab, like a snuff-box, made
of thin silver plate, with a diameter of one
decimetre and such a thickness that it holds

100 grams of water.

The upper surface is
carefully blackened, while the rest is polished
as brilliantly as possible.

In the water is
inserted the bulb of a delicate thermometer,

Fig. 120.—Pouillet’s
and the whole is so mounted that it can be

Pyrheliometer.

turned in any direction so as to point it directly
towards the sun. It is used by first holding a screen between it and the
sun for (say) five minutes, and watching the rise or fall of the mercury
in the thermometer at m. There will usually be some slight change due
to the radiation of surrounding bodies. The screen is then removed, and
the sun is allowed to shine upon the blackened surface for five minutes,
the instrument being continually turned upon the thermometer as an
axis, in order to keep the water in the calorimeter box well stirred.

At the end of the five minutes the screen is replaced and the rise of
the temperature noted. The difference between this and the change of
the thermometer during the first five minutes will give us the amount
by which a beam of sunlight one decimetre in diameter has raised the
temperature of 100 grams of water in five minutes, and were it not for
the troublesome corrections which must be made, would furnish directly
the value of the solar constant.

THE SUN’S LIGHT AND HEAT.

222

341. The second apparatus, Fig. 121,
is the actinometer of Violle, which consists
of two concentric metal spheres, the inner
of which is blackened on the inside, while
the outer one is brightly polished, the space
between the two being filled with water at
a known temperature, kept circulating by
a pump of some kind. The thermoscopic
body in this case, instead of being a box
filled with water, is the blackened bulb of
the thermometer T ; and the observations
may be made either in the same way as with
the pyrheliometer, or simply by noting the
difference between the temperature finally

Fig. 121.—Violle’s

Actinometer.

attained by the thermometer T after it has
ceased to rise in the sun’s rays, and the
temperature of the water circulating in the shell.

342. Correction for Atmospheric Absorption.—The cor-
rection for atmospheric absorption is determined by making observations
at various altitudes of the sun between zenith and horizon. If the rays
were homogeneous (that is, all of one wave-length), it would be com-
paratively easy to deduce the true correction and the true value of the
solar constant. In fact, however, the visible solar spectrum is but a small
portion of the whole spectrum of the sun’s radiance, and, as Langley has
shown, it is necessary to determine the coefficient of absorption sepa-
rately for all the rays of different wave-length.

343. The Bolometer.—This he has done by means of his

“Bolometer,” an instrument which is capable of indicating exceedingly
minute changes in the amount of radiation received by an extremely
thin strip of metal. This strip is so arranged that the least change in
its electrical resistance due to any change of temperature will disturb a
delicate galvanometer. The instrument is far more sensitive than any
thermometer or even thermopile, and has the especial advantage of be-
ing extremely quick in its response to any change of radiation. Fig. 122
shows it so connected with a spectroscope that the observer can bring
to the bolometer, B, rays of any wave-length he chooses. The rays enter
through the collimator lens L, and are then refracted by the prism P to
the reflector M , whence they are sent back to B.

Langley has shown that the corrections for atmospheric absorption
deduced by earlier observers are all considerably too small, and has raised
the received value of the solar constant, from 20 or 25, which was the

THE SUN’S LIGHT AND HEAT.

223
value accepted a few years ago, to 30. We have, however, provisionally
retained the 25, as his new results, though almost certainly correct, have
not yet been universally accepted, and perhaps need verification.

344. A less technical state-
ment of the solar radiation may
be made in terms of thickness of
the quantity of ice which would
be melted by it in a given time.

Since it requires about eighty calo-
ries of heat to melt a kilogram of
ice, it follows that twenty-five calo-
ries per minute per square metre
would liquefy in an hour a sheet
of ice one metre square and about
nineteen millimetres thick. Accord-
ing to this the sun’s heat would
melt about 174 feet of ice annu-
ally on the earth’s equator; or 136 12
feet yearly all over the surface of

Fig. 122.

the earth, if the heat annually re-

Langley’s Spectro-Bolometer, as
used for Mapping the Energy of
ceived were equally distributed in
the Prismatic Spectrum.

all latitudes. (See note at end of
the chapter, page 232.)

345. Solar Heat expressed as Energy.—Since according to
the known value of the “mechanical equivalent of heat” (Physics,
p. 159) a horse-power corresponds to about 10 7 calories per minute,

10
it follows that each square metre of surface (neglecting the air-
absorption) would receive, when the sun is overhead, about two
and one-third horse-power continuously. Atmospheric absorption
cuts this down to about one and one-half horse-power, of which
about one-eighth can be actually utilized by properly constructed
machinery, as, for instance, the solar engines of Ericsson and Mou-
chot (see Langley’s “New Astronomy”). In Ericsson’s apparatus the
reflector, about 11 feet by 16 feet, collected heat enough to work a
three-horse-power engine very well. Taking the earth’s surface as
a whole, the energy received during a year aggregates about sixty
mile-tons for every square foot. That is to say, the heat annually
received on each square foot of the earth’s surface, if employed in a
perfect heat engine, would be able to hoist sixty tons to the height of
a mile.

THE SUN’S LIGHT AND HEAT.

224

346. Solar Radiation at the Sun’s Surface.—If, now, we
estimate the amount of radiation at the sun’s surface itself, we come
to results which are simply amazing and beyond comprehension. It
is necessary to multiply the solar constant observed at the earth

(which is at a distance of 93,000000 miles from the sun) by the
square of the ratio between 93,000000 and 433,250, the radius of
the sun. This square is about 46,000; in other words, the amount
of heat emitted in a minute by a square metre of the sun’s surface
is about 46,000 times as great as that received by a square metre
at the earth. Carrying out the calculations, we find that this heat
radiation at the surface of the sun amounts to over a million calories
per square metre per minute; that it is over 100,000 horse-power per
square metre continuously acting; that if the sun were frozen over
completely to a depth of fifty feet, the heat emitted is sufficient to
melt this whole shell in one minute of time; that if an ice bridge
could be formed from the earth to the sun by a column of ice two
and one-fourth miles square at the base and extending across the
whole 93,000000 of miles, and if by some means the whole of the
solar radiation could be concentrated upon this column, it would
be melted in one second of time, and in between seven and eight
seconds more would be dissipated in vapor. To maintain such a
development of heat by combustion would require the hourly burning
of a layer of the best anthracite coal from sixteen to twenty feet
thick over the sun’s entire surface,—a ton for every square foot of
surface,—at least nine times as much as the consumption of the
most powerful blast furnace in existence. At that rate the sun, if
made of solid coal, would not last 6000 years.

347. Waste Of Solar Heat.—Those estimates are of course
based on the assumption that the sun radiates heat equally in all di-
rections, and there is no assignable reason why it should not do so. On
this assumption, however, so far as we can see, only a minute fraction
of the whole radiation ever reaches a resting-place. The earth receives
about

1
of the whole, and the other planets of the solar system,

2200,000000
with the comets and the meteors, get also their shares; all of them to-
gether, perhaps ten or twenty times as much as the earth. Something
like

1
of the whole seems to be utilized within the limits of the

100,000000
solar system. As for the rest, science cannot yet tell what becomes of it.

A part, of course, reaches distant stars and other objects in interstellar
space; but by far the larger portion seems to be “wasted,” according to
our human ideas of waste.

THE SUN’S LIGHT AND HEAT.

225

348. Experiments with the thermopile, first conducted by

Henry at Princeton in 1845, show that the heat from the edges of the
sun’s disc, like the light, is less than that from the centre—according
to Langley’s measurements about half as much. The explanation
evidently lies in its absorption by the solar atmosphere.

349. The Sun’s Temper-
ature.—While we can measure
with some accuracy the quantity
of heat sent us by the sun, it
is different with its temperature
in respect to which we can only

Fig. 123.

say that it must be very high—
much higher than any temperature attainable by known methods
on the surface of the earth.

This is shown by a number of facts, for instance, by the great
abundance of the violet and ultra-violet rays in the sunlight.

Again, by the penetrating power of sunlight; a large percentage
of the heat from a common fire, for instance, being stopped by a
plate of glass, while nearly the whole of the solar radiation passes
through.

The most impressive demonstration, however, follows from this
fact; viz., that at the focus of a powerful burning-lens all known
substances melt and vaporize, as in an electric arc. Now at the
focus of the lens the limit of the temperature is that which would
be produced by the sun’s direct radiation at a point where the sun’s
angular diameter equals that of the burning-lens itself seen from the
focus, as represented in Fig. 123. An object at F would theoretically

(that is, if there was no loss of heat conducted away by surrounding
bodies and by the atmosphere) reach the same temperature as if
carried to a point where the sun’s angular diameter equals the angle

LF L . In the most powerful burning-lenses yet constructed a body
at the focus is thus virtually carried up to within about 240,000
miles of the sun’s surface, where its apparent diameter would be
about 80◦. Here, as has been said, the most refractory substances
are immediately subdued. If the earth were to approach the sun as
near as the moon is to us, she would melt and be vaporized.

350. Ericsson in 1872 made an exceedingly ingenious and interest-
ing experiment illustrating the intensity of the solar heat. He floated a
calorimeter, containing about ten pounds of water, upon the surface of a

THE SUN’S LIGHT AND HEAT.

226
large mass of molten iron by means of a raft of fire-brick, and found that
the radiation of the metal was a trifle over 250 calories per minute for
each square foot of surface; which is only 1 part of the amount emitted

400
by the same area of the sun’s surface. He estimated the temperature of
the metal at 3000◦ F. or 1649◦ C.

351. Effective Temperature.—The question of the sun’s
temperature is embarrassed by the fact that it has no one temper-
ature; the temperature at different parts of the solar photosphere
and chromosphere must be very different. We evade this difficulty
to some extent by substituting for the actual temperature, as the
object of inquiry, what has been called the sun’s “effective temper-
ature”; that is, the temperature which a sheet of lampblack must
have in order to radiate the amount of heat actually thrown off by
the sun. (Physicists have taken the radiating power of lampblack as
unity.) If we could depend upon the laws1 deduced from laboratory
experiments, by which it has been sought to connect the tempera-
ture of the body with its rate of radiation, the matter would then
be comparatively simple: from the known radiated quantity of heat

(in calories) we could compute the effective temperature in degrees.

But at present it is only by a very unsatisfactory process of extrap-
olation that we can reach conclusions. The sun’s temperature is so
much higher than any which we can manage in our laboratories,
that there is not yet much certainty to be obtained in the matter.

Rosetti, the most recent investigator, whose results seem to be on
the whole the most probable, obtains 10, 000◦ C. or 18, 000◦ F. for
the effective temperature.

352. Constancy of the Sun’s Heat.—It is an interesting and
thus far unsolved problem, whether the total amount of the sun’s
radiation varies perceptibly at different times. It is only certain
that the variations, if real, are too small to be detected by our
present means of observation. Possibly, at some time in the future,

1A number of such laws have been formulated; for instance, the well-known
law of Dulong and Petit (Physics, p. 470). The French physicists Pouillet and

Vicaire, using this formula, have deduced values for the sun’s effective tempera-
ture running from 1500◦ to 2500◦ C. Ericsson and Secchi, using Newton’s law of
radiation (which, however, is certainly inapplicable under the circumstances),
put the figure among the millions. Zöllner, Spörer, and Lane give values ranging
from 25, 000◦ to 50, 000◦ C.

THE SUN’S LIGHT AND HEAT.

227
observations on a mountain summit above the main body of our
atmosphere may decide the question.

It is not unlikely that changes in the earth’s climate such as
have given rise to glacial and carboniferous periods may ultimately
be traced to the condition of the sun itself, especially to changes
in the thickness of the absorbing atmosphere, which, as Langley
has pointed out, must have a great influence in the matter. Since
the Christian era, however, it is certain that the amount of heat
annually received from the sun has remained practically unchanged.

This is inferred from the distribution of plants and animals, which
is still substantially the same as in the days of Pliny.

353. Maintenance of the Solar Heat.—The question at
once arises, if the sun is sending off such an enormous quantity
of heat annually, how is it that it does not grow cold?

(a) The sun’s heat cannot be kept up by combustion. As has
been said before, it would have burned out long ago, even if made
of solid coal burning in oxygen.

(b) Nor can it be simply a heated body cooling down. Huge as it
is, an easy calculation shows that its temperature must have fallen
greatly within the last 2000 years by such a loss of heat, even if it
had a specific heat higher than that of any known substance.

As matters stand at present, the available theories seem to be
reduced to two,—that of Mayer, which ascribes the solar heat to the
energy of meteoric matter falling on the sun; and that of Helmholtz,
who finds the cause in a slow contraction of the sun’s diameter.

354. Meteoric Theory of Sun’s Heat.—The first is based
on the fact that when a moving body is stopped, its mass-energy
becomes molecular energy, and appears mainly as heat. The amount
of heat developed in such a case is given by the formula

M V 2

Q =

,

8339
in which Q is the number of calories of heat produced, M the
mass of the moving body in kilograms, and V its velocity in me-
tres per second; the denominator is the “mechanical equivalent of
heat” (Physics, p. 159) multiplied by 2g expressed in metres; i.e.,

425 × 2 × 9.81.

Now, the velocity of a body coming from any considerable dis-
tance and falling into the sun can be shown to be about 380 miles

THE SUN’S LIGHT AND HEAT.

228
per second, or more than 610 kilometres. A body weighing one kilo-
gram would therefore, on striking the sun with this velocity, produce
about 45,000000 calories of heat,

(610000)2 .

8339

This is 6000 times more than could be produced by burning it, even
if it were coal or solidified hydrogen burning in pure oxygen.

Now, as meteoric matter is continually falling upon the earth, it
must be also falling upon the sun, and in vastly greater quantities,
and an easy calculation shows that a quantity of meteoric matter
equal to 1 of the earth’s mass striking the sun’s surface annually

100
with the velocity of 600 kilometres per second would account for its
whole radiation.

355. Objections to Meteoric Theory of Sun’s Heat.—

There can be no question that a certain fraction of the sun’s heat
is obtained in this way, but it is very improbable that this fraction
is a large one; indeed, it is hardly possible that it can be as much
as one per cent of the whole.

(1) The annual fall on the sun’s surface of such a quantity of meteoric
matter implies the presence near the sun of a vastly greater mass; for,
as we shall see hereafter, only a few of the meteors that approach the
sun from outer space would strike the surface: most of them would act
like the comets and swing around it without touching. Now, if there
were any considerable quantity of such matter near the sun, there would
result disturbances in the motions of the planets Mercury and Venus,
such as observation does not reveal.

(2) Professor Peirce has shown further that if the heat of the sun
were produced in this way, the earth ought to receive from the meteors
that strike her surface about half as much heat as she gets from the sun.

Now the quantity of meteoric matter which would have to fall upon the
earth to furnish us daily half as much heat as we receive from the sun,
would amount to nearly fifty tons for each square mile. It is not likely
that we actually get

1
of that amount. It is difficult to determine

10,000000
the amount of heat which the earth actually does receive from meteors,
but all observations indicate that the quantity is extremely small. The
writer has estimated it, from the best data attainable, as less in a year
than we get from the sun in a second.

356. Helmholtz’s Theory of Solar Contraction.—We
seem to be shut up to the theory of Helmholtz, now almost uni-

THE SUN’S LIGHT AND HEAT.

229
versally accepted: namely, that the heat necessary to maintain the
sun’s radiation is principally supplied by the slow contraction of its
bulk, aided, however, by the accompanying liquefaction and solidi-
fication of portions of its gaseous mass. When a body falls through
a certain distance, gradually, against resistance, and then comes to
rest, the same total amount of heat is produced as if it had fallen
freely, and been stopped instantly. If, then, the sun does contract,
heat is necessarily produced by the process, and that in enormous
quantity, since the attracting force at the solar surface is more than
twenty-seven times as great as terrestrial gravity, and the contract-
ing mass is immense. In this process of contraction each particle
at the surface moves inward by an amount equal to the diminution
of the sun’s radius: a particle below the surface moves less and un-
der a diminished gravitating force; but every particle in the whole
mass, excepting only that at the exact centre of the globe, con-
tributes something to the evolution of heat. In order to calculate
the precise amount of heat evolved by a given shrinkage it would
be necessary to know the law of increase of the sun’s density from
the surface to the centre; but Helmholtz has shown that under the
most unfavorable conditions a contraction in the sun’s diameter of
about two hundred and fifty feet a year (125 feet in the sun’s radius)

would account for the whole annual output of heat. This contraction
is so slow that it would be quite imperceptible to observation. It
would require more than 9000 years to reduce the sun’s diameter
by a single second of arc; and nothing much less would be certainly
detectible by our measurements. If the contraction is more rapid
than this, the mean temperature of the sun must be actually rising,
notwithstanding the amount of heat it is losing. Long observation
alone can determine whether this is really the case or not.

357. Lane’s Law.—It is a remarkable fact, first demonstrated
by Lane of Washington, in 1870, that a gaseous sphere, losing heat by
radiation and contracting under its own gravity, must rise in temperature
and actually grow hotter, until it ceases to be a “perfect gas,” either by
beginning to liquefy, or by reaching a density at which the laws of perfect
gases no longer hold. The kinetic energy developed by the shrinkage of
a gaseous mass is more than sufficient to replace the loss of heat which
caused the shrinkage. In the case of a solid or liquid mass this is not
so. The shrinkage of such a mass contracting under its own gravity on
account of the loss of heat is never sufficient to make good the loss; but
the temperature falls and the body cools. At present it appears that

THE SUN’S LIGHT AND HEAT.

230
in the sun the relative proportions of true gases and liquids are such
as to keep the temperature nearly stationary, the liquid portions of the
sun being of course the little drops which are supposed to constitute the
clouds of the photosphere.

358. Future Duration of the Sun.—If this shrinkage theory
of the solar heat is correct (and there is every reason to accept it),
it follows that in time the sun’s heat must come to an end, and,
looking backwards, we see that there must have been a beginning.

We have not sufficient data to enable us to calculate the future
duration of the sun with exactness, though an approximate esti-
mate can he made. According to Newcomb, if the sun maintains its
present radiation, it will have shrunk to half its present diameter in
about 5,000000 years at the longest. Since when reduced to this size
it must be about eight times as dense as now, it can hardly then
continue to be mainly gaseous, and its temperature must begin to
fall. Newcomb’s conclusion, therefore, is that it is not likely that
the sun can continue to give sufficient heat to support such life on
the earth as we are now acquainted with, for 10,000000 years from
the present time.

359. Age of the Sun.—As to the past of the solar history on
this hypothesis, we can be a little more definite. It is only necessary
to know the present amount of radiation, and the mass of the sun,
to compute how long the solar fire can have been maintained at its
present intensity by the processes of condensation. No conclusion
of geometry is more certain than this,—that the contraction of the
sun to its present size, from a diameter even many times greater
than Neptune’s orbit, would have furnished about 18,000000 times
as much heat as the sun now supplies in a year, and therefore that
the sun cannot have been emitting heat at the present rate for more
than 18,000000 years, if its heat has really been generated in this
manner.

But of course this conclusion as to the possible past duration of the
solar system rests upon the assumption that the sun has derived its heat
solely in this way; and moreover, that it radiates heat equally in all
directions in space,—assumptions which possibly further investigations
may not confirm.

360. Constitution of the Sun.—(a) As to the nature of the
main body or nucleus of the sun, we cannot be said to have cer-

THE SUN’S LIGHT AND HEAT.

231
tain knowledge. It is probably gaseous, this being indicated by its
low mean density and its high temperature—enormously high even
at the surface, where it is coolest. At the same time the gaseous
matter at the nucleus must be in a very different state from gases
as we commonly know them in our laboratories, on account of the
intense heat and the extreme condensation by the enormous force of
solar gravity. The central mass, while still strictly gaseous, because
observing the three physical laws of Boyle, Dalton, and Gay Lussac,
which characterize gases, would be denser than water, and viscous;
probably something like tar or pitch in consistency.1

While this doctrine of the gaseous constitution of the sun is gen-
erally assented to, there are still some who are disposed to consider
the great mass of the sun as liquid.

361. (b) The photosphere is probably a shell of incandescent
clouds, formed by the condensation of the vapors which are exposed
to the cold of space.

362. (c) The photospheric clouds float in an atmosphere con-
taining, still uncondensed, a considerable quantity of the same va-
pors out of which they themselves have been formed, just as in our
own atmosphere the air around a cloud is still saturated with water
vapor. This vapor-laden atmosphere, probably comparatively shal-
low, constitutes the reversing layer, and by its selective absorption
produces the dark lines of the solar spectrum, while by its general
absorption it probably produces the darkening at the limb of the
sun.

But it will be remembered that Mr. Lockyer and others are disposed
to question the existence of any such shallow absorbing stratum, consid-
ering that the absorption takes place in all regions of the solar atmosphere
even to a great elevation.

1The law of Dalton (Physics, p. 181) is, that any number of different gases
and vapors tend to distribute themselves throughout the space which they occupy
in common, each as if the others were absent. The law of Boyle or Mariotte

(Physics, p. 110) is, that at any given temperature the volume of any given
amount of gas varies inversely with the pressure: i.e., pv = p v . The law of Gay

Lussac (Physics, p. 185) is, that a gas under constant pressure expands in volume
uniformly under uniform increment of temperature, so that Vt = V0(1 + at).

This is not true of vapors in presence of the liquids from which they have been
evaporated; for instance, of steam in a boiler.

THE SUN’S LIGHT AND HEAT.

232

363. (d ) The chromosphere and prominences are composed of
the permanent gases, mainly hydrogen and helium, which are min-
gled with the vapors of the reversing stratum in the region near
the photosphere, but usually rise to far greater elevations than do
the vapors. The appearances are for the most part as if the chro-
mosphere was formed of jets of heated hydrogen ascending through
the interspaces between the photospheric clouds, like flames playing
over a coal fire.

364. (e) The corona also rests on the photosphere, and the
peculiar green line of its spectrum (Art. 329) is brightest just at
the surface of the photosphere, in the reversing stratum and in the
chromosphere itself; but the corona extends to a far greater elevation
than even the prominences ever reach, and seems to be not wholly
gaseous, but to contain, besides the hydrogen and the mysterious

“coronium,” dust and fog of some sort, perhaps meteoric. Many
of its phenomena are as yet unexplained, and since it can only be
observed during the brief moments of total solar eclipses, progress
in its study is necessarily slow.

364*. Note to Article 344. The total heat received by the earth
from the sun in any given time is that intercepted by its diametrical
cross-section, i.e., by the area of one of its great circles kept always
perpendicular to the sun’s rays. The quantity of ice which would be
melted annually on this circular plane by the solar rays would be a sheet
having a thickness of 166.5 metres or 546 feet (19mm × 24 × 365 1 =

4

166.5met).

The thickness of the ice which could be melted in a year on a narrow
equatorial belt would be 546ft , or 174ft, since such a belt intercepts the
π
rays that would otherwise fall on a diametrical strip of the same width
upon the circular plane.

If the sun’s heat were uniformly distributed over the earth’s whole
surface, which equals four great circles, (4πR2), it could melt a shell
ft
having a thickness of 546ft , or 136 1 .

4

2

It is true that at the sea-level the solar-constant is much diminished
by atmospheric absorption; and probably does not exceed fifteen calories
per minute directly received from the sun’s rays. But a large part of
the solar heat absorbed by the atmosphere reaches the earth’s surface
indirectly, so that it must not be considered as lost to the earth, because
not directly measurable by the actinometer.

ECLIPSES.

233

CHAPTER X.

ECLIPSES: FORM AND DIMENSIONS OF SHADOWS.—LUNAR

ECLIPSES.—SOLAR ECLIPSES.—TOTAL, ANNULAR, AND

PARTIAL.—ECLIPTIC LIMITS AND NUMBER OF ECLIPSES

IN A YEAR.—THE SAROS.—OCCULTATIONS.

365. The word eclipse (Greek ἔκλειψις) is strictly a medical
term, meaning a faint or swoon. Astronomically it is applied to
the darkening of a heavenly body, especially of the sun or moon,
though some of the satellites of other planets besides the earth are
also “eclipsed” from time to time. An eclipse of the moon is caused
by its passage through the shadow of the earth; an eclipse of the sun,
by the interposition of the moon between the sun and the observer,
or, what comes to the same thing, by the passage of the moon’s
shadow over the observer.

366. Shadows.—If interplanetary space were slightly dusty,
we should see, accompanying the earth and moon and each of the
planets, a long black shadow projecting behind it and travelling
with it. Geometrically speaking, this shadow of a body, the earth
for instance, is a solid—not a surface. It is the space from which
sunlight is excluded. If we regard the sun and other heavenly bodies
as truly spherical, these shadows are cones with their axes in the
line joining the centres of the sun and the shadow-casting body, the
point being always directed away from the sun, because the sun is
always the larger of the two.

367. Dimensions of the Earth’s Shadow.—The length of
the shadow is easily found. In Fig. 124 we have from the similar
triangles OED and ECa, OD : Ea :: OE : EC or l. OD is the
difference between the radii of the sun and the earth, = R − r.

Ea = r, and OE is the distance of the earth from the sun = ∆.

r

1

Hence
l = ∆ ×

=

∆.

R − r

108.5

ECLIPSES.

234

(The fractional factor is constant, since the radii of the sun and
earth are fixed quantities. Substituting the values of the radii, we
find it to be

1

.) This gives 857,200 miles for the length of the

108.5
earth’s shadow when ∆ has its mean value of 93,000000 miles, re-
garding the earth as a perfect sphere and taking its mean radius.

This length varies about 14,000 miles on each side of the mean as
the earth changes its distance from the sun.

Fig. 124.—Dimensions of the Earth’s Shadow.

The semi-angle of the cone (the angle ECb, or ECB in the figure) is
found as follows. Since OEB is exterior to the triangle BEC,

OEB = EBC + BCE,
or

BCE = OEB − EBC.

Now, OEB is the sun’s apparent semi-diameter as seen from the earth,
and EBC is the earth’s semi-diameter as seen from the sun, which is the
same thing as the sun’s horizontal parallax (Art. 83).

Putting S for the sun’s semi-diameter, and p for its parallax, we
have—

Semi-angle at C = S − p.1

From the cone aCb all sunlight is excluded, or would be were it
not for the fact that the atmosphere of the earth by its refraction
bends some of the rays into this shadow. The effect is to make the
shadow a little larger in diameter, but less perfectly dark.

368. Penumbra.—If we draw the lines Ba and Ab, crossing at

C between the earth and the sun, they will bound the penumbra.

Within this space a part, but not the whole, of the sunlight is cut
off: an observer outside of the shadow, but within this cone-frustum,
r

1Also, l =

, an expression sometimes more convenient than the
sin (S − p)

one given above.

ECLIPSES.

235
which tapers towards the sun, would see the earth as a black body
encroaching on the sun’s disc. The semi-angle of the penumbra

EC a is easily shown to be S + p.

369. Although geometrically the boundaries of the shadow and
penumbra are perfectly definite, they are not so optically. If a screen
were placed at M (Fig. 124) perpendicular to the axis of the shadow,
no sharply defined lines would mark the boundaries of either shadow
or penumbra; near the edge of the shadow, the penumbra would be
very nearly as dark as the shadow itself, only a mere speck of the
sun being visible there; and at the outer limit of the penumbra the
shading would be still more gradual.

370. Eclipses of the Moon.—The axis of the earth’s shadow
is always directed to a point exactly opposite the sun. If, then, at
the time of full moon, the moon happens to be near the ecliptic

(that is, not far from one of the nodes of her orbit ), she will pass
into the shadow and be eclipsed. Since, however, the moon’s orbit
is inclined about five and one-fourth degrees to the plane of the
ecliptic, this does not happen very often (seldom more than twice
a year). Ordinarily the moon passes north or south of the shadow
without touching it.

Lunar eclipses are of two kinds,—partial and total: total when
she passes into the shadow completely; partial when she only partly
enters it, going so far to the north or south of the centre of the
shadow that only a portion of her disc is obscured.

We may also have a “penumbral eclipse” when she passes merely
through the penumbra, without touching the shadow. In this case, how-
ever, the loss of light is so gradual and so slight, unless she almost grazes
the shadow, that an observer would notice nothing unusual.

371. Size of the Earth’s Shadow at the Point where the

Moon crosses it.—Since EC in Fig. 125 is 857,000 miles, and the
distance of the moon from the earth is on the average about 239,000
miles, CM must be 618,000 miles, and M N , the semi-diameter of
the shadow at this point, will be 618 of the earth’s radius. This gives

857

M N = 2854 miles, and makes the whole diameter of the shadow a
little over 5700 miles, about two and two-thirds times the diameter
of the moon. But this quantity varies considerably. The shadow is
sometimes more than three times as large as the moon, sometimes
hardly more than twice its size.

ECLIPSES.

236

372. We may reach the same result in another way. Considering
the triangle ECN , Fig. 125, we have the angular semi-diameter of the
cross-section of the shadow where the moon passes through it, as seen
from the earth, represented by M EN .

But

EN a = M EN + ECN ;
whence

M EN = EN a − ECN.

Now EN a is the semi-diameter of the earth as seen from the moon;
that is; it is the moon’s horizontal parallax, for which we write P . Hence,
substituting for ECN its value S − p, we get

M EN = P + p − S.

M EN is called “the radius of the shadow.” The mean value of P is 57 2 ;
of p, 8 .8; and of S, 16 2 , which makes the mean value of M EN =

41 9 . The mean value of the moon’s apparent semi-diameter is 15 40 ,
the ratio between the semi-diameter of the moon and the radius of the
shadow being about 2 2 , as before.

3

Fig. 125.—Diameter of Earth’s Shadow where the Moon crosses it.

In computing a lunar eclipse, this angular value for the “radius of the
shadow,” as it is called, is more convenient than its value in miles. It is
customary to increase it by about 1 part in order to allow for the effect

60
of the earth’s atmosphere, the value ordinarily used being 61 (P + p − S).

60

Some computers, however, use 51 , and others 76 . On account of the

50

75
indistinctness of the edge of the shadow it is not easy to determine what
precise value ought to be employed.

373. Duration of a Lunar Eclipse.—When central, a total
eclipse of the moon may, all things favoring, continue total for about
two hours, the interval from the first contact to the last being about
two hours more. This depends upon the fact that the moon’s hourly
motion is nearly equal to its own diameter. The whole interval from
first contact to last is the time occupied by the moon in moving from

ECLIPSES.

237
a to d (Fig. 126). The totality lasts while it moves from b to c. The
duration of a non-central eclipse varies, of course, according to the
part of the shadow through which the moon passes.

Fig. 126.—Duration of a Lunar Eclipse.

374. Lunar Ecliptic Limit.—The lunar ecliptic limit is the
greatest distance from the node of the moon’s orbit at which the
sun can be consistent with having an eclipse. This limit depends
upon the inclination of the moon’s orbit, which varies a little, and
also upon the radius of the shadow at the time of the eclipse and
the moon’s apparent semi-diameter, which quantities are still more
variable. Hence we recognize two limits, the major and minor. If the
distance of the sun from the node at the time of full moon exceeds
the major limit, an eclipse is impossible; if it is less than the minor,
an eclipse is inevitable. The major limit is found to be 12◦ 5 ; the
minor, 9◦ 30 . Since the sun passes over an arc of 12◦ 5 in less than
thirteen days, it follows that an eclipse of the moon cannot possibly
take place more than thirteen days before or after the time when
the sun crosses the node.

Fig. 127.—Lunar Ecliptic Limit.

375. In Fig. 127 let N E be the ecliptic, and N M the orbit of the
moon, the point N being the node, and the angle at N the inclination
of the moon’s orbit. E is the centre of the earth’s shadow. The sun,
of course, is directly opposite, and its distance from the opposite node
is equal to EN . M is the centre of the moon. Call the semi-diameter

ECLIPSES.

238
of the moon S ; then EM (the greatest possible distance between E
and M which permits an eclipse) equals the sum of the semi-diameters
of the moon and shadow, or S + (P + p − S), and the corresponding
ecliptic limit EN is found by solving the spherical triangle M N E, having
given M E and the angle at N , which is about 5 1 ◦. We must also know

4
one other angle, and with sufficient approximation for such purposes we
may regard the angle at M as a right angle. The solution will give the
value of the ecliptic limit by assigning the proper values to the quantities
involved. The limit is always very nearly eleven times EM , because the
inclination of the moon’s orbit is nearly 1 of a “radian.”

11

376. Phenomena of a Total Lunar Eclipse.—Half an hour
or so before the moon reaches the shadow its eastern limb begins
to be sensibly darkened, and the edge of the shadow itself, when
it is first reached, looks nearly black by contrast with the bright
parts of the moon’s surface. To the naked eye the outline of the
shadow appears reasonably sharp; but with even a small telescope
it is found to be indefinite and hazy, and with a large instrument
and high magnifying power it becomes entirely indistinguishable.

It is impossible to determine the exact moment when the edge of
the shadow reaches any particular point on the moon within half a
minute or so.

Fig. 128.—Light bent into Earth’s Shadow by Refraction.

After the moon has wholly entered the shadow her disc is usually
still distinctly visible, illuminated with a dull, copper-colored light,
which is sunlight, deflected around the earth into the shadow by
the refraction of our own atmosphere, or rather by that portion of
our atmosphere which lies within ten or fifteen miles of the earth’s
surface. Since the ordinary horizontal refraction is 34 , it follows
that light which just grazes the earth’s surface will be bent inwards
by twice that amount, or 1◦ 8 . Now, the maximum “radius of the
shadow” is only 1◦ 2 . In an extreme case, therefore, even when the
moon is exactly central in the largest possible shadow, it receives
some sunlight coming around the edge of the earth, as shown by

Fig. 128. To an observer stationed on the moon, the disc of the

ECLIPSES.

239
earth would appear to be surrounded by a narrow ring of brilliant
sunshine, colored with sunset hues by the same vapors which tinge
terrestrial sunsets, but acting with double power because the light
has traversed a double thickness of our air. If the weather happens
to be clear at this portion of the earth (upon its rim as seen from the
moon), the quantity of light transmitted through the atmosphere
is very considerable, and the moon is strongly illuminated. If, on
the other hand, the weather happens to be stormy in this region,
the clouds cut off nearly all the light. In the lunar eclipse of 1886
the moon was absolutely invisible to the naked eye, a very unusual
circumstance on such an occasion. At the eclipse of Jan. 28, 1888,

Pickering found that the photographic power of the centrally eclipsed
moon was about

1
that of the moon when uneclipsed.

400 000

377. Uses made of Lunar Eclipses.—In astronomical impor-
tance a lunar eclipse cannot be at all compared with a solar eclipse. It
has its uses, however. a. Many dates in chronology are fixed by reference
to certain lunar eclipses. For instance, the date of the Christian era is
determined by a lunar eclipse which happened upon the night that Herod
died. b. Before better methods were devised, lunar eclipses were made
use of to some extent in determining the longitude. Unfortunately, as
has been said (Art. 119), it is impossible to note the critical instants with
any degree of accuracy, on account of the indefiniteness of the moon’s
shadow. c. The study of the spectrum of the eclipsed moon gives us
some data as to the constitution of our own atmosphere. We are thus
enabled to examine light which has passed through a greater thickness
of air than is obtainable in any other way. d. The study of the heat radi-
ated by the moon during the different phases of an eclipse gives us some
important information as to the absorbing power and temperature of its
surface. Observations have been made at Lord Rosse’s observatory of all
the recent lunar eclipses, with this end in view. e. Finally, at the time
when the moon is eclipsed, it is possible to observe its passage over small
stars which cannot be seen at all when near the moon except at such a
time. Observations of these star occultations made at different parts of
the earth, furnish the best possible data for computing the dimensions
of the moon, its parallax, and for determining its precise position in its
orbit at the time of observation. The eclipses of the last few years have
been very carefully observed in this way by concert between the different
leading observatories.

378. Computation of a Lunar Eclipse.—Since all the
phases of a lunar eclipse are seen everywhere at the same abso-

ECLIPSES.

240
lute instant wherever the moon is above the horizon, it follows that
a single computation giving the Greenwich times of the different
phenomena is all that is needed, and can be made and published
once for all. Each observer has merely to correct the predicted time
by simply adding or subtracting his longitude from Greenwich in
order to get the true local time. The computation is very simple,
but lies rather beyond the scope of this work.

ECLIPSES OF THE SUN.

379. Dimensions of the Moon’s Shadow.—By the same
method as used for the shadow of the earth (merely substituting in
the formulæ the radius of the moon for that of the earth), we find
that the length of the moon’s shadow at any time is

1
of its

399.55
distance from the sun, and at new moon averages 232,150 miles. It
varies not quite 4000 miles each way, and so ranges from 236,050 to

228,300. The semi-angle of the moon’s shadow is practically equal
to the semi-diameter of the sun at the earth, or very nearly 16 .

Fig. 129.—The Moon’s Shadow on the Earth.

380. The Moon’s Shadow on the Earth’s Surface.—Since
the mean length of the shadow is less than the mean distance of
the moon from the earth (which is 238,800 miles), it is obvious
that on the average it will not reach the earth. On account of the
eccentricity of the moon’s orbit however, our satellite is much of the
time considerably nearer than this mean distance, and may come
within 221,600 miles from the earth’s centre, or about 217,650 miles
from its surface. The shadow, also, under favorable circumstances,
may have a length of 236,050 miles. Its point may therefore at
times extend nearly 18,400 miles beyond the earth’s surface. The
cross-section of the shadow where the earth’s surface cuts it (at o in

Fig. 129) will then be 167 miles. This is the largest value possible.

Of course, if the shadow strikes obliquely on the surface of the earth,
as it must except when the moon is in the zenith, the shadow spot will
be oval instead of circular, and the length of the oval along the earth’s
surface may much exceed the true cross-section of the shadow.

ECLIPSES.

241

381. The “Negative” Shadow.—Since the distance of the
moon may be as great as 252,970 miles from the earth’s centre, or
nearly 249,000 miles from its surface, while the shadow may be as
short as 228,300 miles, we may have the state of things indicated by
placing the earth at B in the figure. The vertex of the shadow, V ,
will then fall 21,700 miles short of the surface, and the cross-section
of the “shadow produced ” will have a diameter of 206 miles where
the earth’s surface cuts it. When the shadow falls near the edge
of the earth, this cross-section may be as great as 230 miles. The
shadow-spot which is formed by the intersection of the produced
shadow-cone with the earth’s surface is sometimes called the nega-
tive shadow, because in calculating an eclipse its radius comes out
from the formulæ as a minus quantity in case the shadow does not
reach the earth.

382. Total and Annular Eclipses.—To an observer within
the true shadow cone, that is, between V and the moon in Fig. 129,
the sun will be totally eclipsed; but an observer in the produced
cone beyond V will see the moon projected on the sun, leaving an
uneclipsed ring around it. He will have what is called an annular
eclipse. These annular eclipses are considerably more frequent than
total eclipses—nearly in the ratio of three to two.

Fig. 130.—Width of the Penumbra of the Moon’s Shadow.

383. The Penumbra and Partial Eclipses.—The penumbra
can easily be shown to have a diameter on the line CD (Fig. 129) of
very nearly twice the moon’s diameter. We may take it as having
an average diameter at this point of 4400 miles; but as the earth is
often beyond V , its cross-section at the earth is sometimes as much
as 4800 miles. An observer situated within the penumbra observes
a partial eclipse: if he is near the shadow cone, the sun will be
mostly covered by the moon; but if near the outer limit of the
penumbra, the moon will only slightly encroach on the sun’s disc.

While, therefore, total and annular eclipses are visible as such only

ECLIPSES.

242
by an observer within the narrow path traversed by the shadow-
spot, the same eclipse will be visible as a partial one everywhere
within 2000 miles on either side of the shadow path; and the 2000
miles is to be reckoned perpendicularly to the axis of the shadow.

When, for instance, the penumbra falls, as shown in Fig. 130, the
distance BC measured along the earth’s surface will be over 3000
miles, although BF is only 2000.

384. Velocity of the Shadow and Duration of Eclipses.

—The moon advances along its orbit very nearly 2100 miles an
hour, and were it not for the earth’s rotation this is the rate at
which the shadow would pass the observer. The earth, however,
is rotating towards the east in the same general direction as that
in which the shadow moves, and its surface moves at the rate of
about 1040 miles an hour at the equator. An observer, therefore,
on the earth’s equator, with the moon near the zenith, would be
passed by the shadow with a speed of about 1060 miles per hour

(2100 − 1040); and this is its slowest velocity, which is about equal
to that of a cannon-ball.

Fig. 131.—Track of the Moon’s Shadow, Eclipse of July, 1878.

In higher latitudes, where the velocity of the earth’s rotation
is less, the relative speed of the shadow is higher; and where the
shadow falls very obliquely, as it does when an eclipse occurs near
sunrise or sunset, the advance of the shadow along the earth’s sur-
face may become exceedingly swift,—as great as 4000 or 5000 miles

ECLIPSES.

243
per hour. Fig. 131, which we owe to the courtesy of the publish-
ers of Langley’s “New Astronomy,” shows the track of the moon’s
shadow during the eclipse of July 29, 1878.

385. Duration of an Eclipse.—A total eclipse of the sun
observed at a station near the equator under the most favorable
conditions possible (the shadow-spot having its maximum diameter
of 167 miles), may continue total for seven minutes and fifty-eight
seconds. In latitude 40◦ the duration of totality can barely equal six
and one-quarter minutes. The greatest possible excess of the radius
of the moon over that of the sun is only 1 19 .

An annular eclipse may last for 12m 24s at the equator. The
maximum width of the ring of the sun visible around the moon is

1 37 .

In the observation of an eclipse four “contacts” are noted: the
first when the edge of the moon first touches the edge of the sun;
the second, when the eclipse becomes total or annular; the third,
at the cessation of the total or annular phase; and the fourth, when
the moon finally leaves the disc of the sun. From the first contact
to the fourth the duration may be a little over two hours.

Fig. 132.—Solar Ecliptic Limits.

386. The Solar Ecliptic Limits.—It is necessary, in order
to have an eclipse of the sun, that the moon should encroach on
the cone ACBD (Fig. 132), which envelops earth and sun. In this
case the “true” angular distance between the centres of the sun and
moon—that is, their distance as seen from the centre of the earth—
would be the angle M ES in the figure. This is made up of three
angles: M EF , which equals the moon’s semi-diameter, S ; AES,
the sun’s semi-diameter; S; and F EA. This latter angle is equal
to the difference between CF E and F AE. CF E is the moon’s
horizontal parallax (the semi-diameter of the earth seen from the
moon), and F AE or CAE is the sun’s parallax. F EA, therefore,
equals P − p; and the whole angle M ES equals S + S + P − p.

ECLIPSES.

244

This angle may range from 1◦ 34 13 to 1◦ 24 19 , according to the
changing distances1 of the sun and moon from the earth.

The corresponding distances of the sun from the node, calculated
in the same way as the lunar ecliptic limits (taking the maximum
inclination of the moon’s orbit as 5◦ 19 and the minimum as 4◦ 57 ,
according to Neison), give 18◦ 31 and 15◦ 21 for the major and
minor ecliptic limits.

In order that an eclipse may be central (total or annular) at any
part of the earth, it is necessary that the moon should lie wholly
inside the cone ACBD, as at M . In this case the angle M ES will
be S − S + P − p, and the corresponding major and minor central
ecliptic limits come out 11◦ 50 and 9◦ 55 .

387. Phenomena of a Solar Eclipse.—There is nothing of
special interest until the sun is mostly covered, though before that
time the shadows cast by foliage begin to look peculiar. The light
shining through every small interstice among the leaves, instead
of forming a little circle on the earth, makes a little crescent —an
image of the partly covered sun.

Some ten minutes before totality the darkness begins to be felt,
and the remaining light, coming as it does from the edge of the sun
only, is much altered in quality, producing an effect very like that
of a calcium light rather than sunshine. Animals are perplexed,
and birds go to roost. The temperature falls a few degrees, and
sometimes dew appears.

In a few moments, if the observer is so situated that his view
commands a distant western horizon, the moon’s shadow is seen
coming much like a heavy thunder-storm. It advances with almost

1We give herewith in a table the different quantities which determine the
dimensions of the shadows of the earth and moon, as well as the ecliptic limits
and the duration of eclipses.

Greatest.

Least.

Mean.

Apparent semi-diameter of sun

16 18

15 45

16 02

Apparent semi-diameter of moon

16 46

14 44

15 45

Horizontal parallax of the sun

9 .05

8 .55

8 .80

Horizontal parallax of the moon

61 18

53 58

57 38

Inclination of moon’s orbit

5◦ 19

4◦ 57

5◦ 8 40

Sun’s radius, 433,200 miles; earth’s (mean), 3956; moon’s, 1081.5.

ECLIPSES.

245
terrifying swiftness until it envelops him.

For a moment the air appears to quiver, and on every white
surface bands or “fringes,” alternately light and dark, appear. They
are a few inches wide and from a foot to three feet apart, and on
the whole seem to be parallel to the edge of the shadow. Probably
they travel with the wind ; but observations on this point are as
yet hardly decisive. The phenomenon is not fully explained, but is
probably due to irregular atmospheric refraction of the light coming
from the indefinitely narrow strip of the sun’s limb on the point of
disappearing.

388. Appearance of the Corona and Prominences.—As
soon as the shadow arrives, and sometimes a little before it, the
corona and prominences become visible. The stars of the first three
magnitudes make their appearance at the same time.

The suddenness with which the darkness descends upon the ob-
server is exceedingly striking; the sun is so brilliant that even the
small portion which remains visible up to within a very few seconds
of the time of totality so dazzles the eye that it is not prepared
for the sudden transition. In a few moments, however, the vision
becomes accustomed to the changed circumstances, and it is then
found that the darkness is not really very intense. If the totality
is of short duration,—that is, if the diameter of the moon exceeds
that of the sun by less than a minute of arc,—the lower parts of the
corona and chromosphere, which are very brilliant, give a light at
least three or four times as great as that of the full moon. Since the
shadow also in such a case is of small diameter, a large quantity of
light is sent in from the surrounding air, where thirty or forty miles
away the sun is still shining; and what may seem remarkable, this
intrusion of outside light is greatest when the sky is clouded. In
such an eclipse there is not much difficulty in reading an ordinary
watch-face. In an eclipse of long duration (say five or six minutes)

it is much darker, and lanterns are necessary.

389. Observations of an Eclipse.—A total solar eclipse offers
an opportunity of making an immense number of observations of great
importance which are possible at no other time, besides certain others
which can also be made during a partial eclipse. We mention (a) Times
of the four contacts, and direction of the line joining the cusps during
the partial phases. These observations determine accurately the relative
position of the sun and moon at the time, and so furnish the means for

ECLIPSES.

246
correcting the tables of their motion. (b) The search for intra-mercurial
planets. It has been thought likely that there may be one or more plan-
ets between the orbit of Mercury and the sun, and during a total eclipse
they would become visible, if ever. On the whole, however, the observa-
tions, so far made, negative the existence of any body of considerable size
in this region, though in 1878, Professor Watson and Mr. Swift, it was
thought, had discovered one, if not two, such planets. (c) Observations
on the fringes, which have been described as showing themselves at the
commencement of totality. Probably the phenomenon is merely atmo-
spheric and of little importance, but it is not yet sufficiently understood.

(d ) Photometric measurements of the intensity of the light at different
stages of the eclipse and during totality. (e) Telescopic observations of
the details of the prominences and of the corona. (f ) Spectroscopic ob-
servations, both visual and photographic, upon the spectra of the lower
atmosphere of the sun, the prominences, and the corona. (g) Obser-
vations with the polariscope upon the polarization of the light of the
corona, for the purpose of determining the relation between the reflected
and intrinsic light, and perhaps the size of the reflecting particles which
are distributed through the corona. (h) Photography, both of the partial
phases and of the corona.

390. Calculation of a Solar Eclipse.—The calculation of a
solar eclipse cannot be dealt with in any such summary way as that of
a lunar eclipse, owing to the moon’s parallax, which makes the times of
contact and other phenomena different at every different station. The
moon’s apparent path in the sky, relative to the centre of the sun, is
not even a portion of a great circle, nor is it described with a uniform
velocity. Moreover, since the phenomena of a solar eclipse admit of very
accurate observations, it is necessary to take account of numerous little
details which are of no importance in a lunar eclipse.

Certain data for each solar eclipse hold good wherever the observer
may be. These are calculated beforehand and published in the nautical
almanacs; and from them, with the knowledge of his geographical posi-
tion, the observer can work out the results for his own station. But the
calculations are somewhat complicated and lie beyond our scope. The
reader is referred to any work on practical astronomy; Chauvenet and

Loomis treat the matter very fully.

391. Number of Eclipses in a Year.—The least possible
number is two, both central eclipses of the sun. The largest possible
number is seven, five of the sun and two of the moon. The eclipses
each year happen at two seasons (which may be called the “eclipse
months”), half a year apart—about the times, of course, at which

ECLIPSES.

247
the sun in its annual path crosses the two nodes of the moon’s orbit.

If these nodes were stationary, the eclipse months would be always
the same; but because the nodes retrograde around the ecliptic
once in about nineteen years, the eclipse months are continually
changing. The time required by the sun in passing around from a
node to the same node again is 346.62 days, which is sometimes
called the “eclipse year.”

392. Number
of

Lunar

Eclipses.—Representing the ecliptic
by a circle (Fig. 133) with the two
opposite nodes A and a, it is easy to
see first, that there can he but two
lunar eclipses in a year (omitting for
a moment one exceptional case). The
major lunar ecliptic limit is 12◦ 15 ;
hence there is only a space of twice
that amount, or 24◦ 30 , between L
and L , at each “node month,” within Fig. 133.—Number of Eclipses
which the occurrence of a full moon

Annually.

might give a lunar eclipse. Now, in a
synodic month the sun moves along the ecliptic 29◦ 6 , while the
node moves in the opposite direction 1◦ 31 , giving the relative
motion of the sun referred to the node equal to 30◦ 37 ; i.e., the
full-moon points on the circle would fall at a distance of 30◦ 37
from each other. Only one full moon, therefore, can possibly occur
within the lunar ecliptic limits each time that the sun passes the
node.

Since the minor ecliptic limit for the moon is only 9◦ 30 , it may
easily happen that neither of the full moons which occur nearest
to the time when the sun is at the node will fall within the limit.

There are accordingly many years which have no lunar eclipses.

Three lunar eclipses, however, may possibly happen in one cal-
endar year in the following way. Suppose the first eclipse occurs
about Jan. 1, the sun passing the node about that time; the sec-
ond may then happen about June 25 at the other node, a. The
first node, A, will run back during the year, so that the sun will
encounter it again about Dec. 13 at A , and thus a third eclipse
may occur in December of the same year. This actually occurred
in 1787, the dates of the three lunar eclipses being Jan. 3, June 30,

ECLIPSES.

248
and Dec. 24.

393. Number of Solar Eclipses.—Considering now solar
eclipses, we find that there must inevitably be two. Twice the minor
limit (Art. 386) of a solar eclipse (15◦ 21 ) is 30◦ 42 , which, is more
than the sun’s whole motion in a month. One new moon, at least,
therefore, must fall within the limiting distance of the node, and
two may do so, since in the figure, SS is always greater than the
distance between the points occupied by two successive new moons.

If the two new moons in the two eclipse months happen to fall
very near a node, the two full moons, a fortnight earlier and later,
will both be very likely to fall outside the lunar limit. In that case
the year will have only two eclipses, both solar and both central;
i.e., either total or annular. This was the case in 1886.

Again, if in any year two full moons occur when the sun is very
near the node, then since the major solar limit is 18◦ 31 , it may
happen, and often does, that there will be two partial solar eclipses,
one a fortnight before, the other a fortnight after, each of the lunar
eclipses, and so the year will have three eclipses in each eclipse
month—six eclipses in all, two lunar and four solar. A fifth solar
eclipse may also come in near the end of the year, if the node was
passed about Jan. 15, in the same way that sometimes happens with
a lunar eclipse: the year will then have seven eclipses. This was the
case in 1823. The most usual number of eclipses is four or five.

394. Relative Frequency of Solar and Lunar Eclipses.

—Although, taking the whole earth into account, the solar eclipses
are the most numerous, about in the proportion of four to three,
it is not so with the eclipses visible at any given place. A solar
eclipse can be seen only from a limited portion of the globe, while a
lunar eclipse is visible over considerably more than half the earth,
either at the beginning or end, if not throughout its whole duration;
and this more than reverses the proportion between lunar and solar
eclipses for any given station.

395. Recurrence of Eclipses, and the Saros.—It is not
known how early it was discovered that eclipses recur at a regular
interval of eighteen years and eleven and one-third days (ten and
one-third days, if there happen to be five leap years in the interval);
but the Chaldeans knew the period very well, and called it the

Saros (which means “restitution” or “repetition”), and used it in

ECLIPSES.

249
predicting the recurrence of these phenomena. It is a period of 223
synodic months, which is almost exactly equal to nineteen eclipse
years. The eclipse year is 346d.6201, and nineteen of them equal

6585d.78, while 223 months equal 6585d.32.

The difference is only 46 of a day (about 11 hours) in which time

100
the sun moves 28 . If, therefore, an eclipse should occur to-day at
new moon, with the sun exactly at the node, then after 223 months

(18 years 11 days) a new moon will occur again with the sun only

28 west of the node; so that the circumstances of the first eclipse
will be pretty nearly repeated. It would however occur about eight
hours of longitude further west on the earth’s surface, since the 223
months exceed the even 6585 days by 32 of a day, or 7h 42m.

100

396. Number of Recurrences of a Given Eclipse.—It is
usual to speak of eclipses recurring at this regular interval as “repeti-
tions” of one and the same eclipse. A lunar eclipse is usually thus “re-
peated” 48 or 49 times. Beginning as a very small partial eclipse, with
the sun about 12◦ east of the node, it will be a little larger at its next
occurrence eighteen years later; and after 13 or 14 repetitions the sun
will have come so near the node that the eclipse will have become total.

It will then be repeated as a total eclipse 22 or 23 times, after which
it will become partial again with the sun west of the node, and after

13 more returns as a partial eclipse will finally dwindle away and disap-
pear, having thus recurred regularly once in every 223 months during an
interval of 865 1 years.

2

The same thing happens with the solar eclipses, only since the solar
ecliptic limit is larger than the lunar, a solar eclipse has from 65 to 70
returns, occupying some 1200 years. Of these about 25 are only partial
eclipses, the sun being so near the ecliptic limit that the axis of the
shadow does not reach the earth at all. The 45 eclipses in the middle
of the period are central somewhere or other on the earth, about 18 of
them being total, and about 27 annular. These numbers vary somewhat,
however, in different cases.

397. It is to be noticed that the Saros exhibits not only a close
commensurability of the synodic months with the eclipse years, but
also with the nodical 1 and anomalistic months: 242 nodical months

1The nodical month is the time of the moon’s revolution from one of its
nodes to the same node again, and is equal to 27d.21222; the anomalistic month
is the time of revolution from perigee to perigee again, and equals 27d.55460.

See Arts. 454, 455.

ECLIPSES.

250
equal 6585.357 days; 239 anomalistic months equal 6585.549 days.

This last coincidence is important. The moon at the end of the

Saros of 223 months not only returns very closely to its original
position with respect to the sun and the node, but also with respect
to the line of apsides of its orbit. If it was at perigee originally, it
will again be within five hours of perigee at the end of the Saros. If
this were not so, the time of the eclipse might be displaced several
hours by the perturbations of the moon’s motion, to be considered
later, in Chap. XII.

398. Number of Eclipses in a Single Saros.—The total
number is usually about seventy, varying two or three one way or
the other, as new eclipses come in at the eastern limit and go out
at the western. Of the 70, 29 are usually lunar and 41 solar; and
of the solar, 27 are central, 17 being annular and 10 total. (These
numbers are necessarily only approximate.) It appears, therefore,
that total solar eclipses, somewhere or other on the earth, are not
very rare, there being about ten in eighteen years. Since, however,
the shadow track averages less than 100 miles in width, each total
eclipse is visible, as total, over only a very small fraction of the
earth’s whole surface—about 1 in the mean. This gives about one

200
total eclipse in 360 years, in the long run, at any given station.

The total solar eclipses visible in the United States during the present
century are the following:—

June 10, 1806, in New York and New England, duration 4 1 minutes;

2

Nov. 30, 1834, in Arkansas, Missouri, Alabama, and Georgia, duration

2 minutes; July 18, 1860, in Washington Territory and Labrador, 3 min-
utes; Aug. 7, 1869, in Iowa, Illinois, Kentucky, North Carolina, 2 3 min-

4
utes; July 29, 1878, in Wyoming, Colorado, Texas, 2 1 minutes; Jan. 11,

2

1880, in California, duration 32 seconds; Jan. 1, 1889, in California and

Montana, 2 1 minutes; May 27, 1900, from Louisiana to Virginia, 2 min-

4
utes.

399. Occultations of Stars.—In theory, and in the method
of computation, the occultation of a star is precisely like a solar
eclipse, except that the shadow of the moon projected by a star
is a cylinder instead of a cone, since, compared with the distance
of the sun, that of a star is infinite: moreover, the star is a mere
point, so that there is no sensible penumbra. In other words, a star
has neither parallax nor semi-diameter, and these circumstances
somewhat simplify the formulæ.

ECLIPSES.

251

As the moon moves always towards the east, the disappearance
of the star always takes place at the eastern limb, and the reap-
pearance at the western. In the first half of the lunation the eastern
limb is dark and invisible, and the star vanishes without warning.

The suddenness with which it vanishes and reappears has already
been referred to (Art. 255) as proof of the non-existence of a lunar
atmosphere. Observations of this sort determine the moon’s plane
with great accuracy, and when corresponding observations are made
at different places, they supply one of the best possible means of
determining their difference of longitude.

In some cases observers have reported that a star, instead of disap-
pearing instantaneously when struck by the moon’s limb (faintly visible
by earth-shine), has appeared to cling to the limb for a second or two
before vanishing, and in a few instances they have even reported it as
having reappeared and disappeared a second time, as if it had been for a
moment visible through a rift in the moon’s crust. Some of these anoma-
lous phenomena have been explained by the subsequent discovery that
the star was double, or had a faint companion.

CENTRAL FORCES.

252

CHAPTER XI.

CENTRAL FORCES: EQUABLE DESCRIPTION OF AREAS.—

AREAL, LINEAR, AND ANGULAR VELOCITIES.—KEPLER’S

LAWS AND INFERENCES FROM THEM.—GRAVITATION

DEMONSTRATED BY THE MOON’S MOTION.—CONIC SEC-

TIONS AS ORBITS.—THE PROBLEM OF TWO BODIES.—

THE “VELOCITY FROM INFINITY,” AND ITS RELATION

TO THE SPECIES OF ORBIT DESCRIBED BY A BODY

MOVING UNDER GRAVITATION.—THE INTENSITY OF

GRAVITATION.

400. A moving body left to itself, according to Newton’s first
law of motion (Physics, p. 27), moves on forever in a straight line
with a uniform velocity. If we find a body so moving, we may,
therefore, infer that it is either acted on by no force whatever, or,
if forces are acting upon it, that they exactly balance each other.

It has been customary with some writers to speak of a body thus
moving “uniformly in a straight line” as actuated by a “projectile force,”

a very unfortunate expression, which is a survival of the Aristotelian idea
that rest is more “natural” to matter than motion, and that when a body
moves, some force must operate to keep it moving. The mere uniform
rectilinear motion of a material mass in empty space implies no physical
cause, and demands no explanation. Change of motion, either in speed
or in direction—this alone implies force in operation.

401. If a body moves in a straight line,
with swiftness either increasing or decreas-
ing, we infer a force acting exactly in the
line of motion, and accelerating or retard-
ing it. If it moves in a curve, we know that
some force is acting athwart the line of mo-

Fig. 134.—Curvature of
tion. If the velocity in the curve increases,
an Orbit.

we know that the direction of the force that
acts is forward, like ab (Fig. 134), making an angle of less than 90◦
with the “line of motion” a t (the tangent to the path of the body);

CENTRAL FORCES.

253
and vice versa, if the motion of the body is retarded. If the speed
does not alter at all, we know that the force must act along the line
ac, exactly perpendicular to the line of motion.

Here, also, we find many writers, the older ones especially, bringing
in the “projectile force,” and saying that when a body moves in a curve it
does so under the action of two forces, one a force that draws it sideways,
the other the “projectile force” directed along its path. We repeat; this

“projectile force” has no present existence or meaning in the problem.

Such a force may have put the body in motion long ago, but its function
has ceased, and now we have only to do with the action of one single
force,—the deflecting force, which alters the direction of the body’s mo-
tion. Of course it is not intended to deny that the deflecting force may
itself be the resultant of any number of forces all acting together; but a
single force acting athwart a body’s line of motion is sufficient to cause
it to describe a curvilinear orbit, and from such an orbit we can only
infer the necessary existence of one such force.

402. Description of Areas.—

(a) If a body is moving uniformly on
a straight line, and if we connect the
points A, B, C, etc., Fig. 135, which
it occupies at the end of successive
units of time with any point what-
ever, as O, we shall have a series of
triangles, AOB, etc., which will all be

Fig. 135.

Description of Areas in Uniform
equal ; since their bases AB, BC, etc.,

Motion.

are equal and on the same straight
line, and they have a common vertex at O. Calling the line from A
to O its radius vector, and O the “centre,” we may say, therefore,
that when a body is moving undisturbed by any force whatever,
its radius vector, from any centre arbitrarily chosen, describes equal
areas in equal times around that centre. The area enclosed in the
triangle described by the radius vector in a unit of time is called the
body’s “areal (or “areolar ”) velocity,” and in this case is constant.

403. (b) Moreover, any impulse in the line of the radius vector,
either towards or from the centre, leaves unchanged both the plane
of the body’s motion and its areal velocity.

Suppose a body moving uniformly on the line AC (Fig. 136) with
such a velocity that it describes AB, BC, etc., in successive units
of time; then, by the preceding section, the areal velocity will be

CENTRAL FORCES.

254
constant, and measured by the area of any one of the equal triangles

AOB, BOC, or COL. Suppose, now, that at C the body receives
an “impulse” directed along the radius vector towards O—a blow,
for instance, which by itself would make the body describe CK in a
unit of time. The body will now take a new path, which will carry
it to the point D, determined by constructing the “parallelogram
of motions” CKDL, and thus combining the new motion CK with
the former motion CL, according to Newton’s second law (Physics,
p. 27). The new areal velocity, measured by the triangle OCD, will
be the same as before, as is easily shown.

Fig. 136.—Description of Areas under an Impulse directed towards a Centre.

Triangle BOC = triangle COL, because BC = CL, and O is
their common vertex.

Also, triangle COL = triangle COD, because they have the
common base OC, and their summits L and D are on a line which
was drawn parallel to this base in constructing the parallelogram
of motions. Hence triangle BOC = triangle COD, and the areal
velocity remains unchanged.

Also, as may be seen by following out the same reasoning with

CK and CD , the same result would hold true if the impulse had
been directed away from O instead of towards it.

404. This result depends entirely on the fact that the impulse CK
or CK was exactly along the radius vector CO. If it has not been so,
then in constructing the parallelogram of motions to find the points D

CENTRAL FORCES.

255
and D , we should have had to draw LD or LD not parallel to CO, and
the two triangles BOC and COD would necessarily have been unequal.

COD would be greater than BOC if CK were directed ahead of the
radius vector, and less if behind it.

As regards the plane of motion, the point D is on the plane

OCL, because LD was drawn through L parallel to OC. OCL is a
part of the plane which contains the triangles BOC and AOB, and
hence OCD also lies in the same plane.

405. (c) From this obviously follows the important general
proposition that when a body is moving under the action of a force
always directed towards or from a fixed centre, the radius vector will
describe equal areas in equal times; and the path of the body will all
lie in one plane.

Such a force constantly acting is simply equivalent to an indef-
inite number of separate impulses. Now if no single impulse di-
rected along the radius vector can alter the areal velocity or plane
of motion, neither can a succession of them. Hence the proposition
follows.

In case of a continuously acting force the orbit, however, will
become a curve instead of being a broken line.

Observe that this proposition remains true whether the force
is attractive or repulsive, and that it is independent of the law of
the force; that is, the force may vary directly with the distance, or
inversely as the square of the distance, or as the logarithm of it, or
in any conceivable way; it may even be discontinuous, acting only
at intervals and ceasing between times: and still the law holds good.

406. Conversely, if a body moves in this way, describing equal
areas in equal times around a point, it is easily shown that all the
forces acting upon the body must be directed toward that point.

We, however, leave the demonstration to the student.

Since the earth moves very nearly in this way in its orbit around
the sun, we conclude that the only force of any consequence acting
upon the earth is directed towards the sun. We say, “of any con-
sequence,” because there are other small forces which do slightly
modify the earth’s motion, and prevent it from exactly fulfilling the
law of areas.

As a direct consequence of the law of equal areas we have certain

CENTRAL FORCES.

256
laws with respect to the linear and angular velocities of a body
moving under the action of a central force.

407. Law
of

Linear

Velocity.—

Suppose a body moving under the action of
a force always directed towards S (Fig. 137),
and let AB be a portion of its path which
it describes in a second. Draw the tangent

Bb.

Regarding the sector ASB as a tri-
angle (which it will be, sensibly, since the
curvature of the path in one second will be

Fig. 137.

very small) the area of this triangle will be

Linear and Angular

1

Velocities.

(AB × Sb). Now AB, the distance trav-

2
elled in a second, is the linear velocity of the
body (called linear because it is measured with the same units as
any other line; i.e., in miles or in feet per second), and Sb is the
distance from the centre of force to the “line of motion,” as the tan-
gent Bb is called. For Sb, p is usually written; hence in every part

2A
of the same orbit, V (the velocity in miles per second) =

, and
p
is inversely proportional to p. If p were to become zero, V would
become infinite, unless A were zero also.

408. Law of Angular Velocity.—Referring again to the same
figure, the area of ASB is equal to 1 (AS × BS × sin ASB), or

2

A = 1 r

2 1r2 sin ω.

If we draw r to the middle point of AB, then
r1r2 = r2, nearly, since in a second of time the distance would not
change perceptibly as compared with its whole length. ω will also
be a small angle, so that its sine will equal the angle itself expressed
in radians;

2A
hence

1 r2ω = A, and ω =

.

2
r2

Now ω is the angular velocity of the body; that is, the number of

“radians” which it describes in a second of time, as seen from S,
while r is the radius vector.

409. In every case, therefore, of motion under a central force,

I. The Areal velocity (acres per second) is constant ; II. The Linear
velocity (miles per second) varies inversely as the distance from the
centre of force to the body’s line of motion at the moment, which
line of motion is the tangent to the orbit at the point where the

CENTRAL FORCES.

257
body happens to be; III. The Angular velocity (radians, or degrees,
per second) varies inversely as the square of the distance of the body
from the centre of force.

410. The student will remember that it was found by obser-
vation that the sun’s angular velocity varies as the square of its
apparent diameter, and from this (Art. 186) the law of equal areas
was inferred as a fact with respect to the earth’s motion. Newton
was the first to point out that a body moving under the action of
a central force must necessarily observe this law of areas, and, con-
versely, that a body thus observing the law of areas must necessarily
be under the control of a central force.

411. Circular Motion.—In the case of a body moving in a
circle under the action of a central force, the force must be constant,
and (Physics, p. 62) is given by the formula

V 2
f =

,

(a)

r
in which r is the radius of the circle and V the velocity, while f is the
central force measured by the “acceleration” of the body towards
the centre; that is, by the number of units of velocity which the
force would generate in the body in a second of time; just as the
force of gravity is expressed by writing, g = 9.81 metres.

For many purposes it is desirable to have an expression which
shall substitute for V (a quantity not given directly by observation)

the time of revolution, t, which is so given. Since V equals the

2πr
circumference of the circle divided by t, or

, we have at once,
t
by substituting this value for V in equation (a),
r
f = 4π2

.

(b)

t2

This, of course, is merely the equivalent of equation (a), but is
often more convenient.

KEPLER’S LAWS.

412. In 1607–1620 Kepler discovered as facts, without an expla-
nation, three laws which govern the motions of the planets,—laws
which still bear his name. He worked them out from a discussion
of the observations which Tycho Brahe had made through many
preceding years. The three laws are as follows:—

CENTRAL FORCES.

258

I. The orbit of each planet is an ellipse, with the sun in one
of its foci.

II. The radius vector of each planet describes equal areas in
equal times.

III. The “Harmonic law,” so-called. The squares of the periods
of the planets are proportional to the cubes of their mean distances
from the sun; i.e., t2 : t2 = a3 : a3.

1

2

1

2

413. To make sure that the student apprehends the meaning and
scope of this third law, we append a few simple examples of its applica-
tion.

(1) What would be the period of a planet having a mean distance
from the sun of 100 astronomical units; i.e., a distance 100 times that of
the earth?

(Earth’s Dist.)3 : (Planet’s Dist.)3 = (Earth’s Period)2 : (Planet’s Period)2;
i.e.,

13 : 1003 = 12(year) : X2(years),

3
whence

X = 100 2 = 1000 years.

(2) What would be the distance of a planet having a period of 125
years?

(1)2 : 1252 = 13 : X3,

2
whence

X = 125 3 = 25 (Astron. units).

(3) How long would a planet require to fall to the sun?

If the sun were collected in a single point at its centre, a body start-
ing from a point on the planet’s orbit with a slight side-motion, i.e.,
motion at right angles to the radius vector, would describe an extremely
narrow ellipse around the sun, with its perihelion just at the sun, and
the aphelion at the starting-point. Practically it would “fall to the sun,”

and return just as if it had rebounded from a perfectly elastic surface:

the time of “falling” would be just equal to that of returning—the two
making up the whole period of the body in the narrow ellipse. Now the
semi-major axis of this narrow ellipse is evidently one-half the radius of
the planet’s orbit. Hence to find the period in this ellipse which is 2τ (τ
being taken as the time of “falling”), we have
a3 : ( 1 a)3 = t2 : (2τ )2, or 1 : 1 = t2 : 4τ 2;

2

8

1
whence
τ = t

= 0.1768t,
t being the planet’s period.

32

CENTRAL FORCES.

259

In the case of the earth τ = 365 1 × 0.1768 = 64.56 days.

4

(4) What would be the period of a satellite revolving close to the
earth’s surface?

(Moon’s Dist.)3 : (Dist. of Satellite)3 = (27.3 days)2 : X2,
or

603 : 13 = (27.3)2 : X2,

27.3 days
whence

X =

= 1h 24m.

3

60 2

(5) How much would an increase of 10 per cent in the earth’s distance
from the sun increase the length of the year?

Ans. 56.13 days.

(6) What is the distance from the sun of an asteroid which has a
period of 3 1 years?

2

Ans. 2.305 Astron. units.

414. Many surmises were made as to the physical meaning of
these laws. More than one astronomer guessed that a force directed
toward the sun, or emanating from it, might be the explanation.

Newton proved it. He demonstrated the law of equal areas and its
converse as necessary consequences of the laws of motion. He also
proved (Physics, pp. 64–66) that if a body move in an ellipse having
a centre of force at its focus, then the force at different points in the
orbit must vary inversely as the square of the distance from that
centre. And, finally, he showed that, granting the harmonic law, the
force from planet to planet must also vary according to the same
law of inverse squares.

415. The demonstration of this last proposition for circular orbits
is so simple that we give it, merely adding (without proof) that the
proposition is equally true for elliptical orbits, if for r we put a, the
semi-major axis of the orbit.

In a circular orbit, from equation (b), (Art. 411), we have
r
f = 4π2

,
t2
where r and t are the distance and period of a planet. In the same way
the force acting upon a second planet is found from the equation
r1
f1 = 4π2

,
t 2

1
f
r
t 2

1
whence

=

×

.

f1
t2
r1

CENTRAL FORCES.

260

But by Kepler’s third law
t2 : t 2

3

1

:: r3 : r1 ,
t2r 3

1
whence
t 2

1

=

.

r3

Substitute this value of t 2

1
in the preceding equation; we have
f
r
t2r 3

2

1
r1

=

×

=

;
f1
t2
r3r1
r2
i.e.,
f : f

2

1 = r1

: r2,
which is the law of inverse squares.

416. Conversely, the harmonic law is just as easily shown to be a
necessary consequence of the law of gravitation in the case of circular
orbits.

From Art. 411, Eq. (b), we have
r
f = 4π2

;
t2
also, from the law of gravitation,

M
f =

,

M being the mass of the sun.

r2

Hence, equating the two values of f ,

M
r

4π2

= 4π2

,
and
t2 =
r3.

r2
t2

M

Similarly for another planet,

4π2
t 2

3

1

=
r1 .

M

Whence,
t2 : t 2

3

1

= r3 : r1 .

The demonstration for elliptical orbits is a little more complicated,
involving the “law of areas.” It is given in all works on Theoretical

Astronomy, and may be found in Loomis’s “Elements of Astronomy,”

p. 134.

CENTRAL FORCES.

261

417. Correction of Kepler’s Third Law.—The “harmonic
law” as it stands is not exactly true, though the difference is too small
to appear in the observations which Kepler made use of in its discovery.

It would be exactly true if the planets were mere particles of matter; but
as a planet’s mass is a sensible, though a very small fraction of the sun’s
mass, it comes into account. The planet Jupiter, for instance, attracts
the sun as well as is attracted by it. If at the distance r Jupiter is
drawn towards the sun by a force which would give it in a second an
acceleration expressed by M (the sun’s mass being M ), then the sun in
r2
the same time is accelerated towards Jupiter by the quantity m (m being
r2
the mass of Jupiter). The rate at which the two tend to approach each
other is therefore expressed by M+m . Hence, in discussing the motions
r2
of the planet Jupiter around the centre of the sun, instead of writing

M
f =
simply,
r2

M + m
we must put
f =

;
r2
but (in the case of entirely circular motion)

4π2r
f =

.

t2

Hence, we find
t2(M + m) = 4π2r3;
or, as a proportion,
t2(M + m) : t 2

3

1 (M + m1) = r3 : r1 ,
which is strictly true as long as the planet’s motions are undisturbed.

418. Inferences from Kepler’s Laws.—From Kepler’s laws
we are entitled to infer—

First (from the second law), that the force which retains the
planets in their orbits is directed towards the sun.

Second (from the first law), that on any given planet the force
which acts varies inversely as the square of its distance from the
sun.

Third (from the harmonic law), that the force is the same for
one planet as it would be for another in the same place; or, in other

CENTRAL FORCES.

262
words, the attracting force depends only on the mass and distance
of the bodies concerned, and is wholly independent of their physical
conditions, such as their temperature, chemical constitution, etc.

It makes no difference in the motion of a planet around the sun
whether it be made of hydrogen or iron, whether it be hot or cold.

419. Verification of “Gravita-
tion” by Means of the Moon’s Mo-
tion.—When the idea of gravitation
first occurred to Newton (the apple
story is probably apocryphal), he en-
deavored to verify it by comparing the
force which keeps the moon in her orbit
with the force of gravity at the earth’s
surface, reduced in the proper propor-
tion. For lack, however, of an accurate
knowledge of the earth’s dimensions, he
failed at first, there being a discrepancy

Fig. 138.

Verification of the Hypothesis
of about sixteen per cent. He had as-
of Gravitation by Means of
sumed a degree to be exactly sixty miles
the Motion of the Moon.

in length. Some years afterward, when

Picard’s measure of the arc of a meridian in Northern France had
been made and reported to the Royal Society, making a degree
about sixty-nine miles long, he saw at once that the new value would
reconcile the discrepancy; and he resumed his unfinished work and
completed it.

420. At the earth’s surface a body falls about 193 inches in a
second. The distance of the moon being very nearly sixty times the
earth’s radius, if gravity really varies inversely as the square of the

1
distance, a stone at that distance from the earth should fall
or

602

1

193 inches
as far; that is, it ought to fall

= 0.0535 inches,—a

3600

3600
little more than one-twentieth of an inch. Now the distance which
the moon actually does fall towards the earth in a second, i.e., the
deflection of its orbit from a straight line in a second of time, is
easily found; and if the force which keeps the moon in its orbit is
really the same as that which makes bodies fall towards the centre
of the earth, this deflection ought to come out equal to 0in.0535.

Let AE (Fig. 138) be the distance the moon travels in a second

CENTRAL FORCES.

263

2πr

=

, where r is the radius of the moon’s orbit, and t the number
t
of seconds in a month. Then, since AEF is a right-angled triangle,
we have,

AB : AE :: AE : AF (or 2r);

AE2
whence

AB =

.

2r

The calculation is easy enough, though the numbers are rather large.

As a result it gives us AB = 0.0534 inches, which is practically equal
to the thirty-six hundredth part of 193 inches.

If the quantities did not agree in amount, the discrepancy would
disprove the theory, and, as we have said, Newton loyally gave it
up until he was able to show that the apparent discordance was the
result of a mistake in the original data, and disappeared when the
data were corrected. The agreement, however, does not establish
the theory, but only renders it probable. It does not establish it
completely, because it is conceivable that the agreement might be
a case of accidental coincidence, while the forces might really differ
as much in their nature as an electrical attraction and a magnetic.

421. Newton was not satisfied with merely showing that the
principal motions of the planets and the moon could be explained
by the law of gravitation; but he went on to investigate the converse
problem, and to determine what must be the motions necessary
under that law. He found that the orbit of a body moving around
a central mass is not of necessity a circle, or even a nearly circular
ellipse like the planetary orbits, but that it may be a conic section
of any eccentricity whatever—a circle, ellipse, parabola, or even an
hyperbola; but it must be a conic.

422. For the benefit of those of our readers who are not ac-
quainted with conic sections we give the following brief account of
them (Fig. 139):—
a. If a cone of any angle be cut perpendicularly to the axis, the
section will be a circle—M N in the figure.

b. If it be cut by a plane which makes with the axis an angle
greater than the semi-angle of the cone, so that the plane of section
cuts completely across the cone (as EF ), the section, is an ellipse;
the circle being merely a special case of the ellipse. Ellipses, of
course, differ greatly in form, from these which are very narrow to
the perfect circle.

CENTRAL FORCES.

264
c. The parabola is formed by cutting
the cone with a plane parallel to its side;
i.e., making with the axis an angle equal
to the semi-angle of the cone. RP O is
such a plane. As all circles are alike in
form, so are all parabolas, whatever the
angle of the cone at V and wherever the
point P is taken. If the cutting plane is
thus situated, then, no matter what is
the angle of the cone or the place where
the cut is made, the curve will always be
the same in shape, though of course its
size will depend upon a variety of cir-
cumstances. We do not stop to prove
the statement, at first a little surprising;
but it is true.

d. If the cutting plane makes an an-
gle with the axis of the cone less than
the semi-angle at V , so that the cutting
plane gets continually deeper and deeper
into the cone, then the curve is an hy-

Fig. 139.—The Conics.

perbola; so called, because the plane in
this case “shoots over” (ὑπέρ βάλλειν) and intersects the “cone pro-
duced,” cutting out of this second cone a curve precisely like the
curve cut from the original, as at H G K in the figure. The axis of
the hyperbola lies outside of the curve itself, being the line HH in
the figure, and the “centre” of the curve is also outside of the curve
at the middle point of this axis.

423. Philosophically speaking there are therefore but two
species of conic sections,—the ellipse and the hyperbola, with the
parabola for a partition between them. (The circle, as has been said
before, is merely a special case of the ellipse.) Fig. 140 will give the
reader perhaps a better idea of the nature of the curves as drawn on
a plane. In the ellipse the sum of the distances from the two foci,

F N + F N , equals the major axis of the curve; in the hyperbola it
is the difference of these two lines (F N − F N ) that equals the
major axis; in the ellipse the eccentricity is less than unity (zero in
the circle); in the hyperbola it is greater than unity; in the parabola
exactly unity.

CENTRAL FORCES.

265

Fig. 140.—The Relation of the Conics to Each Other.

The general equation of a conic in polar co-ordinates, applying alike
to both the species, is
p
r =

,

1 + e cos V

F C

F C
in which r is the distance F n, or F n , e is the fraction
or

, the

P C

P C
angle V is the angle P F n, P F n , or P F n , and p is the line F Y , F Y
or F Y , called the “semi-parameter.” The word “parameter” means the
cross-measure of a curve, just, as “diameter” means the through-measure
of a curve. If e is zero, the curve is a circle, and r = p. If e < 1, the
curve is an ellipse; if e > 1, the curve is an hyperbola; if e = 1, it is a
parabola.

424. Problem of Two Bodies.—This problem, proposed and
solved by Newton, is the following:—

Given the masses of two spheres and their positions and motions
at any moment; given, also, the law of gravitation: required their
motion ever afterwards, and the data necessary to compute their
place at any future time.

The mathematical methods by which the problem is solved re-
quire the use of the calculus, and must be sought in works on an-
alytical mechanics or theoretical astronomy. Some of the results,
however, are simple and easily stated.

425. (1) In the first place the motion of the centre of gravity of
the two bodies will not be affected by their mutual attraction, but it

CENTRAL FORCES.

266
will move on uniformly through space, as if the bodies were united
into one at that point, and their motions combined under the same
laws which hold good in the case of the collision of inelastic bodies.

Fig. 141.—Motion of Bodies relative to their Centre of Gravity.

The motion of this centre of gravity is most easily worked out graph-
ically as follows: First, in Fig. 141, join the original places of the bodies A and B by a straight line, and mark on it G, the place of the centre of
gravity; then take the positions A and B they would occupy at the end
of a unit of time (if they did not attract each other), and mark the new
position of the centre of gravity G on the line joining them. The line

GG connecting the two positions of the centre of gravity will show the
direction and rapidity of its motion; with reference to this point the two
bodies will have opposite motions proportional to their distances from
it; that is, they will swing around this point as if on a rod pivoted there,
and will either both move towards it along the rod, or from it, with
speeds inversely proportional to their masses. These relative motions
with respect to the centre of gravity are easily found by drawing through

G a line parallel to A B and measuring off on it distances GA and GB
respectively equal to G A and G B . AA and BB will then be the two
motions of A and B relative to their centre of gravity G.

426. The Effect of their Mutual Attraction.—This will
cause them to describe similar conics around this centre of gravity;
the size of their two orbits being inversely proportional to their
masses. The form of the orbits and dimensions will be determined
by the combined mass of the two bodies, and by their velocities
with respect to the common centre of gravity.

427. The Orbit of the Smaller relative to the Centre of
the Larger.—It is convenient (though it is not necessary) to drop
the consideration of the centre of gravity of the two bodies, and
to consider the motion of the smaller one around the centre of the

CENTRAL FORCES.

267
larger one. In reference to that point, it will move precisely as if
its mass had been added to that of the larger body, while itself had
become a mere particle. This relative orbit will in all respects be
like the actual one around the centre of gravity, only magnified in
the proportion of M + m to M ; i.e., if m is 1 of M , the actual

10
orbit around G will be magnified by 11 to produce the relative orbit

10
around M .

Fig. 142.—Elliptical Orbit determined by Projection.

428. The Orbit determined by Projection.—Suppose that
in the figure (Fig. 142) the body P is moving in the direction of
the arrow, and is attracted by S, supposed to be at rest. P will
thenceforward move in a conic, either in an ellipse or hyperbola,
according to its velocity, as we shall see in a moment. S being at
one focus of the curve, the other focus will be somewhere on the
line P N , which makes the same angle with P Q that r (SP ) does

(since it is a property of the conics that a tangent-line at any point
of the curve makes equal angles with the lines drawn from the two
foci to that point). If we can find the place of the second focus F ,
or the length of the line P F in the figure, the curve can at once be
drawn.

Now, it can be proved, though the demonstration lies beyond
our scope, that a, the semi-major axis of the conic, is determined
by the equation

2

1

V 2 = µ

−

,

(Equation 1)

r
a
in which r is the distance SP , V is the velocity, and µ is the at-
tracting mass at S expressed in proper units.

CENTRAL FORCES.

268

(See Watson’s “Theoretical Astronomy,” p. 49; only for µ he writes
k2(1 + m)).

V , r, and µ being given, of course a can be found: we get
r
a = µ

.

(Equation 2)

2µ − rV 2

Then by subtracting r from 2a we shall get r , or the distance

P F , if the curve is an ellipse. If it is a hyperbola, a will come out
negative; and to find r we must take r = 2a + r and measure it
off to F , on the other side of the line of motion. In either case,
however, we easily find the other focus, and the line drawn through
the foci will be the line of apsides; a point half-way between the
foci will be the centre of the curve, and any line drawn through this
centre will be a diameter. Having the two foci and the major axis

2a, i.e., AA the curve can at once be drawn.

429. Expression for a in Terms of the “Velocity from

Infinity,” or “Parabolic Velocity.”—The expression for a ad-
mits of a more convenient and very interesting form. It is shown in
analytical mechanics that if, under the law of gravitation, a particle
falls towards an attracting body whose mass is µ, from one distance
s to another distance r, its velocity is given by the simple equation

1

1
w2 = 2µ

−

.1

(Equation 3)

r
s

If in this equation s be made infinite, w does not also become
infinite (that is, a body falling from an infinite distance towards the
sun will not acquire an infinite velocity until it actually reaches the

1If the difference between s and r is called h, this equation becomes

1

1
h
w2 = 2µ

−

= 2µ

.

r
r + h
r2 + rh

Now if h is very small as compared with r, this gives

2µ
w2 =
h,
r2
which is the same as the usual expression for the velocity of a falling body at

2µ
the earth’s surface, viz., V 2 = 2gh, 2g being replaced by the fraction

.

r2

CENTRAL FORCES.

269
centre of the sun, and r becomes zero); but we get in this case

2µ
w2 =

.

r

This special value of w is usually called “the velocity from infinity
for the distance r,” or the “parabolic velocity” (for a reason which
will appear very soon). U is generally used as its symbol; therefore

2µ
rU 2

U 2 =

; whence µ =

.

(Equation 4)

r

2

Substituting this value of µ in equation 2, we get
r

U 2
a =

.

(Equation 5)

2

U 2 − V 2

430. Relation between the Velocity and the Species of

Conic Described.—From equation 5 it is obvious how the velocity
determines whether the orbit will be an ellipse or a hyperbola. If

V 2 is less than U 2, the denominator of the fraction will be positive,
a will also be positive, and the curve will be an ellipse; i.e., if the
velocity of the body P , at the distance r from the central body S, be
less than the velocity acquired by the body falling from infinity to
that point, the body will move around S permanently in an ellipse.

If, on the other hand, V 2 is greater than U 2, the denominator
will become negative, a will also come out negative, and the orbit
will be a hyperbola. In this case P , after once moving past S at
the perihelion point, will go off never to return; and it will recede
towards a a different region of space from that out of which it came,
because the two legs of the hyperbola never become parallel. There
will in this case be no permanent connection between the two bodies.

They simply pass each other with a little graceful recognition of
each other’s presence by a curvature in their paths, and then part
company for ever.

If V 2 exactly equals U 2, the denominator of the fraction becomes
zero, a comes out infinite, and the curve is a parabola. In this case,
also, the body will never return; but it will recede from the sun
ultimately towards the same point on the celestial sphere as that
from which it appeared to come, since the two legs of the parabola
tend to parallelism. Obviously, if a body were thus moving in a
parabola, the slightest increase of its velocity would transform the

CENTRAL FORCES.

270
orbit into an hyperbola, and the least diminution into an ellipse; the
bearing of which remark will become evident when we come to deal
with comets.

r

U 2

431. Again, since
a =

,

2

U 2 − V 2
all bodies having the same velocity V , at the same distance r from
the centre of force, will have major axes of the same length for their
orbits, no matter what may be their direction of motion.

They will have the same period also, the expression for the period
being

3

2πa 2
t = √

. (Watson, p. 46, Equation 28.)

µ

If, therefore, a body moving around the sun were to explode at
any point, all of its particles which did not receive a velocity greater
than the “parabolic velocity” would come around to the same point
again, and these which were projected with equal velocities would
come around and meet at the same moment, however widely differ-
ent their paths might be.

Fig. 143.—Confocal Conics described under Different Velocities of Projection.

CENTRAL FORCES.

271

432. Fig. 143 represents the orbits which would be described by five
bodies projected at O with different velocities along the line OV , the
distance OS or r being taken as unity, as well as the parabolic velocity

U 2. The squares of the velocities are assumed as given below, with the
resulting values of a and r .

V 2

1

= 1 ; whence a

; and r

.

4

1 = 2

3

1 = 1

3

This places the empty focus at F1.

For the next larger ellipse

V 2

2

= 1 ; a

2

2 = 1; r2 = 1.

In the same way

V 2

3

= 3 ; a

4

3 = 2; r3 = 3.

V 2

4

= 1; a4 = ∞; r4 = ∞.

(Parabola.)

V 2

5

= 2; a5 = − 1 ; r

2

5 = −2.

(Hyperbola.)

433. Fig. 144 shows how three bodies projected at P with equal
velocities, but in different directions, indicated by the arrows, describe
three different ellipses; all, however, having the same period, and the
same length of semi-major axis; namely, a = 2r; V 2 being taken equal
to 3 U 2.

4

Fig. 144.—Ellipses of the Same Periodic Time.

For a fourth body, V 2 is taken as = 1 U 2, and with the direction

2
of motion perpendicular to r. This body will move in a perfect circle,
a coming out equal to r, when V 2 = 1 U 2. In order to have circular

2
motion, both conditions must be fulfilled; namely, V 2 must equal 1 U 2,

2
and the direction of motion must be perpendicular to the radius vector.

CENTRAL FORCES.

272

434. Velocity of a Planet
at Any Point in its Orbit.—If

AA (Fig. 145) be the major axis of
a planet’s orbit, and KK the diam-
eter of a circle described around S
with AA as radius, then the veloc-
ity of a planet at any point, N , on its
orbit is equal to that which it would
have acquired by falling to N from
the point n on the circumference
of the circle.

The demonstration
is not difficult and may be found

Fig. 145.—Van der Kolk’s Theorem.

in No. 1426 of the “Astronomische

Nachrichten.”

435. Projectiles
near
the

Earth.—A good illustration of the
principles
stated
above
is
obtained
by considering the motion of bodies
projected horizontally from the top of a
tower near the earth’s surface, supposing
the air to be removed so there will be
no resistance to the motion.

The “parabolic velocity” due to the
earth’s attraction equals 6.94 miles per
second at the earth’s surface; i.e., a body

Fig. 146.—Projectiles near the
falling from the stars to the surface of

Earth.

the earth, drawn by the earth’s attraction
only, would have acquired this velocity on reaching the earth’s surface.

First. If a body be projected with a very small velocity, it would fall
nearly straight downwards. If the earth were concentrated at the point
in its centre so that the body should not strike its surface, it would move
in a very long narrow ellipse having the centre of the earth at the further
focus, and would return to the original point after an interval of 29.9
minutes.

Second. With a larger velocity the orbit would be a wider ellipse
with a longer period, C being still at the remoter focus.

Third. V = U

1 , or about 4.9 miles per second. In this case the

2
orbit of the body would be a perfect circle, and the period would be 1 h.

24.7 m. It will be remembered that we found that if the earth’s rotation
were 17 times as rapid, thus completing a revolution in 1h 24m.7, the
centrifugal force at the equator would become equal to gravity (Art. 154).

CENTRAL FORCES.

273

Also, Art. 413 (4), this same time, 1h 24m.7, was found from Kepler’s
third law as the period of a satellite revolving close to the earth’s surface.

Fourth. V = U = 6.94 miles. In this case the projectile would go off
in a parabola, never to return.

Fifth. V > 6.94. In this case, also, the body would never return, but
would pass off in a hyperbola.

436. Intensity of Solar Attraction.—The attraction be-
tween the sun and the earth from some points of view looks like
a very feeble action.

It is only able, as has been before stated

(Art. 278), to bend the earth out of a rectilinear course to the ex-
tent of about one-ninth of an inch in a second, while she is travelling
nearly nineteen miles; and yet if it were attempted to replace by
bonds of steel the invisible gravitation which holds the earth to the
sun, we should find the surprising result that it would be necessary
to cover the whole surface of the earth with wires as large as tele-
graph wires, and only about half an inch apart from each other, in
order to get a metallic connection that could stand the strain. This
ligament of wires would be stretched almost to the breaking point.

The attraction of the sun for the earth expressed as tons of force

(not tons of mass, of course) is 3,600000 millions of millions of tons

(36 with seventeen ciphers); and similar stresses act through the
apparently empty space in all directions between all the different
pairs of bodies in the universe.

436*. Note to Art. 429. —It is worth noting that U 2 (the square
of the parabolic velocity at any point) is simply twice the gravitation
potential due to the sun’s attraction at that point. The “potential” may
be defined as the energy which would be acquired by a mass of one
unit, in falling to the point in question from a place where the potential

(and attraction) is zero, i.e., from infinity (Physics, pp. 26 and 28).

Now 1 mV 2 is the general expression for the kinetic energy of a mass,

2
m, moving with velocity V ; if in this expression we make m = 1, and

V = U , we shall have, for the case in hand, Energy = 1 U 2, which equals

2
the Potential at the point.

THE PROBLEM OF THREE BODIES.

274

CHAPTER XII.

THE PROBLEM OF THREE BODIES.—DISTURBING FORCES:

LUNAR PERTURBATIONS AND THE TIDES.

437. The problem of two bodies is completely solved; but if,
instead of two spheres attracting each other, we have three or more,
given completely in respect to their positions, masses, and veloci-
ties, the general problem of finding their subsequent motions and
predicting their positions at any future date transcends the present
power of our mathematics.

This problem of three bodies is in itself just as determinate and
capable of solution as that of two. Given the initial data,—that is,
the positions, masses, and motions of the three bodies at a given
instant,—then their motions for all the future, and the positions
they will occupy at any given date, are absolutely predetermined.

The difficulty of the problem lies simply in the inadequacy of our
present mathematical methods, and it is altogether probable that
some time in the future this difficulty will be overcome—very pos-
sibly by the invention of new functions and numerical tables which
shall bear some such relation to our present tables of logarithms,
sines, etc. as these do to the old multiplication table of Pythagoras.

438. But while the general problem of three bodies is thus in-
tractable, all the special cases of it which arise in the consideration
of the moon’s motion and in the motions of the planets have been
solved by special methods of approximation. Newton himself led the
way; and the strongest proof of the truth of his theory of gravitation
lies in the fact that it not only accounts for the regular elliptic mo-
tions of the heavenly bodies, but also for the apparent irregularities
of these motions.

439. The Disturbing Force.—In the case where two bodies
are revolving around their common centre of gravity, and the third
body is either very much smaller than the central one, or very re-
mote, the motion of the two will be but slightly modified by the

THE PROBLEM OF THREE BODIES.

275
action of the third; and in such a case the small differences be-
tween the actual motion and the motion as it would be if the third
body were not present, are technically called “disturbances” and

“perturbations,”1 and the force which produces them is called the

“disturbing force.” This disturbing force is not the attraction of
the disturbing body, but only a component of that attraction, and
usually only a small fraction of it.

The disturbing force of the attracting body depends upon the dif-
ference of its attraction upon the two bodies it disturbs; difference
either in amount or in direction, or in both. For instance, if the sun
attracted the earth and moon exactly alike (i.e., equally and along
parallel lines), it would not disturb their relative motions in the
least, no matter how powerful its attraction might be. The sun’s
maximum disturbing force on the moon, as we shall see, is only
about one eighty-ninth of the earth’s attraction; and yet the sun’s
attraction for the moon is actually much greater than that of the
earth.

Since the sun’s mass is 330,000 times that of the earth, and its dis-
tance just about 389 times that of the moon from the earth, its attraction
on the moon equals the earth’s attraction × 330,000 = 2.18; i.e., the sun’s

3892
attraction on the moon is more than double that of the earth.

440. Why the Sun does not take the Moon away from
the Earth.—If at the time of new moon, when the moon is between
the earth and sun, the sun attracts the moon more than twice as
much as the earth does, it is a natural question why the sun does
not draw the moon away entirely, and rob us of our satellite. It
would do so if it were the case of a “tug of war”; that is, if earth
and sun were fixed in space, pulling opposite ways upon the moon
between them. But it is not so; neither sun nor earth has any
foothold, so to speak; but all three bodies are free to move, like chips
floating on water. The sun attracts the earth almost as much as he
does the moon, and both earth and moon fall towards him freely;

1The student will bear in mind that these terms (“perturbations” and “dis-
turbances”) are mere figures of speech; that philosophically the purely elliptical
motion of two mutually attracting bodies alone in space is no more “regular ”

than the (at present) incomputable motion of three or more attracting bodies.

We have in mind a theologian of some note who once maintained that the “per-
turbations” in the solar system are a consequence of Adam’s fall. Hence the
caution.

THE PROBLEM OF THREE BODIES.

276
though of course this falling motion towards the sun is continually
combined with whatever other motion the earth or moon possesses.

The only effective disturbance is produced by the fact that, in the
case considered, the new moon, being nearer the sun than the earth
is by about 1 part of the whole distance, falls towards the sun a

389
trifle faster than the earth, and so on that account the curvature of
its orbit toward the earth is, for the time being, diminished.

At the half-moon the two bodies are equally attracted towards
the sun, but on converging lines; and so as they fall towards the sun
they approach each other slightly; and for this reason, at quadra-
ture, the moon’s orbit is a little more curved towards the earth than
it would be otherwise.

441. Diagram of the Disturbing Force.—A very simple di-
agram enables us to find graphically the disturbing force produced
by a third body.

(What follows applies verbatim et literatim to either of the two dia-
grams of Fig. 147.)

Fig. 147.—Determination of the Disturbing Force by Graphical Construction.

Let E be the earth, M the moon, and S the disturbing body

(the sun in this case); and let the sun’s attraction on the moon be
represented by the line M S. On the same scale the attraction of
the sun on the earth will be represented by the line EG, G being a
point so taken that EG : M S = M S2 : ES2; that is,—

The sun’s attraction on the earth is to the sun’s attraction on the
moon as the square of the sun’s distance from the moon is to the square
of the sun’s distance from the earth, according to the law of gravita-
tion. (M S has to do double duty in this proportion: in the first ratio it
represents a force; in the second, a distance.)

M S2

From this proportion

EG = M S ×

.

ES2

THE PROBLEM OF THREE BODIES.

277

In figure (a) the moon is nearer to the sun than the earth is, and so

EG comes out less than M S. In figure (b) the reverse is the case, and
therefore in this case EG is larger than M S.

Now if the force represented by the line M S were parallel and
equal to that represented by EG, there would be no disturbance, as
has been said. If, then, we can resolve the force M S into two com-
ponents, one of which is equal and parallel to EG, this component
will be innocent and harmless, and the other one will make all the
disturbance.

To effect this resolution, draw through M the line M K parallel
and equal to EG. Join KS, and draw M L parallel and equal to
it. M L is then the disturbing force on the same scale as M S; i.e.,
the line M L shows the true direction of the disturbing force, and
in amount the disturbing force is equal to the sun’s attraction for

M L
the moon multiplied by the fraction

. The diagonal of the

M S
parallelogram M LSK is M S, which represents the resultant of the
two forces M K and M L, that form its sides.

For the sake of clearness the lines which represent forces in the figures
are indicated by herring-bone markings.

442. At first it seems a little strange that in figure (b) the
disturbing force should be directed away from the sun; but a little
reflection justifies the result. If E and M were connected by a rod,
and the E-end of the rod were pulled towards the right more swiftly
than the M -end, it is easy to see that the latter would he relatively
thrown to the left, as the figure shows.

443. The sun is the only body that sensibly disturbs the moon.

The planets, of course, act upon the moon to disturb it, but their
mass is so small compared with that of the sun, and their distances
so great, that in no case is their direct action sensible. It is true,
however, that some of the lunar perturbations are affected by the
existence of one or two of the planets. While they cannot disturb
the moon directly, they do so indirectly: they disturb the earth in
her orbit sufficiently to make the sun’s action different from what it
would be if the planets did not exist. In this way the planets Venus,

Mars, and Jupiter make themselves felt in the lunar theory. There
are also a few small disturbances that depend upon the fact that
the earth is not a perfect sphere.

THE PROBLEM OF THREE BODIES.

278

444. Since the distance of the sun is nearly 400 times that of the
moon from the earth, the construction of the disturbing force M L,

Fig. 147, admits of considerable simplification. It is only necessary
to drop the perpendicular M P upon the line that joins the earth and
the sun, and take the point L upon this line, so that EL equals three
times EP . The line M L so determined will then very approximately

(but not exactly) be the true disturbing force.

To prove this relation, let M S, in Fig. 147, be D, ES = R, M E = r,
and EP = p, also R = D + p, very nearly, p being negative when

M S2

D3

M S > ES. EG was taken equal to M S ×

=

.

ES2

R2

Now,

D3

R3 − D3

(D + p)3 − D3

EL = GS = (ES − EG) = R −

=

=

.

R2

R2

(D + p)2

Developing this expression, we have

3D2p + 3Dp2 + p3

EL =

.

D2 + 2Dp + p2

Since p is very small compared with D, all the terms except the first
nearly vanish both in numerator and denominator, and we have

3D2p

EL =

= 3p (very nearly).

D2

445. Resolution of the Disturbing Force into Compo-
nents.—In discussing the effect of the disturbing force it is more
convenient to resolve it into three components known as the radial,
the tangential, and the orthogonal. The first of these acts in the
direction of the radius vector, tending to draw the moon either to-
wards or from the earth. The second, the tangential, operates to
accelerate or retard the moon’s orbital velocity.

Fig. 148.—Radial and Tangential Components of the Disturbing Force.

THE PROBLEM OF THREE BODIES.

279

Fig. 148 exhibits these two components at different points of the
moon’s orbit.

The orthogonal component has no existence in cases where the
disturbing body lies in the plane of the disturbed orbit; but when-
ever it lies outside of that plane, the disturbing force M L will gen-
erally also lie outside of the orbit-plane, and will have a component
tending to draw the moving body out of the plane of its orbit. The
motion of the moon’s node and the changes of the inclination of its
orbit are due to this component of the sun’s disturbing force, which
could not be conveniently represented in the figure.

446. The radial force in the case of the moon’s orbit is a maxi-
mum at syzygies and quadratures; in fact, at quadratures the whole
disturbing force is radial, the tangential and orthogonal components
both vanishing. At syzygies (new moon and full moon) the radial
force is negative; that is, it draws the moon from the earth, dimin-
ishing the earth’s attraction by about one eighty-ninth1 of its whole
amount.

At quadrature or half-moon the radial force is positive; and since

L then falls at E, it is represented by the line QE, and is just half
what it is at syzygies; that is, it equals about one one hundred and
seventy-eighth of the earth’s attraction.

It becomes zero at four points 54◦ 44 on each side of the line of
syzygies.

This angle is found from the condition that, the disturbing force

M1L1, etc., in Fig. 148, must be perpendicular to the radius EM1 at this
point, which gives us EP1 : P1M1 :: P1M1 : P1L1. But P1L1 = 2EP1;

P

√

1M1
therefore P

2

2

1M1

= 2EP1 , and

= tan M1EL1 =

2.

EP1

447. The tangential component starts at zero at the time of full
moon, rises to a maximum at the critical angle of 45◦ (having at that
point a value of 1 of the earth’s attraction), and disappears again

119
at quadratures. During the first and third quadrants this force is

M S

1At syzygies M L = N L0 = 2 × EN (Fig. 147); but EN =

. Therefore

389

2

M L =
of the sun’s attraction on the moon. Now the sun’s attraction

389
is 2.18 times the earth’s; hence N L0 = the earth’s attraction multiplied by

2 × 2.18

1

=

.

389

89.2

THE PROBLEM OF THREE BODIES.

280
negative; that is, it retards the moon’s motion; in the second and
fourth it is positive and accelerates the motion.

448. Lunar Perturbations.—So far it has been all plain sail-
ing, for nothing beyond elementary mathematics is required in de-
termining the disturbing force at any point in the moon’s orbit; but
to determine what will be the effect of this continually varying force
after the lapse of a given time, upon the moon’s place in the sky is
a problem of a very different order, and far beyond our scope. The
reader who wishes to follow up this subject must take up the more
extended works upon theoretical astronomy and the lunar theory.

A few points, however, may be noted here.

449. In the first place, it is found most convenient to consider
the moon as never deviating from an elliptical orbit, but to consider
the orbit itself as continually changing in place and form, writhing
and squirming, so to speak, under the disturbing forces; just as if
the orbit were a material hoop with the moon strung upon it like
a bead and unable to get away from it, although she can be set
forward and backward in her motion upon it.

450. In the next place, it is found possible to represent nearly
all the perturbations by periodical formulæ—the same values recur-
ring over and over again indefinitely at regular intervals. This is
because the sun, moon, and earth keep coming back into the same,
or nearly the same, relative positions, and this leads to recurring
values of the disturbing force itself, and also of its effects.

451. Third, the number of these separate perturbations which
have to be taken account of is very large. In the computation of
the moon’s longitude in the American Ephemeris about seventy dif-
ferent inequalities are reckoned in, and about half as many in the
computation for the latitude. Theoretically the number is infinite,
but only a certain number produce effects sensible to observation.

It is of no use to compute disturbances that do not displace the
moon as much as one-tenth of a second of arc; i.e., about 500 feet
in her orbit.

452. Fourth, in spite of all that has been done, the lunar theory
is still incomplete, or in some way slightly erroneous. The best ta-
bles yet made begin to give inaccurate results after fifteen or twenty
years, and require correction. The almanac place of the moon at

THE PROBLEM OF THREE BODIES.

281
present is not unfrequently “out” as much as 3 or 4 of arc; i.e.,
about three or four miles. Astronomers are continually at work
on the subject, but the computations by our present methods are
exceedingly tedious and liable to numerical error.

453. The principal effects of the sun’s disturbing action on the
moon are the following:–

First: Effect on Length of Month.—Since the radial component
of the disturbing force is negative more than half the way round

(54◦ 44 on each side of the line of syzygies) and is twice as great at
syzygies as the positive component is at quadrature, the net result
is that, taking the whole month through, the earth’s attraction for
the moon is lessened by nearly

1
part. The effect of this is to

360
increase the major axis of the moon’s orbit, and to make the month
a little longer than it otherwise would be; that is, if the sun were
taken away, the month would immediately shorten by about three
hours.

454. Second: The Revolution of the Line of

Apsides.—This is due mainly to the radial com-
ponent of the disturbing force. When the moon
comes to apogee at the time of new or full moon,
the diminution of the earth’s effective attraction
for the moon causes it to move on farther than
it would otherwise do before turning the corner,
so to speak, the consequence being that the line

Fig. 149.

Advance of the
of apsides advances in the line of the moon’s mo-

Apsides of the
tion. When the moon passes perigee at that time,

Moon’s Orbit.

the effect is reversed, and the apsides regress; but
since the disturbing force is greater at apogee than at perigee, in
the long run the advancing motion predominates. When perigee or
apogee is passed at the time of quadrature, the line of apsides is also
disturbed; but the disturbances thus produced exactly balance each
other in the long run. The net result, as has been stated before

(Art. 238), is that the line of apsides completes a direct revolution
once in about nine years (8.855 years—Neison). It does not move
forward steadily and uniformly, but its motion is made up of alter-
nate advance and regression. Fig. 149 illustrates this motion of the
moon’s apsides.

For a fuller discussion of the subject, see Herschel’s “Outlines of

Astronomy,” sections 677–689; or Airy’s “Gravitation,” pp. 89–100.

THE PROBLEM OF THREE BODIES.

282

455. Third: The Regression of the Nodes.—The orthogonal
component generally (not always) tends to draw the moon towards
the plane of the ecliptic. Whenever this is the case at the time
when the moon is passing a node, the effect (as is easily seen from

Fig. 150) of such a force P1O1, acting upon the moon at P1, is to
shift the node backward from N1 to N2, the moon taking the new
path P1b1N2. As the moon is approaching the node, the inclination
of its orbit is also increased; but as the moon leaves the node, it
is again diminished, the path N2P2 being bent at P2 back to P2b2,
parallel to P1a1: so that while by both operations the node is made
to recede from N1 to N3, the inclination suffers very little change,
if the orthogonal component remains the same on both sides of the
node.

Since the orthogonal component vanishes twice a year,—when
the sun is at the nodes of the moon’s orbit,—and also twice a
month,—when she is in quadrature,—the rate at which the nodes
regress is extremely variable. In the long run it makes its back-
ward revolution once in about nineteen years (Arts. 249 and 391).

[18.5997 years.—Neison.]

See Herschel’s “Outlines of Astronomy,” section 638 seqq.

Fig. 150.—Regression of the Nodes of the Moon’s Orbit.

456. Fourth: The Evection.—This is an irregularity which at
the maximum puts the moon forward or backward about 1 1 ◦

4

(1◦ 16 27 .01—Neison), and has for its period the time which is oc-
cupied by the sun in passing from the line of apsides of the moon’s
orbit to the same line again; i.e., about a year and an eighth. This
is the largest of the moon’s perturbations, and was earliest discov-
ered, having been detected by Hipparchus about 150 years b.c.,
and afterwards more fully worked out by Ptolemy, though of course

THE PROBLEM OF THREE BODIES.

283
without any understanding of its cause. It was the only lunar per-
turbation known to the ancients. It depends upon the alternate
increase and decrease of the eccentricity of the moon’s orbit, which
is always a maximum when the sun is passing the moon’s line of
apsides, and a minimum when the sun is at right angles to it.

This inequality may affect the time of an eclipse by nearly six
hours, making it anywhere from three hours early to three hours
late, as compared with the time at which it would otherwise occur;
it was this circumstance which called the attention of Hipparchus
to it.

See Herschel’s “Outlines of Astronomy,” sections 748 seqq.

457. Fifth: The Variation.—This is an inequality due mainly
to the tangential component of the disturbing force. It has a period
of one month, and a maximum amount of 39 30 .70, attained when
the moon is half-way between the syzygies and quadratures, at the
so-called “octants.” At the first and third octants the moon is 39 12
ahead of her mean place (about an hour and twenty minutes); at
the second and fourth she is as much behind. This inequality was
detected by Tycho Brahe, though there is some reason for believing
that it had been previously discovered by the Arabian astronomer,

Aboul Wefa, five centuries earlier. This inequality does not affect the
time of an eclipse, being zero both at the syzygies and quadratures,
and therefore was not detected by the Greek astronomers.

See Herschel’s “Outlines of Astronomy,” sections 705 seqq.

458. Sixth: The Annual Equation.—The one remaining in-
equality which affects the moon’s place by an amount visible to
the naked eye, is the so-called “annual equation.” When the earth
is nearer the sun than its mean distance, the sun’s disturbing force
is, of course, greater than the mean, and the month is lengthened a
little; during that half of the year, therefore, the moon keeps falling
behindhand; and vice versa during the half when the sun’s dis-
tance exceeds the mean. The maximum amount of this inequality
is 11 9 .00, and its period one anomalistic year.

See Herschel’s “Outlines of Astronomy,” sections 738 seqq.

There remains one lunar irregularity among the multitude of
lesser ones, which is of great interest theoretically, and is still a
bone of contention among mathematical astronomers; namely,—

THE PROBLEM OF THREE BODIES.

284

459. Seventh: The Secular Acceleration of the Moon’s Mean

Motion.—It was found by Halley, early in the last century, by a com-
parison of ancient with modern eclipses, that the month is now cer-
tainly shorter than it was in the days of Ptolemy, and that the short-
ening has been progressive, apparently going on continuously,—in
sæcula sæculorum,—whence the name. In 100 years the moon, ac-
cording to the results of Laplace, gets in advance of its mean place
about 10 , and the advance increases with the square of the time,
so that in a thousand years it would gain nearly 1000 , and in 2000
years 4000 , or more than a degree. The moon at present is sup-
posed to be just about a degree in advance of the position it would
have held if it had kept on since the Christian era with precisely
the rate of motion it then had. If this acceleration were to continue
indefinitely, the ultimate result would be that the moon would fall
upon the earth, as the quickened motion corresponds to a shortened
distance.

460. It was nearly 100 years after Halley’s discovery before

Laplace found its explanation in the decreasing eccentricity of the
earth’s orbit. Under the action of the other planets this orbit is now
growing more nearly circular, without, however, changing the length
of its major axis. Thus its area becomes larger, and the earth’s aver-
age distance from the sun becomes greater (although the mean dis-
tance, technically so-called, does not change, the “mean distance”

being simply half the major axis). As a result of this rounding up of
the earth’s orbit, the average disturbing force of the sun is therefore
diminished, and this diminution allows the month to come nearer
the length it would have if there were no sun to disturb the motion;
that is to say, the month keeps shortening little by little, and it will
continue to do so until the eccentricity of the earth’s orbit begins
to increase again, some 25,000 years hence.

461. But the theoretical amount of this acceleration, about 6 in a
century, does not agree with the value obtained by comparing the most
ancient and modern eclipses, which is about 12 ; and this value, again,
does not agree with the one derived by comparing modern observations
of the moon with these made by the Arabians about a thousand years
ago, which, according to recent investigations by Professor Newcomb,
indicate an acceleration of only about 8 .

So long as the actual acceleration was considered to be 12 , it was
generally supposed that the discrepancy between the theoretical and ob-

THE PROBLEM OF THREE BODIES.

285
served result is due to a retardation of the earth’s rotation by the friction
of the tides, and a consequent lengthening of the day. Evidently if the
day and the seconds become a little longer, there will be fewer of them
in each month or year, and the apparent effect of such a change would
be to shorten all really constant astronomical periods by one and the
same percentage. As matters stand to-day it is hardly possible to assert
with confidence that there is any real discrepancy to be accounted for
between the theoretical and observed values, the latter being consider-
ably uncertain. In Newcomb’s “Popular Astronomy” (pp. 96–102) there
will be found an interesting and trustworthy discussion of the subject.

Questions like this, and those relating to the remaining discrepancies
between the lunar tables and the observed places of our satellite, lie on
the very frontiers of mathematical astronomy, and can be dealt with only
by the ablest and most skilful analysts.

THE TIDES.

462. Just as the disturbing force due to the sun’s attraction
affects the motions of the moon in her orbit, so the disturbing forces
due to the attractions of the moon and sun acting upon the fluids
of the earth’s surface produce the tides. These consist of the regular
rise and fall of the water of the ocean, the average interval between
successive high waters being 24h 51m, which is precisely the same as
the average interval between two successive passages of the moon
across the meridian. This coincidence, maintained indefinitely, of
itself makes it certain that there must be some causal connection
between the moon and the tides. As some one has said, the odd 51
minutes is the moon’s “ear-mark.”

463. Definitions.—When the water is rising, it is “flood ” tide;
when falling, it is “ebb.” It is “high water ” at the moment when the
tide is highest, and “low water ” when it is lowest. “Spring tides” are
the highest tides of the month (which occur near the times of new
and full moon), while “neap tides” are the smallest, which occur
when the moon is in quadrature. The relative heights of the spring
and neap tides are about as 7 to 4. At the time of spring tides
the interval between the corresponding tides of successive days is
less than the average, being only about 24h 38m, and then the tides
are said to “prime.” At neap tides the interval is 25h 6m, which is
greater than the mean, and the tides “lag.”

The “establishment” of a port is the mean interval between the

THE PROBLEM OF THREE BODIES.

286
time of high water at that port and the next preceding passage
of the moon across the meridian. At New York, for instance, this

“establishment” is 8h 13m, although the actual interval varies about

22 minutes on each side of the mean at different times of the month.

That the moon is largely responsible for the tides is also shown
by the fact that the tides, at the time when the moon is in perigee,
are nearly twenty per cent higher than those which occur when she is
in apogee. The highest tides of all happen when a new or full moon
occurs at the time the moon is in perigee, especially if this occurs
about January 1st, when the earth is nearest to the sun. Since, as
we shall see, the “tide-raising” force varies inversely as the cube of
the distance, slight variations in the distance of the moon and sun
from the earth make much greater variations in the height of the
tide—greater nearly in the ratio of 3 to 1.

464. The Tide-Raising Force.—This is the difference be-
tween the attractions of the sun and moon (mainly the latter) on
the main body of the earth, and the attractions of the same bodies
on particles at different parts of the earth’s surface. The tide-raising
force is but a very small part of the whole attraction.

Fig. 151.—The Moon’s Tide-Raising Force on the Earth.

The amount of this disturbing force for a particle at any point on
the earth’s surface can be found approximately by the same geomet-
rical construction which was used for the lunar theory (Art. 441).

Draw a line from the moon through the centre of the earth. At the
points A and B, Fig. 151, where the moon is directly over head or
under foot, the tide-raising force is directly opposed to gravity, and
equals nearly 1 of the moon’s whole attraction, since the line Aa

30
represents the disturbing force on the same scale as the line from A
to the moon represents the moon’s attraction, and this line, AM ,
is about sixty times the earth’s radius, while Aa is just double it,
because Ca has to be taken equal to 3 × CA (Art. 444).

Since the moon’s mass is only about 1 of the earth’s, and its

80
distance is sixty radii of the earth, this lifting force under the moon,

THE PROBLEM OF THREE BODIES.

287
expressed as a fraction of the earth’s gravity, equals

1 × 1 × 1 =

1

;

30

80

3600

8640000
i.e., a body weighing four thousand tons loses about one pound of
its weight when the moon is over head or under foot.

At D and E, anywhere on the circle of the earth’s surface which
is 90◦ from A and B, the moon’s disturbing force increases the
weight of a body by just half this amount, the disturbing force
being measured by the lines DC and EC. At a point F , situated
anywhere on a circle drawn around either A or B with a radius of

54◦ 44 , the weight of a body is neither increased nor decreased, but
it is urged towards A or B with a horizontal force expressed by the
line F f , which force is equal to about

1
of its weight.

12 000000

In the same way the tidal forces at G and H are expressed by
the lines Gg and Hh.

465. The same result for the lifting-force directly under the moon
may be obtained more exactly as follows. The distance from the moon
to the centre of the earth is sixty times the earth’s radius, and therefore
the distance from the moon to the points A and B respectively will be

59 and 61. The moon’s attraction at A, C, and B, expressed as fractions
of the earth’s gravity, will be as follows:—

1

Attraction of moon on particle at A = g × 80 = 0.0000035910 × g.

592

1

Attraction of moon on particle at C = g × 80 = 0.0000034723 × g.

602

1

Attraction of moon on particle at B = g × 80 = 0.0000033593 × g.

612

1

Hence

A − C = 0.0000001187 g =
g.

8,424,000

1

C − B = 0.0000001130 g =
g.

8,835,000

This is more correct than the preceding, which is based on an ap-
proximation that considers the moon’s distance as very large compared
with the earth’s radius, while it is really only sixty times as great, and
sixty is hardly a “very large” number in such a case.

Attempts have been made to observe directly the variations in the
force of gravity produced by the moon’s action, but they are too small
to be detected with certainty by any experimental method yet contrived.

THE PROBLEM OF THREE BODIES.

288

Both Darwin and Zöllner found that other causes which they could not
get rid of produced disturbances more than sufficient to mask the whole
action of the moon.

466. It is worth while to note in this connection that the maximum
lifting-force due to the attraction of a distant body varies inversely as
the cube of its distance, as is easily shown, thus:—calling D the distance
of the disturbing body from the earth’s centre, and r the earth’s radius,
we have

M

M

Attraction at A

=

;
attraction at C =

.

(D − r)2

D2

1

1

Tide-raising force at A = M

−

(D − r)2

D2

2Dr − r2

= M

D2(D2 − 2Dr + r2)

2Dr − r2

= M

D4 − 2D3r + D2r2

2r

= M

, nearly,

D3
when r is a small fraction of D.

467. It is very apt to puzzle the student that the moon’s action
should be a lifting force at B as well as at A (Fig. 151). He is likely to
think of the earth as fixed, and the moon also fixed and attracting
the water upon the earth, in which case, of course, the moon’s
attraction, while it would decrease gravity at A, would increase it
at B.

The two bodies are not fixed, how-
ever. Let him think of the three parti-
cles at A, C, and B, Fig. 152, as un-
connected with each other, and falling
freely towards the moon; then it is obvi-
ous that they would separate; A would
fall faster than C, and C than B. Now

Fig. 152.—The Statical
imagine them connected by an elastic

Theory of the Tides.

cord. It is obvious that they will still
draw apart until the tension of the cord prevents any further sepa-
ration. Its tension will then measure the “lifting force” of the moon
which tends to draw both the particles A and B away from C.

THE PROBLEM OF THREE BODIES.

289

468. The Sun’s Action.—This is precisely like that of the
moon, except that the sun’s distance, instead of being only sixty
times the earth’s radius, is nearly 23,500 times that quantity. Since
the tide-raising power varies as the cube of the distance inversely,
while the attracting force varies only with the inverse square, it
turns out that although the sun’s attraction on the earth is nearly

200 times as great as that of the moon, its tide-raising power is only
about two-fifths as much. When the sun is over head or under foot,
his disturbing force diminishes gravity by about

1

.

19 600000

469. Statical Theory of the Tides.—If the earth were
wholly composed of water, and if it kept always the same face to-
wards the moon (as the moon does towards the earth), so that every
particle on the earth’s surface was always subjected to the same dis-
turbing force from the moon, then, leaving out of account the sun’s
action, a permanent tide would be raised upon the earth, distorting
it into a lemon-shaped form with the point towards the moon. It
would be permanently higher water at the points A and B (Fig. 152)

directly under the moon, and low water all around the earth on the
circle 90◦ from these points, as at D and E. The difference of the
level of the water at A and D would in this case be about two feet.

The sun’s action would produce a similar tide superposed upon
the lunar tide and having about two-fifths of the same elevation, if
the two tide summits should coincide, the resulting elevation of the
high water would be the sum of the two separate tides. If the sun
were 90◦ from the moon, the waves would be in opposition, and the
height of the tide would be decreased, the solar tide partly filling
up the depression at the low water due to the moon’s action.

Suppose now the earth to be put in rotation. It is easy to see
that these tidal waves would tend to move over the earth’s surface,
following the moon and sun at a certain angle dependent on the
inertia of the water, and with a westward velocity precisely equal
to that of the earth’s eastward rotation,—about a thousand miles
an hour at the equator. But it is also evident that on account
of the varying depth of the ocean, and the irregular form of the
shores, the tides could not maintain this motion, and that the actual
result must become exceedingly complicated. In fact, the statical
theory becomes utterly unsatisfactory in regard to what actually
takes place, and it is necessary to depend almost entirely upon the
results of observation, using the theory merely as a guide in the

THE PROBLEM OF THREE BODIES.

290
discussion of the observations.

Yet while this statical theory of the tides worked out by Newton is
certainly inadequate, and in some respects incorrect, it easily furnishes
the explanation of some of the most prominent of the peculiarities of the
tides.

470. The Priming and Lagging of the Tides.—About the
time of new and full moon, as has been stated before (Art. 463), the
interval between the corresponding tides of successive days is about thir-
teen minutes less than the average of 24h 51m, while a week later it is
about as much longer. The reason is found in the combination of the
solar and lunar tides.

Fig. 153.

Fig. 154.

Priming and Lagging of the Tide.

On the days of new and full moon the two tides coincide, and the
tide wave has its crest directly under the moon, or rather at the normal
distance behind the moon which corresponds to the “establishment” of
the port of observation.

At quadrature the crest of the solar tide will be just 90◦ from the
crest of the lunar wave, but it will leave the summit of the combined wave
just where it would be if there were no solar wave at all: evidently there
is no possible reason why the smaller wave at S and S should displace
the crest of the wave at L (Fig. 153) towards the right that would not
also require its displacement towards the left; it will therefore simply
lower the wave at L without displacing it one way or the other. But
when the solar tide wave SS (Fig. 154) has its crest at S1 and S , 45◦

1
from L and L , as it will do about three days after new or full moon, then
its combination with the lunar wave will make the crest of the combined
wave take position at a point X between the two crests, and about half
an hour of time ahead (west ) of the lunar tide; so that at that time of
the month high water will occur about half an hour earlier than if there
were no solar tide (since the tide waves travel westward). And this half-
hour has to be gained by diminishing the interval between the successive
tides for the three preceding days. Similar reasoning shows that when
the solar tide crest falls at S2 and S , the combined tide wave will be

2
east of the lunar wave, and come later into port.

THE PROBLEM OF THREE BODIES.

291

471. Effect of the Moon’s Dec-
lination and Diurnal Inequality.—In
high latitudes on the Pacific Ocean, twice
a month, when the moon is farthest north
or south of the celestial equator, the two
tides of the day are very different in magni-
tude. When the moon’s declination is zero,
there is no such difference: nor is there ever

Fig. 155.—The Diurnal
any difference at ports which are near the

Inequality.

earth’s equator.

Fig. 155 makes it clear why it should be so. When the moon’s decli-
nation is zero, things are as in Fig. 152 (Art. 469), and the two tides of the same day are sensibly equal at ports in all latitudes. When the moon
is at her greatest northern declination, say 28◦, the two tide summits will
be at A and A in Fig. 155; the tide which occurs at B when the moon
is overhead will be great, while the tide in the corresponding southern
latitude at B will be small. The tides which occur twelve hours later
will be small at the northern station, then situated at C, and large at
the southern station, then at C . For a port on the equator at E or Q
there will be no such difference. In the Atlantic Ocean the difference is
hardly noticeable, because, as we shall see very soon, the tides in that
ocean are mainly (not entirely) due to tide waves propagated into it from
the Pacific and Indian Oceans around the Cape of Good Hope.

472. The Wave Theory of the Tides.—If the earth were
entirely covered with deep water, except a few little islands project-
ing here and there to serve for observing stations, the tide waves
would run around the globe regularly; and if the depth of the ocean
were over thirteen miles, the tide crests, as can be shown, would
follow the moon at an angle of just 90◦. It would be high water
just where the statical theory would give low water. If the depth
were (as it really is) much less than thirteen miles, the tide wave in
the ocean could not keep up with the moon, and the result would
be a very complicated one. The real state of the case is still worse.

The continents of North and South America, with the southern
antarctic continent, make a barrier almost complete from pole to
pole, leaving only a narrow passage at Cape Horn; and the vary-
ing depth of the water, and the irregular contours of the shores are
such that it is quite impossible to determine by theory what the
course and character of the tide wave must be. We must depend
upon observation; and observations are inadequate, because, with

THE PROBLEM OF THREE BODIES.

292
the exception of a few islands, our only possible tide stations are on
the shores of continents where local circumstances largely control
the phenomena.

473. Free and Forced Oscillations.—If the water in the
ocean is suddenly disturbed (as for instance, by an earthquake),
and then left to itself, a “free” wave will be formed, which, if the
horizontal dimensions of the wave are large as compared with the
depth of the water, will travel at a rate depending solely on the
depth. The velocity of such a free wave is given by the formula

√
v =
gh; that is, it is equal to the velocity acquired by a body in
falling through half the depth of the ocean.

Thus a depth of 25 feet gives a velocity of

19+ miles per hour.

100 “

“

“

“

“

39

“

“

“

10,000 “

“

“

“

“

388

“

“

“

40,000 “

“

“

“

“

775

“

“

“

67,200 (12 3 miles)

“

“ 1000

“

“

“

4

90,000 “

“

“

“

“ 1165

“

“

“

Observations upon the waves caused by certain earthquakes in

South America and Japan have thus informed us that between the
coasts of these countries the Pacific averages between two and one-
half and three miles in depth.

474. Now, as the moon in its diurnal motion passes across the

American continent each day, and comes over the Pacific Ocean, it
starts such a “parent” wave in the Pacific, and the wave once started
moves on nearly (but not exactly) like an earthquake wave. Not
exactly, because the velocity of the earth’s rotation being about 1050
miles an hour at the equator, the moon runs relatively westward
faster than the wave can naturally follow, and so for a while slightly
accelerates it. A second tidal wave is produced daily twelve hours
later when the moon passes underneath. The tidal wave is thus,
in its origin, a forced oscillation, while in its subsequent travel it is
pretty nearly a free one.

475. Co-Tidal lines.—These are lines drawn upon the surface
of the ocean connecting those places which have their high water
at the same moment of Greenwich time. They mark the crest of
the tide wave for each hour of Greenwich time; and if we could
draw them with certainty upon the globe, we should have all nec-
essary information as to the motion of the wave. Unfortunately we

THE PROBLEM OF THREE BODIES.

293

Lines.

Cotidal
of

—Map

156.

Fig.

THE PROBLEM OF THREE BODIES.

294
can obtain no direct knowledge as to the position of these lines in
mid-ocean; we only get a few points here and there on the coasts
and on islands, so that a great deal necessarily remains conjectural.

Fig. 156 is a reduced copy of such a map, borrowed with some mod-
ifications from that given in Guyot’s “Physical Geography.”

476. Course of Travel of the Tidal Wave.—On studying the
map we find that the main or “parent” wave starts twice a day in the

Pacific, off Callao, on the coast of South America. This is shown on the
chart by a sort of oval “eye” in the co-tidal lines, just as a mountain
summit is shown on a topographical chart by an “eye” in the contour
lines. From this point the wave travels northwest through the deepest
water of the Pacific, at the rate of about 850 miles per hour, reaching

Kamtchatka in about ten hours. To the west and southwest the water
is shallower and the travel slower,—only 400 to 600 miles per hour,—so
that the wave arrives at New Zealand in about twelve hours. Passing on
by Australia, and combining with the small wave which the moon raises
directly in the Indian Ocean, the resultant tide crest reaches the Cape
of Good Hope in about twenty-nine hours, and enters the Atlantic. Here
it combines with the tide wave, twenty-four hours younger, which has

“backed” into the Atlantic around Cape Horn, and it is modified also
by the direct tide produced by the moon’s action upon the waters of the

Atlantic. The resultant tide crest then travels northward through the

Atlantic at the rate of nearly 700 miles per hour. It is about forty hours
old when it first reaches the coast of the United States in Florida, and
our coast is so situated that it arrives at all the principal ports within
two or three hours of that time. It is forty-one or forty-two hours old
when it arrives at New York and Boston. To reach London it has to
travel around the northern end of Scotland and through the North Sea,
and is nearly sixty hours old when it arrives at that port and the ports
of the German Ocean,—Hamburg, etc.

In the great oceans there are thus three or four tide crests travelling
simultaneously, following each other nearly in the same track, but with
continual minor changes, owing to the variations in the relative positions
of the sun and moon and their changing distances and declinations. If
we take into account the tides in rivers and sounds, the number of simul-
taneous tide crests must be at least six or seven; that is, the high water
at the extremity of its travel, up the Amazon River, for instance, must
be at least three or four days old, reckoned from its birth in the Pacific.1

1We are greatly indebted to Loomis’s discussion of the subject in his “Ele-
ments of Astronomy.”

THE PROBLEM OF THREE BODIES.

295

477. Tides in Rivers.—The tide wave ascends a river at a
rate which depends upon the depth of the water, the amount of
friction, and the swiftness of the stream. It may, and generally
does, ascend until it comes to a rapid, where the velocity of the
water is greater than that of the wave. In shallow streams, however,
it dies out earlier.

Contrary to what is usually supposed, it often ascends to an elevation
far above that of the highest crest of the tide wave at the river’s mouth.

In the La Plata and Amazon it goes up to an elevation at least one
hundred feet above the sea-level. The velocity of the tide wave in a river
seldom exceeds ten or twenty miles an hour, and is usually less.

Fig. 157.—Increase in Height of Tide on approaching the Shore.

478. Height of Tides.—In mid-ocean the difference between
high and low water is usually between two and three feet, as ob-
served on isolated deep-water islands in the Pacific; but on the
continental shores the height is usually much greater. As soon as
the tide wave touches bottom, so to speak, the velocity is dimin-
ished and the height of the wave is increased, something as in the
annexed figure (Fig. 157). Theoretically the height varies inversely
as the fourth root of the depth. Thus, where the water is 100 feet
deep, the tide wave should be twice as high as at the depth of 1600
feet.

Where the configuration of the shore forces the wave into a cor-
ner, it sometimes becomes very high. At Annapolis, on the Bay of

Fundy, tides of seventy feet are not uncommon, and an altitude of

100 feet is said to be occasionally attained.

At Bristol, England, in the mouth of the Severn the tide rises fifty
feet, and sometimes ascends the river (as it also does the Seine, in

France, and the Amazon) as a breaking wave, called the “bore” or “eiger”

(French, mascaret ), with a nearly vertical front five or six feet in height,
crested with foam, and very dangerous to small vessels. On the east
coast of Ireland, opposite to Bristol, the tide ranges only about two feet.

In mid-ocean the water has no progressive motion, but near the land
it has, running in at the flood to fill up the bays and cover the flats, and
then running out again at the ebb. The velocity of these tidal currents
must not be confounded with that of the tide wave itself.

THE PROBLEM OF THREE BODIES.

296

479. Reflection and Interference.—The tide wave when it
reaches the shore is not entirely destroyed, especially if the coast is
bold and the water deep; but is partly reflected, and the reflected
wave goes back into the ocean to meet and modify the new tide wave
which is coming in. Of course, in such a case we get “interferences,”

so that on islands in the Pacific only a few hundred miles apart we
find great differences in the heights of the tides. At one place the
direct waves and the waves reflected from the shores of Asia and

South America may conspire to give a tide of three or four feet, or
nearly double its normal value, while at another they nearly destroy
each other.

There are places, also, which are reached by tides coming by two
different routes. Thus on the east coast of England and Scotland the
tide waves come both around the northern end of Scotland and through
the Straits of Dover. In some places on this coast we have, therefore, a
tide of nearly double height, while at others not very far away there will
be hardly any tide at all; and at intermediate points there are sometimes
four distinct high waters in twenty-four hours. As a consequence of this
reflection and interference of the tide waves it follows that if the tide-
raising power were suddenly abolished, the tides would not immediately
cease, but would continue to run for several days, and perhaps weeks,
before they gradually died out.

480. Effect of the Varying Pressure of the Barometer,
and of the Wind.—When the barometer at a given port is lower
than usual, the level of the water is generally higher than the aver-
age, at the rate of about one foot for every inch of the mercury in
the barometer; and vice versa when it is higher than usual.

When the wind blows into the mouth of a harbor, it drives in
the water of the ocean by its surface friction, and may raise the
water several feet. In such cases the time of high water, contrary to
what might at first be supposed, is delayed, sometimes as much as
fifteen or twenty minutes.

This result depends upon the fact that the water runs into the
harbor for a longer time than it would do if the wind were not
blowing.

The normal depth of the water on the bar is reached
before the predicted time, so that at the predicted time the water
is deeper than it would be if there were no wind, but the maximum
depth is not attained until some time later. Of course, the results
are the opposite when the wind blows out of the harbor: the time

THE PROBLEM OF THREE BODIES.

297
of high water comes earlier, and the depth of water on the bar at
the predicted time of high water is less than it otherwise would be.

481. Tides in Lakes and Inland Seas.—These are small and
difficult to detect. Theoretically, the range between high and low water
in a land-locked sea should bear about the same ratio to the rise and fall
of the tide in mid-ocean that the length of the sea does to the diameter
of the earth. Variations in the direction of the wind and the barometric
pressure cause continual oscillations in the water-level which, even in a
quiet lake, are much larger than the true tides; so that it is only by
taking a long series of observations, and discussing them with reference
to the moon’s position in the sky, that it is possible to separate the real
tide from the effects of other causes. In Lake Michigan, at Chicago, a
tide of about one and three-quarters inches has thus been detected, the

“establishment” of the port being about thirty minutes. In Lake Erie,
at Buffalo and Toledo, the tide is about three-quarters of an inch. On
the coasts of the Mediterranean the tide averages about eighteen inches,
attaining a height of three or four feet at the head of some of the bays.

482. The Rigidity of the Earth.—Sir W. Thomson has en-
deavored to make the tides the criterion of the rigidity of the earth’s
core. Evidently if the solid parts of the earth were fluid, there would
be no observable tide anywhere, since the whole surface would rise
and fall together. If the earth were semi-solid, so to speak (that is,
viscous, and capable of yielding more or less to the forces tending
to change its form), the tides would be observable, but to a less
degree than if the earth’s core were rigid. And with this further
peculiarity—since a viscous body requires time to change its form,
waves of short period would be observable upon the semi-solid earth
nearly to their full extent, while those of long period would almost
entirely disappear, owing to the slow yielding of the earth’s crust.

Now the actual tide wave, as observed, is really made up of a multi-
tude of component tide waves of different periods, ranging from half
a day upwards. According to the “principle of forced vibrations”

every regularly recurring periodic change in the forces which act
on the surface of the ocean must produce a tide of greater or less
magnitude, and of exactly corresponding period.

We have, for instance, the semi-diurnal, solar, and lunar tides; then
the two monthly tides due to the change in the moon’s distance and
declination, and the two annual tides due to the changes of the sun’s
distance and declination, not to speak of the nineteen-year tide due to

THE PROBLEM OF THREE BODIES.

298
the revolution of the moon’s nodes.

A thorough analytical discussion of thirty-three years’ tidal ob-
servations at different parts of the world has been made under the
direction of Sir W. Thomson by Mr. George Darwin, with the result
that not only do the short waves show themselves, but the waves
of long period are found to manifest themselves with almost their
full theoretical value. Thomson’s conclusion is that the earth as a
whole “must be more rigid than steel, but perhaps not quite so rigid
as glass.” This result is at variance with the prevalent belief of ge-
ologists that the core of the earth is a molten mass, and has led to
much discussion which we cannot deal with here.

483. Effect of the Tides on the Earth’s Rotation.—If the
tidal motion consisted merely in the upward and downward motion
of the particles of the ocean to the extent of two feet or so twice a
day, it would involve a very trifling expenditure of energy; and this
is the case with the mid-ocean tide. But near the land this almost
insensible mere oscillatory motion is transformed into the bodily
travelling of immense masses of water, which flow in upon the shal-
lows and then out again to sea with a great amount of fluid friction;
and this involves the expenditure of a very considerable amount of
energy which is dissipated as heat. From what sources does this
energy come? The answer is that it must he derived mainly from
the earth’s energy of rotation, and the necessary effect is to diminish
that energy by lessening the speed of the rotation. Compared with
the earth’s whole stock of rotational energy, however, the loss of it
by tidal friction, even in a century, is very small, and the effect on
the length of the day is extremely slight.

The reader will recall the remarks upon the subject of the secular
acceleration of the moon’s mean motion a few pages back (Art. 461).

While it is certain that the tidal friction tends to lengthen the
day, it does not follow that the day really grows longer. There are
counteracting causes:—for example, the earth’s radiation of heat
into space, and the consequent shrinkage of her volume.

As matters stand we do not know whether, as a fact, the day
is really longer or shorter than it was a thousand years ago. The
change, if any has really occurred, can hardly be as great as

1
of

1000
a second.

THE PROBLEM OF THREE BODIES.

299

484. Effect of the Tide on the Moon’s Motion.—Not only
does the tide diminish the earth’s energy of rotation directly by
the tidal friction, but, theoretically, it also communicates a minute
portion of that energy to the moon. It will be seen that a tidal
wave, situated as in Fig. 158, would slightly accelerate the moon’s
motion, the attraction of the moon by the tidal protuberance F
being slightly greater than that of the tide wave at F —a difference
tending to draw it along in its orbit a little, thus increasing the
major axis of the moon’s orbit. The tendency is therefore to make
the moon recede from the earth, and to lengthen the month.

Fig. 158.

Effect of the Tide on the Moon’s Motion.

Upon this interaction between the tides and the motions of the
earth and moon Professor George Darwin has founded his theory
of “tidal evolution”; namely, that the satellites of a planet, having
separated from it millions of years ago, have been made to recede
to their present distances by just such an action.

An excellent popular statement of the theory will be found in the
closing chapters of Ball’s “Story of the Heavens.” The original papers of

Mr. Darwin in the “Philosophical Transactions” are of course intensely
mathematical.

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

300

CHAPTER XIII.

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.—THE

PTOLEMAIC, TYCHONIC, AND COPERNICAN SYSTEMS.—

THE ORBITS AND THEIR ELEMENTS.—PLANETARY PER-

TURBATIONS.

485. For the most part, the stars keep their relative configu-
rations unchanged, however much they alter their positions in the
sky from hour to hour. The “dipper” remains always a “dipper”

in whatever part of the heavens it may be. But while this is true
of the stars in general, certain of the heavenly bodies, and among
them those that are the most conspicuous of all, form an exception.

The sun and moon continually change their places, moving east-
ward among the stars; and certain others, which to the eye appear
as very brilliant stars, also move,1 though not in quite so simple a
way.

486. These bodies were named by the Greeks the “planets”;
that is, “wanderers.” The ancient astronomers counted seven of
them. They reckoned the sun and moon, and in addition Mercury,

Venus, Mars, Jupiter, and Saturn.

Venus and Jupiter are at all times more brilliant than any of the
fixed stars. Mars at times, but not usually, is nearly as bright as

Jupiter; and Saturn is brighter than all but a very few of the stars.

Mercury is also bright, but seldom seen, because always near the
sun.

At present the sun and moon are not reckoned as planets; but
the roll includes, in addition to the five other bodies known by
the ancients, the earth itself, which Copernicus showed should be
counted among them, and also two new bodies of great magnitude

(though inconspicuous because of their distance) which have been

1When we speak of the motion of the planets, the reader will understand
that the diurnal motion is not taken into account. We speak of their motions
among the stars.

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

301
discovered in modern times; then there is in addition a host of
so-called “asteroids” which circulate in the otherwise vacant space
between the planets Mars and Jupiter.

487. The list of the planets in the order of distance from the sun
stands thus at present: Mercury, Venus, the Earth, Mars, Jupiter,

Saturn, Uranus, and Neptune; and between Mars and Jupiter, in
the place where a planet would naturally be expected to revolve,
there are at present known nearly 300 little planets, which probably
represent a single one, somehow “spoiled in the making,” so to
speak, or burst into fragments.

The planets are all dark bodies, shining only by reflected
sunlight,—globes which, like the earth, revolve around the sun in
orbits nearly circular, moving all in the same direction, and (with
some exceptions among the asteroids) nearly in the common plane
of the ecliptic and sun’s equator. All of them but the inner two and
the asteroids are also attended by “satellites.” Of these the earth
has one (the moon), Mars two, Jupiter four, Saturn eight, Uranus
four, and Neptune one; i.e., so far as at present known; for although
it is hardly probable, it is not at all impossible that others may yet
be found.

488. Relative Distances of Planets from the Sun:

Bode’s Law.—There is a curious approximate relation between
the distances of the planets from the sun, which makes it easy to
remember them. It is usually known as Bode’s Law, because Bode
first brought it prominently into notice in 1772, though Titius of

Wittenberg seems to have discovered and enunciated it some years
earlier. The law is this: Write a series of 4’s. To the second 4 add

3; to the third add 3 × 2, or 6; to the fourth, 3 × 4, or 12; and so
on, doubling the added number each time, as in the accompanying
scheme.

4

4

4

4

4

4

4

4

4

3

6

12

24

48

96

192

384

4

7

10

16

[28]

52

100

196

388

Â

Ã

❤

+

Ä

❦

1

Å

Æ

Ç

È

The resulting numbers, divided by 10, are pretty nearly the true
mean distances of the planets from the sun, in terms of the radius
of the earth’s orbit. In the case of Neptune, however, the law breaks
down utterly, and is not even approximately correct.

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

302

For the present, at least, the law is to be regarded as a mere
coincidence, there being so far no reasonable explanation of any
such numerical relation.

The general expression for the nth term of the series is 4 + 3 × 2(n−2);
but it does not hold good of the first term, which is simply 4, instead of
being 5 1 , i.e., (4 + 3 × 2−1), as it should be.

2

489. Table of Names, Distances, and Periods.

Name.

Symbol. Distance. Bode. Diff.

Sid. Period.

Syn.

Period.

Mercury

Â

0.387

0.4

−0.013

88d or

3m

116d
m

Venus . .

Ã

0.723

0.7

+0.023 224.7d or

7 1

584d

2

❤
d

Earth . .

+

1.000

1.0

0.000 365 1
or 1y

. . .

4
m

Mars . . .

Ä

1.523

1.6

−0.077 687d
or 1y 10 1

780d

2

Mean

2.650

2.8

−0.150

3y.1 to 8y.0
various

Asteroid

Jupiter .

Å

5.202

5.2

+0.002

11y.9

399d

Saturn .

Æ

9.539

10.0

−0.461

29y.5

378d

Uranus .

& Ç

19.183

19.6

−0.417

84y.0

370d
d

Neptune

È

30.054

38.8

−8.746 !

164y.8

367 12

The column headed “Bode” gives the distance according to Bode’s
law; the column headed “Diff.,” the difference between the true distance
and that given by Bode’s law.

490. Fig. 159 shows the smaller orbits of the system (including the
orbit of Jupiter) drawn to scale, the radius of the earth’s orbit being
taken as one centimetre. On this scale the diameter of Saturn’s orbit
would be 19cm.08, that of Uranus would be 38cm.36, and that of Neptune,

60cm.11. The nearest fixed star on the same scale would be about a mile
and a quarter away. It will be seen that the orbits of Mercury, Mars,
and Jupiter are quite distinctly “out of centre” with respect to the sun.

This is intentional and correct. The dotted half of each orbit is that
which lies below, i.e., south of, the plane of the ecliptic. The place of
perihelion of each planet’s orbit is marked with a P . The orbits of five
of the asteroids, including the nearest and the most remote, as well as
the most eccentric, are also given.

Periods.—The sidereal period of a planet is the time of its rev-
olution around the sun from a star to the same star again, as seen
from the sun. The synodic period is the time between two successive
conjunctions of the planet with the sun, as seen from the earth. The

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

303
sidereal and synodic periods are connected by the same relation as
the sidereal and synodic months (Art. 232); namely,—

1

1

1

=

−

,

S

P

E
in which E, P , and S are respectively the periods of the earth and
of the planet, and the planet’s synodic period, and the numerical

1

1
difference between
and
is to be taken without regard to sign.

P

E

The two last columns of the table in Article 489 give the approxi-
mate periods, both sidereal and synodic, for the different planets.

Fig. 159.—Plan of the Orbits of the Planets inside of Saturn.

491. Apparent Motions.—As viewed from a distant point
on the line drawn through the sun, perpendicular to the plane of
the ecliptic, the planets would be seen to travel in their nearly
circular orbits with a regular motion. As seen from the earth the
apparent motion is much more complicated, being made up of their
real motion around the sun combined with an apparent motion due
to the earth’s own movement.

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

304

492. Law of Relative Motion.—The motion of a body rela-
tive to the earth can be very simply stated. It is always the same,
as if the body had, combined with its own motion, another motion,
identical with that of the earth, but reversed.

Fig. 160.—The Relative Motions of Two Bodies.

The proof of this is simple. Let E, Fig. 160, be the earth, and P
the planet, its direction and distance being given by the line EP . Let E
have a motion which will take it to E in a unit of time, and P a motion
which will take it to P in the same time. Then at the end of a unit of
time the distance and direction of P from E will be given by the line

E P . But if we suppose E to remain at rest, and give to P a motion P e
equal to EE but opposite in direction, and combine this motion with

P P by drawing the parallelogram of motions, we shall get P
for the
resulting place of P at the end of the unit of time; and because the line

EP
is parallel and equal to E P (as follows from the construction), the
point P , as seen from E, would occupy, in the celestial sphere, precisely
the same position as P seen from E ; since all parallel lines pierce the
sphere at one and the same optical point (Art. 7).

If, therefore, the earth moves in a circle, every body really at rest
will appear to move in a circle of the same size as the earth’s orbit,
but keeping in such a part of its circle as always to have its motion
precisely opposite to the earth’s own real motion at the moment.

We shall have occasion to use this principle very frequently.

493. Effect of the Combination of the Earth’s Motion
with that of the Planet.—As a consequence, the apparent “geo-
centric” motion of a planet must be made up of two motions,—that
of a body moving once a year around the circumference of a circle
equal to the earth’s orbit, while the centre of this circle itself goes
around the sun upon the real orbit of the planet, and with a peri-
odic time equal to that of the planet. Jupiter, for instance, appears

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

305
to move as in Fig. 161, making eleven loops in each revolution, the
smaller circle having a diameter of about one-fifth that of the larger
one, upon which its centre moves, since the diameter of Jupiter’s
orbit is about five times that of the earth’s.

494. Direct
and

Retrograde

Motions of the Planets and Sta-
tionary Points.—As a consequence
of this looped motion we have the
peculiar back-and-forth movement of
the planets among the stars which has
been described. Starting from the time
when the sun is between us and the
planet,—the time of superior conjunc-
tion,1 as it is called, because the planet
is then above the sun, i.e., further from

Fig. 161.

the earth,—the planet moves eastward

Geocentric Motion of Jupiter
from 1708 to 1720. (Cassini.)

among the stars for a certain time, con-
tinually increasing its longitude (and
also its right ascension) until at last its apparent motion slackens
and it becomes stationary. The distance of this stationary point
from the sun depends upon the size of the planet’s orbit compared
with that of the earth.

Then it reverses its motion and moves westward, or “retro-
grades,” for a while, the middle of the arc of retrogression being
passed at the time when the earth and planet are in line with the
sun, and on the same side of it. If the planet is one of the outer
ones, it will then be opposite to the sun in the sky like the full
moon, and is said to be “in opposition.” If the planet is one of
the inferior planets (Venus or Mercury), it will then be in “inferior
conjunction,” as it is called, between the earth and sun.

After the planet has completed its arc of retrogression, it again

1We give Fig. 162 to illustrate the meaning of the different terms, Opposi-
tion, Quadrature, Inferior and Superior Conjunction, and Greatest Elongation.

E is the position of the earth, the inner circle being the orbit of an inferior
planet, while the outer circle is the orbit of a superior planet. In general, the
angle P ES (the angle at the earth between lines drawn from the earth to the
planet and to the sun) is the planet’s elongation at the moment. For a superior
planet it can have any value from zero to 180◦; for an inferior it has a maximum
value that the planet cannot exceed, depending upon the diameter of its orbit.

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

306
becomes stationary, turns upon its course, and once more advances
eastward among the stars, until the synodic period is completed by
its re-arrival at superior conjunction.

Fig. 162.—Planetary Configurations and Aspects.

Both in the number of degrees passed over, and in the time spent
in this motion, the eastward or “direct ” motion always exceeds the
retrograde. In the case of the remoter planets the excess is small—
from 3◦ to 10◦; in the case of the nearest ones, Mars and Venus, it
is from 16◦ to 18◦.

As observed with a sidereal clock, all the planets come later
to the meridian each night when moving direct, since their right
ascension is then increasing; but vice versa, of course, when they
are retrograding.

495. Motion of the Planets with Respect to the Sun’s

Place in the Sky. Change of Elongation.—The visibility of a
planet depends mainly upon its angular distance, or “elongation,”

from the sun, because when near the sun the planet will be above
the horizon only by day, and cannot usually be seen. As regards
their motions, considered from this point of view, there is a marked
difference between the inferior planets and the superior.

496. Behavior of a Superior Planet.—The superior plan-
ets drop always steadily westward with respect to the sun’s place in
the heavens, continually increasing their western elongation, or de-
creasing their eastern: they therefore invariably come earlier to the

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

307
meridian every successive night, as observed by a time-piece keeping
solar time.

Beginning at superior conjunction, the planet is then moving east-
ward among the stars with its greatest speed; but even then its eastward
motion is not so great as the sun’s, and so the planet relatively falls west-
ward. After a while it will have fallen behind by 90◦, and will then be in
western quadrature, and on the meridian at sunrise; at the end of half
its synodic period it will have lost 180◦, and will be just opposite the sun
at sunset, being then at its least possible distance from the earth, and
at its greatest brilliance. At this time the difference between the times
of its daily culminations is also the greatest possible, and may be as
much (in the case of Mars) as six minutes, by which amount it arrives at
the meridian earlier each successive night. After opposition the planet is
higher in the sky each night at sunset until it reaches eastern quadrature,
when it is 90◦ east of the sun, and therefore on the meridian at sunset.

Thence it drops back, falling more and more slowly westwards towards
the sun, until the synodic period is completed by a new conjunction.

497. Motion of an Inferior Planet.—The inferior planets
appear, on the other hand, to vibrate across the sun, moving out
equal distances on each side of it, but making the westward swing
much quicker than the eastern.

The reason of this difference is obvious from Fig. 162. Matters
take place with respect to the earth, sun, and planet as if the earth
were at rest, and the planet revolving around the sun once in a
synodic (not sidereal) period. Now, since the distance between the
points of greatest elongation, V and V , is less through inferior
conjunction I, than from V around to V through C, the time ought
to be correspondingly shorter, as it is.

At superior conjunction the planet is moving eastward faster than
the sun.

Accordingly, it creeps out to the east from the sun’s rays,
becoming visible in the twilight as an evening star. As long as its direct
motion is greater than the sun’s it keeps receding from it until it reaches
its “greatest eastern elongation,” as it is called, which could in no case be
greater than 90◦, even if the planet’s distance from the sun were almost
equal to that of the earth. (In the case of Venus it is actually about 47◦,
while for Mercury it ranges from 18◦ to 28◦.) Then as its eastern motion
slackens the sun begins to overtake it, and when the planet becomes
really stationary as regards its motion among the stars, it appears to
be slipping westward towards the sun at the rate of about a degree a
day. At the stationary point it begins really to retrograde, and adds its

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

308
motion to the sun’s advance, so that from that point it rushes swiftly
towards the inferior conjunction. It passes this and runs out quickly
on the western side, becoming a morning star, and reaching its western
elongation in just the same number of days that it took to drop from
eastern elongation to inferior conjunction. When the elongation has been
gained, the planet turns around to pursue the sun, gradually gains upon
it, and at last overtakes it again at the next superior conjunction, having
completed its synodic period.

499. Motions in Latitude.—If the planets’ orbits lay pre-
cisely in the same plane with each other and with the earth’s orbit,
they would always keep in the ecliptic. But in fact, while they
never go far from that circle, they do deviate north or south to
the extent of 5◦ or 6◦, and Mercury sometimes as much as 8◦; their
paths among the stars are consequently loops and kinks like those
of Fig. 163, which represent forms actually observed.

Fig. 163.—Loops of Regression.

500. The Ptolemaic System.—The ancient astronomers, for
the most part, never doubted the fixity of the earth, and its position
in the centre of the celestial universe, though there are some reasons
to think that Pythagoras may have done so. Assuming this and the
actual diurnal revolution of the heavens, Ptolemy, who flourished
at Alexandria about 140 a.d., worked out the system which bears
his name. His Μεγάλη Σύνταξις (or Almagest in Arabic) was for
fourteen centuries the authoritative “Scripture of Astronomy.” He
showed that all the apparent motions of the planets could be ac-
counted for by supposing each planet to move around the circumfer-
ence of a circle called the “epicycle,” while the centre of this circle,
sometimes called the “fictitious planet,” itself moved on the circum-
ference of another and larger circle called the “deferent.” It was
as if the real planet was carried on the end of a crank-arm which
turned around the fictitious planet as a centre, in such a way as to

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

309
point towards or from the earth at times when the planet is in line
with the sun.

In the case of the superior planets the revolution in the epicycle was
made once a year, so that the “crank-arm” was always parallel to the
line joining earth and sun, while the motion around the deferent occupied
what we now call the planet’s period. Fig. 164 represents the Ptolemaic

System, except that no attention is paid to dimensions, the “deferents”

being spaced at equal distances. It will be noticed that the epicycle-radii
which carry at their extremities the planets Mars, Jupiter, and Saturn
are all always parallel to the line that joins the earth and sun. In the
case of Venus and Mercury this was not so. Ptolemy supposed that the
deferent circles for these planets lay between the earth and the sun, and
that the “fictitious planet” in both cases revolved in the deferent once a
year, always keeping exactly between the earth and the sun: the motion
in the epicycle in this case was completed in the time of the planet’s
period, as we now know it. He ought to have seen that, for these two
planets, the deferent was really the orbit of the sun itself, as the ancient

Egyptians are said to have understood.

Fig. 164.—The Ptolemaic System.

501. To account for some of the irregularities of the planets’ motions
it was necessary to suppose that both the deferent and epicycle, though
circular, are eccentric, the earth not being exactly in the centre of the
deferent, nor the “fictitious planet” in the exact centre of the epicycle. In
after times, when the knowledge of the planetary motions had become
more accurate, the Arabian astronomers added epicycle upon epicycle
until the system became very complicated. King Alphonso of Spain is
said to have remarked to the astronomers who presented to him the

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310

Alphonsine tables of the planetary motions, which had been computed
under his orders, that “if he had been present at the creation he would
have given some good advice.”

502. Some of the ancient astronomers attempted to account for the
planetary and stellar motions in a mechanical way by means of what
were called the “crystalline spheres.” The planet Jupiter, for instance,
was supposed to be set like a jewel on the surface of a small globe of
something like glass, and this itself was set in a hollow made to fit it in
the thick shell of a still larger sphere which surrounded the earth. Thus
the planets were supported and carried by the motions of these invisible
crystalline spheres; but this idea, though prevalent, was by no means
universally accepted.

503. Copernican

System.—Copernicus

(1473–1543)

as-
serted the diurnal rotation of the earth on its axis, and showed
that it would fully account for the apparent diurnal revolution of
the stars. He also showed that nearly all the known motions of
the planets could be accounted for by supposing them to revolve
around the sun, with the earth as one of them, in orbits circular,
but slightly out of centre. His system, as he left it, was nearly that
which is accepted to-day, and Fig. 159 may be taken as represent-
ing it. He was, however, obliged to retain a few small epicycles to
account for certain of the irregularities.

So far, no one dared to doubt the exact circularity of celestial
orbits. It was metaphysically improper that heavenly bodies should
move in any but perfect curves, and the circle was regarded as the
only perfect one. It was left for Kepler, some sixty-five years later
than Copernicus, to show that the planetary orbits are elliptical,
and to bring the system substantially into the form in which we
know it now.

504. Tychonic System.—Tycho Brahe, who came between

Copernicus and Kepler, found himself unable to accept the Coperni-
can system for two reasons. One reason was that it was unfavorably
regarded by the clergy, and he was a good churchman. The other was
the scientific objection that if the earth moved around the sun, the fixed
stars all ought to appear to move in a corresponding manner (Art. 492),
each star describing annually an oval in the heavens of the same appar-
ent dimensions as the earth’s orbit itself, seen from the star. Technically
speaking, they ought to have an “annual parallax.” His instruments were
by far the most accurate that had so far been made, and he could detect

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311
no such parallax; hence he concluded, not illogically, but incorrectly, that
the earth must be at rest. He rejected the Copernican system, placed
the earth at the centre of the universe, according to the then received
interpretation of Scripture, made the sun revolve around the earth once
a year, and then (this was the peculiarity of his system) made all the
planets except the earth revolve around the sun.

This theory just as fully accounts for all the motions of the planets
as the Copernican, but breaks down absolutely when it encounters the
aberration of light, and the annual parallax of the stars, which we can
now detect with our modern instruments, although Tycho could not with
his. The Tychonic system never was generally accepted; the Copernican
was very soon firmly established by Kepler and Newton.

505. Elements of a Planet’s Orbit.—Those elements are the
numerical quantities which must be given in order to describe the
orbit with precision, and to furnish the means of finding the planet’s
place in the orbit at any given time, whether past or future. They
are seven in number, as follows:—

1. The semi-major axis, a.

2. The eccentricity, e.

3. The inclination to the ecliptic, i.

4. The longitude of the ascending node,

.

5. The longitude of the perihelion, π.

6. The epoch, E.

7. The period P , or daily motion µ.

Fig. 165.—The Elements of a Planet’s Orbit.

506. Of these, the first five pertain to the orbit itself, regarded
as an ellipse lying in space with one focus at the sun, while two are
necessary to determine the planet’s place in the orbit.

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312

The semi-major axis, a (CA in Fig. 165), defines the Size of
the orbit, and may be expressed either in “astronomical units” (the
earth’s mean distance from the sun is the astronomical unit) or in
miles.

The Eccentricity defines the orbit’s Form. It is a mere numer-
c
ical quantity, being the fraction

(usually expressed decimally),
a
obtained by dividing the distance between the sun and the centre
of the orbit by the semi-major axis. In some computations it is
convenient to use, instead of the decimal fraction itself, the angle φ
which has e for its sine; i.e., φ = sin−1 e.

The third element, i, is the Inclination between the plane of the
planet’s orbit and that of the earth. In the figure it is the angle

KN O, the plane of the ecliptic being lettered EKLM .

The fourth element,

(the Longitude of the ascending node),
defines what has been called the “aspect ” of the orbit-plane; i.e.,
the direction in which it faces. The line of nodes is the line N N
in the figure, the intersection of the two planes of the orbit and
ecliptic; and the angle

SN is the longitude of the ascending node,
the line S
being the line drawn from the sun to the first of Aries.

The planet, moving around its orbit in the plane ORBT , and in the
direction of the arrow, passes from the lower or southern side of the
plane of the ecliptic to the northern at the point n, which, as seen
from S, is in the same direction as N .

The fifth, and last, of the elements which belong strictly to the
orbit itself is π, the so-called Longitude of the perihelion, which
defines the direction in which the major axis of the ellipse (the line

P A) lies on the plane ORBT . It is not strictly a longitude, but
equals the sum of the two angles
and ω; i.e.,

SN (in the
plane of the ecliptic) + N SP (in the plane of the orbit). It is quite
sufficient to give ω alone, and in the case of cometary orbits this is
usually done.

507. If we regard the orbit as an oval wire hoop suspended in
space, these five elements completely define its position, form, and
size. The plane of the orbit is fixed by the elements numbered three
and four, the position of the orbit in this plane by number five, the
form of it by number two, and finally its magnitude, by number
one.

The student will recollect that the general equation of curves of

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313
the second degree (the conics) in analytical geometry contains five
constants, and therefore that number of data is enough to define
such a curve completely.

508. To determine where the planet will be at any subsequent
date we need two more elements.

Sixth. The Periodic Time,—we must have the sidereal period,

P , or else the mean daily motion, µ, which is simply 360◦ divided
by the number of days in P .

Seventh.

And finally; we must have a starting-point, the

“Epoch,” so-called; i.e., the longitude of the planet as seen from
the sun, at some given date, usually Jan. 1st, 1850 or 1900, or else
some precise date at which the planet passed the perihelion or node.

509. If it were not for perturbations caused by the mutual interac-
tion between the planets, these elements would never change, and could
be used directly for computing the planet’s place at any date in the past
or in the future; but, excepting a and P , they do change on account
of such interaction, and accordingly it is usual to add in tables of the
planetary elements, columns headed ∆

, ∆π, ∆i, and ∆e, giving the
amount by which the quantities

, π, i, and e respectively change in a
century.

510. If Kepler’s harmonic law were strictly true, we should not need
both a and P , because we should have

3

(Earth’s Period)2 : P 2 :: 13 : a3, or P = a 2 ,

P being expressed in years and a in astronomical units. But since the
exact form of the equation is

P 2

2

3

3

1 (1 + m1) : P2 (1 + m2) :: a1

: a2 (Art. 417),
it is necessary, in cases where the highest attainable accuracy is required,
to regard P and a as independent, and give them both in the tables.

511. Geocentric Place.—Our observations of a planet’s place
are necessarily “geocentric,” or earth-centred; they give us, when
properly corrected for refraction and parallax, the planet’s right
ascension and declination as seen from the centre of the earth, and
from them, if desired, the corresponding geocentric longitude and
latitude are easily obtained by the method explained in Article 180.

It often happens that we want the place at some moment of time
when the planet could not be directly observed, as, for instance, in the

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

314
day time. If we have a series of observations of the planet made about
that time, the place for the exact moment is readily deduced by a process
of interpolation, and with an accuracy actually exceeding that of any
single observation of the series.

Graphically it is done by simply plotting the observations actually
made. Suppose, for instance, we want the right ascension of Mars for

8 a.m. on June 3, and have meridian-circle observations made at 10
o’clock p.m. on June 1, at 9h 55m on June 2, at 9h 50m on June 3,
and so on. We first lay off the times of observation as abscissas along
a horizontal line taken as the time-scale, and then lay off the observed
right ascensions as ordinates at points corresponding to the times. Then
we draw a smooth curve through the points so determined, and from
this curve we can read off directly, the right ascension corresponding to
any desired moment: similarly for the declination. Of course, what can
be done graphically can be done still more accurately by computation.

512. Heliocentric Place.—The heliocentric place of a planet
is the place as seen from the sun; and when we know the longitude of
the node of a planet’s orbit and its inclination, as well as the planet’s
distance from the sun, this heliocentric place can at once be deduced
from the geocentric by a trigonometrical calculation. The process
is rather tedious, however, and its discussion lies outside the scope
of this work.

(The reader is referred to Watson’s “Theoretical Astronomy,” p. 86.

An elementary geometrical treatment of the reduction is also given in

Loomis’s “Treatise on Astronomy,” p. 211.)

513. Determination of a Period of a Planet.—This can
be done in two ways:

First. By observation of its node-passage. When the planet is
passing its node, it is in the plane of the ecliptic, and the earth being
also always in that plane, the planet’s latitude, both geocentric and
heliocentric, will be zero, no matter what may be the place of the
earth in its orbit. (At any other point of the planet’s orbit except
the node its apparent latitude would not be thus independent of
the earth’s place, but would vary according to its distance from the
earth.) If, then, we observe the planet at two successive passages of
the same node, the interval between the moments when the latitude
becomes zero will be the planet’s period,—exactly, if the node is
stationary; very approximately, even if the node is not absolutely
stationary, as none of the nodes actually are.

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315

There are two difficulties with this method.

(a) In the case of Uranus the period is eighty-four years, and in that
of Neptune 164 years—too long to wait.

(b) Since the orbits all cross the ecliptic at a very small angle, so
that the latitude remains near zero for a number of days, it is extremely
difficult to determine the precise minute and second when it is exactly
zero; and slight errors in the declinations observed will produce great
errors in the result.

514. Second. By the mean synodic period of the planet. The
synodic period is the interval between two successive oppositions or
conjunctions of the planet, the opposition being the moment when
the planet’s longitude differs from that of the sun by 180◦.

This angle between the planet and sun cannot well be measured di-
rectly, but we can make with the meridian circle a series of observations
both of the planet’s right ascension and declination for several days be-
fore and after the date of opposition, and reduce the observations to
latitude and longitude. The sun will be observed, of course, at noon,
and the planet near midnight; but from the solar observations we can
deduce the longitudes of the sun corresponding to the exact moments
when the planet was observed. From these, we find the difference of
longitude between the planet and the sun at the time of each planetary
observation; and finally from these differences of longitude, we find the
precise moment when that difference was exactly 180◦, or the moment
of opposition. This can be ascertained within a very few seconds of time
if the observations are good.

Since the orbits are not strictly circular, the interval between
two successive observations will not be the mean synodic period,
but only an approximation to it; but when we know it nearly, we
can compare oppositions many years apart, and by dividing the
interval by the known number of entire synodic periods (which is
easily determined when we know the approximate length of a single
period) we get the mean synodic period very closely,—especially
if the two oppositions occur at about the same time of the year.

Having the synodic period, the true sidereal period at once follows
from the equation

1

1

1

=

−

.

P

E

S

515. To find the Distance of a Planet in Terms of the

Earth’s Distance.—When we know the planet’s sidereal period,
this is easily done by means of two observations of the planet’s

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

316

“elongation” taken at an interval equal to its periodic time. The

“elongation” of a planet is the difference between its longitude and
that of the sun, and a series of meridian-circle observations of sun
and planet will furnish these differences of longitude for any selected
moment included within the term of observation.

Fig. 166.—Determination of the Distance of a Planet from the Sun.

To find the distance of the planet Mars, for instance, we must
therefore have two observations separated by an interval of 687 days.

Suppose the earth to have been at A (Fig. 166) at the moment of
the first observation. Then at the time of the second observation she
will be at the point C, the angle ASC being that which the earth
will describe in the next 43 1 days, which is the difference between

2
two complete years (or 730 1 days) and the 687-day interval between

2
the two observations.

The angles SCM and SAM are the “elongations” of the planet
from the sun, and are given directly by the observations. The two
sides SA and SC are also given, being the earth’s distance from
the sun at the dates of observation. Hence we can easily solve the
quadrilateral, and find the length of SM , as well as the angle ASM .

This angle determines the planet’s heliocentric longitude at M , since
we know the direction of SA, the longitude of the earth at the time of
observation.

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317

The student can follow out for himself the process by which, from
two elongations of Venus, SAV and SBV , observed at an interval of 225
days, the distance of Venus from the sun (or SV ) can be obtained.

516. In order that this method may ap-
ply with strict accuracy it is necessary that
at the moment of observation M should be in
the same plane as A, S, and C; that is, at
the node. If it is not so, the process will give
us, not the true distance of the planet itself

Fig. 167.

from the sun, but that of the “projection” of
this distance on the plane of the ecliptic; i.e., the distance from the
sun to the point m (Fig. 167), where the perpendicular from the planet
would strike that plane. But when we have determined Am and the
angle mAM , the planet’s geocentric latitude, we easily compute M m;
and from Sm and M m we get the true distance SM and the heliocentric
latitude of the planet M Sm.

517. From a series of pairs of observations distributed around
the planet’s orbit it would evidently be possible to work out the
orbit completely. It was in this way that Kepler showed that the
orbits of the planets are ellipses, and deduced their distances from
the sun; and his third, or harmonic law, was then discovered simply
by making a comparison between the distances thus found and the
corresponding periods.

518. Mean Distance of an Infe-
rior Planet by Means of Observa-
tions of its Greatest Elongation.—

By observing from the earth the great-
est elongation SEV (Fig. 168) of one
of the inferior planets, its distance from
the sun can very easily be deduced if
we regard the orbit as a circle; for the
triangle SV E will be right-angled at V ,

Fig. 168.

and SV = SE × sin SEV .

Distance of an Inferior Planet
determined by Observations

In the case of Venus the orbit is so
of its Greatest Elongation.

nearly circular that the method answers
very well, the greatest elongation never differing much from 47◦. Mer-
cury’s orbit is so eccentric that the distance thus obtained from a single
elongation might be very wide of the true mean distance. Since the
greatest elongation, SEM , varies all the way from 18◦ to 28◦, it would

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

318
be necessary to observe a great many elongations, and take the average
result.

519. Deduction of the Orbit of a Planet from Three

Observations.—When one has command of a great number of ob-
servations of a planet running back many years, and can select such
as are convenient for his purpose, as Kepler could from Tycho’s
records, it is comparatively easy to find the elements of a planet’s
orbit; but when a new planet is discovered, the case is different.

The problem first arose practically in 1801, when Ceres, the first of
the asteroids, was discovered by Piazzi in Sicily, observed for a few
weeks and then lost in the sun’s rays at conjunction, before other
astronomers could be notified of the discovery, in these days of slow
communication, made slower and more uncertain by war.

Gauss, then a young man at Göttingen, attacked the problem,
and invented the method which, with slight modifications, is now
universally used in such cases.

We do not propose to enter into details, but simply say that three
absolutely accurate observations of a planet’s right ascension and
declination are sufficient to determine its orbit. Three observations,
made only as accurately as is now possible, with intervals of two or
three weeks between them, will give a very good approximation to
the orbit; and it can then be corrected by further observations.

520. Since there are five independent coefficients in the equation
of a conic, it is necessary to have five conditions in order to determine
them. Three such conditions are given by the observations themselves;
viz., the three directions of the planet as seen from the earth at the given
moments of observation; a fourth is given by the fact that the sun must
be in the focus of the orbit; and finally, the fifth is supplied by the “law
of equal areas,” since the areas described by the radius vector between
the first and second observations, and the second and third, must be
proportional to the corresponding intervals of time.

(The student is referred to Gauss’s “Theoria Motus,” or to Watson’s

“Theoretical Astronomy,” or to Oppolzer’s great work on “The Deter-
mination of Orbits,” for the full development of the subject.)

521. Planetary Perturbations.—The attraction of the plan-
ets for each other disturbs their otherwise elliptical motion around
the sun. As in the case of the lunar theory the disturbing forces are,
however, always relatively small, but not for the same reason. The

THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.

319
sun’s disturbing force is small because its distance from the moon is
nearly four hundred times that of the earth. In the planetary theory
the disturbing bodies are often nearer to the disturbed than is the
sun itself, as, for instance, in the disturbance of Saturn by Jupiter
at certain points of their orbits; but the mass of the disturbing body
in no case is as great as

1
part of the sun’s mass, and for this rea-

1000
son the disturbing force arising from planetary attraction is never
more than a small fraction of the sun’s attraction.

The greatest disturbing force which occurs in the planetary system

(except in the case, of some of the asteroids) is that of Jupiter on Saturn
at the time when the planets are nearest: it then amounts to 1 of the

128
sun’s attraction. When these two planets are most remote from each
other, it amounts to

1 . There is no other case where the disturbing

357
force is as much as

1
of the sun’s attraction (again excepting the

1000
asteroids disturbed by Jupiter).

522. In any special case the disturbing force can he worked out
on precisely the same principles that lie at the foundation of the
diagram by which the sun’s disturbing force upon the moon was
found (Art. 441, Fig. 147); but the resulting diagram will look very
differently, because the disturbing body is relatively very near the
disturbed orbit.

The planetary perturbations which result from the “integration”

of effects of the disturbing forces, i.e., from their continual action
through long intervals of time, divide themselves into two great
classes,—the Periodic and the Secular.

523. Periodic Perturbations.—These are such as depend on
the positions of the planets in their orbits, and usually run through
their course in a few revolutions of the planets concerned. For the
most part they are very small. Those of Mercury never amount to
more than 15 , as seen from the sun. Those of Venus may reach
about 30 , those of the earth about 1 , and these of Mars about

2 . The mutual disturbances between Jupiter and Saturn are much
larger, amounting respectively to 28 and 48 ; while those of Uranus
are again small, never exceeding 3 , and those of Neptune are not
more than half as great as that. In the case of the asteroids, which
are powerfully disturbed by Jupiter, the periodical perturbations
are enormous, sometimes as much as 5◦ or 6◦.

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320

524. Long Inequalities.—The periodic inequalities of the plan-
ets are so small, because, as a rule, there is a nearly complete compensa-
tion effected at every few revolutions, so that the accelerations balance
the retardations. The line of conjunction falls at random in different
parts of the orbits, and when this is the case, no considerable displace-
ment of either planet can take place. But when the periodic times of
two planets are nearly commensurable, their line of conjunction will fall
very near the same place in the two orbits for a considerable number of
years, and the small unbalanced disturbance left over at each conjunc-
tion will then accumulate in the same direction for a long time. Thus,
five revolutions of Jupiter roughly equal two of Saturn; and still more
nearly, seventy-seven of Jupiter equal thirty-one of Saturn, in a period
of 913 years. From this comes the so-called “long inequality” of Jupiter
and Saturn, amounting to 28 in the place of Jupiter and 48 in that of

Saturn, and requiring more than 900 years to complete its cycle.

In the case of the earth and Venus there is a similar “long inequality”

with a period of 235 years, amounting, however, to less than 3 in the
positions of either of the planets; between Uranus and Neptune there is
a similar inequality with a period of over 4000 years, but this also is very
small.

525. Secular Inequalities.—These are inequalities which de-
pend not on the position of the planets in their orbits, but on the
relative position of the orbits themselves, with reference to each
other,—the way, for instance, in which the lines of nodes and ap-
sides of two neighboring orbits lie with reference to each other.

Since the planetary orbits change their positions very slowly, these
perturbations, although in the strict sense of the word periodic also,
are very slow and majestic in their march, and the periods involved
are such as stagger the imagination. They are reckoned in myriads
and hundreds of thousands of years. From year to year they are
insignificant, but with the lapse of time become important.

526. Secular Constancy of the Periods and Mean Dis-
tances.—It is a remarkable fact, demonstrated by Lagrange and

La Place about 100 years ago, that the mean distances and periods
are entirely free from all such secular disturbance. They are subject
to slight periodic inequalities having periods of a few years, or even
a few hundred years: but in the long run the two elements never
change. They suffer no perturbations which depend on the position
of the orbits themselves, but only such as depend on the positions
of the planets in their orbits.

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321

527. Revolution of the Nodes and Apsides.—The nodes
and perihelia, on the other hand, move on continuously. The lines
of apsides of all the planets (Venus alone excepted) advance, and
the nodes of all without exception (except possibly some of the
asteroids), regress on the ecliptic.

The quickest moving line of apsides—that of Saturn’s orbit—com-
pletes its revolution in 67,000 years, while that of Neptune requires

540,000. The swiftest line of nodes is that of Uranus, which completes its
circuit in less than 37,000 years, while the slowest—that of Mercury—
requires 166,000 years.

528. The Inclinations of the Orbits.—These are all slowly
changing—some increasing, and others decreasing; but as La Place
and Leverrier have shown, all the changes are confined within nar-
row limits for all the larger planets: they oscillate, but the oscilla-
tions are never extensive.

It is not certain that this is so with the asteroids, some of which have
inclinations to the ecliptic of 25◦ and 30◦: it is possible that some of
these inclinations may change by a very considerable amount.

529. The Eccentricities.—These also are slowly changing in
the same way as the inclinations, some increasing and some decreas-
ing; and their changes also are closely restricted. The periods of the
alternate increase and decrease are always many thousand years in
length.

The asteroids are again to be excepted; the eccentricities of their
orbits may change considerably.

530. Stability of the Planetary System.—About the end of
the eighteenth century La Place and Lagrange succeeded in proving
that the mutual attraction of the planets could never destroy the
system, nor even change the elements of the orbit of any one of the
larger planets to an extent which would greatly alter its physical
condition.

The nodes and apsides revolve continuously, it is true, but that
change is of no importance. The distances from the sun and the
periods do not change at all in the long run; while the inclinations
and eccentricities, as has just been said, confine their variations
within narrow limits.

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322

531. The “Invariable Plane” of the Solar System.—There
is no reason, except the fact that we live on the earth, for taking the
plane of the earth’s orbit (the plane of the ecliptic) as the fundamental
plane of the solar system. There is, however, in the system an “invariable
plane,” the position of which remains forever unchanged by any mutual
action among the planets, as was discovered by La Place in 1784. This
plane is defined by the following conditions,—that if from all the planets
perpendiculars be drawn to it (i.e., to speak technically, if the planets
be “projected” upon it), and then if we multiply each planet’s mass by
the area which the planet’s projected radius vector describes upon this
plane in a unit of time, the sum of these products will be a maximum.

The ecliptic is inclined about 1 1 ◦ to this invariable plane, and has its

2
ascending node nearly in longitude 286◦.

532. La Place’s Equations for the Inclinations and Ec-
centricities.—La Place demonstrated the two following equations, viz.:

√

√

(1)

(m a × e2) = C.

(2)

(m a × tan2 i) = C .

Equation (1) may be thus translated: Multiply the mass of each planet
by the square root of the semi-major axis of its orbit, and by the square
of its eccentricity; add these products for all the planets, and the sum
will be a constant quantity C, which is very small. It follows that no
eccentricity can become very large, since e2 in the equation is essentially
positive: there can therefore be no counterbalancing of positive and neg-
ative eccentricities; and if the eccentricity of one planet increases, that
of some other planet or planets must correspondingly decrease.

The second equation is the same, merely substituting tan2 i for e2, i
being the inclination of the planet’s orbit to the invariable plane.

The constant in this case also is small, though of course not the same
as in the preceding equation. These two equations, taken in connection
with the invariability of the periods of the major axes of the planetary
orbits, have been called the “Magna Charta” of the stability of the solar
system.

533. It does not follow that because the mutual attractions of
the planets cannot seriously derange the system it is, therefore, of
necessity securely stable. There are many other conceivable actions
which might end in its ultimate destruction; such, for instance, as
that of a resisting medium, or the entrance into the system of bodies
coming from without.

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323

CHAPTER XIV.

THE PLANETS: METHODS OF FINDING THEIR DIAMETERS,

MASSES,

ETC.—THE

“TERRESTRIAL

PLANETS”

AND

ASTEROIDS.—INTRA-MERCURIAL

PLANETS

AND

THE

ZODIACAL LIGHT.

In discussing the individual peculiarities of the planets, we have
to consider a multitude of different data; for instance, their diame-
ters, their masses, and densities, their axial rotation, their surface-
markings, their reflecting power or “albedo,” and their satellite sys-
tems.

534. Diameter.—The apparent diameter of a planet is ascer-
tained by measurement with some kind of micrometer (Art. 73).

For this purpose the “double-image” micrometer has an advantage
over the wire micrometer because of the effect of irradiation.

When we bring two wires to touch
the two limits of the planet in the field
of view of the telescope, Fig. 169, a,
the bright image of the planet is al-
ways measured too large, because ev-

Fig. 169.

ery bright object appears to extend it-

Micrometer Measures of a
self somewhat into the dark surround-

Planet’s Diameter.

ing space, by its physiological action
upon the retina of the eye. This is known as irradiation—well exem-
plified at the time of new moon, when the bright crescent appears to be
much larger than the “old moon” faintly visible by earth-shine. With
small instruments this error is often considerable, varying with the per-
sonal equation of the observer, but it may be reduced to some extent
by using as bright an illumination of the field of view as the object will
bear.

With the double-image micrometer, the observer in measuring has
to bring in contact two discs of equal brightness, as in Fig. 169, b; and
in this case the irradiation almost vanishes at the point of contact.

The diameter thus measured is, of course, only the apparent

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324
diameter, to be expressed in seconds of arc, and varies with every
change of distance. To get the real diameter in linear units, we have

∆ × D

Real diameter =

,

206265
in which ∆ is the distance of the planet from the earth, and D the
diameter in seconds of arc. If ∆ is given only in astronomical units,
the diameter comes out, of course, in terms of that unit. To get
the diameter in miles, we must know the value of this unit in miles;
that is, the sun’s distance from the earth.

535. Extent of Surface and Volume.—Having the diameter,
the surface, of course, is proportional to its square, and is equal to
s 2
the earth’s surface multiplied by

, in which s is the semi-
ρ
diameter of the planet and ρ that of the earth.

s 3

The volume equals
in terms of the earth’s volume. (The
ρ
student must be on his guard against confounding the volume or
bulk of a planet with its mass.)

The nearer the planet, other things being equal, the more ac-
curately the above data can be determined. The error of 0 .1 in
measuring the apparent diameter of Venus, when nearest, counts
for less than thirteen miles in the real diameter of the planet; while
in Neptune’s case it would correspond to more than 1300 miles.

The student must not be surprised, therefore, at finding consid-
erable discrepancies in the data given for the remoter planets by
different authorities.

536. Mass of a Planet which has a Satellite.—In this case
its mass is easily and accurately found by observing the period and
distance of the satellite. We have the fundamental equation
r3

(M + m) = 4π2

,
t2
in which M is the mass of the planet, m that of its satellite, r the
radius of the orbit of the satellite, and t its period.

The formula is derived as follows: From the law of gravitation the
accelerating force which acts on the satellite is given by the equation

M + m
f =
r2

THE PLANETS THEMSELVES.

325

(Art. 417), in which M is the mass of the planet and m that of the
satellite. From the law of circular motion (Art. 411, Eq. b) we have
r
f = 4π2

;
t2
whence (equating the two values of f ) we have

M + m
r

= 4π2

;
r2
t2
and finally
r3

(M + m) = 4π2

.

t2

This demonstration is strictly good only for circular orbits; but the equa-
tion is equally true, and can be proved for elliptical orbits, if for r we
put a, the semi-major axis of the satellite’s orbit.

For many purposes a proportion is more convenient than this
equation, since the equation requires that M , m, r, and t be ex-
pressed in properly chosen units in order that it may be numerically
true. Converting the equation into a proportion, we have
r3
r3

(M + m) : (M

1

1 + m1) =

:

;
t2
t21
or, in words, the united mass of a body and its satellite is to the
united mass of a second body and its satellite as the cube of the
distance of the first satellite divided by the square of its period is to
the cube of the distance of the second satellite divided by the square
of its period. This enables us at once to compare the masses of any
two bodies which have attendants revolving around them.

The mass of the moon is so considerable as compared with that
of the earth (about 1 ) that it will not do to neglect it; but in all

80
other cases the satellite is less than

1
of the mass of its primary,

1000
and need not be taken into account.

537. Examples.—(1) Required the mass of the sun compared
with that of the earth. The proportion is

(93000000)3

(238000)3

(S + earth) : (E + moon) =

:

.

(365 1 )2

(27.4)2

4

The quantities in the last term of the proportion are of course the dis-
tance and period of the moon; and it is to be remembered that for the

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326
period of the moon we must use, not the actual sidereal period, but the
period as it would be if the moon’s motion were undisturbed,—a period
about three hours longer.

(2) Compare the mass of the earth with that of Jupiter, whose re-
motest satellite has a period of 16 3 days, and a distance of 1,160,000

4
miles. We have

(238000)3

1160000)3

(E + moon) : (J + satellite) =

:

,

(27.4)2

(16 3 )2

4
which gives the mass of Jupiter about 316 times as great as that of the
earth and moon together.

538. It is customary to express the mass of a planet as a certain
fraction of the sun’s mass, and the proportion is simply

R3
r3

Sun : Planet =

:

,

T 2
t2
r

2

3

T
whence

Planet’s mass = Sun’s mass ×

×

,

R
t
where T and R are the planet’s period and distance from the sun.

Since R and r can both be determined in astronomical units without
any necessity for knowing the length of that unit in miles, the masses
of the planets in terms of the sun’s mass are independent of any
knowledge of the solar parallax. But to compare them with the
earth, we must know this parallax, since the moon’s distance from
the earth, which enters into the equations, is found by observation
in miles or in radii of the earth, and not in astronomical units.

In order to make use of the satellites for this purpose we must
determine by micrometrical observations their distances from the
planets and their periods.

539. Mass of a Planet which has no Satellite.—When a
planet has not a satellite, the determination of its mass is a very dif-
ficult and troublesome problem, and can be solved only by finding
some perturbation produced by the planet, and then ascertaining,
by a sort of “trial and error” method, the mass which would pro-
duce that perturbation. Venus disturbs the earth and Mercury, and
from these perturbations her mass is ascertained. Mercury disturbs

Venus, and also one or two comets which come near him, and in
this way we get a rather rough determination of his mass.

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327

540. Density.—The density of a body as compared with the
earth is determined simply by dividing its mass by its volume; i.e.,
m

Density =

.

s 3
ρ

For example, Jupiter’s diameter is about eleven times that of the
s
earth (i.e.

= 11), so that his volume is 113, or 1331 times the
ρ
earth’s. His mass, derived from satellite observations, is about 316 times
the earth’s. The density, therefore, equals 316 , or about 0.24, of the

1331
earth’s density, or about 1 1 times that of water, the earth’s density

3
being 5.58 (Art. 171).

541. The Surface Gravity.—The force of gravity on a
planet’s surface as compared with that on the surface of the earth
is important in giving us an idea of its physical condition. If r is
the radius of the planet in terms of the earth’s radius, then
m
m
s

Surface gravity, or G, =

, =

×

.

2

3
s
s
ρ
ρ
ρ
i.e., it equals the planet’s density, multiplied by its diameter ex-
pressed in terms of the earth’s diameter.

316

For Jupiter, therefore, G =

= 11 × density = 11 × 0.24 = 2.64

112
nearly. That is, a body at Jupiter would weigh 2.6 as much as one at
the earth’s surface.

542. The

Planet’s

Oblateness.—The

“oblateness”

or

“polar-compression” is the difference between the equatorial and
polar diameters divided by the equatorial diameter. It is, of course,
determined, when it is possible to determine it at all, simply by
micrometric measurements of the difference between the greatest
and least diameters. The quantity is always very small and the
observations delicate.

543. The Time of Rotation, when it can be determined, is
found by observing the passage of some spot visible in the telescope
across the central line of the planet’s disc. In reducing the observa-
tions to find the interval between such transits, account has to be
taken of the continual change in the direction of the line which joins

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328
the planet and the earth, and also of the variations in the distance,
which will alter the time taken by light in coming to the earth from
the body.

544. The Inclination of the Axis is deduced from the same
observations which are used in obtaining the rotation period. It is
necessary to determine with the micrometer the paths described by
different spots as they move across the planet’s disc. It is possible to
ascertain it with accuracy for only a very few of the planets: Mars,

Jupiter, and Saturn are the only ones that furnish the needed data.

545. The Surface Peculiarities and Topography of the
surface are studied by the telescope. The observer makes drawings
of any markings which he may see, and by their comparison is at
last able to discriminate between what is temporary and what is
permanent on the planet. Mars alone, thus far, permits us to make
a map of its surface.

546. Spectroscopic Peculiarities and Albedo.—The char-
acteristics of the planet’s atmosphere can be to some extent studied
by means of the spectroscope, which in some few cases shows the
presence of water-vapor and other absorbing media, by dark bands
in the planet’s spectrum. The “albedo,” or reflecting power of a
planet’s surface is determined by photometric observations, com-
paring it with a real or artificial star, or with some other planet.

547. The Satellite System of a Planet.—The principal data
to be ascertained are the distances and periods of the satellites, and
the observations are made by measuring the apparent distances and
directions of the satellites from the centre of the planet with the wire
micrometer (Art. 73). Observations made at the times when the
satellite is near its elongation are especially valuable in determining
the distance.

If the planet and earth were at rest, the satellite’s path would appear
to be an ellipse, unaltered in dimensions during the whole series of ob-
servations; but since the earth and planet are both moving, it becomes
a complicated problem to determine the satellite’s true orbit from the
ensemble of observations.

548. With the exception of the moon and the outer satellite of Sat-
urn, all the satellites of the planetary system move almost exactly in
the plane of the equator of the primary; and all but the moon and the
seventh satellite of Saturn (Hyperion) move in orbits almost perfectly

THE PLANETS THEMSELVES.

329
circular. Tisserand has recently shown that the equatorial protuber-
ance of a planet compels any satellite which is not very remote from its
primary to move nearly in the equatorial plane, but the almost perfect
circularity of the orbits is not yet explained. When there are a number of
satellites in a system, interesting problems arise in connection with their
mutual disturbances; and in a few cases it becomes possible to determine
a satellite’s mass as compared with that of its primary. In several in-
stances satellites show peculiar variations in their brightness, which are
supposed to indicate that they make an axial rotation in the time of one
revolution around the primary, in the same way as our moon does.

549. Humboldt’s

Classification
of
the

Planets.—

Humboldt has divided the planets into two groups: the terrestrial
planets, so called, and the major planets. The terrestrial planets
are Mercury, Venus, the earth, and Mars.

They are bodies of
the same order of magnitude, ranging from 3000 to 8000 miles in
diameter, not very different in density (though Mercury is much
denser than either of the others), and are probably roughly alike
in physical constitution, and covered with water and air. But we
hasten to say that the differences in the amount of heat and light
which they receive from the sun, and in the force of gravity upon
their surfaces, and probably in the density of their atmospheres, are
such as to bar any positive conclusions as to their being the abode
of life resembling the forms of life with which we are acquainted on
the earth.

Fig. 170.—Relative Sizes of the Planets.

THE PLANETS THEMSELVES.

330

550. The four major planets, Jupiter, Saturn, Uranus, and

Neptune, are much larger bodies (ranging in diameter between

30,000 and 90,000 miles), are much less dense, and so far as we
can make out, present to us only a surface of cloud, and may not
have anything solid about them. There are some reasons for sus-
pecting that they are at a high temperature; in fact, that Jupiter is
a sort of semi-sun, but this is by no means yet certain.

As for the multitudinous asteroids, the probability is that they
represent a single planet of the terrestrial group which, as has been
intimated, failed for some reason in its evolution, or else has been
broken to pieces. All of them united would not make a planet one-
half the mass of the earth.

Fig. 170 shows the relative sizes of the different planets.

In what follows, all the numerical data, so far as they depend
on the solar parallax, are determined on the assumption that that
parallax is 8 .80, and that the sun’s mean distance is 92,897000
miles.

MERCURY.

551. There is no record of the discovery of the planet. It has
been known from remote antiquity; and we have recorded observa-
tions running back to b.c. 264.

For a time the ancient astronomers seem to have failed to recognize it
as the same body on the eastern and western sides of the sun, so that the

Greeks had for a time two names for it,—Apollo when it was morning
star, and Mercury when it was evening star. According to Arago, the

Egyptians called it Set and Horus, and the Hindoos also gave it two
names.

It is so near the sun that it is comparatively seldom seen with
the naked eye; but when near its greatest elongation it is easily
enough visible as a brilliant star of the first magnitude low down
in the twilight, perhaps not quite so bright as Sirius, but certainly
brighter than Arcturus. It is usually visible for about a fortnight at
each elongation, and is best seen in the evening at such eastern elon-
gations as occur in March and April. In Northern Europe it is much
more difficult to observe than in lower latitudes, and Copernicus is
said never to have seen it. Tycho, however, obtained a considerable
number of observations.

THE PLANETS THEMSELVES.

331

552. It is exceptional in the solar system in a great variety of
ways. It is the nearest planet to the sun, receives the most light and
heat, is the swiftest in its movement, and (excepting some of the
asteroids) has the most eccentric orbit, with the greatest inclination
to the ecliptic. It is also the smallest in diameter (again excepting
asteroids), has the least mass, and the greatest density of all the
planets.

553. Distance, Light, and Heat.—Its mean distance from
the sun is 36,000000 miles, but the eccentricity of its orbit is so
great (0.205), that the sun is seven and one-half millions of miles
out of its centre, and the actual distance of the planet from the sun
ranges all the way from 28,500000 to 43,500000, while its velocity in
its orbit varies from thirty-five miles a second at perihelion to only
twenty-three at aphelion. On the average it receives 6.7 times as
much light and heat as the earth; but the heat received at perihelion
is to that at aphelion in the ratio of 9 to 4. For this reason there
must be two seasons in its year due to the changing distance, even
if the equator of the planet is parallel to the plane of its orbit,
which would preclude seasons like our own. If the planet’s equator
is inclined at an angle like the earth’s, then the seasons must be
very complicated.

554. Period.—The sidereal period is very nearly 88 days, and
the synodic period, or the time from conjunction to conjunction
again, is about 116 days. The greatest elongation ranges from 18◦
to 28◦, and occurs about twenty-two days before and after the in-
ferior conjunction, or about thirty-six days before and after the
superior conjunction. The planet’s arc of retrogression is about 12◦

(considerably variable), and the stationary point is very near the
greatest elongation.

555. Inclination.—The inclination of the orbit to the ecliptic
is about 7◦, but the greatest geocentric latitude (that is, the planet’s
greatest distance from the ecliptic as seen from the earth) is never
quite so great.

556. Diameter, Surface, and Volume.—The apparent di-
ameter ranges from 5 to about 13 , according to its distance from
us; the least distance from the earth being about 57,000000 miles

(93 − 36), while the greatest is about 129,000000 (93 + 36). The real
diameter is very near 3000 miles, not differing from that more than

THE PLANETS THEMSELVES.

332
fifty miles either way. It is not easy to measure, and the “proba-
ble error” is perhaps rather larger than would have been expected.

With this diameter, its surface is 1 of the earth’s, and its volume

7

1 .

18.5

557. Mass, Density, and Surface Gravity.—Its mass is
very difficult to determine, since it has no satellite, and the val-
ues obtained by La Place, Encke, Leverrier, and others, range all
the way from 1 of the earth’s mass to 1 .

9

15

The most recent, and perhaps the most reliable, determination, al-
though the largest of all, is that of Backlund, by means of the pertur-
bations of Encke’s comet. He makes its mass about 1 that of the earth.

8

This gives us a mean density about 2 1 times that of the earth, or nearly

4

12 1 times that of water, a little less than that of the metal mercury. So

2
far as known, the earth stands next to it in density, but with a wide
interval, so that Mercury is altogether exceptional in this respect, as has
been said before. The result, however, is liable to a considerable error,
on account of the uncertainty of the planet’s mass, Leverrier’s estimate
of the mass and density being only about one-half as great as Backlund’s.

The superficial gravity is about 5 that at the earth’s surface.

6

558. Its Albedo, or reflecting power, as determined by Zöllner
is very low—only 0.13, somewhat inferior to that of the moon.

In 1878 Mr. Nasmyth observed the planet in the same field of view
with Venus; and although Mercury was then not much more than half
as far from the sun as Venus, and therefore four times as brightly illu-
minated, it appeared to be less luminous in the telescope. “Venus was
like silver, Mercury like zinc or lead.”

In the proportion of light given out at its different phases, it
behaves like the moon, flashing out strongly near the full, as if it
had a surface of the same rough structure as that of our satellite.

559. Telescopic Appearance and Phases.—Seen by the
telescope, the planet looks like a little moon, showing phases pre-
cisely similar to those of our satellite. At inferior conjunction the
dark side is towards us; at superior conjunction the illuminated
surface. At the greatest elongation it appears like a half-moon. Be-
tween superior conjunction and greatest elongation it is gibbous,
while between inferior conjunctions and the elongations it shows
the crescent phase. Fig. 171 illustrates the phases of both Mercury
and Venus.

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333

Fig. 171.—Phases of Mercury and Venus.

For the most part, Mercury can be observed only by daylight; but
when proper precautions are taken to screen the object-glass of the tele-
scope from the direct sunlight, the observation is not difficult. The sur-
face presents very little of interest. There are no markings well enough
defined to give us any trustworthy determination of the planet’s rotation,
or of its geography. Occasionally the terminator (i.e., the line which sep-
arates the illuminated and unilluminated portions of the planet) appears
to be a little irregular, instead of a true oval, as it should be; and at times
when the phase is crescent-form, the points, or cusps, are a little blunted;
from this some astronomers have inferred the existence of high mountains
upon the planet. Schröter, a German astronomer, the contemporary of
the elder Herschel, deduced from his observations a rotation period for
the planet of 24 h. 5 m.; but later observers, with instruments certainly
far more perfect, have not been able to verify his results, and they are
now considered as of little weight.

560. Atmosphere.—The evidence upon this subject is not
conclusive. Its atmosphere, if it has one, must, however, be much
less dense than that of Venus. No ring of light is seen surround-
ing the disc of the planet when it enters the limb of the sun at
the time of a transit, while in the case of Venus such a ring, due
to the atmospheric refraction, is very conspicuous. On the other
hand, Huggins and Vogel, who have examined the spectrum of the
planet, report that certain lines in the spectrum, due to the pres-
ence of water-vapor, were decidedly stronger than in the spectrum
of the air (illuminated by sunshine), which formed the background
for the planet, making it probable that it has an atmosphere con-
taining water-vapor like the atmosphere of the earth, but probably
less extensive and dense.

561. Transits.—Usually at the time of inferior conjunction the
planet passes north or south of the sun, the inclination of its orbit
being 7◦; but if the conjunction occurs when the planet is very near

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334
its node, it will cross the disc of the sun and be visible upon it us a
small black spot—not, however, large enough to be seen without a
telescope, as Venus can under similar circumstances.

At this time we have the best opportunity for measuring the diameter
of the planet; but unless special precautions are taken, the measured
diameter under these circumstances is likely to be too small, on account
of the irradiation of the surrounding background, which encroaches upon
the planet’s disc.

Since the planet’s nodes are in longitudes 227◦ and 47◦, and
are passed by the earth on May 7 and November 9, the transits
can occur only near these days. If the orbit of the planet were
strictly circular, the “transit limit” (corresponding to an ecliptic
limit) would be 2◦10 ; but at the May transits the planet is near
its aphelion and much nearer the earth than ordinarily, so that the
limit is diminished, while the November limit is correspondingly
increased. The May transits are in fact only about half as numerous
as the November transits.

562. Interval between Transits.—Twenty-two synodic pe-
riods of Mercury are pretty nearly equal to 7 years; 41 still more
nearly equal 13 years; and 145 almost exactly equal 46 years. Hence,
after a November transit, a second one is possible in 7 years, prob-
able in 13 years, and practically certain in 46. For the May transits
the repetition after 7 years is not possible, and it often fails in

13 years.

The transits of the present century are the following:—

May Transits.—1832, May 5; 1845, May 8; 1878, May 6; 1891, May 9.

November Transits.—1802, Nov. 9; 1815, Nov. 11; 1822, Nov. 5; 1835,

Nov. 7; 1848, Nov. 10; 1861, Nov. 12; 1868, Nov. 5; 1881, Nov. 7; 1894,

Nov. 10.

The first transit of Mercury ever observed was by Gassendi, Nov. 7,

1631.

The transits of Mercury are of no particular astronomical importance,
except as giving accurate determinations of the planet’s place, by means
of which its orbit can be determined. Newcomb has also recently made
an investigation of all the recorded transits, for the purpose of testing the
uniformity of the earth’s rotation. They indicate no perceptible change
in the length of the day.

VENUS.

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335

563. The next planet in order from the sun is Venus, the bright-
est and most conspicuous of all, the earth’s twin sister in magnitude,
density, and general constitution, if not also in age, as to which
we have no knowledge. Like Mercury, it had two names among
the Greeks,—Phosphorus as morning star, and Hesperus as evening
star. It is so brilliant that it is easily seen by the naked eye in the
daytime for several weeks when near its greatest elongation; some-
times it is bright enough to catch the eye at once, but usually it is
seen by daylight only when one knows precisely where to look for
it.

(It is not, however, the ”Star of Bethlehem,” though it has of late
been frequently taken for it.)

564. Distance, Period, and Inclination of Orbit.—Its
mean distance from the sun is 67,200000 miles. The eccentricity
of the orbit is the smallest in the planetary system (only 0.007), so
that the greatest and least distances of the planet from the sun dif-
fer from the mean only 470,000 miles each way. Its orbital velocity
is twenty-two miles per second.

Its sidereal period is 225 days, or seven and one-half months,
and its synodic period 584 days—a year and four months. From
superior conjunction to elongation on either side is 220 days, while
from inferior conjunction to elongation is only 71 or 72 days. The
arc of retrogression is 16◦.

The inclination of its orbit is only 3 1 ◦.

2

565. Diameter, Surface, and Volume.—The apparent di-
ameter ranges from 67 at the time of inferior conjunction to only

11 at the superior. This great difference depends, of course, upon
the enormous change in the distance of the planet from the earth.

At inferior conjunction the planet is only 26,000000 miles from us

(93 − 67). No other body ever comes so near the earth except the
moon, and occasionally a comet. Its greatest distance at superior
conjunction is 160,000000 miles (93 + 67), so that the ratio between
the greatest distance and the least is more than 6 to 1.

The real diameter of the planet is 7700 (± 30) miles. Its surface,
as compared with that of the earth, is ninety-five per cent; its volume
ninety-two per cent.

566. Mass, Density, and Gravity.—By means of the pertur-
bations she produces upon the earth, the mass of Venus is found to

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336
be seventy-eight per cent of the earth’s; hence her density is eighty-
six per cent, and her superficial gravity eighty-three per cent of the
earth’s.

567. Phases.—The telescopic appearance of the planet is strik-
ing on account of her great brilliance. When about midway between
greatest elongation and inferior conjunction she has an apparent di-
ameter of 40 , so that, with a magnifying power of only forty-five,
she looks exactly like the moon four days old, and of precisely the
same apparent size.

Very few persons, however, would think so on their first view through
the telescope, for a novice always underrates the apparent size of a tele-
scopic object: he instinctively adjusts his focus as if looking at a picture
only a few inches away, instead of projecting the object visually into the
sky.

Fig. 172.—Telescopic Appearances of Venus.

According to the theory of Ptolemy, Venus could never show
us more than half her illuminated surface, since according to his
hypothesis she was always between us and the supposed orbit of the
sun (Art. 500). Accordingly, when in 1610 Galileo discovered that
she exhibited the gibbous phase as well as the crescent, it was a
strong argument for Copernicus. Galileo announced his discovery
in a curious way, by publishing the anagram,—

“Haec immatura a me jam frustra leguntur; o. y.”

Some months later he furnished the translation,—

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337

“Cynthiæ figuras æmulatur mater amorum,”

which is formed by merely transposing the letters of the anagram.

His object was to prevent any one from claiming to have anticipated
him in this discovery, as had been done with respect to his discovery
of the sun spots.

Fig. 172 represents the disc of the planet as seen at four points in
its orbit. 1, 3, and 5 are taken at superior conjunction, greatest elon-
gation, and near inferior conjunction respectively, while 2 and 4 are at
intermediate points.

568. Maximum Brightness.—The planet attains its maxi-
mum brilliance thirty-six days before and after inferior conjunction,
at a distance of about 38◦ or 39◦ from the sun, when its phase is
like that of the moon about five days old. It then casts a strong
shadow, and, as has already been said, is easily visible by day with
the naked eye.

569. Surface Markings.—These are not at all conspicuous.

Near the limb of the planet, which is always much brighter than the
central parts (as is also the case with Mercury and Mars), they can
never be well seen, although sometimes when Venus was in the crescent
phase, intensely bright spots have been reported near the cusps, as at
a and b in No. 4, Fig. 172. These may perhaps be ice-caps like those
which are seen on Mars. Near the “terminator,” which is less brilliant
and less sharply defined than the limb, irregular darkish shadings are
sometimes seen, such as are indicated by the dotted lines in the figures,
but without any distinct outline. They may be continents and oceans
dimly visible, or they may be mere atmospheric objects; observations do
not yet decide.

570. Rotation of the Planet.—From the observation of such
markings Schröter deduced the rotation period of 23h 21m, which
has since been partially confirmed by one or two other observers,
and may be approximately correct. De Vico concluded that the
planet’s equator makes an angle of about 54◦ with the plane of its
orbit. This, however, is extremely doubtful. If the bright spots
which have been referred to in Art. 569 are really polar ice-caps,
the equator cannot be very greatly inclined. (See Clerke’s “History
of Astronomy,” p. 299.)

No sensible difference has been ascertained between the different di-
ameters of the planet. If it were really as much flattened at the poles as

THE PLANETS THEMSELVES.

338
the earth is, there should be a difference of 0 .2 between the polar and
equatorial diameters as measured at the time of the planet’s transit.

571. Mountains.—From certain irregularities occasionally ob-
served upon the terminator, and especially from the peculiar
blunted form of one of the cusps of the crescent, various observers
have concluded that there are numerous high mountains upon the
surface of the planet. Schröter assigned to some of these near the
southern pole the extravagant altitude of twenty-five or thirty miles,
but the evidence is entirely insufficient to warrant any confidence
in the conclusion.

572. Albedo.—According to Zöllner the Albedo of the planet
is 0.50, which is about three times that of the moon, and almost
four times that of Mercury. It is, however, exceeded by the reflecting
power of the surfaces of Jupiter and Uranus, while that of Saturn
appears to be about the same. This high reflecting power probably
indicates that the surface is mostly covered with cloud, as few rocks
or soils could match it in brightness.

573. Evidences of Atmosphere.—When the planet is near
the sun, the horns of the crescent extend notably beyond the diam-
eter, and when very near the sun, a thin line of light has been seen
by several observers, especially Professor Lyman of New Haven, to
complete the whole circumference. This is due to refraction of sun-
light by the planet’s atmosphere, a phenomenon still better seen
as the planet is entering upon the sun’s disc at a transit, when the
black disc is surrounded by a beautiful ring of light. From the obser-
vations of the transit of 1874, Watson concluded that the planet’s
atmosphere must have a depth of about fifty-five miles, that of the
earth being usually reckoned at forty miles. Other observers in
different ways have come to substantially the same results. Its at-
mosphere is probably from one and a half to two times as extensive
and dense as our own, and the spectroscope shows evidence of the
presence of water-vapor in it.

Lights on Dark Portion.—Many observers have also reported
faint lights as visible at times on the dark portion of the planet’s
disc. These cannot be accounted for by reflection, but must orig-
inate on the planet’s surface; they recall the Aurora Borealis and
other electrical manifestations on the earth.

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339

574. Satellites.—No satellite is known, although in the last
century a number of observers at various times thought they had
found one.

In most cases they observed small stars near the planet, which we can
now identify by computing the place occupied by the planet at the date
of observation. It is not, however, impossible that the planet may have
some very minute and near attendants like those of Mars, which may
yet be brought to light by means of the great telescopes of the future, or
by photography. Of course the extreme brilliance of the planet, and the
fact that the necessary observations can be made only in strong twilight,
render the discovery of such objects, if they exist, very difficult.

575. Transits.—Occasionally Venus passes between the earth
and the sun at inferior conjunction, giving us a so-called “transit.”

She is then visible (even to the naked eye) as a black spot on the
disc, crossing it from east to west.

As the inclination of the planet’s orbit is but 3 1 ◦, the “transit

2
limit ” is small (only 1 3 ◦), and the transits are therefore very rare

4
phenomena. The sun passes the nodes of the orbit on June 5 and

December 7, so that all transits must occur on or near those dates.

When Venus crosses the sun’s disc centrally, the duration of the
transit is about eight hours. Taking the mean diameter of the sun
as 32 , or 1 of a circumference, and the planet’s synodic period

375
as 584 days, the geocentric duration of a central transit should be

277 × 1 × 584d, which equals 0.332 days, or 7h 58m.

723

675

When the transit track lies near the edge of the disc, the duration
is of course correspondingly shortened.

576. Recurrence of Transits.—Five synodic, or thirteen sid-
ereal, revolutions of Venus are very nearly equal to eight years, the
difference being only a little more than one day; and still more
nearly, in fact almost exactly, 243 years are equal to 152 synodic,
or 395 sidereal, revolutions. If, then, we have a transit at any time,
we may have another at the same node eight years earlier or later.

Sixteen years before or after it would be impossible, and no other
transit can occur at the same node until after the lapse of two hun-
dred and thirty-five or two hundred and forty-three years.

If the planet crosses the sun nearly centrally, the transit will not
be accompanied by another at an eight-year interval, but the planet
will pass either north or south of the sun’s disc, at the conjunctions
next preceding and following. If, however, as is now the case, the

THE PLANETS THEMSELVES.

340
transit path is near the northern or southern edge of the sun, then
there will be a companion transit across the opposite edge of the
disc eight years before or after. Thus, if we have a pair of June
transits, separated by an eight-year interval, it will be followed by
another pair at the same node in 243 years; and a pair of December
transits will come in about half-way between the two pairs of June
transits. After a thousand years or so from the present time the
transits will cease to come in pairs, as they have been doing for

2000 years.

577. Transits of Venus have occurred or will occur on the fol-
lowing dates:—

December 7, 1631.

June 5, 1761.

December 4, 1639.

June 3, 1769.

December 9, 1874.

June 8, 2004.

December 6, 1882.

June 6, 2012.

The special interest in these transits consists in the use that has
been made of them for the purpose of finding the sun’s parallax, a
subject which will be discussed later on (Chap. XVI.).

The first observed transit, in 1639, was seen by
only two persons,—Horrox and Crabtree, in Eng-
land. The four which have occurred since then
have been extensively observed in all parts of the
world where they were visible, by scientific expedi-
tions sent out for the purpose by different nations.

The transits of 1769 and 1882 were visible in the

Fig. 173.

United States. Fig. 173 shows the track of Venus

Transit of Venus
across the sun’s disc at the two transits of 1874

Tracks.

and 1882.

MARS.

This planet is also prehistoric as to its discovery. It is so con-
spicuous in color and brightness, and in the extent and apparent
capriciousness of its movement among the stars, that it could not
have escaped the notice of the very earliest observers.

578. Orbit.—Its mean distance from the sun is 141,500000
miles, and the eccentricity of the orbit is so considerable (0.093)

THE PLANETS THEMSELVES.

341
that this distance varies about 13,000000 miles. The light and heat
which it receives from the sun is somewhat less than half of that
received by the earth. The inclination of its orbit is small, 1◦51 .

mo

The planet’s sidereal period is 687 days, or 1y 10 1

, which gives it

2
an average orbital velocity of fifteen miles per second. Its synodic
mo
period is 780 days, or 2y 1 2

. It is the longest in the solar system,

3
that of Venus (584 days) coming next. Of the 780 days, it moves
eastward during 710, and retrogrades during 70, through an arc of

18°.

579. At opposition its average distance from the earth is

48,600000 miles (141,500000 miles minus 92,900000 miles). When
the opposition occurs near the planet’s perihelion, this distance is
reduced to 35,500000 miles; if near aphelion, it is increased to over

61,000000. At superior conjunction the average distance from the
earth is 234,400000 miles (141,500000 plus 92,900000).

The apparent diameter and brilliancy of the planet, of course,
vary enormously with these great changes of distance.

If we put R for the planet’s distance from the sun, and ∆ for its dis-
tance from the earth, its brightness, neglecting the correction for phase,

1
should equal

. We find from this that, taking the brightness at

R2∆2
conjunction as unity (at which time the planet is about as bright as
the pole-star), it is more than twenty-three times brighter at the aver-
age opposition, and fifty-three times brighter if the opposition occurs at
the planet’s perihelion. At an unfavorable opposition Mars, as has been
said, may be 61,000000 miles distant, and its brightness then is only
about twelve times as great as at conjunction,—the difference between
favorable and unfavorable oppositions being more than four to one.

These favorable oppositions occur always in the latter part of August

(at which date the sun passes the line of apsides of the planet), and at
intervals of fifteen or seventeen years. The last was in 1877, and the
next will be in 1892. A reference to Fig. 159 will show how great is the
difference between the planet’s opposition distance from the earth under
varying circumstances.

580. Diameter, Surface, and Volume.—The apparent di-
ameter of the planet ranges from 3 .6 at conjunction, to 24 .5 at a
favorable opposition. Its real diameter is very closely 4200 miles,—
the error may be twenty miles one way or the other. This makes its
surface 0.28, and its volume 0.147 (equal to 1 ) of the earth’s.

7

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342

581. Mass, Density, and Gravity.—Observations upon its
satellites give its mass as 1 compared with that of the earth. This

9.4
makes its density 0.73 and superficial gravity 0.38; that is, a body
which weighs 100 pounds on the earth would have a weight of 38
pounds on the surface of Mars.

582. Phases.—Since the orbit of the planet is outside that of
the earth, it never comes between us and the sun, and can never
show the crescent phase; but at quadrature enough of the unillumi-
nated portion is turned towards the earth to make the disc clearly
gibbous like the moon three or four days from full. Fig. 174 shows
its maximum phase accurately drawn to scale.

583. The

“Albedo”

of
the

Planet.—

According to Zöllner’s observations this is 0.26,
which is considerably higher than that of the moon

( 1 ), and just double that of Mercury.

6

584. Rotation.—The planet’s time of rota-
tion is 24h 37m 22s.67. This very exact determi-
nation has been made by Kaiser and Bakhuyzen,

Fig. 174.

by comparing drawings of the planet which were

Greatest Phase
made more than 200 years ago by Huyghens with
of Mars.

others made recently.

It is obvious that observations made a few days or weeks apart
will give the time or rotation with only approximate accuracy.

Knowing it thus approximately, we can then determine, without
fear of error, the whole number of rotations between two observa-
tions separated by a much longer interval of time. This will give a
second and closer approximation to the true period; and with this
we can carry our reckoning over centuries, and thus finally deter-
mine the period within a very minute fraction of a second. The
number given is not uncertain by more than 1 of a second, if so

50
much.

585. The Inclination of the Planet’s Equator to the

Plane of its Orbit.—This is very nearly 24◦ 50 (26◦ 21 to the
ecliptic), not very different from the inclination of the earth’s equa-
tor; so far, therefore, as depends upon that circumstance, its seasons
should be substantially the same as our own.

THE PLANETS THEMSELVES.

343

586. Polar Compression.—There is a slight but sensible flat-
tening of the planet at the poles. The earlier observers found for the
polar compression values as large as 1 , and even 1 . These large

40

10
values, however, are inconsistent with the existence of any extensive
surface of liquid upon the planet, and more recent observations of
the writer show the polar compression to be about 1 ; which is

220
almost exactly what would be expected from a planet constituted
as we suppose Mars to be.

587. Telescopic

Ap-
pearance
and

Surface-

Markings.—The
fact
that
we are able to determine the
time of rotation so accurately
of course implies the existence
of identifiable markings upon
the surface. Viewed through a

Fig. 175.—Telescopic Views of Mars.

powerful telescope, the planet’s
disc, as a whole, is ruddy,—in fact, almost rose color,—very bright
around the limb, but not at the “terminator,” if there is any consid-
erable phase. The central portions of the disc present greenish and
purplish patches of shade, for the most part not sharply defined,
though some of the markings have outlines reasonably distinct. On
watching the planet for only a few hours even, the markings pass
on across the disc, and are replaced by others. Some of them are
permanent, and recur at regular intervals with the same form and
appearance, while others appear to be only clouds which for a time
veil the surface below, and then clear away.

By comparing drawings made when the same side is turned to-
wards the earth, it is possible in a short time to ascertain what
features really belong to the planet’s geography.

The polar ice-caps, brilliant white patches near the poles, form
a marked feature in the planet’s telescopic appearance. These are
believed to be ice-caps from the fact that the one which is near the
pole that happens to be turned towards the sun continually dimin-
ishes in size, while the other increases, the process being reversed
with the seasons of the planet.

588. The principal peculiarity of the surface of Mars appears to
be the way in which land and water are intermeshed. There seem to
be few great oceans and continents, but there are narrow arms of the

THE PLANETS THEMSELVES.

344
sea, like the Baltic and the Red Sea, penetrating and dividing the
land masses. As to the nomenclature of the principal “Areographic”

features, Mr. Proctor in his map has for the most part assigned
the names of astronomers who have taken special interest in the
study of the planet. Schiaparelli, on the other hand, has taken his
names mostly from classical geography. Fig. 176 is a reduced copy
of his map of the planet, drawn on Mercator’s Projection. The
student will remember that in maps of this sort the polar regions
are extended beyond their due proportion, and also that the parts
near the poles can never be seen from the earth nearly as well as
those near the equator, and hence that our knowledge of the details
in that part of the planet is very limited. (See note to Art. 588,
p. 355.)

Fig. 176.—Schiaparelli’s Map of Mars.

589. Atmosphere.—At one time it was supposed that the at-
mosphere of the planet is very dense, but more recent observations
have shown that this cannot be the case. The probability is that
its density is considerably less than that of our own atmosphere.

Dr. Huggins has found with the spectroscope unequivocal evidence
of the presence of aqueous vapor. Although the planet receives from
the sun less than half the amount of heat and light per unit of sur-
face that the earth does, the climate appears, for some reason not
yet discovered, to be much more mild than would be expected. So
far as we can judge, the water on the planet is never frozen except
very near the poles. The earth, as seen from Mars, would present
much more extensive snow-caps.

THE PLANETS THEMSELVES.

345

590. Satellites.—There are two satellites, which were discov-
ered in August, 1877, by Professor Hall at Washington with the
then new 26-inch telescope. They are exceedingly minute, and can
be seen only with the most powerful instruments. The outer one,

Deimos, is at a distance of 14,600 miles from the centre of the planet,
and has a period of 30h 18m, while the inner one, Phobos, is at a
distance of only 5800 miles, and its month is but 7h 39m long, not
one-third of the day of Mars. Owing to this fact it rises in the west
every night for the “Marticoli ” (if there are any people there) and
sets in the east, after about 11h.

Deimos does not do this; it rises in the east like other stars, but
its orbital eastward motion among the stars is so nearly equal to its
diurnal motion westward, that it is nearly 132 hours between two
successive risings. This is more than four of its months, so that it
undergoes all its changes of phase four times in the interval.

Of course, both the satellites are frequently eclipsed,—the inner
one at every revolution; and it also transits across the sun’s disc at
every new moon, as seen from some point or other on the planet’s
surface.

Their orbits appear to be exactly circular, and they move exactly
in the plane of the planet’s equator; and they keep so, maintained in
their relation to the equator by the action of the “equatorial bulge”

upon the planet.

591. As givers of moonlight they do not amount to much. Their
diameters are too small to be measured with any micrometer; but
from their apparent “magnitude” (i.e., brightness), as seen from
the earth, and assuming that their surfaces have the same reflective
power as that of the planet, Professor Pickering has estimated the
diameter of Phobos, which is the larger one, as about seven miles,
and that of Deimos, as five or six. The light given by Phobos to
the inhabitants of Mars would be about 1 of our moonlight; that

60
of Deimos about

1

.

1200

The period of Phobos is by far the shortest period in the solar
system. Next to it is that of Mimas, the inner satellite of Saturn,
h
which, however, is nearly three times as long,—22 1 . This rapid-

2
ity of revolution raises important questions as to the theory of the
development of the solar system, and requires modification of the
views which had been held up to the time of their discovery. If
the nebular hypothesis is true, a shortening of the satellite’s period,

THE PLANETS THEMSELVES.

346
or a lengthening of the planet’s day, must have occurred since the
satellite came into being, since that hypothesis will not account for
the existence of a satellite having a period shorter than the diurnal
rotation of its primary. Fig. 177 is a diagram of the satellite orbits
as they appeared from the earth in 1888. It is reduced from the

American Nautical Almanac for that year.

Fig. 177.—Orbits of the Satellites of Mars.

THE ASTEROIDS, OR MINOR PLANETS.

592. These are a group of small planets circulating in the space
between Mars and Jupiter. The name “asteroid ” was suggested by

Sir William Herschel early in the century, when the first ones were
discovered. The later term planetoid is preferred by many.

It was very early noticed that there is a break in the series of
the distances of the planets from the sun. Kepler, indeed, at one
time thought he had discovered the true law1 and the real reason

1His supposed law was as follows: Imagine the sun surrounded by a hollow
spherical shell, on which lies the orbit of the earth. Inside of this shell inscribe
a regular icosahedron (the twenty-sided regular solid), and within that inscribe
a second sphere. This sphere will carry upon it the orbit of Venus. Inside
of the sphere of Venus inscribe an octahedron (the eight-sided solid), and the
sphere which fits within it will carry Mercury’s orbit. Next, working outwards
from the earth’s orbit, circumscribe around the earth’s sphere a dodecahedron,
circumscribing around it another sphere, and this will carry upon it the orbit
of Mars. Around the sphere of Mars circumscribe the tetrahedron, or the reg-
ular pyramid. The corners of this solid project very far, so that the sphere
circumscribed around the tetrahedron will be at a very great distance from the
sphere of Mars. It carries the orbit of Jupiter. Finally, the cube, or hexahedron,
circumscribed around the orbit of Jupiter, gives us in the same way the orbit of

THE PLANETS THEMSELVES.

347
why the planets’ distances are what they are. This theory of his
was broached twenty-two years, however, before he discovered the
harmonic law, and he probably abandoned it when he discovered
the elliptical form of the planets’ orbits. At any rate, in later life he
suggested that it was likely that there was a planet between Mars
and Jupiter too small to be seen.

The impression that such a planet existed gained ground when

Bode published the law which bears his name in 1772, and it was
still further deepened when, nine years later, in 1781, Uranus was
discovered, and the distance of the new planet was found to con-
form to Bode’s law. An association of twenty-four astronomers,
mainly German, was immediately formed to look for the missing
planet, who divided the zodiac between them and began the work.

Singularly enough, however, the first discovery was made, not by a
member of this association, but by Piazzi, the Sicilian astronomer of

Palermo, who was then engaged upon an extensive star-catalogue.

On January 1, 1801, he observed a seventh-magnitude star which by
the next evening had unquestionably moved, and kept on moving.

He observed it carefully for some six weeks, when he was taken ill;
before he recovered, it had passed on towards superior conjunction,
and was lost in the rays of the sun. He named it Ceres, after the
tutelary goddess of Sicily.

When at length the news reached Germany in the latter part of

March it created a great excitement, and the problem now was to redis-
cover the lost planet. The association of planet-hunters began the search
in September, as soon as its elongation from the sun was great enough to
give any prospect of success. During the summer Gauss devised his new
method of computing a planetary orbit, and computed the ephemeris of
its path. Very soon after receiving his results, Baron Von Zach rediscov-
ered Ceres on December 31, and Dr. Olbers on the next day, just one
year after it was first found by Piazzi.

593. In March, 1802, Dr. Olbers, who in looking for Ceres had
carefully examined the small stars in the constellation of Virgo,
on going over the ground again, found a second planet, which he

Saturn. We thus obtain a series of distances not enormously incorrect (though
by no means agreeing with fact even as closely us does Bode’s law); and, more-
over, the theory had the great advantage to Kepler’s mind of accounting for the
fact that there are (so far as was then known) but seven planets, there being
possible but five regular solids.

THE PLANETS THEMSELVES.

348
named Pallas, a body of about the same brightness as Ceres. Two
having now been found, and Pallas having a very eccentric and much
inclined orbit, he conceived the idea that they were fragments of a
broken planet, and that other planets of the same group could prob-
ably be found by searching near the intersection of their two orbits.

Juno, the third, was discovered by Harding at Lilienthal (Schröter’s
observatory) in 1804; and Vesta, the largest and brightest of the
whole group (sometimes visible to the naked eye), was found by

Olbers himself in 1807. The search was kept up for several years
after this, but no more planets were found because they did not
look after small enough stars.

The fifth, Astræa, was discovered in 1845 by Hencke, an amateur
astronomer who for fifteen years had been engaged in studying the
smaller stars in hopes of just the reward he captured. In 1846 no
asteroid was found (the discovery of Neptune was glory enough for
that year), but in 1847 three more were brought to light; and since
then not a year has passed without adding from one to twenty to
the number. The list at the end of 1888 counts up 281, and there is
no prospect of diminution in the rate of discovery, though the new
ones are mostly very small,—stars of the twelfth and thirteenth
magnitudes, which require a large telescope to make them even
visible. All the brighter ones have evidently been already picked
up.

They have been discovered by comparatively a few observers. Four
persons have found more than 20 each: Palisa, of Vienna, who stands
far in advance of all others, has alone discovered 68, and Dr. Peters
of Clinton, N.Y., 52; Luther of Düsseldorf, 24; and the late Professor

Watson, 22. The German astronomers have thus far discovered 102; the

American, 78; the French, 66; the English, 19; and the Italians, 16.

594. Method of Search.—The asteroid-hunter selects certain
portions of the sky, usually near the ecliptic, and prepares charts
covering two or three square degrees, on which he sets down all the
stars his telescope will show. This is a very laborious operation,
which in the future is likely to be very much facilitated by photog-
raphy. The chart once made, he goes over the ground from time
to time, comparing every object with the map and looking out for
interlopers. If he finds a star which is not on his chart, the proba-
bility is that it is a planet, though it may be a variable star which
was invisible when the chart was made. He very soon settles the

THE PLANETS THEMSELVES.

349
question, however, by measuring with the micrometer the distance
of the new star from some of its neighbors, keeping up the process
for an hour or two. If it is a planet, it will move perceptibly in that
length of time.

Of course, great care must be taken to be sure that it is a new
planet, and not one of the multitude already known. Generally it is
possible to decide very quickly which of the known planets will be in the
neighborhood, and a rough computation will commonly decide at once
whether the planet is new or not. Not always, however, and mistakes in
this regard are not very unusual.

These minor planets are all named, the names being derived
from mythology and legend. They are also designated by numbers,
and the symbol for each planet is the number written in a circle.

Thus, for Ceres the symbol is 1

○; for Hilda, (153); and so on.

A full list of them, with the elements of their orbits, is published
yearly in the “Annuaire du Bureau des Longitudes,” Paris.

595. Their Orbits.—The mean distance of the different aster-
oids from the sun varies greatly, and of course the periods are corre-
spondingly different. Medusa, (149), has the smallest mean distance

(2.13, or 198,000000 miles), and the shortest period, 3y 40d. Thule,

(279), is the remotest, with a mean distance of 4.30 or 400,000000
miles, and a period of 8y 313d. According to Svedstrup, the mean
distance of the “mean asteroid” is 2.65 (246,000000 miles), and its
period about 4 1 years. Its distance from the earth at time of oppo-

3
sition would be, of course, 1.65, or 153,000000 miles.

The inclinations of their orbits average about 8◦; but Pallas, 2

○,
has an inclination of 35◦, and Euphrosyne, (31), of 26 1 ◦.

2

Several of the orbits are extremely eccentric. Æthra, (132), has
an almost cometary eccentricity of 0.38, and nine or ten others have
eccentricities exceeding 0.30. They are distributed quite unequally
in the range of distance, there being, as Kirkwood has pointed out,
very few at such distances that their periods would be exactly com-
mensurable with that of Jupiter.

596. Diameter and Surface.—Very little is known as to the
real size of these bodies. The four first discovered, and a few of
the newer ones, show a minute but sensible disc in very powerful
telescopes. Vesta, which is the brightest of the whole group, and is
just visible to the naked eye when near opposition, may be from two

THE PLANETS THEMSELVES.

350
hundred to four hundred miles in diameter. Pickering, by photo-
metric measurements (assuming the reflecting power of the planet’s
surface to be the same as that of Mars), finds a diameter of 319
miles. The other three of the original four are perhaps two-thirds
as large. As for the rest, it is hardly possible that any one of them
can be as much as 100 miles in diameter, and the smallest, such as
are now being discovered in such numbers, are probably less than
ten miles through,—nothing more than “mountains broke loose.”

The surface area of one of the smaller ones would hardly make a
large Western farm.

597. Mass, Density, etc.—As to the individual masses and
densities we have no certain knowledge. It is probable that the
density does not differ much from the density of the crust of the
earth, or the mean density of Mars. If this is so, the mass of Vesta
might possibly be as great as

1
or

1
of the earth. On such

15000

20000
a planet the force of superficial gravity would be about 1 to 1 of

25

30
gravity on the earth, and a body projected from the surface with a
velocity of about 2000 feet a second—that of an ordinary rifle-ball—
would fly off into space and never return to the planet, but would
circulate around the sun as a planet on its own account. On the
smallest asteroids, with a diameter of about ten miles, it would be
quite possible to throw a stone from the hand with velocity enough
to send it off into space.

598. Aggregate Mass.—Although we can only estimate very
roughly the masses of the individual members of the flock, it is
possible to get some more certain knowledge of their aggregate mass.

Leverrier found a motion in the line of apsides of the orbit of Mars
which indicates that the whole amount of matter thus distributed
in the space between Mars and Jupiter is equal to about one-fourth
of the mass of the earth. This quantity of matter collected into a
body of the same density as Mars would make a planet of about

5000 miles in diameter.

The united masses of these which are already known would make only
a very small fraction of such a body. Up to August, 1880, the united
bulk of the asteroids then discovered was estimated at

1
part of the

4000
earth’s bulk, with a mass probably about

1
of the earth’s. Presum-

6000
ably, therefore, the number of these bodies remaining undiscovered is
exceedingly great—to be counted by thousands, if not by millions. Most
of them, of course, must be much smaller than those which are already
known.

THE PLANETS THEMSELVES.

351

599. Forms, Variations of Brightness, and Atmosphere.

—We have no definite knowledge on this point, but Dr. Olbers ob-
served in the case of Vesta certain fluctuations in her brightness
which seemed to him to indicate that she is not a globe, but an
angular mass,—a splinter of rock. This, however, is not confirmed
by the more recent photometric observations of Müller or Pickering.

Müller examined seven of the asteroids, and found their changes of
brightness very regular. Four of them, one of which is Vesta, behaved
precisely like Mars, as if their surfaces were comparatively smooth, while
three others, Ceres and Pallas among them, behaved like the moon and
the planet Mercury, as if having a rough surface with very little at-
mosphere, and nearly cloudless. Some of the earlier observers reported
evidences of an extensive halo around them. Later observations do not
confirm it, and it is not likely that they carry much air with them.

600. Origin.—With respect to this we can only speculate.

Two views have been held, as has been already intimated. One
is, that the material, which according to the nebular hypothesis
ought to have been concentrated to form a single planet of the class
to which the earth belongs, has failed to be so collected, and has
formed a flock of small separate masses. It is now very generally
believed that the matter which at present forms the planets was
once distributed in rings, like the rings of Saturn. If so, this ring
next outside of Mars would necessarily suffer violent perturbations
from the nearness of the enormous planet Jupiter, and so would be
under very different conditions from any of the other rings. This,
as Peirce has shown, might account for its breaking up into many
fragments.

The other view is that a planet about the size of Mars has bro-
ken to pieces. It is true, as has been often urged, that this theory
in its original form, as presented by Olbers, cannot be correct. No
single explosion of a planet could give rise to the present assemblage
of orbits, nor is it possible that even the perturbations of Jupiter
could have converted a set of orbits originally all crossing at one
point (the point of explosion) into the present tangle. The smaller
orbits are so small that however turned about they lie wholly in-
side the larger, and cannot be made to intersect them. If, however,
we admit a series of explosions, this difficulty is removed; and if
we grant an explosion at all, there seems to be nothing improba-
ble in the hypothesis that the fragments formed by the bursting

THE PLANETS THEMSELVES.

352
of the parent mass would carry away within themselves the same
forces and reactions which caused the original bursting; so that they
themselves would be likely enough to explode at some time in their
later history.

At present opinion is divided between these two theories.

601. The number of these bodies already known is so great, and
the prospect for the future is so indefinite, that astronomers are at their
wits’ end how to take care of this numerous family. To compute the orbit
and ephemeris of one of these little rocks is more laborious (on account
of the great perturbations produced by Jupiter) than to do the same for
one of the major planets; and to keep track of such a minute body by
observation is far more difficult. Until recently, the German Jahrbuch
has been publishing the ephemerides of such as came within the range of
observation each year; but this cannot be kept up much longer, and the
probability is that hereafter only the larger ones, or those which present
some remarkable peculiarity in their orbits, will be followed up. One
little family of them, however, is “endowed.” Professor Watson, at his
death, left a fund to the American National Academy of Sciences to bear
the expense of taking care of the twenty-two which he discovered.

INTRA-MERCURIAL PLANETS AND THE ZODIACAL

LIGHT.

It is very probable, indeed almost certain, that there are masses
of matter revolving around the sun within the orbit of Mercury.

602. Motion of the Perihelion of Mercury’s Orbit.—

Leverrier, in 1859, from a discussion of all the observed transits
of Mercury, found that the perihelion of its orbit has a movement
of nearly 38 a century. This is more than can be accounted for by
the action of the known planets, and, so far as known, could be ex-
plained only by the attraction of a planet, or ring of small planets,
revolving inside this orbit nearly in its plane, with a mass about
half as great as that of Mercury itself.

We say “so far as known,” because an alternative hypothesis has
been proposed, viz., that the law of gravitation, though strictly true
for bodies at rest is not absolutely so for bodies in motion; that when
bodies are moving towards each other the attraction is less by a minute
fraction than if they were at rest. The hypothesis is known as the electro-
dynamic theory of gravitation, but has at present very little to support
it. If, however, it were true, then the peculiar motion of the apsides of

THE PLANETS THEMSELVES.

353

Mercury’s orbit would be a necessary consequence.

Subsequent investigations by a number of mathematicians have
fully confirmed Leverrier’s results; Mercury’s orbit is beyond ques-
tion affected as it would be if there were an intra-Mercurial planet,
or a number of them.

603. Dr. Lescarbault’s Observation: Vulcan.—A certain
country physician, living some eighty miles from Paris, Dr. Lescarbault,
on the publication of Leverrier’s result, announced that he had actually
seen this planet crossing the sun nine months before, on the 26th of

March of that year, 1859. He was visited by Leverrier, who became
satisfied of the genuineness of his observations, and the doctor was duly
congratulated and honored as the discoverer of “Vulcan,” which name
was assigned to the supposed new planet. An interesting account of the
matter may be found in Chambers’ “Descriptive Astronomy”; and in
many of the works published from twenty to twenty-five years ago, as
well as in some more recent ones, “Vulcan” is assigned a place in the
solar system, with a distance of about 13,000000 miles and a period of

19+ days. Lescarbault described it as having an apparent diameter of
about 7 , which would make it over 2500 miles in diameter.

604. Nevertheless, it is nearly certain that Vulcan does not exist.

There are various opinions which we need not here discuss as to the ex-
planation of this pseudo-discovery. But the planet, if real, ought since

1859 to have been visible on the sun’s face at certain definite times which

Leverrier calculated and published; and it has never been seen, though
very carefully looked for. Small, round, dark objects have from time to
time been indeed reported on the sun’s disc, which in the opinion of the
observers at the time were not sun spots; but most of these observa-
tions were made by amateurs with comparatively little experience, with
small telescopes, and with no measuring apparatus by which they could
certainly determine whether or not the spot seen moved like a planet.

In most of these cases photographs or simultaneous observations made
elsewhere by astronomers of established reputation, and having adequate
apparatus, have proved that the problematical “dots” were really noth-
ing but ordinary small sun spots, and the probability is that the same
explanation applies to the rest.

605. Eclipse Observations.—A planet large enough to be
seen distinctly on the sun by a 2 1 -inch telescope, such as Lescar-

2
bault used, would be a conspicuous object at the time of a solar
eclipse, and most careful search has been made for the planet on
such occasions; but so far, although stars of the third and fourth

THE PLANETS THEMSELVES.

354
magnitudes, and even of the fifth, have been clearly seen by the
observers within a few degrees of the eclipsed sun, no planet has
been found.

One apparent exception occurred in 1878. During the eclipse of that
year, Professor Watson observed two star-like objects (of the fourth mag-
nitude), which he thought at the time could not be identified with any
known stars consistently with his observations. Mr. Swift, also, at the
same eclipse, reported the observations of two bright points very near
the sun; but these from his statement could not (both) have been iden-
tical with Watson’s stars. Later investigations of Dr. Peters have shown
that the assumption of a very small and very likely error in Professor

Watson’s circle-readings (which were got in a very ingenious, but rather
rough way, without the use of graduations) would enable his stars to be
identified with θ and ζ Cancri, and it is almost certain that these were
the stars he saw. Mr. Swift’s observations remain unexplained. With
this exception, the eclipse observations all give negative results, and as-
tronomers generally are now disposed to consider the “Vulcan question”

as settled definitely and adversely.

606. At the same time it is extremely probable that there are
a number, and perhaps a very great number, of intra-Mercurial
asteroids.

A body two hundred miles in diameter near the sun
would have an angular diameter of only about 1 , as seen from the

2
earth, and would not be easily visible on the sun’s disc, except with
very large telescopes. It would not be at all likely to be picked up
accidentally. Objects with a diameter of not more than forty or
fifty miles would be almost sure to escape observation, either at a
transit or during a solar eclipse.

607. Zodiacal Light.—This is a faint, soft beam of light that ex-
tends both ways from the sun along the ecliptic. In the evening it is best
seen in February, March, and April, because the portion of the ecliptic
which lies east of the sun’s place is then most nearly perpendicular to
the western horizon. During the autumnal months the zodiacal light is
best seen in the morning sky for a similar reason. In our latitudes it
can seldom be traced more than 90◦ or 100◦ from the sun; but at high
elevations within the tropics it is said to extend entirely across the sky,
forming a complete ring, and there is said to be in it at the point exactly
opposite to the sun a patch a few degrees in diameter of slightly brighter
luminosity, called the “Gegenschein” or “counter-glow.”

The portions of this object near the sun are reasonably bright, and
even conspicuous at the proper seasons of the year; but the more distant

THE PLANETS THEMSELVES.

355
portions in the neighborhood of the “counter-glow” are so extremely
faint that it is only possible to observe them at a distance from cities
and large towns, in places where the air is free from smoke, and where
the darkness of the sky is not affected by the general illumination due
to gas and electric lights.

608. The cause of the phenomenon is not certainly known, but at
present the theory most generally accepted attributes it to sunlight re-
flected by myriads of small meteoric bodies which are revolving around
the sun nearly in the plane of the ecliptic, forming a thin, flat sheet like
one of Saturn’s rings, and extending far beyond the orbit of the earth. It
may be that the denser portion of this meteoric ring within the orbit of

Mercury is the cause of the motion of the perihelion of that planet which

Leverrier detected; it is for this reason that we deal with the subject here
rather than in connection with meteors. While this theory, however, is
at present more generally accepted than any other, it cannot he said to
be established. Some are disposed to consider the zodiacal light as a
mere extension of the sun’s corona, whatever that may be.

608*. (Note to Art. 588.) The Canals of Mars.—According
to Schiaparelli (and his observations are at least partially confirmed by
others) a most characteristic feature of the planet’s surface is a series
of long, straight, narrow “canals” connecting the larger bodies of water.

These canals were first seen and recognized in 1877. In 1881 they were
seen again, and at that time nearly all of them double. If there is not
some fallacy in the observation, the problem as to the nature of these
canals, and the cause of their “gemination,” is a very important and
perplexing one. It is hoped that at the next favorable opposition in 1892
it may find its solution.

THE MAJOR PLANETS.

356

CHAPTER XV.

THE PLANETS CONTINUED.—THE MAJOR PLANETS: JUPITER,

SATURN, URANUS, AND NEPTUNE.

JUPITER.

609. While this planet is not so brilliant as Venus at her best,
it stands next to her in this respect, being on the average about five
times brighter than Sirius, the brightest of the fixed stars. Jupiter,
moreover, being a “superior” planet, is not confined, like Venus, to
the neighborhood of the sun, but at the time of opposition is the
chief ornament of the midnight sky.

610. Orbit.—The orbit presents no marked peculiarities. The
mean distance of the planet from the sun is 483,000000 miles. The
eccentricity of the orbit being nearly 1 (0.04825); the greatest and

20
least distances vary by about 21,000000 miles each way, making the
planet’s greatest and least distances from the sun 504,000000 and

462,000000 miles respectively. The average distance of the planet
from the earth at opposition is 390,000000, while at conjunction
it is 576,000000 miles. The minimum opposition distance is only

369,000000, which is obtained when the opposition occurs about

October 6, Jupiter being in perihelion when its heliocentric longi-
tude is about 12◦. At an aphelion opposition (in April) the distance
is 42,000000 miles greater; that is, 411,000000.

The relative brightness of Jupiter at an average conjunction and
at the nearest and most remote oppositions is respectively as the
numbers 10, 27, and 18. The average brightness at opposition is,
therefore, more than double that at conjunction; and at an October
opposition the planet is fifty per cent brighter than at an April one.

The differences are considerable, but far less important than in the
case of Mars, Venus, and Mercury.

The inclination of the orbit to the ecliptic is small,—only 1◦ 19 .

THE MAJOR PLANETS.

357

611. Period.—The sidereal period is 11.86 years, and the syn-
odic is 399 days (a number easily remembered), a little more than a
year and a month. The planet’s orbital velocity is about eight miles
a second.

612. Dimensions.—The planet’s apparent diameter varies
from 50 at an October opposition (or 45 1
at an April one) to

2

32 at conjunction. The form, however, of the planet’s disc is not
truly circular, the polar diameter being about 1 part less than

17
the equatorial, so that the eye notices the oval form at once. The
equatorial diameter in miles is 88,200, the polar being 83,000. Its
mean1 diameter, therefore, is 86,500,—almost eleven times that of
the earth.

This makes its surface 119 times, and its volume 1300 times, that
of the earth. It is by far the largest of the planets in the system; in
fact, whether we regard its bulk or its mass, larger than all the rest
put together.

613. Mass, Density, etc.—Its mass is very accurately known,
both by the motions of its satellites, and the perturbations of the
asteroids. It is

1
of the sun’s mass, or very nearly 316 times that

1048
of the earth. Comparing this with its volume, we find its density

0.24, less than 1 the density of the earth, and almost precisely the

4
same as that of the sun. Its mean superficial gravity comes out 2.64
times that of the earth; that is, a body on Jupiter would weigh 2 58
times as much as upon the surface of the earth; but on account of
the rapid rotation of the planet and its ellipticity there is a very
considerable difference between the force of gravity at the equator
and at the pole, amounting to 1 of the equatorial gravity. (On the

5
earth the difference is only 1 .)

190

614. Phases and Albedo.—Its orbit is so much larger than
that of the earth that the planet shows no sensible phases, even
at quadrature, though at that time the edge farthest from the sun
shows a slight darkening.

The reflecting power, or Albedo, of the planet’s surface is very
high,—0.62 according to Zöllner, that of white paper being only

2a + b
a + b

1The mean diameter of an oblate spheroid is
not

. Of the three

3

2
axes of symmetry which cross at right angles at the planet’s centre, one is the
axis of rotation, and both the others are equatorial.

THE MAJOR PLANETS.

358

0.78. The centre of the disc of this planet (and the same is also
true of Saturn) is considerably brighter than the limb—just the
reverse, as will be remembered, from the condition of things upon
the moon, and upon Mars, Venus, and Mercury. This peculiarity of
a darkened limb, in which Jupiter resembles the sun, has suggested
the idea that it is to some extent self-luminous. This, however, is
not a necessary consequence, as a nearly transparent atmosphere
overlying a uniformly reflecting surface would produce the same
effect.

The light which the planet omits, if it emits any, must be very
feeble as compared with sunlight, since the satellites, when they are
eclipsed by entering the shadow, become totally invisible.

615. Axial Rotation.—The planet rotates on its axis in about

9h 55m. The time can be given only approximately, not because it
is difficult to find and observe distinct markings on the planet’s
disc, but simply because different results are obtained from differ-
ent spots, according to their nature and their distance from the
planet’s equator. Speaking generally, spots near the equator indi-
cate a shorter day than those in higher latitudes, and certain small,
sharply defined, bright, white spots, such as are often seen, give a
quicker rotation than the dark markings in the same latitude.

For instance, a white spot observed near the equator in 1886 for
several months gives 9h 50m 4s, while another one near latitude 60◦ gave

9h 55m 12s. The great red spot has given values ranging from 9h 55m 34s.9

(in 1879) to 9h 55m 40s.7 (in 1886),—the time of rotation as determined
in each case, being certainly accurate within half a second. The progres-
sive increase has been regular and unmistakable, and is not due to any
possible uncertainty in the observations.

616. The Axis of Rotation and the Seasons.—The plane
of the equator is inclined only 3◦ to that of the orbit, so that as
far as the sun is concerned there can be no seasons. The heat and
light received from the sun by Jupiter are, however, only about 1

27
as intense as the solar radiation at the earth, its distance being 5.2
times as great.

617. Telescopic Appearance.—Even in a small telescope the
planet is a beautiful object. When near opposition a magnifying
power of only 40 makes its apparent size equal to that of the full
moon (though, as remarked in connection with Venus, no novice

THE MAJOR PLANETS.

359
would receive that impression), and with a telescope of 8 or 10
inches aperture, and with a magnifying power of 300 or 400, the
disc is covered with an infinite variety of beautiful and interesting
details which rapidly shift under the observer’s eye in consequence
of the planet’s swift rotation. The picture is rich in color, also,
browns and reds predominating, in contrast with olive-greens and
occasional purples; but to bring out the colors well and clearly re-
quires large instruments. For the most part the markings are ar-
ranged in streaks more or less parallel to the planet’s equator, as
shown by Fig. 178. With a small telescope the markings usually re-
duce to two dark and comparatively well-defined belts, one on each
side of the equator, occupying about the same regions of latitude
that the trade-wind zones do upon the earth; and very likely in

Jupiter’s case similar aerial currents have something to do with the
appearance, though upon Jupiter, as has been already said, the so-
lar heat is a comparatively unimportant factor. The markings upon
the planet are almost, if not entirely, atmospheric, as is proved by
the manner in which they change their shapes and relative posi-
tions. They are cloud forms. It is hardly probable that we ever
see anything upon the solid surface of the planet underneath, nor
is it even certain that the planet has anything solid about it. In

Fig. 178, the upper left-hand figure is from a drawing by Trouvelot
made in February, 1872; the second is by Vogel in 1880. The small
one below represents the planet as seen in a small telescope.

Fig. 178.—Telescopic Views of Jupiter.

THE MAJOR PLANETS.

360

618. The Great Red Spot.—While most of the markings on
the planet are evanescent, it is not so with all. There are some which
are at least “sub-permanent,” and continue for years, not without
change indeed, but with only slight changes. The “great red spot”

is the most remarkable instance so far. It seems to have been first
observed by Prof. C. W. Pritchett of Glasgow, Missouri, in July,

1878, as a pale, pinkish, oval spot some 13 in length by 3 in width

(30,000 miles by 7000). Within a few months it had been noticed
by a considerable number of other observers, though at first it did
not attract any special attention, since no one thought of it as likely
to be permanent. The next year, however, it was by far the most
conspicuous object on the planet. It was of a clear, strong brick-red
color, with a length fully one-third the diameter of the planet and
a width about one fourth of its length.

Fig. 179.—Jupiter’s “Red Spot.” From Drawings by Mr. Denning. 1880–85.

For two or three years it remained without much change: in 1882–83
it gradually faded out: in 1885 it had become a pinkish oval ring, the
central part being apparently occupied with a white cloud. In 1886 it
was again a little stronger in color, and the same in 1887,—an object not
difficult to see with a large telescope, but the merest ghost of what it
was in 1880. The present year (1888) its appearance is about the same
as in 1887,—perhaps a little paler. During the ten years its form and
size have varied very little. It lies at the southern edge of the southern
equatorial belt, in latitude about 35◦, and for some reason the belt seems
to be “notched out” for it, so that there has been always a narrow white

THE MAJOR PLANETS.

361
streak separating the belt from the spot. Even when the spot was palest
and hard to see its place was always evident at once from the indentation
in the outline of the belt.

Such phenomena suggest abundant matter for speculation which
would be out of place here. It must suffice to say that no satisfactory
explanation of the phenomena has yet been presented. The unquestion-
able fact before mentioned (Art. 615), that the time of rotation of the
spot has changed by more than 5s in the ten years, greatly complicates
the subject. Fig. 179, from the drawings of Mr. Denning, represents the
appearance of the spot at four different dates; viz., 1, 1880, Nov. 19;

2, 1882, Oct. 30; 3, 1884, Feb. 6; 4, 1885, Feb. 25.

619. Temperature and Physical Constitution.—The ra-
pidity of the changes upon the visible surface implies the expendi-
ture of a considerable amount of heat, and since the heat received
from the sun is too small to account for the phenomena which we
see, Zöllner, thirty years ago, suggested that it must come from
within the planet, and that in all probability Jupiter is at a tem-
perature not much short of incandescence,—hardly yet solidified to
tiny considerable extent. Mr. Proctor has given special currency to
these views among English readers. The idea that Jupiter might be
such a “semi-sun” is not at all new. Buffon, Kant, Nasmyth, and

Bond all entertained and discussed it; but it is only since the inves-
tigations of Zöllner that it has become an accepted item of scientific
belief. (See Clerke’s “History of Astronomy,” p. 335 seqq.)

620. Atmosphere.—As to the composition of the planet’s at-
mosphere, the spectroscope gives us rather surprisingly little infor-
mation. We get from the planet a good solar spectrum with the
solar lines well marked, but there are no well-defined absorption
bands due to the action of the planet’s atmosphere. There are,
however, some shadings in the lower red portion of the spectrum
that are probably thus caused. The light, for the most part, seems
to come from the upper surface of the planet’s envelope of clouds
without having penetrated to any depth.

621. Satellite System.—Jupiter has four satellites,—the first
heavenly bodies ever discovered —the first revelation of Galileo’s
telescope. His earliest observation of them was on Jan. 7, 1610,
and in a very few weeks he had ascertained their true character,
and determined their periods with an accuracy which is surprising
when we consider his means of observation. The number of the

THE MAJOR PLANETS.

362
heavenly bodies was now no longer seven, and the discovery excited
among churchmen and schoolmen a great deal of angry incredulity
and vituperation. Galileo called them “the Medicean stars.”

They are now usually known as the first, second, etc., in the or-
der of distance from the primary, but they also have names which are
sometimes used; viz., Io, Europa, Ganymede, and Callisto. Their
relative distances range between 262,000 and 1,169,000 miles, being
very approximately 6, 9, 15, and 26 radii of the planet. The dis-
tance of the first from the surface of the planet is almost exactly the
same as that of our own moon from the surface of the earth. Their
h
h
sidereal periods range between 1d 18 1 and 16d 16 1 (accurate values

2

2
in distances and periods are given in the table in the Appendix).

The orbits are almost exactly circular, and lie in the plane of the
planet’s equator.

The satellites slightly disturb each other’s motions, and from these
disturbances their masses can be ascertained in terms of the planet’s
mass. The third, which is much the largest, has a mass of about

1

11000
of the planet’s, a little more than double the mass of our own moon.

The mass of the first satellite appears to be a little less than 1 as much.

5

The second is somewhat larger than the first, and the fourth is about
half as large as the third; i.e., it has about the mass of our own moon.

The densities of the first and fourth appear to be not very different from
that of the planet itself, while the densities of the second and third are
considerably greater.

622. Relation between Mean Motions and Longitudes of
the Satellites.—In consequence of their mutual interaction a curious
relation (discovered by La Place) exists between the mean motions of
the first three satellites. The mean motion is of course 360◦ divided by

T (T being the satellite’s period). It appears that the mean motion of
the first plus twice the mean motion of the third equals three times that
of the second, or

1

2

3

+

=

.

T1

T3

T2

A similar relation holds for their longitudes:

L1 + 2L3 = 3L2 + 180◦;
so that they cannot all three come into opposition or conjunction with the
sun at once. These relations are permanently maintained by their mutual
attractions: exactly in the long run, though there are slight perturbations
produced by the fourth satellite which disturb the arrangement slightly

THE MAJOR PLANETS.

363
for short periods.

The fourth satellite does not come into any such
arrangement.

623. Diameters, etc.—The diameter of the first satellite is a
little more than 2400 miles; the second is almost exactly the size
of our own moon, i.e., between 2100 and 2200 miles; and the third
and fourth have diameters, respectively, of 3600 and 3000 miles,
the third, Ganymede, being much larger than either of his sisters.

When Jupiter is in opposition, the fourth satellite is sometimes
nearly 10 1 away from the planet, or 1 of the moon’s diameter; and

2

3
in very clear air can be seen by a sharp eye without telescopic aid.

The third, though much larger, never goes more than 6 from the
planet, and it is perhaps doubtful whether it is ever seen with the
naked eye, unless when the fourth happens to be close beside it. A
good opera-glass will easily show them all as minute points of light.

624. Brightness.—Since the sunlight of Jupiter is only 1 as

27
intense as ours, the moonlight made by the satellites is decidedly
inferior to our own, although their reflective power appears to be
higher than that of the lunar surface. They differ among themselves
considerably in this respect. The fourth satellite is of an especially
dark complexion, so that it often looks perfectly black when it passes
between us and the planet, and is projected on the disc. The others,
under similar circumstances, show light or dark according as they
have a dark or light portion of the planet for a background. Even
the fourth, when crossing the disc, is always seen bright while very
near the planet’s limb.

625. Markings upon the Satellites.—The satellites show
sensible discs when viewed with a large telescope, and all of them
but the second sometimes show dark markings upon the surface.

These markings, however, are only visible under the most favorable
circumstances, and it has not been possible to determine whether
they are atmospheric or really geographical, nor has it yet been
possible to deduce from them the satellites’ periods of rotation.

626. Variability.—Galileo noticed variations in the brightness
of the satellites at different times, and subsequent observers have
confirmed his result. In the case of the fourth satellite there seems
to be a regular variation depending upon the place of the satellite
in its orbit, and suggesting that in its axial rotation it behaves like
our own moon, keeping always the same side next its primary. In

THE MAJOR PLANETS.

364
addition it shows other irregular changes in its luminosity: so also
do the other satellites according to nearly all authorities, though
it is singular that one or two of the best observers do not find
any such irregularity indicated by their instrumental1 photometric
observations.

627. Eclipses and Transits.—The satellites’ orbits are so
nearly in the plane of the planet’s orbit that, excepting the fourth,
they all pass through the shadow of the planet, and suffer eclipse
at every revolution. At conjunction, also, they cast their shadows
upon the planet, and these shadows can easily be seen in the tele-
scope as black dots on the planet’s disc, the satellites themselves,
which cross the disc about the same time, being much more difficult
to observe. The fourth satellite escapes eclipse when Jupiter is far
from the node of its orbit. Thus, during 1888 and in the first half
of 1889, there are no eclipses of Callisto at all.

Exactly at opposition or conjunction the planet’s shadow lies
straight behind it out of our sight, so that we cannot at that time
observe the eclipses of the satellites, but only their transits across
the disc. Before and after these times, however, the shadow lies one
side of the planet.

Fig. 180.—Eclipses of Jupiter’s Satellites, at Western Elongation.

When the planet is at quadrature and the condition of things
is as represented in Fig. 180 (which is drawn to scale), the shadow

1Clerke’s “History of Astronomy,” p. 339.

THE MAJOR PLANETS.

365
projects so far to one side of the planet that the whole eclipse of
all the satellites, except the first, takes place clear of the planet’s
disc,—both the disappearance and reappearance of the satellite be-
ing visible.

628. “Equation of Light.”—The most important use that
has been made of these eclipses has been to ascertain the time re-
quired by light in traversing the distance between us and the sun,
the so-called “equation of light.” It was in 1675 that Roemer, the

Danish astronomer (the inventor of the transit instrument, meridian
circle, and prime vertical instrument,—a man nearly a century in
advance of his day), found that the eclipses of the satellites showed
a peculiar variation in their times of occurrence, which he explained
as due to the time taken by light to pass through space. His bold
and original suggestion was rejected by most astronomers for more
than fifty years,—until long after his death,—when Bradley’s dis-
covery of aberration (Art. 225) proved the correctness of his views.

629. If the planet and earth re-
mained at an invariable distance the
eclipses of the satellites would recur
with unvarying regularity (their dis-
turbances being very slight), and the
mean interval could be determined,
and the times tabulated. But if we
thus predict the times of eclipses for
a synodic period of the planet, then,
beginning at the time of opposition,
it will be found that as the planet

Fig. 181.

recedes from the earth, the eclipses

Determination of the Equation
fall constantly more and more be-
of Light.

hindhand, and by precisely the same
amount for all four of the satellites. The difference between the tab-
ulated and observed time continues to increase until the planet is
near conjunction, when the eclipses are more than sixteen minutes
late.

From the insufficient observations at his command, Roemer made
the difference twenty-two minutes.

After the conjunction, the eclipses quicken their pace and exactly
make up all the loss; so that when opposition is reached once more,

THE MAJOR PLANETS.

366
they are again on time.

It is easy to see from Fig. 181 that at opposition the planet is
nearer the earth than at conjunction by just twice the radius of the
earth’s orbit; i.e., J B −J A = 2SA. The whole apparent retardation
of the eclipses between opposition and conjunction, should therefore
be exactly twice the time required for light to come from the sun to
the earth. This time is very nearly 500 seconds, or 8m 20s.

Early in the century Delambre, from all the satellite eclipses of which
he could then secure observations, found it to be 493s. A few years ago
a redetermination by Glasenapp of Pulkowa made it 501s, from fifteen
years’ observation of the eclipses of the first satellite. Probably this value
is much nearer the truth than Delambre’s.

630.

Photometric Observations of the Eclipses.—The
eclipses are gradual phenomena, the obscuration of the satellite
proceeding continuously from the time it first strikes the shadow
of the planet until it entirely vanishes. The moment at which the
satellite seems to disappear depends, therefore, on the state of the
air and of the observer’s eye, and upon the power of his telescope.

The same is true of the reappearance; so that the observations are
doubtful to the extent of from half a minute for the first satellite

(which moves quickly), to a full minute for the fourth. Professor

Pickering has proposed to substitute for this comparatively indef-
inite moment of disappearance or reappearance, the instant when
the satellite has lost or regained just half its normal light, and he
determines this instant by a series of photometric comparisons with
one of the neighboring uneclipsed satellites, or with the planet itself.

These comparisons are made with a special photometer devised for
the purpose, and planned with reference to rapid reading: by merely
turning a small button, the observer is immediately able to make the
image of the uneclipsed satellite appear to be of the same brightness as
the satellite which is disappearing, and the observations can be repeated
very rapidly with the help of special contrivances for recording the times
and readings. It is found that this instant of “half-brightness” can be
deduced from the set of photometric readings with an error not much
exceeding a second or two. Observations of this kind have now been
going at Cambridge (U. S.)1 for several years. A similar plan has also

1Professor Pickering has more recently (August, 1888) applied photography
to these observations with most gratifying success. A series of pictures is taken,

THE MAJOR PLANETS.

367
been devised by Cornu, and is being carried out at the Paris Observatory
under his direction.

A series of such observations covering the planet’s whole period of
twelve years, ought to give us a much more accurate determination of
the light-equation than we now have.

631. Until 1849 our only knowledge of the velocity of light was
obtained by observations of Jupiter’s satellites. By assuming as
known the earth’s distance from the sun, the velocity of light follows
when we know the time occupied by light in coming from the sun.

At present, however, the case is reversed: we can determine the
velocity of light by two independent experimental methods, and
with a surprising degree of accuracy; and then, knowing the velocity
and the light-equation, we can deduce the distance of the sun.

SATURN.

632. The Orbit and Period.—Saturn is the remotest of the
ancient planets, its mean distance from the sun being 9.54 astro-
nomical units, or 886,000000 miles. This distance varies by nearly

50,000000 miles on account of the eccentricity of its orbit (0.056),
which is a little greater than that of Jupiter.

Its nearest approach to the earth at a December opposition (the
longitude of its perihelion being 90◦ 4 ) is 744 millions of miles, and
its greatest distance at a May conjunction is 1028 millions. It is so
far from the sun that these changes of distance do not so greatly
affect its apparent brightness, as in the case of the nearer planets,
the whole range of variation from this cause being less than two
to one; that is, at the nearest of all oppositions, the planet is not
twice as bright as at the remotest of all conjunctions. The changing
phases of the rings make quite as great a difference as the variations
of distance.

The orbit is inclined to the ecliptic about 2 1 ◦.

2

The sidereal period of the planet is twenty-nine and one-half
years, the synodic period being 378 days.

The planet itself is unique among the heavenly bodies. The
great belted globe carries with it a retinue of eight satellites, and is
each with an exposure of 10s, the time being recorded on a chronograph, and
they determine with great precision the moment when the satellite’s brightness
had any special value, say fifty per cent of its maximum.

THE MAJOR PLANETS.

368
surrounded by a system of rings unlike anything else in the universe
so far as known, the whole constituting the most beautiful and most
interesting of all telescopic objects.

633. Diameter,

Volume,
and Surface.—The apparent
mean diameter of the planet varies from 20 to 14 according to
the distance. We say mean diameter because this planet is more
flattened at the pole than any other, its ellipticity being nearly ten
per cent, though different observers vary somewhat in their results.

The equatorial diameter of the planet is about 75,000 miles, and
its polar about 68,000, the mean being very nearly 73,000, or a lit-
tle more than nine times that of the earth. Its surface is therefore
about eighty-two times, and its volume 760 times that of the earth.

634. Mass, Density, and Gravity.—Its mass is only ninety-
five times the earth’s mass, from which follows the remarkable fact
that the density of Saturn is only one-eighth that of the earth, or
only about five-sevenths that of water. It is by far the least dense of
all the planets. The superficial gravity is 1.2.

635. Axial Rotation.—It revolves upon its axis in about

10h 14m according to a determination of Professor Hall, made in

1876 by means of a white spot which suddenly appeared upon its
surface, and continued visible for some weeks. Although the surface
of the planet is beautifully marked with belts which often show del-
icate rose-colored tints, it is seldom that any well-defined markings
present themselves by which the rotation can be determined.

The inclination of the axis is about 28◦.

636. Surface, Albedo, and Spectrum.—As in the case of

Jupiter, the edges of the disc are not quite so brilliant as the central
portions, so that the belts appear to fade out near the limb. These
belts are less distinct and less variable than those of Jupiter; and
are arranged as shown in Fig. 182, with a very brilliant zone at the
equator, though the engraving much exaggerates the contrast. The
planet’s pole is marked by a darkish cap of greenish hue.

According to Zöllner, the Albedo, or reflecting power of the sur-
face is 0.52, almost precisely the same as that of Venus, but a little
inferior to that of Jupiter. The spectrum of the planet is the solar
spectrum without any evidence of the presence of water-vapor, so
far as can be made out, but with certain unexplained dark bands
in the red and orange similar to these observed in the spectrum of

THE MAJOR PLANETS.

369

Jupiter. The darkest of these bands, however, are not seen in the
spectrum of the ring; this might have been expected, since the ring
probably has but little atmosphere.

637. The Rings.—The most remarkable peculiarity of Saturn
is his ring-system. The planet is surrounded by three, thin, flat,
concentric rings like circular discs of paper pierced through the cen-
tre. Two of them are bright, while the third, the one nearest to
the planet, is dusky and comparatively difficult to see. They are
generally referred to by Struve’s notation as A, B, and C, A being
the exterior one.

Fig. 182.—Saturn and his Rings.

For nearly fifty years this appendage of Saturn was a complete
enigma to astronomers, Galileo, in 1610, saw with his little telescope

THE MAJOR PLANETS.

370
that the planet appeared to have something attached to it on each
side, and he announced the discovery that “the outermost planet
is triple,”—“ultimam planetam tergeminam observavi.” Not long
afterwards the rings were edgewise to the earth so that they became
invisible to him; and in his perplexity he inquired “whether Saturn
had devoured his children, according to the legend.” Huyghens,
in 1655, was the first to solve the problem and explain the true
structure of the rings. Cassini,1 twenty years later, discovered that
the ring was double,—composed of two concentric portions with a
narrow black rift of division between them.

The third, or dusky ring, C, is an American discovery, and
was first brought to light by W. C. Bond at Cambridge, U. S., in

November, 1850. About two weeks later, but before the news had
been published in England, it was also discovered independently by

Dawes.

For a while there was some question whether it was not really a new
formation; but an examination of old drawings shows that Herschel and
several other astronomers had previously seen it where it crosses the
planet, although without recognizing its character.

638. Dimensions of the Rings.—The outer ring, A, has an
exterior diameter of 168,000 miles, and is a little more than 10,000 miles
wide. The division between it and ring, B, is about 1600 miles in width,
and apparently perfectly uniform all around. Ring B is about 16,500
miles wide, and is much brighter than A, especially at its outer edge.

At the inner edge it becomes less brilliant, and is joined without any
sharp line of demarcation by ring C, which is sometimes known as the

“gauze” or “crape” ring, because it is only feebly luminous and is semi-
transparent, allowing the edge of the planet to be seen through it. The
innermost ring is nearly, perhaps not quite, as wide as the outer one, A.

There is thus left a clear space of from 9000 to 10,000 miles in width
between the planet’s equator and the inner edge of the gauze ring, the
whole ring system having an external diameter of 168,000 miles, and a
width of between 36,000 and 37,000.

1In consequence of a misunderstanding of some expressions used by Ball,
an English astronomer who observed Saturn in 1665–66, the discovery of the
division between the rings was for a time attributed to him, and statements to
that effect will be found in a number of important books. The original drawings
belonging to his paper in the Philosophical Transactions have, however, recently
been found, and show that he did not see the division at all, nor, indeed, even
understand that the appendage was a ring.

THE MAJOR PLANETS.

371

The thickness of the rings is very small indeed, probably not
exceeding 100 miles. If we were to construct a model of them on
the scale of 10,000 miles to the inch, so that the outer one would
be nearly seventeen inches in diameter, the thickness of an ordinary
sheet of writing paper would be about in due proportion. This
extreme thinness is proved by the appearances presented when the
plane of the ring is directed towards the earth, as it is once in every
fifteen years. At that time the ring becomes invisible except to the
most powerful telescopes.

639. Phases of the Rings.—The rings are parallel to the
equator of the planet, which is inclined about 27◦ to its orbit, and
about 28◦ to the plane of the ecliptic, the two nodes of the ring
being in longitude 166◦ and 346◦, in the constellations of Aquarius
and Leo. Now in the planet’s revolution around the sun, the plane
of the planet’s equator and of the rings always keeps parallel to itself

(as shown in Fig. 183), just as does the plane of the earth’s equa-
tor. Twice, therefore, in the planet’s revolution, when the plane
of the ring passes through the earth, we see it edgewise; and twice
at its maximum width, when it is at the points half-way between
the nodes. The angle of inclination being 28◦, the apparent width
of the ring at the maximum is just about half its length. The last
disappearance of the rings was in February, 1878; the next will be
in the autumn of 1891. Near the time of disappearance the ring
appears simply as a thin needle of light projecting on each side of
the planet to a distance nearly equal to its diameter. Upon this the
satellites are threaded like beads when they pass between us and
the planet.

Fig. 183.—The Phases of Saturn’s Rings.

THE MAJOR PLANETS.

372

640. Irregularities of Surface and Structure.—When the
rings are edgewise we find that there are notable irregularities upon
them. They are not truly plane, nor quite of even thickness through-
out.

The same thing is indicated by certain peculiarities sometimes re-
ported in the form of the shadow cast by the planet on the rings; but
caution must be used in accepting and interpreting such observations,
because illusions are very apt to occur from the least indistinctness of
vision or feebleness of light. The writer has usually found that the better
the seeing, the fewer abnormal appearances were noted, and the experi-
ence of the Washington observers is the same.

It can hardly be doubted that the details of the rings are contin-
ually changing to some extent. Thus the outer ring, A, is occasion-
ally divided into two by a very narrow black line known as “Encke’s
division,” although more usually there is merely a darkish streak
upon it, not amounting to a real “crack” in the surface.

641. Structure of the Rings.—It is now universally admit-
ted that the rings are not continuous sheets of either solid or liquid
matter, but are composed of a swarm of separate particles, each a
little independent moon pursuing its own path around the planet.

The idea was suggested long ago, by J. Cassini in 1715, and by

Wright in 1750, but was lost sight of until Bond revived it in con-
nection with his discovery of the dusky ring. Professor Benjamin

Peirce soon afterwards demonstrated that the rings could not be
continuous solids; and Clerk Maxwell finally showed that they can
be neither solid nor liquid sheets, but that all the known conditions
would be answered by supposing them to consist of a flock of sep-
arate and independent bodies, moving in orbits nearly circular and
in one plane,—in fact, a swarm of meteors.

642. Stability of the Ring.—If the ring were solid it would cer-
tainly not be stable, and the least disturbance would bring it down upon
the planet; nor is it certain that even the swarm-like structure makes it
forever secure. It is impossible to say positively that the rings may not
after a time be broken up. A few years ago there was much interest in a
speculation which Struve published in 1851. All the measures which he
could obtain up to that date appeared to show that a change was actu-
ally in progress, and that the inner edge of the ring was extending itself
towards the planet. His latest series of measurements (in 1885) does not,
however, confirm this theory. They show no considerable change since

THE MAJOR PLANETS.

373

1850, and the measurements of other observers agree with his in this
respect.

The researches of Professor Kirkwood of Indiana make it probable
that the divisions in the ring are due to the perturbations produced by
the satellites. They occur at distances from the planet where the period
of a small body would be precisely commensurable with the periods of
a number of the satellites. It will be remembered that similar gaps are
found in the distribution of the asteroids, at points when the period of
an asteroid would be commensurable with that of Jupiter.

643. Satellites.—Saturn has eight1 of these attendants. The
largest of them was discovered by Huyghens in 1655. It appears as
a star of the ninth magnitude, and is easily observable with a three-
inch telescope. Four others were discovered by Cassini before 1700,
two by Sir William Herschel near the end of the last century, and
one, Hyperion, the latest addition to the planet’s family, by Bond
of Cambridge, in September, 1848, and independently by Lassell at

Liverpool two days later.

The range of the system is enormous. Iapetus has a distance of

2,225000 miles, with a period of 79 days, nearly as long as that of

Mercury. There is a remarkable variation in the brightness of this
satellite. On the western side of the planet it is fully twice as bright
as upon the eastern, which practically demonstrates that, like our
own moon, it keeps the same face towards the planet at all times,
one-half of its surface being much more brilliant than the other.

Mimas, the nearest and smallest of the satellites, coasts around
the edge of the ring at a distance from it of only 34,000 miles, or

118,000 from the planet’s centre, having a period of only 22 1 hours.

2

This satellite is so small and so near the planet that it can be seen
only by very large telescopes and under favorable conditions.

Titan, as its name suggests, is by far the largest of the family. Its

1Until Herschel’s time it was customary to distinguish the satellites as first,
second, etc., in order of distance from the planet; but as Herschel’s new satel-
lites were within the orbits of those which were known before, their discovery
confused matters, and the confusion became worse confounded when the eighth
appeared. They are now usually designated by names assigned by Sir John Her-
schel as follows, beginning with the most remote, namely: Iap˘
etus (Hyperion),

Titan; Rhea, Dione, Tethys; Enceladus, Mimas. It will be noticed that these
names, leaving out Hyperion, which was undiscovered when they were assigned,
form a line and a half of a regular Latin pentameter.

THE MAJOR PLANETS.

374
distance is about 770,000 miles, and its period a little less than 16
days. It is probably 3000 or 4000 miles in diameter, and according
to Stone, its mass is

1
of Saturn’s.

4600

644. Peculiar Behavior of Hyperion.—Hyperion has a dis-
tance of 934,000 miles, and a period of 21 1 days. Under the action of

4

Titan its orbit is rendered considerably eccentric, and its line of apsides
always keeps itself in the line of conjunction with Titan, retrograding in
a way which at first seemed to defy theoretical explanation, but turns
out to be only a “new case in celestial mechanics,” and a necessary result
of the disturbance by Titan.

The orbit of Iapetus is inclined about 10◦ to the plane of the rings, but
all the other satellites move exactly in their plane, and all the five inner
ones move in orbits sensibly circular. The orbits of Iapetus, Hyperion,
and Titan have a slight eccentricity. It is not at all impossible or even
improbable that other minute satellites may yet be discovered in the
great gap between Titan and Iapetus.

URANUS.

645. As the satellites of Jupiter were the first heavenly bodies
to be “discovered,” so Uranus was the first “discovered” planet, all
the other planets then known having been known from prehistoric
antiquity. On March 13, 1781, the elder Herschel, in sweeping over
the heavens systematically with a seven-inch reflector made by him-
self, came upon an object which, by its disc, he saw at once was not
an ordinary star. In a day or two he had ascertained that it moved,
and announced the discovery as that of a comet. After a short
time, however, it became obvious from the computations of Lexell,
that its orbit was nearly circular, that its distance was enormous,
and that its path did not at all resemble that ordinarily taken by a
comet; and within a year its planetary character was recognized and
it was formally admitted as a new member of the solar system. The
name of Uranus, suggested by Bode, finally prevailed over other ap-
pellations (Herschel himself called it the Georgium Sidus, in honor
of the king), with the symbol Ç or

. The former is still generally
used by English astronomers.

The discovery of a new planet, a thing then utterly unprecedented,
caused great excitement. The king knighted Herschel, gave him a pen-
sion, and furnished him with the funds for constructing his great forty-
foot reflector of four feet aperture, with which he afterwards discovered

THE MAJOR PLANETS.

375
the two inner satellites of Saturn. It was found on reckoning back from
the date of Herschel’s discovery that the planet had been several times
before observed as a star by astronomers who narrowly missed the honor
which fell to the more fortunate and diligent Herschel.

Twelve such
observations had been made by Lemonnier alone.

646. Orbit.—The mean-distance of Uranus from the sun is
very nearly 1800 millions of miles, and the eccentricity a trifle less
than that of Jupiter’s orbit, amounting to about 83,000000. The
inclination of the orbit to the plane of the ecliptic is very slight,
only 46 . The planet’s periodic time is 84 years, and the synodic
period (from opposition to opposition) 369d 8h. The orbital-velocity
is 4 1 miles per second.

3

647. Appearance and Magnitude.—Uranus is distinctly
visible to the naked eye on a dark night as a small star of the
so-called sixth magnitude. It is so remote, its orbit having a di-
ameter more than 19 times that of the earth’s, that there is very
little change in its appearance, and it makes no practical difference
whether it is at opposition or quadrature.

In the telescope it shows a sea-green disc of about 4 in apparent
diameter, corresponding to a real diameter of 32,000 miles.

Its
surface is about 16 times, and its volume about 66 times greater
than that of the earth, so that the earth compares in size with

Uranus about as the moon does with the earth. The mass of Uranus
is 14.6 times that of the earth, and its density and surface-gravity
are respectively 0.22 and 0.90.

648. Albedo and Light.—The reflecting power of the planet’s
surface is very high, its albedo, according to Zöllner, being 0.64, even
exceeding that of Jupiter. It is to be remembered, however, that
sunlight at Uranus is only 1 as intense as at the earth, and only

368
about 1 as intense as at Jupiter; so that the disc of the planet

14
does not appear in the telescope even nearly as bright as a piece
of Jupiter’s disc of the same apparent size. The greenish blue tint
of the planet is accounted for by the fact that its spectrum shows
certain conspicuous dark bands in its lower portion, bands perhaps
identical with these which are visible in the spectrum of Saturn,
but much more intense.

The F line is also specially prominent
in the spectrum of Uranus. These facts probably indicate a dense
atmosphere.

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376

649. Polar Compression, Belts, and Rotation.—The disc
of the planet shows a decided ellipticity—about 1 according to the

14

Princeton observations of 1883, which agree nearly with those of

Schiaparelli. There are also sometimes visible upon the planet’s
disc certain extremely faint bands or belts, much like the belts of

Jupiter viewed with a very small telescope. What is exceedingly
singular, however, is that the trend of these belts seems to indicate
a plane of rotation not coinciding with the plane of the satellites’

orbits. Nearly all the observers who have seen them at all find that
they are inclined to the satellites’ orbit-plane at an angle of from 15◦
to 40◦. Now unless there is some error in Tisserand’s investigations
upon the motions of satellites, it is certain that the plane of these
orbits must of necessity nearly coincide with the planet’s equator.

Probably the error lies in judging the direction of the belts, which
at the best are at the very limit of visibility.

One or two observers have assigned to the planet rotation peri-
ods ranging from 9h to 12h; but it cannot be said that any determi-
nation of this element yet made is to be trusted.

650. Satellites.—The planet has four satellites; viz., Ariel,

Umbriel, Titania, and Oberon; Ariel being the nearest to the planet.

The two brightest of them, Oberon and Titania, were discovered by

Sir William Herschel a few years after the discovery of the planet.

He observed them sufficiently to obtain a reasonably correct deter-
mination of their distances and periods.

It is not certain that he saw either of the other two, though he thought
he had found six satellites in all, and a few years ago a popular writer on
astronomy actually credited the planet with eight satellites,—the four
whose names have been given, and four others which Herschel supposed
he had seen.

Ariel and Umbriel were first certainly discovered by Lassell in 1851,
and have since been satisfactorily observed by numerous large telescopes.

They are telescopically the smallest bodies in the solar system, and the
most difficult to see. In real size, they are, of course, much larger than
the satellites of Mars or many of the asteroids, very likely measuring
from 200 to 500 miles in diameter; but they are ten times as far away as
the asteroids, and illuminated by a sunlight not 1 as brilliant as theirs.

50

651. Satellite Orbits.—The orbits of the satellites are sensibly
circular, and all lie in one plane, which, as has been said, ought to be,
and probably is, coincident with the plane of the planet’s equator. They

THE MAJOR PLANETS.

377
are very close-packed also, Oberon having a distance of only 375,000
miles, with a period of 13d 11h, while Ariel has a period of 2d 12h, at
a distance of 223,000 miles. Titania, the largest and brightest of them,
has a distance of 280,000 miles, somewhat greater than that of the moon
from the earth, with a period of 8d 17h. Under favorable circumstances
this satellite can be just seen with a telescope of eight or nine inches
aperture.

652. Plane of Revolution.—The most remarkable thing
about this satellite system remains to be mentioned. The plane
of their orbits is inclined 82◦.2 to the plane of the ecliptic, and in
that plane they revolve backwards; or we may say, what comes to
the same thing, that their orbits are inclined to the ecliptic at an
angle of 97◦.8, in which case their revolution is to be considered as
direct.

When the line of nodes of their orbit plane passes through the earth,
as it did in 1840 and 1882, the orbits are seen edgewise and appear as
straight lines. On the other hand, in 1861, they were seen almost in plan
as nearly perfect circles, and will be seen so again in 1903. The year

1882–83 was a specially favorable time for determining the inclination
of the orbits and the position of the nodes, as well as for measuring the
polar compression of the planet.

NEPTUNE.

653. The discovery of this planet is justly reckoned as the great-
est triumph of mathematical astronomy. Uranus failed to move
precisely in the path which the computers predicted for it, and was
misguided by some unknown influence to an extent which a keen eye
might almost see without telescopic aid. The difference between its
observed place and that prescribed for it had become in 1845 nearly
as much as the “intolerable” quantity of 2 of arc.

Near the bright star Vega there are two little stars which form with
it a small equilateral triangle, the sides of the triangle being about 1 3 ◦

4
long. The northern one of the two little stars is the beautiful double-
double star

Lyræ, and can be seen as double by a keen eye without a
telescope, the two companions being about 3 1 apart. Now the distance

2
between the computed place of Uranus and its actual position was, when
at its maximum, just a little more than half of the distance between these
components of

Lyræ, that only a keen eye can separate. One would
almost say that such a minute discrepancy between observation and the-
ory was hardly worth minding, and that to consider it “intolerable” was

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378
what a Scotchman would call “sheer pernickittyness.”

But just these minute discrepancies constituted the data which
were found sufficient for calculating the position of a hitherto un-
known planet, and bringing it to light. Leverrier wrote to Galle, in
substance: “Direct your telescope to a point on the ecliptic in the
constellation of Aquarius, in longitude 326◦, and you will find within
a degree of that place a new planet, looking like a star of about the
ninth magnitude, and having a perceptible disc.” The planet was
found at Berlin on the night of Sept. 23, 1846, in exact accordance
with this prediction, within half an hour after the astronomers be-
gan looking for it, and only about 52 distant from the precise point
that Leverrier had indicated.

654. So far as the mathematical operations are concerned, the
honor is to be equally divided between two then young men,—

Leverrier of Paris, and Adams of Cambridge, England. Each took
up the problem, and by perfectly independent and considerably
different methods arrived at substantially the same solution, and
each promptly communicated the result (Adams some months ear-
lier than Leverrier) to a practical astronomer provided with the
necessary apparatus for actually detecting the planet.

Adams, who was then a graduate of three years’ standing, a fel-
low and a tutor in his college, communicated his results to Challis, his
professor of astronomy at Cambridge, in the autumn of 1845. Challis
at once consulted Airy, the Astronomer Royal, but between them the
matter rather lay in abeyance for some months, until a notice appeared
of a preliminary paper by Leverrier, which indicated that he also had
reached substantially the same conclusions as Adams. Then, at the ur-
gent suggestion of Airy, Challis decided to begin the search at once, and
to capture the planet by siege, so to speak. If he had had such star-
maps as we now possess of the regions where the planet lay concealed,
it would have been comparatively an easy operation; but as he had not,
he decided to go over a space 10◦ wide by 30◦ long, and to go over it
three times. The positions of all fixed stars would of course be the same
at each of the three observations, but a planet would change its place in
the meantime, and so would be surely detected.

He began his work on July 29, including in his sweep all stars down
to the tenth magnitude.

When, on Oct. 1, he learned of the actual
discovery of the planet, he had recorded the positions of something over

3000 stars, and was preparing to map them. He had already secured,
as it turned out, three observations of the planet on July 30, Aug. 12,

THE MAJOR PLANETS.

379
and Sept. 29, and of course it was only the question of a few weeks more
or less when the discussion of the observations would have brought the
planet to light.

But while, this rather deliberate process was going on in England,

Leverrier had revised his work, making a second approximation, and
had communicated his results to Galle, at Berlin, substantially as above
indicated. The Berlin astronomers had the great advantage of a new star-
chart by Bremiker, covering that very region of the sky, and therefore
did not need to enter upon any such tedious campaign as that begun by

Challis. In less than half an hour they found a new star, not indicated
on the map, and showing a sensible disc, just as Leverrier had predicted;
and within twenty-four hours its motion proved it to be the planet.

655. Computed Elements Erroneous.—Both Adams and

Leverrier, besides computing the planet’s position in the sky had
deduced elements of its orbit, and a value for its mass, which turned
out to be considerably erroneous. The reason was that they had as-
sumed that the mean distance of the new planet from the sun would
follow Bode’s law, a supposition which, as it turned out, is not even
roughly true, although it was entirely warranted by the existing
facts, since all the then known planets, not excepting Uranus, obey
it with reasonable exactness. This assumption of an erroneous mean
distance of thirty-eight astronomical units, instead of the true dis-
tance of thirty, carried with it errors in all the other elements of
the orbit; and the computed elements are so wide of the truth that
great authorities have maintained that the actual Neptune was not
at all the Neptune of Leverrier and Adams, but an entirely differ-
ent planet; and even that the discovery was a “happy accident.”

It was not an accident at all, however. While the data and meth-
ods employed were not competent to determine the planet’s orbit
accurately, they were sufficient to determine the direction of the
unknown body, which was the one thing needed to insure its dis-
covery. The computers informed the searchers precisely where to
point their telescopes, and could do so again were a new case of the
same kind to appear.

656. Old Observations of Neptune.—After a few weeks’ ob-
servation of the new planet it became possible to compute an approxi-
mate orbit; and reckoning back by means of this approximate orbit, the
approximate place on any given date for many years preceding could be
found. On examining the observations of stars made by different as-

THE MAJOR PLANETS.

380
tronomers in these regions of the sky, there were found several instances
in which they had observed the planet; a star of the ninth magnitude
in the proper place for Neptune being recorded in their star-catalogues,
while the place is now vacant. These old observations, thus recovered,
were of great use in determining the planet’s orbit with accuracy.

657. The Orbit of Neptune.—The planet’s mean distance
from the sun is a little more than 2800,000000 of miles, instead of
being over 3600,000000, as it should be according to Bode’s law.

The orbit, instead of being considerably eccentric, as it appeared to
be from the computation of Adams and Leverrier, is more nearly
circular than any other in the system except that of Venus, its ec-
centricity being only

9

. Even this small fraction, however, makes

1000
a variation of over 50,000000 of miles in the planet’s distance from
the sun at different parts of its orbit. The inclination of the orbit
is about 1 3 ◦. The period of the planet is about 164 years, instead

4
of 217, as it should have boon according to Leverrier’s computed
orbit. The orbital velocity is about 3 1 miles a second.

3

658. The Solar System as seen from Neptune.—At Nep-
tune’s distance the sun itself has an apparent diameter of only a
little more than 1 of arc,—only about the diameter of Venus when
nearest us, and too small to be seen as a disc by the eye, if there
are eyes on Neptune. The light and heat received from it are only

1
part of what we get at the earth. Still, we must not imagine

900
that, as compared with starlight or even moonlight, the Neptunian
sunlight is feeble.

Assuming Zöllner’s estimate that sunlight at the earth is 618,000
times as intense as the light of the full moon, we find that the sun,
even at Neptune, gives a light equal to 687 full moons. This is about
seventy-eight times the light of a standard candle at one metre distance,
or about the light of a thousand candle power electric arc at a distance
of 10 1 feet—abundant for all visual purposes. In fact, as seen from

2

Neptune, the sun would look very much like a large electric arc-lamp
at a distance of a few feet. We call special attention to this, because
erroneous statements are not unfrequently met with that “at Neptune
the sun would be only a first magnitude star.” It would really give about

44,000000 times the light of such a star.

659. From Neptune the four terrestrial planets would be hopelessly
invisible, unless with powerful telescopes and by carefully screening off
the sunlight. Mars would never reach an elongation of 3◦ from the sun;

THE MAJOR PLANETS.

381
the maximum elongation of the earth would be about 2◦, and that of

Venus about 1 1 ◦. Jupiter, attaining an elongation of about 10◦, would

2
probably be easily seen somewhat as we see Mercury. Saturn and Uranus
would be conspicuous, though the latter is the only planet of the whole
system that can be better seen from Neptune than it can be from the
earth.

660. The Planet itself.—Neptune appears in the telescope
as a small star of between the eighth and ninth magnitudes, abso-
lutely invisible to the naked eye, though easily seen with a good
opera-glass. It shows a greenish disc, having an apparent diame-
ter of about 2 .6, which varies very little, since the entire range of
variation in the planet’s distance from us is only about 1 of the

15
whole. The real diameter of the planet is about 35,500 miles (but
the probable error of this must be fully 500 miles); the volume is a
little more than ninety times that of the earth. Its mass, as deter-
mined by means of its satellite, is about 17 times that of the earth,
and its density 0.20.

The planet’s albedo, according to Zöllner, is about forty-six per
cent, a trifle lower than that of Saturn and Venus, and considerably
below that of Jupiter and Uranus. There are no visible markings
upon its surface, and nothing is known as to its rotation. The spec-
trum of the planet appears to be precisely like that of Uranus. The
light is so feeble that the ordinary lines of the solar spectrum are
difficult to make out, but there are a number of heavy, dark bands,
which indicate the existence of a dense atmosphere, through which
the light, reflected by the cloud-covered surface of the planet, is
transmitted,—an atmosphere which appears to be identically the
same on both Uranus and Neptune, while some of its constituents
are probably present in Jupiter and Saturn, as shown by the prin-
cipal dark band in the red. It is not possible as yet to identify the
substance which produces these bands.

It will be seen that Uranus and Neptune form a “pair of twins”

very much as the earth and Venus do; being nearly alike in magni-
tude, density, and other characteristics.

661. Satellite.—Neptune has one satellite, discovered by Las-
sell within a month after the discovery of the planet itself.

Its
distance is 223,000 miles, and its period is 5d 21h 2m.7. Its orbit is
inclined 34◦ 53 , and it moves backward in it; i.e., clockwise, from
east towards the west, like the satellites of Uranus. It is a very small

THE MAJOR PLANETS.

382
object, appearing of about the same brightness as Oberon, the outer
satellite of Uranus. From its brightness, as compared with that of

Neptune itself, we estimate that its diameter is about the same as
that of our own moon, though perhaps a little larger.

Since Neptune is so far from the sun, and the planet has no other
satellite of any size (none certainly comparable with this one), its motion
must be practically undisturbed and very nearly uniform. It has been
therefore proposed to make it a test of the uniformity of other motions
in the solar system, such as the length of the day and the length of the
month. It revolves rapidly enough to admit of very precise observations
by large telescopes. It is, of course, possible that the planet has other
undiscovered satellites, but if so, they must be very minute.

662. Trans-Neptunian Planet.—Perhaps the breaking down
of Bode’s law at Neptune may be regarded as an indication that the
system terminates with him, and that there is no remoter planet. If
such a planet exists, however, it is sure to be found sooner or later,
either by means of its disturbing action upon Uranus and Neptune, or
else by the methods of the asteroid hunters, although, of course, its slow
motion will render its detection in this way difficult.

In 1877, Professor Todd, from a graphical investigation of the
outstanding differences between the computed and observed places of

Uranus (after allowing for Neptune’s action), concluded that an undis-
covered planet very probably exists, and that its longitude was then
about 170◦. He made a careful but unsuccessful search for it with the

Washington telescope, going over a region about 40◦ long by 2◦ wide
with a power of 400, hoping to recognize the planet by its disc.

663. In the Appendix, we give tables containing the most accurate
data of the planetary system at present available, but with renewed
cautions to the student that these data are of very different degrees of
accuracy.

The distances (in astronomical units), and the periods of the planets

(except perhaps some of the asteroids) are known with extreme preci-
sion; probably the very last figure of the table may be trusted. The
other elements of their orbits are also known very closely, if not quite
so precisely as the distances and periods. The masses, in terms of the
sun’s mass, stand next to the orbit elements in order of precision, with
an error probably not exceeding one per cent (except, however, in the
case of Mercury, the mass of which remains still very uncertain).

The ratio of the earth’s mass to the sun’s is however less accurately
known, being at present subject to an uncertainty of at least two per
cent. This is because its determination involves a knowledge of the solar

THE MAJOR PLANETS.

383
parallax (Art. 278*), the cube of which appears in the formula for the
ratio of the masses.

Of course all the masses of the planets expressed in terms of the
earth’s mass are subject to the same uncertainty in addition to all other
possible causes of error.

When we come to the diameters, volumes, and densities of the plan-
ets, the data become less and less certain, as has been pointed out before.

For the nearer and larger planets, say Venus, Mars, and Jupiter, they
are reasonably satisfactory, for the remoter ones less so, and the figures
for the density of the distant planets,—Mercury, Uranus, and Neptune,
for instance,—are very likely subject to errors of ten or twenty per cent,
if not more.

664. We borrow from Herschel’s “Outlines of Astronomy” the fol-
lowing illustration of the relative magnitudes and distances of the mem-
bers of our system. “Choose any well-levelled field. On it place a globe
two feet in diameter. This will represent the sun; Mercury will be rep-
resented by a grain of mustard-seed on the circumference of a circle 164
feet in diameter for its orbit; Venus, a pea on a circle of 284 feet in di-
ameter; the Earth also, a pea on a circle of 430 feet; Mars, a rather large
pin’s-head on a circle of 654 feet; the asteroids, grains of sand in orbits
of 1000 to 1200 feet; Jupiter, a moderate-sized orange in a circle nearly
half a mile across; Saturn, a small orange on a circle of four-fifths of a
mile; Uranus, a full-sized cherry or small plum upon the circumference
of a circle more than a mile and a half; and finally Neptune, a good-sized
plum on a circle about two miles and a half in diameter. . . . To imitate
the motions of the planets in the above-mentioned orbits, Mercury must
describe its own diameter in 41 seconds; Venus, in 4m 14s; the Earth, in

7 minutes; Mars, in 4m 48s; Jupiter, in 2h 56m; Saturn, in 3h 13m; Uranus,
in 2h 16m; and Neptune, in 3h 30m.” We may add that on this scale the
nearest star would be on the opposite side of the globe, at the antipodes,

8000 miles away.

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

384

CHAPTER XVI.

THE DETERMINATION OF THE SUN’S HORIZONTAL PAR-

ALLAX

AND

DISTANCE.—TRANSITS

OF

VENUS

AND

OPPOSITIONS OF MARS.—GRAVITATIONAL METHODS.—

DETERMINATION BY MEANS OF THE VELOCITY OF

LIGHT.

665. This problem, from some points of view, is the most fun-
damental of all that are encountered by the astronomer. It is true
that it is possible to deal with many of the subjects that present
themselves in the science without the necessity of employing any
units of length and mass but these that are purely astronomical,
leaving for subsequent determination the relation between these
units and the more familiar ones of ordinary life: we can get, so
to speak, a map of the solar system, correct in proportion, though
without a scale of miles. But to give the map its real meaning and
use, we must find the scale finally, if not at first, and until this
is done we can form no true conceptions of the actual dimensions,
masses, and distances of the heavenly bodies.

The problem of finding the true value of the astronomical unit
is difficult, because of the great disproportion between the size of
the earth and the distance of the sun. The relative smallness of
the earth limits the length of our available “base line,” which is less
than

1
part of the distance to be determined by it. It is as if a

12000
person confined in an upper room with a wide prospect were set to
determine the distance of objects ten miles or more away, without
going outside the limits of his single window. It is hopeless to look
for accurate results by direct methods, such as answer well enough
in the moon’s case, and astronomers are driven to indirect ones.

666. Historical.—Until nearly 1700 no even reasonably accurate
knowledge of the sun’s distance had been obtained. Aristarchus, by a
very ingenious though inaccurate method, had found, as he thought, that
the distance of the sun was nineteen times as great as that of the moon

(it is really 390 times as great), and Hipparchus, combining this deter-

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

385
mination of Aristarchus with his own knowledge of the moon’s distance,
estimated the sun’s parallax at 3 , which is more than twenty times too
large. This value was accepted by Ptolemy, and remained undisputed
for twelve centuries, until Kepler, from Tycho’s observations of Mars,
satisfied himself that the sun’s parallax could not exceed 1 ; i.e., that
the sun’s distance must be at least as great as twelve or fifteen millions
of miles. Between 1670 and 1680 Cassini proposed to determine the so-
lar parallax by observations of Mars; for by that time the distance of

Mars from the earth at any moment could be very accurately computed
in astronomical units, so that the determination of the parallax of Mars
would make known that of the sun. Observations in France, compared
with observations made in South America by astronomers sent out for
the purpose, showed that the parallax of Mars could not exceed 25 , or
that of the sun, 10 . Cassini set it at 9 .5, corresponding to a distance
of 86,000000 of miles,—giving the first reasonable approach to the true
dimensions of the solar system.

In 1677, and more fully in 1716, Halley explained how transits of

Venus might be utilized to furnish a far more accurate determination
of the solar parallax than was possible by any method before used. He
died before the transits of 1761 and 1769 occurred, but they were both
observed, the first not very satisfactorily, but the second with perfect
success, and in the most widely separated parts of the globe. The results,
however, were by no means as accordant as had been expected. Various
values of the sun’s parallax were deduced, ranging all the way from 8 12
to 9 , according to the observations used, and the way they were treated
in the discussion. Towards the end of the century, La Place adopted and
used for a while the value 8 .81, while Delambre preferred 8 .6.

667. In 1822–24 Encke collected all the transit observations
that had been published, and discussed them as a whole in an ex-
tremely thorough manner, deducing as a final result from the two
transits of 1761 and 1769, 8 .5776, corresponding to a distance of

95 1 millions of miles. The decimal is very imposing, and the dis-

2
cussion by which it was obtained was unquestionably thorough and
laborious, so that his value stood unquestioned and classical for
many years.

The first note of dissent was heard in 1854, when Hansen, in pub-
lishing certain researches upon the motion of the moon, announced
that Encke’s parallax was certainly too small; he afterwards fixed
the figure at 8 .97, but the correction of certain numerical errors in
his work reduced this result to 8 .92.

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

386

Three or four years later Leverrier found a value of 8 .95 from the
so-called lunar equation of the sun’s motion; and in 1862 Foucault, from
a new determination of the velocity of light, combined with the constant
of aberration, got the value 8 .86. A re-discussion of the old transit of

Venus observations was then made by Stone, of England, who deduced
from them a value of 8 .943. The value of 8 .95 was adopted by the

British Nautical Almanac, and used in it until the issue of 1882. The
corresponding distance of 91 1 millions of miles found its way into nu-

2
merous text-books, and, though known to be erroneous, still holds its
place in some of them.

In 1867 Newcomb made a discussion of all the data then avail-
able, and obtained the value 8 .848 (or 8 .85 practically), which
value is now (1888) used in all the Nautical Almanacs except the

French. They use 8 .86, the result of an investigation published by

Leverrier in 1872.

668. The observations of Gill on the planet Mars in 1877, and
the new determinations of the velocity of light by Michelson and

Newcomb in this country, as well as the investigations of Neison and
others upon the so-called “parallactic inequality” of the moon, all
point, however, to a somewhat smaller value. Professor Newcomb
says (in 1885), “All we can say at present is that the solar parallax
is probably between 8 .78 and 8 .83, or if outside these limits, that
it can be very little outside.”

It is not, however, thought worth while to change the constant used
in the almanacs until the final reduction of the transits of 1874 and 1882
has been made, and until certain experiments and investigations now
in progress have been finished. The difference between 8 .80 and 8 .85
is of no practical account for almanac purposes, and the change would
involve alterations in a number of the tables.

Accepting Clarke’s value of the earth’s equatorial radius (Art. 145),
viz., 6,378,206.4m or 3963.3 miles, we find that a solar parallax of

8 .75 corresponds to 23,573 radii of the earth = 93,428000 miles.

8 .80

“

“ 23,439

“

“

“

“

= 92,897000

“

8 .85

“

“ 23,307

“

“

“

“

= 92,372000

“

8 .90

“

“ 23,196

“

“

“

“

= 91,852000

“

669. Methods of finding the Solar Parallax and Dis-
tance.—We may classify them as follows:—

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

387

I. Ancient Methods.

(A) Method of Aristarchus [0].

(B) Method of Hipparchus [0].

II. Geometrical and Trigonometrical Methods, in which we at-
tempt to find by angular measurements the parallax, either of
the sun itself or of one of the nearer planets.

(A) The direct method [0].

(B ) Observations of the displacement of Mars among the
stars at the time of opposition.

(a) Declination observations from two or more stations in
widely different latitudes made with meridian circles or
micrometer [25].

(b) Observations made at a single station near the equator,
by measuring the distance of the planet east or west from
neighboring stars, using the heliometer [90].

(C ) Declination observations of Venus [20].

(D ) Observations of one of the nearer asteroids in the same
way as Mars.

(a) Meridian observations at two stations in widely different
latitudes [20].

(b) Heliometer observations at an equatorial station [75].

(E ) Observations of the transits of Venus at widely separated
stations.

(a) Observations of the contacts.

(1) Halley’s method—the “method of durations” [40].

(2) Delisle’s method—observation of absolute times [50].

(b) Heliometer measurements of the position of the planet
on the sun [75].

(c) Photographic methods—various [20 to 75].

III. Gravitational methods.

(A) By the moon’s parallactic inequality [70].

(B ) By the lunar equation of the sun’s motion [40].

(C ) By the perturbations produced by the earth on Venus
and Mars [70]; (ultimately [95]).

IV. By the Velocity of Light, combined with

(A) The light equation [80].

(B ) The constant of aberration [90].

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

388

The figures in brackets at the right are intended to express roughly
the relative value of the different methods, on the scale of 100 for a
method which would insure absolute accuracy.

670. Of the Ancient methods, that of Aristarchus is so inge-
nious and simple that it really deserved to be successful. When the
moon is exactly at the half phase, the angle at M (Fig. 184) must
be just 90◦, and the angle AEM must equal M SE. If, then, we can
find how much shorter the arc N M (from new to half moon) is than

M F (from half moon to full), half the difference will measure AM ,
and give the angle at S. Aristarchus concluded that the first quarter
of the month was just about twelve hours shorter than the second,
so that AM was measured by six hours’ motion of the moon, or a
little less than 4◦. Hence he found SE, the distance of the sun, to
be about nineteen times EM —a value absurdly wrong, Since SE is
in fact nearly 390 times EM . The real difference between the two
quarters of the month is only about half an hour, instead of twelve
hours.

The
difficulty
with
the
method is that, owing to the
ragged and broken character of
the lunar surface, it is impossible
to observe the moment of half
moon with sufficient accuracy.

Fig. 184.

Aristarchus’ Method of Determining

671. The estimate of Hip-
the Sun’s Distance.

parchus was based upon the er-
roneous calculation of Aristarchus
that the sun’s distance is 19 times the moon’s, and the solar parallax,
therefore, 1 of the moon’s parallax.

19

The “radius of the earth’s shadow,” where the moon cuts it at a
lunar eclipse, is given, as Hipparchus knew, by the formula ρ = P + p − S

(Art. 372), or P +p = ρ+S. Assuming that P = 19p, we have 20p = ρ+S.

Now S, the sun’s semi-diameter, is about 15 ; and from the duration of
lunar eclipses Hipparchus found ρ to be about 40 ; hence he obtained
for p, the solar parallax, a value a little less than 3 , which, as has
been already mentioned, was accepted by Ptolemy, and by succeeding
astronomers for more than 1500 years. (Wolf’s “History of Astronomy,”

p. 175.)

672. Of the Geometrical Methods, A, the “direct method”

consists in observing the sun’s apparent declination with the merid-

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

389
ian circle at two stations widely differing in latitude, just as we
observe the moon when finding its parallax (Art. 239). Theoret-
ically, observations of this sort might give the value of the sun’s
parallax within 1
or so, but the method is practically worthless,

2
because the errors of observation are large as compared with the
quantity to be determined. The sun’s limb is a very bad object
to point on, and besides, its heat disturbs the adjustments of the
instrument, thus rendering the observations still more inaccurate.

673. The first of the two methods of observing the planet Mars
is precisely the same as this direct method of observing the sun;
but the distance of Mars at a “near opposition” is only a little more
than 1 that of the sun, so that any error of observation affects the

3
final result by only about 1 as much; and, moreover, Mars is a very

3
good object to observe, so that the errors of observation themselves
are much lessened. The planet’s distance from the earth having
been found in astronomical units by the method of Art. 515, the
determination of its distance in miles will fix the value of this unit,
and so give us directly the sun’s distance and parallax.

The method requires two observers working at a distance from
each other with different instruments, which is a serious disadvan-
tage.

For some unexplained reason, observations of this sort seem almost
invariably to give too large a result for the solar parallax, averaging
between 8 .90 and 8 .98. The red color of the planet may possibly have
something to do with this by affecting the astronomical refraction. This
method, in 1680, was the first to give a reasonable approximation to the
sun’s true distance, as has been mentioned before.

The planet Venus can be observed in the same way, and has been
once so observed by Gillis, 1849–52, at Santiago, Chili, in co-operation
with the Washington observers, but the result was not very satisfactory.

Fig. 185.—Effect of Parallax on the Right Ascension of Mars.

674. Heliometer Observations of Mars (Method b).—It is
possible, however, for a single observer to obtain better results than
can be got by two or more using the preceding method. Suppose

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

390
that the orbital motion of Mars is suspended for a while at oppo-
sition, and that the planet is on or near the celestial equator; and
also that the observer is at a station, O, on the earth’s equator.

When Mars is rising at Me, Fig. 185, the horizontal parallax OMeC
depresses the planet; that is, he appears from O to be further east
than he would if seen from C, the centre of the earth; so that the
parallax then increases the planet’s right ascension. Twelve hours
later, when he is setting, the parallax will throw him towards the
west, diminishing his right ascension by the same amount. If, then,
when the planet is rising, we measure carefully its distance west
of a star S, which is supposed to be just east of it (the distance

MeS in Fig. 186), and then measure the distance MwS from the
same star again when it is setting, the difference will give us twice
the horizontal parallax. The earth’s rotation will have performed
for the observer the function of a long journey in transporting him
from one station to another 8000 miles away in a straight line.

675. Of course the observations are
not practically limited to the moment
when the planet is just rising, nor is it
necessary that the star measured from
should be exactly east or west of the
planet. Measures from a number of the
neighboring stars, S1, S2, S3, and S4
would fix the positions Me and Mw with
more accuracy than measures from S
alone. Nor will the planet stop in its
orbit to be observed, nor will it have
a declination of zero, nor can the ob-
server command a station exactly on

Fig. 186.

Micrometric Comparison of
the earth’s equator. But these varia-

Mars with Neighboring Stars.

tions from the ideal conditions do not
at all affect the principles involved; they simply complicate the cal-
culations slightly without compromising their accuracy.

The method has the very great advantage that all the observa-
tions are made by one person, and with one instrument, so that, as
far as can be seen, all errors that could affect the result are very
thoroughly eliminated.

676. The most elaborate determination of the solar parallax yet
made by this method is that of Mr. Gill, who was sent out for the purpose

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

391
by the Royal Astronomical Society in 1877 to Ascension Island in the

Atlantic Ocean. His result, from 350 sets of measurements, gives a solar
parallax of 8 .783 + 0 .015,—a result probably very close to the truth,
though possibly a little small. In 1892 and 1894 favorable oppositions of

Mars will occur again, and it is quite likely that the observations will be
repeated on a scale even more extensive.

Venus cannot be observed in this way, since either her rising or setting
is in the daytime, when the small stars cannot be seen near her; but the
nearer asteroids can be utilized by this method. As they are more distant,
however, than Mars, a given error in observation produces a larger final
error in the result.

677. The Heliometer.—The heliometer, the instrument em-
ployed in these measures, is one of the most important of the modern
instruments of precision. As its name implies, it was first designed to
measure the diameter of the sun, but it is now used to measure any
distance ranging from a few minutes up to one or two degrees, which
it does with the same accuracy as that with which the filar micrometer
measures distances of a few seconds. It is a “double image” micrometer,
made by dividing the object-glass of a telescope along its diameter, as
shown in Fig. 187. The two halves are so mounted that they can slide
by each other for a distance of three or four inches, the separation of
the centres being accurately measured by a delicate scale, or by a mi-
crometer screw operated and read by a suitable arrangement from the
eye-end. The instrument is mounted equatorially with clock-work, and
the tube can be turned in its cradle so as to make the line of division
of the lenses lie in any desired direction. When the centres of the two
halves of the object-glass coincide, the whole acts as a single lens, giving
but one image of each object or star in the field of view. As soon as the
centres are separated, each half of the object-glass forms its own image.

To measure the distance from Mars to a star,
the telescope tube is turned so that the line of cen-
tres points in the right direction, and then the semi-
lenses are separated until one of the two images of
the star comes exactly in the centre of one of the
images of Mars; this can be done in two positions
of the semi-lens A with respect to B, as shown by
the figure. We may either make S0 (the image of
the star formed by semi-lens B) coincide with M1

Fig. 187.—The
formed by A, or make S2 coincide with M0. The

Heliometer.

whole distance from 1 to 2 then measures twice the
distance between M and S.

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

392

678. Transit of Venus Observations.—At the time when

Venus passes between us and the sun, her distance from the earth
is only some 26,000000 of miles, so that her horizontal parallax is
nearly four times as great as that of the sun itself. At this time her
apparent displacement upon the sun’s disc, due to a change of the
observer’s station upon the earth, is the difference between her own
parallax due to this displacement, and that of the sun itself; and
this difference is greater than the sun’s parallax nearly in the ratio
of 3 to 1, or, more exactly, of 723 to 277. The object, then, of the
observations of a transit is to obtain in some way a measure of the
angular displacement of Venus on the sun’s disc, corresponding to
the known distance between the observer’s stations upon the earth.

679. Halley’s Method.—The method
proposed by Halley, who in 1677 brought to
notice the great advantages presented by a
transit of Venus for determining the sun’s
parallax, was as follows: Two stations are
chosen upon the earth’s surface, as far sepa-
rated in latitude as possible. From them we
observe the duration of the transit; that is,
the interval of time between its beginning

Fig. 188.—Contacts in
a Transit of Venus.

and end, both of which must be visible at
both stations. If the clock runs correctly during the few hours dur-
ing which the transit lasts, this is all that is necessary. We do not
need to know its error in reference to Greenwich time, nor even in re-
spect to the local time, except roughly. This was a great advantage
of the method in those days, before the era of chronometers, when
the determination of the longitude of a place was a very difficult and
uncertain operation. The observation to be made is simply to note
the clock time at which “contact ” occurs, there being four of these
contacts,—two exterior and two internal, at the points marked 1,

2, 3, 4, in Fig. 188. Halley depended mainly on the two internal
contacts, which he supposed could be observed with an error not
exceeding one second of time.

680. Computation of the Parallax.—Having the durations
of the transits at the two stations, and knowing the hourly angular
motion of Venus, we have at once and very accurately the length
of the two chords described by Venus upon the sun, expressed in
seconds of arc. We also know the sun’s semi-diameter in seconds,

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

393
and hence in the triangles (Fig. 189) Sab and Sde, we can compute
the length (in seconds still) of Sb and Se, the difference of which,
be, is the displacement sought, due to the distance between the
stations on the earth.1 The virtual base line is, of course, not the
distance between B and E as a straight line, because that line is
not perpendicular to the line of sight from the earth to Venus, nor
to the plane of the planet’s orbit, but the true value to be used is
easily found. Calling this base line β, we have

277
r
p = (be) ×

,

723
β
r being the radius of the earth.

The rotation of the earth, of course, comes in to shift the places
of E and B during the transit, but this can easily be allowed for.

Fig. 189.—Halley’s Method.

681. The Black Drop.—Halley expected, as has been said,
that it would be possible to observe the instant of internal contact
within a single second of time, but he reckoned without his host. At
the transits of 1761 and 1769, at most of the stations the planet at
the time of internal contact showed a “ligament” or “black drop,”

like Fig. 190, instead of presenting the appearance of a round disc
neatly touching the edge of the sun; and the time of real contact
was thus made doubtful by 10s or 15s.

1In order that the method may be practically successful, it is necessary that
the transit track should lie near the edge of the sun’s disc, for two reasons. It is
desirable that the duration should not be more than three or four hours, while
for a central transit it lasts eight hours (Art. 575). Moreover, if the two chords were near the centre of the disc, any small error in the length of either chord
would produce a great error in the computed distance between them. When
they lie as in the figure (which has been the case in all recent transits), the reverse is true: a considerable error in the observed length of one of the chords
affects their computed distance only slightly.

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

394

This “ligament” depends upon the fact that the
optical edge of the image of a bright body is not, and
in the nature of things cannot be, absolutely sharp in
the eye or in the telescope. In the eye itself we have
irradiation. In the telescope we have the difficulty
that even in a perfect instrument the image of a lu-
minous point or line has a certain width (which with

Fig. 190.—The

Black Drop.

a given magnifying power is less for a large instru-
ment). Moreover a telescope is usually more or less
imperfect, and practically adds other defects of definition, so that when-
ever the limbs of two objects approach each other in the field of view of
a telescope we have more or less distortion due to the overlapping of the
two “penumbras of imperfect definition,”—the same sort of effect that is
obtained by putting the thumb and finger in contact, holding them up
within two or three inches of the eye and then separating them: as they
separate, a “black ligament” will be seen between them.

With modern telescopes, and by
great care in preventing the sun’s
image from being too bright, so as
to diminish irradiation in the eye as
far as possible, the black drop was
reduced to reasonably small propor-
tions in 1874 and 1882, and practice
beforehand with an “artificial tran-
sit” enabled the observer in some de-
gree to allow for its effect. But a
new difficulty appeared, from which
there seems to be absolutely no way
of escape,—the planet’s atmosphere
causes it to be surrounded by a lumi-

Fig. 191.—Atmosphere of Venus as
nous ring as it enters upon the sun’s
seen during a Transit. (Vogel, 1882.)

disc, and thus renders the time of
the contact uncertain by at least five or six seconds. In both the tran-
sits of 1874 and 1882, differences of that amount continually appeared
among the results of the best observers. Fig. 191 shows the appearances
due to this cause as observed by Vogel in 1882.

682. Delisle’s Method.—Halley’s method requires the use of
polar stations, uncomfortable and hard to reach, and also that the
weather should permit the observer to see both the beginning and
end of the transit.

Delisle’s method, on the other hand, utilizes two stations near

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

395
the equator, taken on a line roughly parallel to the planet’s motion.

It requires also that the observers should know their longitude ac-
curately, so as to be able to determine the Greenwich time at any
moment; but it does not require that they should see both the be-
ginning and end of the transit; observations of either phase can be
utilized: and this is a great advantage. Suppose, then (Fig. 192),
that the observer W on one side of the earth notes the moment of
internal contact in Greenwich time, the planet then being at V1.

When E notes the contact (also in Greenwich time) the planet will
be at V2, and the angle at D will be the angular diameter of the
earth as seen from D; i.e., simply twice the sun’s parallax. Now the
angle D is at once determined by the time occupied by Venus in
passing from V1 to V2, since in 584 days (the synodic period) she
moves completely round from the line DW to the same line again.

If the time from V1 to V2 were twelve minutes, we should find the
angle at D to be about 18 .

Fig. 192.—Delisle’s Method.

683. Heliometer Observations.—Instead of observing sim-
ply the times of contact, and leaving the rest of the transit unuti-
lized, as in the two preceding methods, it is possible to make a
continuous series of measurements of the distance and direction of
the planet from the nearest point of the sun’s limb. These measure-
ments are best made with the heliometer (Art. 677), and give the
means of determining the planet’s apparent position upon the sun’s
disc at any moment with extreme precision. Such sets of measure-
ments, made at widely separated stations, will thus furnish accurate
determinations of the apparent displacement of the planet on the
sun’s disc, corresponding to known distances on the earth, and so
will give the solar parallax.

During the transit of 1882 extensive series of observations of
this sort were made by the German parties, two of which were in
the United States,—one at Hartford, Conn., and the other at San

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

396

Antonio, Texas. The results have not yet (August, 1888) been pub-
lished, but they will soon appear, and it is understood that they
are considerably more accordant than these obtained by any other
method of observation.

684. Photographic Observations.—The heliometer mea-
surements cannot be made very rapidly. Under the most favorable
circumstances a complete set requires at least fifteen minutes, so
that the whole number obtainable during the seven or eight hours
of the transit is quite limited. Photographs, on the other hand,
can be made with great rapidity (if necessary, at the rate of two a
minute), and then after the transit we can measure at leisure the
position of the planet on the sun’s disc as shown upon the plate. At
first sight this method appears extremely promising, and in 1874
great reliance was placed upon it. Nearly all the parties, some fifty
in number, were provided with elaborate photographic apparatus of
various kinds. On the whole, however, the results, upon discussion,
appear to be no more accordant than those obtained by other meth-
ods, so that in 1882 the method was generally abandoned, and used
only by the American parties, who employed an apparatus having
some peculiar advantages of its own.

685. English, German, and French Methods.—In 1874, the

English parties used telescopes of six or seven inches aperture, and mag-
nified the image of the sun formed by the object-glass by a combination
of lenses applied at the eye-end. There were no special appliances for
eliminating the distortion produced by the enlarging lenses, nor for as-
certaining the exact orientation of the picture (that is, the direction of
the image upon the plate with reference to north and south), nor for
determining its scale.

The Germans and Russians employed a nearly similar apparatus, but
with the important difference that at the principal focus of the object-
glass they inserted a plate of glass ruled with squares. These, squares are
photographed upon the image of the sun, and furnish a very satisfactory
means of determining the scale and distortion, if any, of the image. The
object-glasses used by the English and the Germans had a focal length
of seven or eight feet. The French employed object-glasses with a focal
length of some fourteen feet, the telescope being horizontal, while the
rays of the sun were reflected into it by a plane mirror; instead of glass
plates they used the old-fashioned metallic daguerreotype plates, in order
to avoid any possible “creeping” of the collodion film, which was feared in
the more modern wet-plate process. The French plates furnish, however,
no accurate orientation of the picture.

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

397

Fig. 193.—American Apparatus for Photographing the Transit of Venus.

686. The American Apparatus.—The Americans used a sim-
ilar plan, with some modifications and additions. The telescope lenses
employed were five inches in diameter and forty feet in focal length, so
that the image directly formed upon the plate was about 4 1 inches in

2
diameter, and needed no enlargement. The telescope was placed hori-
zontal and in the meridian, its exact direction being determinable by a
small transit instrument which was mounted in such a manner that it
could look info the photograph telescope, as into a collimator, when the
reflector was removed. The reflector itself was a plane mirror of unsil-
vered glass driven by clock-work. Fig. 193 shows the arrangement of the
apparatus. In front of the photographic plate, and close to it, was sup-
ported a glass plate ruled with squares called the “reticle plate,” and in
the narrow space between this and the photograph plate was suspended
a plumb-line of fine silver wire, the image of which appeared upon the
plate, and gave the means of determining the orientation of the image
with extreme precision. If the reflector were, and would continue to be,
perfectly plane through the whole operation, the method could not fail to
give extremely accurate results; but the measurements and discussion of
the observations seem to show that this mirror was actually distorted to
a considerable extent by the rays of the sun. On the whole the American
plates do not appear to be much more trustworthy than those obtained
by other methods Fig. 194 is a reduced copy of one of the photographs
made at Princeton during the transit of 1882. The black disc near the
middle, with a bright spot in the centre, is the image of a metal disc
cemented to the reticle to mark the centre lines of the reticle plate; 192
plates were taken during the transit, and at some of the stations where
the weather was good the number was much greater—nearly 300 in some
cases.

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

398

The difficulties to be encoun-
tered are numerous.

Photo-
graphic irradiation, or the spread
of the image on the plate, slight
distortion of the image by the
lenses or mirrors employed, ir-
regularities of atmospheric re-
fraction, uncertainty as to the
precise scale of the picture,—all
these present themselves in a very
formidable manner. It is obvious
why this should be so, when we
recall that on a four-inch picture
of the sun’s disc,

1
of an inch

10 000

Fig. 194.—Photograph of Transit of
corresponds to about 1 of a sec-

20

Venus.

ond of arc, and the whole uncer-
tainty as to the solar parallax does not amount to as much as that. An
image of the sun, therefore, in which the position of Venus upon the
sun’s disc cannot be determined accurately without an error exceeding

1
of an inch, is of very little value. Imperfections that would be of

10 000
no account whatever in plates taken for any other purpose make them
practically worthless for this.

Still there is reason to hope that considering the enormous number
of photographs made in 1874 and 1882 (certainly not less than 5000 in
all), the result to be obtained from such a mass of material will prove to
be worth something.

Gravitational Methods.—These are too recondite to permit
any full explanation here. We can only indicate briefly the principles
involved.

687. (1) The first of these methods is by the moon’s parallactic
inequality, an irregularity in the moon’s motion which has received
this name, because by means of it the sun’s parallax can be deter-
mined. It depends upon the fact that the sun’s disturbing action
upon the moon differs sensibly from what it would be if its distance,
instead of being less than 400 times that of the moon from the earth,
were infinitely great.

The disturbing action upon the half of the moon’s orbit which lies
nearest the sun is greater than on the opposite half of the orbit. The
retarding action of the tangential force, therefore, during the first
quarter after new moon, is perceptibly greater than the acceleration

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

399
produced during the second quarter (Art. 447), so that at the first
and third quarters respectively, the moon is a little more than 2
behind and ahead of the place she would occupy if the tangential
forces were equal in all four quadrants of the orbit—as they would
be if the sun’s distance were infinite. This puts the moon about
four minutes of time behindhand at the first quarter, and as much
ahead at the third; and if the centre of the moon could be observed
within a fraction of a second of arc (as it could if she were a mere
point of light like a star), the observations would give a very accu-
rate determination of the sun’s distance. The irregularities of the
moon’s limb, however, and the worse fact, that at the first quarter
we observe the western limb, while at the third quarter it is the
eastern one which alone is observable, make the result somewhat
uncertain, though the method certainly ranks high.

688. (2) The “lunar equation of the sun’s motion” is, it will be re-
membered, an apparent slight monthly displacement of the sun, amount-
ing to about 6 .3, and due to the fact that both earth and moon re-
volve around their common centre of gravity. It is generally made use of

(Art. 243) to determine the mass of the moon as compared with that of
the earth, using as a datum the assumed known distance of the sun; but
if we consider the mass of the moon as known (determined by the tides,
for instance), then we can find the sun’s parallax1 in terms of the lunar
equation.

The method is not a good one, since the solar parallax, 8 .8, is greater
than the quantity by means of which it is determined.

689. (3) The third method (by the earth’s perturbations of

Venus and Mars) is one of the most important of the whole list.

It depends upon the principle that if the mass of the earth, as
compared with that of the sun, be accurately known, then the dis-
tance of the sun can be found at once. The reader will remember
that in Art. 278 the mass of the sun was found by comparing the
distance which the earth falls towards the sun in a second (as mea-
sured by the curvature of her orbit) with the force of gravity at the

1Putting L for the maximum value of the lunar equation (about 6 .3 of
arc), P for the sun’s parallax, and R and r for the distance of the moon and
the semi-diameter of the earth respectively, we have the equation
r

E + m

1

P = L

= L

(81) nearly.

R
m

60

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

400
earth’s surface; and in the calculation the sun’s distance enters as
a necessary datum. Now, if we know independently the sun’s mass
compared with the earth’s, the distance becomes the only unknown
quantity, and can be found from the other data.

In the same way as in Art. 536 we have

D3

(S + E) = 4π2

,

T 2
in which S and E are the masses of the sun and earth, D is the mean
distance of the earth from the sun, and T the number of seconds in a
year. Also we have for the force of gravity at the earth’s surface,

E
g =

,
or

E = gr2,
r2
in which r is the earth’s radius.

Dividing the preceding equation by this we, get

S + E

4π2

D3

=

;

E
gT 2
r2

S + E
gT 2r2
whence

D3 =

.

E

4π2

S

If we put

= M , this becomes

E

M + 1

D3 =
gT 2r2.

4π2

In this equation everything in the second term is known when we
have once found M , or the ratio between the masses of the sun and
earth; g is found by pendulum observations on the earth, T is the length
of the year in seconds, and r is the earth’s radius.

Now, the disturbing force of the earth upon its next neighbors,

Mars and Venus, depends directly upon its mass as compared with
the sun’s mass, and the ratio of the masses can be determined when
the perturbations have been accurately ascertained; though the cal-
culation is, of course, anything but simple. But the great beauty of
the method lies in this, that as time goes on, and the effect of the
earth upon the revolution of the nodes and apsides of the neigh-
boring orbits accumulates, the determination of the earth’s mass in
terms of the sun’s becomes continually and cumulatively more pre-
cise. Even at present the method ranks high for accuracy,—so high

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

401
that Leverrier, who first developed it, would have nothing to do with
the transit of Venus observations in 1874, declaring that all such old-
fashioned ways of getting at the sun’s parallax were relatively of no
value. The method is the “method of the future,” and two or three
hundred years hence will have superseded all the others,—unless
indeed it should appear that bodies at present unknown are inter-
fering with the movements of our neighboring planets, or unless it
should turn out that the law of gravitation is not quite so simple as
it is now supposed to be.

690. The Physical Method.—The physical, or “photo-tachy-
metrical” method, as it has been dubbed, depends upon the funda-
mental assumption that light travels in interplanetary space with
the same velocity as in vacuo. This is certainly very nearly true,
and probably exactly so, though we cannot yet prove it.

By the recent experiments of Michelson and Newcomb in this
country, following the general method of Foucault, the velocity of
light has been ascertained with very great precision and may be
taken as 299,860 kilometres, or 186,330 miles, with a probable error
which cannot well be as great as twenty-five miles either way.

691. Sun’s Distance from the Equation of Light.—(1)

“The equation of light ” is the time occupied by light in travelling
between the sun and earth, and is determined by observation of the
eclipses of Jupiter’s satellites (Art. 629). By simply multiplying the
velocity of light by this time (499s ± 2s) we have at once the sun’s
distance; and that independent of all knowledge as to the earth’s
dimensions. The reader will remember, however, that the determi-
nation of this “light-equation” is not yet so satisfactory as desirable
on account of the indefinite nature of the eclipse observations in-
volved.

692. From the Constant of Aberration.—(2) When we
know the velocity of light we can also derive the sun’s distance from
the “constant of aberration,” and this constant, 20 .492, derived
from star observations (Art. 225), is known with a considerably
higher percentage of accuracy than the light-equation.

Calling the constant α, we have

U
tan α =

,

V
where U is the velocity of the earth in its orbit, and V the velocity

DETERMINATION OF SUN’S PARALLAX AND DISTANCE.

402
of light. Now U equals the circumference of the earth’s orbit divided
by the length of the year; i.e.,

2πD

U =

;

T

2πD
hence
tan α =

,

V T
tan α
and

D =

(V T ) .

2π

On the whole it seems likely at present that the value of the sun’s
distance thus derived is the most accurate of all. Using α = 20 .492
and V = 186, 330 miles, we have D = 92, 975, 500 miles, and taking
the earth’s equatorial radius as 3963.296 miles (Clarke, 1878), we
get 8 .793 as the sun’s equatorial horizontal parallax.

693. The reader will notice that the geometrical methods give
the distance of the sun directly, apart from all hypothesis or assump-
tion, except as to the accuracy of the observations themselves, and
of their necessary corrections for refraction, etc.: the gravitational
methods, on the other hand, assume the exactness of the law of
gravitation; and the physical method assumes that light travels in
space with the same velocity as in our terrestrial experiments. The
near accordance of the results obtained by the different methods
shows that these assumptions must be very nearly correct, if not
absolutely so.

COMETS.

403

CHAPTER XVII.

COMETS: THEIR NUMBER, MOTIONS, AND ORBITS.—THEIR

CONSTITUENT PARTS AND APPEARANCE.—THEIR SPEC-

TRA.—THEIR PHYSICAL CONSTITUTION, AND ORIGIN.

694. From time to time bodies of a very different character
from the planets make their appearance in the heavens, remain
visible for some weeks or months, move over a longer or shorter
path among the stars, and then vanish. These are the Comets, or

“hairy stars,” as the word means, since the appearance of such as
are bright enough to be visible to the naked eye is that of a star sur-
rounded by a hazy cloud, and usually carrying with it a streaming
trail of light. Some of them have been magnificent objects,—the
nucleus, or central star, as brilliant as Venus and visible even by
day, while the cloudy head was nearly as large as the sun itself,
and the tail extended from the horizon to the zenith,—a train of
shining substances long enough to reach from the earth to the sun.

The majority of comets, however, are faint, and visible only with a
telescope.

695. Superstitions.—In ancient times these bodies were re-
garded with great alarm and aversion, being considered from the as-
trological point of view as always ominous of evil. Their appearance was
supposed to presage war, or pestilence, or the death of princes. These
notions have survived until very recent times with more or less vigor,
but, it is hardly necessary to say, without the least reason. The most
careful research fails to show any effect upon the earth produced by a
comet, even of the largest size. There is no observable change of temper-
ature or of any meteorological condition, nor any effect upon vegetable
or animal life.

696. Number of Comets.—Thus far we have on our lists
about 650 different comets. About 400 of these were recorded pre-
vious to 1600, before the invention of the telescope, and must, of

COMETS.

404
course, have been bright enough to attract the attention of the
naked eye. Since that time the number annually observed has in-
creased very greatly, for only a few of these bodies, perhaps one in
ten, are visible without telescopic aid. Their total number must be
enormous. Not unfrequently from five to eight are discovered in a
single year, and there is seldom a day when one or more is not in
sight.

While telescopic comets, however, are thus numerous, brilliant
ones are comparatively rare. Between 1500 and 1800 there were,
according to Newcomb, 79 visible to the naked eye, or about one
in three and three-fourths years. Humboldt enumerates 43 within
the same period as conspicuous; during the first half of the present
century there were 9 such, and since 1850 there have been 11. Since,
and including, 1880 we have had 7,—a remarkable number for so
short a time,—and two of them, the principal comet of 1881 and
the great comet of 1882, were unusually fine ones. In August, 1881,
for a little time two comets were conspicuously visible to the naked
eye at once and near together in the sky, a thing almost if not quite
unprecedented.

697. Designation of Comets.—The more remarkable ones
generally bear the name of their discoverer, or of some astronomer
who made important investigations relating to them,—as for in-
stance, Halley’s, Encke’s, and Donati’s comets. They are also des-
ignated by the year of discovery, with a Roman number indicating
the order of discovery in the year. A third method of designation
is by year and letter, the letters denoting the order in which the
comets of a given year pass the perihelion. Thus Donati’s comet
was both comet F and comet VI, 1858. Comet I is, however, not
necessarily comet A, though it usually is so. In some cases the
comet bears the name of two discoverers. Thus the Pons–Brooks
comet of 1883 is a comet which was discovered by Pons in 1812, and
at its return in 1883 was discovered by Brooks.

698. The Discovery of Comets.—As a rule these bodies are
first seen by comet-hunters, who make a business of searching for them.

For such purposes they are usually provided with a telescope known as
a “comet-seeker,” having an aperture of from four to six inches, with
an eye-piece of low power, and a large field of view. When first seen, a
comet is usually a mere roundish patch of faintly luminous cloud, which,
if really a comet, will reveal its true character within an hour or two by

COMETS.

405
its motion. Some observers have found a great number of these bodies.

Messier discovered twelve between 1794 and 1798, and Pons twenty-seven
between 1800 and 1827.

699. Duration of Visibility, and Brightness.—The time
during which they are visible differs very much. The great comet of

1811 was observed for seventeen months, the longest time on record.

The comet of 1861 was observed for a year, and the great comet of

1882 for five months. In some cases, when a comet does not happen
to be discovered until it is receding from the sun, it is seen only for
a week or two.

As to their brightness they also differ widely. The great majority
can be seen only with a telescope, although a considerable number
reach the limit of naked-eye vision at that part of their orbit where
they are most favorably situated. A few, as has been said above,
become conspicuous; and a very few, perhaps four or five in a cen-
tury, are so brilliant that they can be seen by the naked eye in full
sunlight, as was the case with the great comets of 1843 and 1882.

700. Their Orbits.—The ideas of the ancients as to the motions
of these bodies were very vague. Aristotle and his school believed them
to be nothing but earthly exhalations inflamed in the upper regions of the
air, and therefore meteorological phenomena rather than astronomical.

Ptolemy accordingly omits all notice of them in the Almagest.

Tycho Brahe was the first to show that they are more distant than
the moon by comparing observations of the comet of 1577 made in dif-
ferent parts of Europe. Its position among the stars at any moment,
as seen from his observatory at Uranienburg, was sensibly the same as
that observed at Prague, more than 400 miles to the south. It followed
that its distance must be much greater than that of the moon, and that
its real orbit must be of enormous size, cutting through interplanetary
space in a manner absolutely incompatible with the old doctrine of the
crystalline spheres. He supposed the path to be circular, however, as
befitted the motion of a celestial body.

Kepler supposed that comets moved in straight lines; and he seems
to have been half disposed to consider them as living creatures, travelling
through space with will and purpose, “like fishes in the sea.”

Hevelius first, nearly a hundred years later, suggested that the or-
bits are probably parabolas, and his pupil Doerfel proved this to be the
case in 1681 for the comet of that year. The theory of gravitation had
now appeared, and Newton soon worked out and published a method
by which the elements of a comet’s orbit can be determined from the

COMETS.

406
observations.

Immediately afterwards Halley, using this method and
computing the parabolic orbits of all the comets for which he could find
the needed observations, ascertained that a series of brilliant comets hav-
ing nearly the same orbit had appeared at intervals of about seventy-five
years. He concluded that these were different appearances of one and
the same comet, the orbit not being really parabolic but elliptical, and
he predicted its return, which actually occurred in 1759—the first of the

“periodic comets.”

Fig. 195.

The Close Coincidence of Different Species of Cometary Orbits within the

Earth’s Orbit.

701. Determination of a Comet’s Orbit.—Strictly speak-
ing, the orbit of a comet being always a conic section, like that of
a planet, requires only three perfect observations for its determina-
tion; but it seldom happens that the observations1 can be made so
accurately as to enable us to distinguish an orbit truly parabolic
from one slightly hyperbolic, or from an ellipse of long period. The

1Observations for the determination of a comet’s place are usually made
with an equatorial, by measuring the apparent distance between the comet and
some neighboring “comparison star” with some form of micrometer, as indicated
in Art. 129. If the star’s place is not already accurately known, it is afterwards specially observed with the meridian circle of some standard observatory: this
observation of comparison stars forms quite an item in the regular work of such
an institution.

COMETS.

407
plane of the orbit and its perihelion distance, can be made out with
reasonable accuracy from such observations as are practically ob-
tainable, but the eccentricity, and the major axis with its corre-
sponding period, can seldom be determined with much precision
from the data obtained at a single appearance of a comet.

The reason is that a comet is visible only in that very small
portion of its orbit which lies near the earth and sun, and, as the
figure shows (Fig. 195), in this portion of the orbit, the long ellipse,
the parabola, and the hyperbola almost coincide. Moreover, from
the diffuse nature of a comet it is not possible to observe it with the
same accuracy as a planet.

Comets which really move in parabolas or hyperbolas visit the
sun but once, and then recede, never to return; while these that
move in ellipses return in regular periods, unless disturbed.

It will be understood, that in a catalogue of comets’ orbits, those
which are indicated as parabolic are not strictly so. All that can be
said is that during the time while the comet was visible, its posi-
tion did not deviate from the parabola given by an amount sensible
to observation. The chances are infinity to one against a comet’s
moving exactly in a parabola, since the least retardation of its ve-
locity would render the orbit elliptical, and the least acceleration,
hyperbolic, according to the principles explained in Article 430.

702. Relative Numbers of Parabolic, Elliptical, and Hy-
perbolic Comets.—The orbits of about 270 comets have been thus
far computed. Of this number about 200 are sensibly parabolic, and
six appear to be hyperbolic, although the eccentricity exceeds unity
by so small a quantity as to leave the matter somewhat doubtful.

There are also a number of comets which, according to the best com-
putations, appear to have orbits really elliptical, but with periods
so long that their elliptical character cannot be positively asserted.

About fifty have orbits which are certainly and distinctly oval; and
twenty-six of these have periods which are less than one hundred
years. Thirteen of these periodic comets have already been actually
observed at more than one return.

As to the rest of the twenty-six, some of them are expected to return
again within a few years, and some of them have been lost,—either in
the same way as the comet of Biela, of which we shall soon speak, or
by having their orbits so changed by perturbations that they no longer
come near enough to the earth to be observed. Table III. of the Appendix

COMETS.

408
gives the elements of these thirteen comets taken from the “Annuaire du

Bureau des Longitudes” for 1888. It will be observed that the shortest
period is that of Encke’s comet, which is only three and one-half years,
while the period of Halley’s comet exceeds seventy-six.

There are three comets with computed periods ranging between sev-
enty and eighty years, whose returns are looked for within the next forty
years. There is also one comet with a period of thirty-three years which
is due to return in 1899.

703. Fig. 196 shows the or-
bits of several of the comets
of short period,—from three to
eight years. (It would cause con-
fusion to insert all of them.) It
will be seen that in every case the
comet’s orbit comes very near to
the orbit of Jupiter, and when
the orbit crosses that of Jupiter,
one of the nodes is always near
the place of apparent intersection

(the node being marked on the
comet’s orbit by a short cross-
line).

If Jupiter were at that

Fig. 196.—Orbits of Short-period
point of its orbit at the time

Comets.

when the comet was passing, the
two bodies would really be very near to each other. The fact, as
we shall see, is a very significant one, pointing to a connection be-
tween these bodies and the planet. It is true for all the comets
whose periods are less than eight years—for those not inserted in
the diagram as well as those that are. The orbits of the seventy-
five-year comets are similarly related to the orbit of Neptune, and
the thirty-three-year comet passes very close to the orbit of Uranus.

704. Recognition of Elliptic Comets.—Modern observa-
tions are so much more accurate than those made two centuries
ago that it is now sometimes possible to determine the eccentricity
and period of an elliptic comet by means of the observations made
at a single appearance. Still, as a general rule, it is not safe to
pronounce upon the ellipticity of a comet’s orbit until it has been
observed at least twice, nor always then. A comet possesses no “per-
sonal identity,” so to speak, by which it can be recognized merely

COMETS.

409
by looking at it,—no personal peculiarities like those of the planets

Jupiter and Saturn. It is identifiable only by its path.

When the approximate parabolic elements of a new comet’s orbit
have been computed, we examine the catalogue of preceding comets to
see if we can find others which resemble it; that is, which have nearly
the same inclination and longitude of the node with the same perihelion
distance and perihelion longitude. If so, it is probable that we have to do
with the same comet in both cases. But it is not certain, and investi-
gations, often very long and intricate, must be made to see whether an
elliptical orbit of the necessary period can be reconciled with the observa-
tions, after taking into account the perturbations produced by planetary
action. These perturbations are extremely troublesome to compute, and
are often very great, since the comets not unfrequently pass near to the
larger planets. In some such cases the orbit is completely altered. Even
if the result of this investigation appears to show that the comets are
probably identical, we are not yet absolutely safe in the conclusion, for
we have what are known as—

705. Cometary Groups.—These are groups of comets which
pursue nearly the same orbits, following along one after another at
a greater or smaller interval, as if they had once been united, or
had come from some common source. The existence of such groups
was first pointed out by Hoek of Utrecht in 1865. The most re-
markable group of this sort is the one composed of the great comets
of 1668, 1843, 1880, and 1882, and there is some reason to sus-
pect that the little comet visible on the picture of the corona of the

Egyptian eclipse (Art. 328) also belongs to it. The bodies of this
group have orbits very peculiar in their extremely small perihelion
distance (they actually go within half a million miles of the sun’s
surface), and yet, although their elements are almost identical, they
cannot possibly all be different appearances of one and the same
comet.

So far as regards the comets of 1668 and 1843, considered alone,
there is nothing absolutely forbidding the idea of their identity: pertur-
bations might account for the differences between their two orbits. But
the comets of 1880 and 1882 cannot possibly be one and the same; they
were both observed for a considerable time and accurately, and the ob-
servations of both are absolutely inconsistent with a period of two years
or anything like it. In fact, for the comet of 1882 all of the different
computers found periods ranging between 600 and 900 years.

There are about half a dozen other such comet-groups now known.

COMETS.

410

706. Perihelion Distances.—These vary greatly.

Eight
comets have a perihelion distance less than six millions of miles;
about seventy-four per cent of all that have been observed lie within
the earth’s orbit; about twenty-four per cent lie outside, but within
twice the earth’s distance from the sun; and six comets have been
observed with a perihelion distance exceeding that limit.

A single one, the comet of 1729, had a perihelion distance exceeding
four astronomical units,—as great as the mean distance of the remoter
asteroids. It must have been an enormous comet to be visible from such
a distance. It is one of the six hyperbolic comets.

Obviously, however, the distribution of comets as determined by ob-
servation, depends not merely on the existence of the comets themselves,
but upon their visibility from the earth. Those comets which approach
near the orbit of the earth have the best chance of being seen, because
their conspicuousness increases as they approach us, so that we must not
lay too much stress on the apparent crowding of the perihelia within the
earth’s orbit.

The perihelia are not distributed equally in all directions from
the sun, but more than sixty per cent are within 45◦ of what is
called “the sun’s way”; i.e., the line in space along which the sun is
travelling, carrying with it its attendant systems.

707. Orbit Planes.—The inclinations of the comets’ orbits
range all the way from 0◦ to 90◦. The ascending nodes are dis-
tributed all around the ecliptic, with a decided tendency, however,
to cluster in two regions having a longitude of about 80◦ and 270◦.

708. Direction of Motion.—With the two exceptions of Hal-
ley’s comet, and the comet of the Leonid meteors (Art. 786), the
elliptical comets which have periods less than one hundred years all
move in the direction of the planets; and the same is true of the
six hyperbolic comets. Of the other comets, a few more move ret-
rograde than direct, but there is no decided preponderance either
way.

709. It is hardly necessary to point out that the fact that the
comets move for the most part in parabolas, and that the planes of
their orbits have no evident relation to the plane of the planetary
motions, tends to indicate (though it falls short of demonstrating)

that they do not in any proper sense belong to the solar system itself,
but are merely visitors from interstellar space. They come towards

COMETS.

411
the sun with almost precisely the velocity they would have if they
had simply dropped towards it from an infinite distance, and they
leave it with a velocity which, if no force but the sun’s attraction
operates upon them, will carry them back to an unlimited distance,
or until they encounter the attraction of some other sun. With one
remarkable exception, their motions appear to be just what might
be expected of ponderable masses moving in empty space under the
law of gravitation.

710. Acceleration of Encke’s Comet.—The one exception
referred to is in the case of Encke’s comet which, since its first
discovery in the last century (it was not, however, discovered to be a
periodic comet until 1819), has been continually quickening its speed
and shortening its period at the rate of about two hours and a half
in each revolution; as if it were under the action of some resistance
to its motion. No perturbation of any known body will account
for such an acceleration, and thus far no reasonable explanation
has been suggested as even possible, except the one mentioned—the
resistance of an interplanetary medium which retards its motion just
as air retards a rifle bullet. At first sight it seems almost paradoxical
that a resistance should accelerate a comet’s speed; but referring to

Article 429 we see that since the semi-major axis of a comet’s orbit
is given by the equation
r

U 2
a =

,

2

U 2 − V 2
any diminution of V will also diminish a; and it can be shown that
this reduction in the size of the orbit will be followed by an increase
of velocity above that which the body had in the larger orbit. It
is accelerated by being thus allowed to drop nearer to the sun, and
gains its speed in moving inwards under the sun’s attraction.

711. Another action of such a retarding force is to diminish the
eccentricity of the body’s orbit, making it more nearly circular. If the
action were to go on without intermission, the result would be a spiral
path winding inward towards the sun, upon which the comet would ulti-
mately fall. For many years the behavior of Encke’s comet was quoted as
an absolute demonstration of the existence of the “luminiferous ether.”

Since, however, no other comets show any such action (unless perhaps

COMETS.

412

Winnecke’s1 comet—No. 5 in the table in the appendix), and moreover,
since according to the investigations of Von Asten and Backlund the
acceleration of Encke’s comet itself seems suddenly to have diminished
by nearly one-half in 1868, there remains much doubt as to the theory
of a resisting medium. It looks rather more probable that this acceler-
ation is due to something else than the luminiferous ether—perhaps to
some regularly recurring encounter of the comet with a cloud of meteoric
matter. The fact that the planets show no such effect is not surprising,
since, as we shall see, they are enormously more dense than any comet,
so that the resistance that would bring a comet to rest within a single
year would not sensibly affect a body like our earth in centuries. The

“resisting medium,” if it exists at all, must have much less retarding
power than the residual gas in one of Crookes’s best vacuum tubes.

712. Physical Characteristics of Comets.—The orbits of
these bodies are now thoroughly understood, and their motions are
calculable with as much accuracy as the nature of the observations
permit; but we find in their physical constitution some of the most
perplexing and baffling problems in the whole range of astronomy,—
apparent paradoxes which as yet have received no satisfactory ex-
planation. While comets are evidently subject to gravitational at-
traction, as shown by their orbits, they also exhibit evidence of
being acted upon by powerful repulsive forces emanating from the
sun. While they shine, in part at least, by reflected light, they are
also certainly self-luminous, their light being developed in a way not
yet satisfactorily explained. They are the bulkiest bodies known,
in some cases thousands of times larger than the sun or stars; but
they are “airy nothings,” and the smallest asteroid probably rivals
the largest of them in actual mass.

713. Constituent Parts of a Comet.—(a) The essential part
of a comet—that which is always present and gives it its name—
is the coma or nebulosity, a hazy cloud of faintly shining matter,
which is usually nearly spherical or oval in shape, though not always
so.

(b) Next we have the nucleus, which, however, is not found in
all comets, but commonly makes its appearance as the comet ap-

1Oppolzer, in 1880, found that according to his computations Winnecke’s
comet was accelerated precisely in the same way as Encke’s, but by less than
half the amount. His result, however, is not confirmed by the recent work of

Härdtl, who finds no acceleration at all.

COMETS.

413
proaches the sun. It is a bright, more or less star-like point near the
centre of the coma, and is the object usually pointed on in deter-
mining the comet’s place by observation. In some cases the nucleus
is double or even multiple; that is, instead of a single nucleus there
may be two or more near the centre of a comet. Perhaps three
comets out of four present a nucleus during some portion of their
visibility.

(c) The tail or train, is a streamer of light which ordinarily
accompanies a bright comet, and is often found even in connection
with a telescopic comet. As the comet approaches the sun, the tail
follows it much as the smoke and steam from the locomotive trail
after it. But that the tail does not really consist of matter simply
left behind in that way, is obvious from the fact that as the comet
recedes from the sun, the train precedes it instead of following. It is
always directed away from the sun, though its precise position and
form is to some extent determined by the comet’s motion. There is
abundant evidence that it is a material substance in an exceedingly
tenuous condition, which in some way is driven off from the comet
and then repelled by some solar action. (See also Art. 736.)

(d ) Envelopes and Jets.—In the case of a very brilliant comet,
its head is often veined by short jets of light which appear to be
continually emitted by the nucleus; and sometimes instead of jets
the nucleus throws off a series of concentric envelopes, like hollow
shells, one within the other. These phenomena, however, are not
usually observed in telescopic comets to any marked extent.

714. Dimensions of Comets.—The volume of a comet is of-
ten enormous—sometimes almost beyond conception, if the tail be
included in the estimate of bulk.

As a general rule the head or coma of a telescopic comet is from

40,000 to 100,000 miles in diameter. A comet less than 10,000 miles
in diameter is very unusual; in fact, such a comet would be almost
sure to escape observation. Many, however, are much larger than

100,000 miles. The head of the comet of 1811 at one time measured
nearly 1,200000 miles,—more than forty per cent larger than the
diameter of the sun itself. The comet of 1680 had a head 600,000
miles across. The head of Donati’s comet of 1858 was 250,000 miles
in diameter. The head of the great comet of 1882 was not so bulky
as many others, having had a diameter of only 150,000 miles; but
its tail was at one time 100,000000 miles in length.

COMETS.

414

Fig. 197.—Naked-eye View of Donati’s Comet, Oct. 4, 1858. (Bond.)

715. Contraction of a Comet’s Head as it approaches
the Sun.—It is a very singular fact that the head of a comet con-
tinually and regularly changes its diameter as it approaches to and
recedes from the sun; and what is more singular yet, it contracts
when it approaches the sun, instead of expanding, as one would
naturally expect it to do under the action of the solar heat. No
satisfactory explanation is known. Perhaps the one suggested by

Sir John Herschel is as plausible as any,—that the change is optical
rather than real; that near the sun a part of the cometary matter
becomes invisible, having been evaporated, perhaps, by the solar
heat, just as a cloud of fog might be.

The change is especially conspicuous in Encke’s comet. When this
body first comes into sight, at a distance of about 130,000000 miles from
the sun, it has a diameter of nearly 300,000 miles. When it is near

COMETS.

415
the perihelion, at a distance from the sun of only 33,000000 miles, its
diameter shrinks to 12,000 or 14,000 miles, the volume then being less
than

1
of what it was when first seen. As it recedes it expands,

10 000
and resumes its original dimensions. Other comets show a similar, but
usually less striking, change.

716. Dimensions of the Nucleus.—This has a diameter
ranging in different comets from 6000 or 8000 miles in diameter

(Comet III, 1845) to a mere point not exceeding 100 miles. Like
the head, it also undergoes considerable and rapid changes in diam-
eter, though its changes do not appear to depend in any regular way
upon the comet’s distance from the sun, but rather upon its activity
at the time. They are usually associated with the development of
jets and envelopes.

717. Dimensions of a Comet’s Tail.—The tail of a large
comet, as regards simple magnitude, is by far its most imposing
feature.

The length is seldom less than 10,000000 to 15,000000
miles; it frequently reaches from 30,000000 to 50,000000, and in
several cases has been known to exceed 100,000000. It is usually
more or less fan-shaped, so that at the outer extremity it is millions
of miles across, being shaped roughly like a cone projecting behind
the comet from the sun, and more or less bent like a horn. The
volume of such a train as that of the comet of 1882, 100,000000
miles in length, and some 200,000 miles in diameter at the comet’s
head, with a diameter of 10,000000 at its extremity, exceeds the
bulk of the sun itself by more than 8000 times.

718. The Mass of Comets.—While the volume of comets is
enormous, their masses appear to be insignificant. Our knowledge
in this respect is, however, thus far entirely negative; that is, while
in many cases we are able to say positively that the mass of a
particular comet cannot have exceeded a limit which can be named,
we have never been able to fix a lower limit which we know it must
have reached; it has in no case been possible to perceive any action
whatever produced by a comet on the earth or any other body of
the planetary system, from which we can deduce its mass; and this,
although they have frequently come so near the earth and other
planets that their own orbits have been entirely transformed, and
if their masses had been as much as

1
of the earth’s, they

100 000
would have produced very appreciable effects upon the motion of

COMETS.

416
the planet which disturbed them.

Lexell’s comet of 1770, Biela’s comet on more than one occasion,
and several others, have come so near the earth that the length of their
periods of revolution have been changed by the earth’s attraction to the
extent of several weeks, but in no instance has the length of the year
been altered by a single second. One might be tempted to think that
comets were possessed of matter without attracting power; but attraction
is always mutual, and since the comets move according to the law of
gravitation, and themselves suffer perturbation from attraction, there is
no escape from the conclusion that, enormous as they are in volume,
they contain very little matter. Some have gone so far as to say that a
comet properly packed could be carried about in a hat-box or a man’s
pocket, which, of course, is an extravagant assertion. The probability
is that the total amount of matter in a comet of any size, though very
small as compared with its bulk, is yet to be estimated at many millions
of tons. The earth’s mass (Art. 132, 4) is expressed in tons by 6 with
twenty-one ciphers following (6000 millions, of millions, of millions of
tons). A body, therefore, weighing only one-millionth as much as the
earth would contain 6000 millions of millions of tons. The atmosphere of
the earth alone constitutes about

1
the earth’s mass, and contains

250 000
more than twenty-four millions of millions of tons.

719. The late Professor Peirce based his estimate of a comet’s
mass upon the extent of the nebulous envelope which it carries with
it, assuming (what may be doubted, however) that this envelope
is gaseous, and is held in equilibrium by the attraction of solid
matter in and near the nucleus; and on this assumption he came
to the conclusion that the matter in and near the nucleus of an
average comet must be equivalent in mass to an iron ball as much
as 100 miles in diameter. This would be about

1
of the earth’s

300 000
mass. While this estimate is not intrinsically improbable, it cannot,
however, be relied upon. We simply do not know anything about a
comet’s mass, except that it is exceedingly small as compared with
that of the earth.

720. Density.—This must necessarily be almost inconceivably
small. If a comet 40,000 miles in diameter has a mass equal to

1

250 000
of the earth’s mass, its mean density is a little less than

1
of that

7000
of the air at the earth’s surface,—much lower than that of the best
air-pump vacuum. Near the centre of the comet the density would
probably be greater than the mean; but near its exterior very much
less. As for the density of its tail, when such a comet has one, that,

COMETS.

417
of course, must be far lower yet, and much below the density of
the residual gas left in the best vacuum we can make by any means
known to science.

This estimate of the density of a comet is borne out by the fact
that small stars can be seen through the head of a comet 100,000
miles in diameter, and even very near its nucleus, with hardly any
perceptible diminution of their lustre. In such cases the writer has
noticed that the image of a star is rendered a little indistinct; and
recent observations of several astronomers have shown a very small
apparent displacement of the star, such as might be ascribed to a
slight refraction produced by the gaseous matter of the comet.

Students often find difficulty in conceiving how bodies of so infinites-
imal density as comets can move in orbits like solid masses, and with
such enormous velocities. They forget that in a vacuum a feather falls
as freely and as swiftly as a stone. Interplanetary space is a vacuum far
more perfect than any air-pump could produce, and in it the rarest and
most tenuous bodies move as freely as the densest.

721. The reader, however, must bear in mind that, although
the mean density of a comet (that is, the quantity of matter in
a cubic mile) is small, the density of the constituent particles of a
comet need not necessarily be so. The comet may be composed of
small, heavy bodies, widely separated, and there is some reason for
thinking that this is the case; that, in fact, the head of a comet is
a swarm of meteoric stones; though whether these stones are many
feet in diameter, or only a few inches, or only a few thousandths of
an inch, like particles of dust, no one can say. In fact, it now seems
quite likely that the greatest portion of a comet’s mass is made
up of such particles of solid matter, carrying with them a certain
quantity of enveloping gas.

722. Light of Comets.—There has been much discussion
whether these bodies shine by light reflected or intrinsic. The fact
that they become less brilliant as they recede from the sun, and
finally disappear while they are in full sight simply on account of
faintness,1 and not by becoming too small to be seen, shows that

1If a comet shone with its own independent light, like a star or a nebula,
then, so long as it continued to show a disc of sensible diameter, the intrinsic
brightness of this disc would remain unchanged: it would only grow smaller as
it receded from the earth, not fainter.

COMETS.

418
their light is in some way derived from the sun. The further fact
that the light shows traces of polarization also indicates the pres-
ence of reflected sunlight. But while the light of a comet is thus in
some way attributable to the sun’s action, the spectroscope shows
that it does not consist, to any considerable extent, of mere reflected
sunlight, like that of the moon or a planet.

723. If a comet shone by mere reflected light, or by any light the
intensity of which is proportional inversely to the square of the sun’s dis-
tance (as would naturally be the case if the light were excited directly by
the sun’s radiation, and proportional to it), we should have its apparent

1
brightness at any time equal to the quantity

, in which D and ∆

D2∆2
are the comet’s distances from the sun and from the earth respectively.

The brightness of a comet does, in fact, generally follow this law roughly,
but with many and striking exceptions. The light of a comet often varies
greatly and almost capriciously, shining out for a few hours with a splen-
dor seven or eight fold multiplied, and then falling hack to the normal
state or even below it. The Pons–Brooks comet in 1883 furnished several
remarkable instances of this sort (Clerke, p. 418).

724. The Spectra of Comets.—The spectrum of most
comets consists of a more or less faint continuous spectrum (which
may be due to reflected sunlight, though it is usually too faint to
show the Fraunhofer lines) overlaid by three bright bands,—one in
the yellow, one in the green, and the third in the blue. These bands
are sharply defined on the lower, or less refrangible, edge, and fade
out towards the blue end of the spectrum. A fourth band is some-
times visible in the violet. The green band, which is much the
brightest of the three, in some cases is crossed by a number of fine,
bright lines, and there are traces of similar lines in the yellow and
blue bands. This spectrum is absolutely identical with that given
by the blue base of an ordinary gas or candle flame; or better, by
the blue flame of a Bunsen burner consuming ordinary illuminating
gas. Almost beyond question it indicates the presence in the comet
of some gaseous hydrocarbon, which in some way is made to shine;
either by a general heating of the whole body to the point of lumi-
nosity (which is hardly probable), or by electric discharges within
it, or by local heatings due to collisions between the solid masses
disseminated through the gaseous envelope; or possibly phosphores-
cence due to the action of sunlight: or none of these surmises may
be correct, and we may have to seek some other explanation not yet
suggested.

COMETS.

419

It is not at all certain that the temperature of the comet, consid-
ered as a whole, is very much elevated. Nor will it do to suppose that
because the spectrum reveals the presence in the comet of gaseous
hydrocarbon, this substance, therefore, composes the greater part
of the comet’s mass. The probability is that the gaseous portion of
the comet is only a small percentage of the whole quantity of matter
contained in it.

725. Metallic Lines in Spectrum.—When a comet ap-
proaches very near to the sun, as did Wells’s comet in 1882, and a
few weeks later the great comet of that year, the spectrum shows
bright metallic lines in addition to the hydrocarbon bands. The
lines of sodium and magnesium are most easily and certainly rec-
ognizable. As for the other lines—a multitude of which were seen
by Ricco (of Palermo) for a few hours, in the spectrum of the great
comet of 1882—they are probably due to iron; though that is not
certain, for they were not seen long enough to be studied thoroughly.

Fig. 198.—Comet Spectra.

(For convenience in engraving, the dark lines of the solar spectrum in the
lowest strip of the figure are represented as bright.)

726. Anomalous Spectra.—While most comets show the hy-
drocarbon spectrum, occasionally a different spectrum of bands ap-
pears. Fig. 198 shows the spectra of three comets compared with
the solar spectrum and with that of hydrocarbon gas.

The first, the spectrum of Tebbutt’s comet of 1881, is the usual
one. The other two are unique. Brorsen’s comet, at its later returns,

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420
showed the ordinary comet spectrum, and it might perhaps be considered
possible that an error was made in fixing the position of the bands at the
first observation. But the peculiar spectrum of comet C, 1877, hardly
permits such an explanation. It was observed at Dunecht on the same
night, by the same observers and with the same spectroscope, as another
comet which gave the usual spectrum; so that in this case it hardly seems
possible that the anomalous result can be a mistake, though the spectrum
itself as yet remains unidentified and unexplained.

Indeed, from some points of view, it is strange that comets, coming
as they do from such widely separated regions of space, do not show an
almost infinite variety of spectra; a priori we should expect differences
rather than uniformity.

It is maintained by Mr. Lockyer that the spectrum of a comet changes
as it varies its distance from the sun, the bands altering in appearance
and shifting their position. But the evidence of this is not yet conclusive.

727. Development of Jets and Envelopes.—When a comet
is first seen at a great distance from the sun it is ordinarily a mere
roundish, hazy patch of faint nebulosity, a little brighter near the
centre.

Fig. 199.—Head of Donati’s Comet, Oct. 6, 1858. (Bond.)

As the comet draws near the sun it brightens, and the central
condensation becomes more conspicuous and sharply defined, or
star-like. Then, on the side next the sun, the newly formed nucleus
begins to emit jets and streamers of light, or to throw off more or
less symmetrical envelopes, which follow each other concentrically
at intervals of some hours, expanding and growing fainter as they

COMETS.

421
ascend, until they are lost in the general nebulosity which forms the
head. During these processes the nucleus continually changes in
brilliancy and magnitude, usually growing smaller and brighter just
before the liberation of each envelope. When jets are thrown off, the
nucleus seems to oscillate, moving slightly from side to side; but no
evidences of a continuous rotation have ever been discovered. The
two figures, 199 and 200, represent the heads of two comets which behaved quite differently. Fig. 199 is the head of Donati’s comet as
seen on Oct. 5, 1858. This comet was characterized by the quiet,
orderly vigor of its action. It did very little that was anomalous
or erratic, but behaved in all respects with perfect propriety. The
system of envelopes in the head of this comet was probably the most
symmetrical and beautiful ever seen. Fig. 200 is from a drawing by

Common of the head of Tebbutt’s comet in 1881. This comet, on
the other hand, was always doing something outré, throwing off
jets, breaking into fragments, and, in fact, continually exhibiting
unexpected phenomena.

728. Formation of the Tail.—

The material which is projected from
the nucleus of the comet, as if re-
pelled by it, is also repelled by the sun,
and driven backward, still luminous,
to form the train. (At least, this is
the appearance.) Fig. 201 shows the
manner in which the tail is thus sup-
posed to be formed.1

The researches of Bessel, Norton,
and especially the late investigations
of the Russian Bredichin, have shown

Fig. 200.—Tebbutt’s Comet,

1881. (Common.)

that this theory—that the tail is com-
posed of matter repelled by both the
comet and the sun—not only accounts for the phenomena in a gen-
eral way, but for almost all the details, and agrees mathematically
with the observed position and magnitude of the tail on different

1Other theories of comets’ tails have been presented, and have had a certain
currency,—theories according to which the tail is a mere “luminous shadow”

of the comet, so to speak, or a swarm of meteors. But all these theories break
down in the details. They fail to account for the phenomena of jets, envelopes,
etc., in the head of the comet, and they furnish no mathematical determination
of the outlines and curvature of the tail.

COMETS.

422
dates.

The repelled particles are still subject to
the sun’s gravitational attraction, and the
effective force acting upon them is there-
fore the difference between the gravitational
attraction and the electrical (?) repulsion.

This difference may or may not be in fa-
vor of the attraction, but in any case, the
sun’s attracting force is, at least, lessened.

The consequence is that those repelled par-
ticles, as soon as they get a little away from
the comet, begin to move around the sun in
hyperbolic1 orbits which lie in the plane of

Fig. 201.—Formation of a
the comet’s orbit, or nearly so, and are per-

Comet’s Tail by Matter
expelled from the Head.

fectly amenable to calculation.

The tail is simply an assemblage of these
repelled particles, and, according to theory, ought, therefore, to be a sort
of flat, hollow, horn-shaped cone, as represented by Fig. 202, open at the
large end, and rounded and closed at the smaller one, which contains the
nucleus.

729. Curvature of the Tail.—The
cone is curved as shown, because the par-
ticles repelled still retain their original or-
bital motion, so that they will not be ar-
ranged along a straight line drawn from the
sun through the comet, but along a curve
convex to the direction of the comet’s mo-

Fig. 202.

tion; but the stronger the repulsion, the less

A Comet’s Tail as a
will be the curvature. Fig. 203 shows how

Hollow Cone.

the tail ought to lie as the comet rounds the
perihelion of its orbit. According to this theory, the tail should be

1Referring to the formula for the semi-major axis of an orbit, viz.,
r

U 2
a =

,

2

U 2 − V 2
we see that a repulsive force acting from the sun diminishes U (which measures
the sun’s attraction), and the consequence is that if the unrepelled particles are
describing a parabola (in which case U 2 = V 2) then for the repelled particles
the denominator will become negative (U having been made smaller than V by
the repulsive action), and thus a will also become negative, so that the orbit
for a repelled particle will be a hyperbola.

COMETS.

423
hollow, and in the case of comets when at their brightest it usually
seems to be so, the centre being darker than the edges.

Fig. 203.—A Comet’s Tail at Different Points in its Orbit near Perihelion.

730. The

Central

Stripe
in
a

Comet’s Tail.—Very often, there is a pe-
culiar straight, dark stripe through the axis
of the tail as shown in Figs. 199 and 200 of
the head of Donati’s and Tebbutt’s comets.

It might be mistaken for the shadow of the

Fig. 204.

nucleus if it were pointed exactly away from

Bright Centred Tail of

Coggia’s Comet. June,
the sun; but it is not, usually making an an-

1874.

gle of several degrees with the direction of a
true shadow. Sometimes, however, and not very unfrequently, the
tail has a bright centre instead of a dark one, perhaps on account
of the feebleness of the comet’s own repulsive action; in fact, this
seems to be usually the case when the comet has reached a consid-
erable distance from the sun in receding from it, and often it is so
when the comet is approaching the sun, but is still remote, as in
the case of Coggia’s comet shown in Fig. 204. In such cases the tail
is generally faint and ill-defined at the edge, with a central spine of
light, and in some cases it becomes apparently a mere slender ray,
of less diameter than the head of the comet itself. This, however,
is unusual. The explanation of this kind of tail requires a slight
modification of the theory, so far as to admit that the particles at
first repelled by the front of the comet are afterwards attracted by
it, though still repelled by the sun.

731. Tails of Three Different Types.—Bredichin has found
that the tails of comets may be grouped under three types:—

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424

1. The long, straight rays. They are formed of matter upon
which the sun’s repulsive action is from twelve to fifteen times as
great as the gravitational attraction, so that the particles leave the
comet with a relative velocity of at least four or five miles a second;
and this velocity is continually increased as they recede, until at
last it becomes enormous, the particles travelling several millions of
miles in a day. The straight rays which are seen in the figure of the
tail of Donati’s comet, tangential to the tail, are streamers of this
first type; as also was the enormous tail of the comet of 1861.

2. The second type is the
curved, plume-like train, like
the principal tail of Donati’s
comet. In this type the repul-
sive force varies from 2.2 times
gravity (for the particles on the
fixed edge of the tail) to half
that amount for those which
form the inner edge. This is by
far the most common type of
cometary train.

3.

A few comets show
tails
of
the
third
type,—
short, stubby brushes violently
curved, and due to matter of
which the repulsive force is only
a fraction of gravity,—from 2 1

10
to 2 1 .

2

732. According
to

Bredichin,
the
tails
of
the
first type are probably composed
of hydrogen, those of the second

Fig. 205.—Bredichin’s Three Types of
type of some hydrocarbon gas,

Cometary Tails.

and those of the third of iron
vapor, with probably an admixture of sodium and other materials.

There has been no opportunity since Bredichin published this result
to test the matter spectroscopically for tails of the first and third types,
by looking for the lines of hydrogen and iron. The hydrogen tails are
almost always very faint, and the tails of the third class are uncommon.

Tails of the second type, which are brightest and most usual, do show a
hydrocarbon spectrum throughout their entire length, and so far confirm

COMETS.

425
his view.

The reason for this conclusion of Bredichin is that he supposes the
repulsive force to be a surface action, the same for equal surfaces of any
kind of matter; the effective accelerating force, therefore, measured by
the velocity it would produce, would depend upon the ratio of surface to
mass in the particles acted upon, and so, in his view, should be inversely
proportional to their molecular weights. Now the molecular weights of
hydrogen, of hydrocarbon gases, and of the vapor of iron, bear to each
other just about the required proportion.

733. Nature of the Repulsive Force.—As to this we have
so far no absolute knowledge. While the old “corpuscular theory of
light” held its ground, many thought that the apparent repulsion
was due to the actual impact of the light corpuscles. Since the aban-
donment of this theory, others have tried to find it in the “impulse”

of the light and heat waves of the ether, without, however, explain-
ing how such waves could produce any such impulsive action. No
experiments show any such carrying power of light or any pressure
produced by its impact, although when Crookes first invented his
radiometer he seems to have thought he had found it. On the whole,
opinion at present strongly inclines to the view long ago suggested
by numerous speculators, but specially worked out and enforced by

Zöllner; namely, that the force is electrical ; and some authorities of
such eminence as Dr. Huggins and the late Professor Peirce have
asserted it positively. The difficulty is that we have no evidence
that the sun is electrically charged, nor do we know how it could
acquire a charge. At the same time, the unquestionable magnetic
effects produced upon the earth by solar disturbances rather favor
the belief that there must be also a purely electric reaction between
the sun and its attendant bodies.

A singular theory has been proposed by Zenker, that the repulsion
is due to the reaction produced by rapid evaporation on the surface of
the little solid and liquid particles of which he supposed a comet to
consist: this evaporation would, of course, be most rapid on the side
of the particles next the sun, and would cause a recoil in a manner
analogous to that by which the so-called spheroidal state of liquids is
produced on a heated surface. Ranyard has suggested that the cometary
particles may consist principally of minute liquid drops or frozen “hail-
stones” of certain hydrocarbons which evaporate rapidly at a very low
temperature (such as rhigoline and its congeners).

COMETS.

426

734. State of the Matter composing the Tail.—This also
is a subject of speculation rather than of knowledge. Perhaps the
simplest supposition is that we have to do with gaseous matter
rarefied even beyond the limits of the gas contained in Crookes’s
tubes,—so rarefied that since its molecules no longer suffer frequent
collisions with each other, it has thus lost all the peculiar mechan-
ical characteristics of a gaseous mass, and become a mere cloud of
separate particles each particle consisting, however, of but a single
molecule. Spectroscopically such a cloud would still be gaseous, but
from a mechanical point of view extremes would have met, and this
most tenuous gas would have become a cloud of finely powdered
solid.

735. What becomes of the Matter thrown off in

Comets’ Tails.—To this we have no certain answer at present;
but if the theory which has been stated is true, it is clear that most
of the matter so repelled from comets can never be re-gathered by
the nucleus, but must be dissipated in space.

Whenever a planet meets any of the particles, it picks them up, of
course, as it picks up meteors; and Newton long ago suggested, what has
of late been forcibly dwelt upon by Dr. Sterry Hunt, that in this way the
atmospheres of the planets may be supplied with material to take the
place of the carbon which has been absorbed and fixed by the processes
of crystallization and of life. Otherwise it would seem that the processes
now going on upon the earth’s surface must necessarily in the course of
time deprive the atmosphere of all its carbonic acid.

If this view is correct, it follows that such comets as have tails lose a
portion of their substance every time that they visit the sun. It is quite
conceivable, also, that the processes by which light is excited in the head
of a comet may use up and render unfit for future shining, a portion of
its material; so that, as a periodic comet grows old, it may become both
smaller and less luminous, until finally it ceases to be observable.

736. Anomalous Tails and Streamers.—It is not very un-
usual for comets to show tails of two different types at the same
time, as, for instance, Donati’s comet. But occasionally stranger
things happen, and the great comet of 1744 is reported to have had
six tails diverging like a fan. Winnecke’s comet of 1877 threw out
a tail laterally, making an angle of about 60◦ with the normal tail,
and having the same length,—about 1◦. Pechüle’s comet of 1880

(a small one), besides the normal tail, had another of about the

COMETS.

427
same dimensions directed straight towards the sun: streamers of
considerable length so directed are not very infrequent. The great
comet of 1882 presented a number of peculiarities, which will be
mentioned in the more particular description of that body, which is
to follow. Most of these anomalies are as yet entirely unexplained.

737. Nature of Comets.—It is obvious from what has been
said that we have little certain knowledge on this subject; but per-
haps on the whole the most probable hypothesis is the one which
has been hinted at repeatedly,—that a comet is, as Professor New-
ton expresses it, nothing but a “sand-bank ”; i.e., a swarm of solid
particles of unknown size and widely separated (say pin-heads sev-
eral hundred feet apart), each particle carrying with it an envelope
of gas, largely hydrocarbon, in which gas light is produced, either
by electric discharges between the particles, or by some other light-
evolving action1 due to the sun’s influence.

This hypothesis derives its chief plausibility from the modern
discovery of the close relationship between meteors and comets, to
be discussed in the next chapter.

738. Origin of Periodic Comets.—It is obvious that the
comets which move in parabolic orbits are, as has been said already,
mere visitors to the solar system, and not citizens of it: but as to
those which now move in elliptical orbits around the sun, returning
as regularly as planets, it is a question whether we are to regard
them as native-born, or only as naturalized. Did they originate in
the system, or are they captives?

739. Planets’ Families of Comets.—It is quite clear that in
some way or other many of them owe their present status in the
system to Jupiter, Saturn, and the other planets. In Article 703 we

1Some have ascribed the light to the collisions between the little stones
of which they assume the comet to be made up, forgetting that, although the
absolute velocity of the comet is extremely great, the relative velocities of its
constituent masses with reference to each other must be very slight—far too
small apparently to account for any considerable rise of temperature or evolution
of light in that way.

It is perhaps worth considering whether gases in the
mass may not become sensibly luminous at a much lower temperature than has
usually been supposed. It would seem not improbable a priori that at every
temperature, radiations of every wave-length must be omitted in some degree;
i.e., that at any temperature above the absolute zero no body is absolutely non-
luminous.

COMETS.

428
called attention to the fact that, without exception, all the short-
period comets (i.e., those having periods ranging from three to eight
years), pass very near to Jupiter’s orbit at some point in their paths;
and they are now recognized and spoken of as Jupiter’s family of
comets,—sixteen of them in all, at present known.

Nine of the sixteen are in the table of comets whose returns have been
actually observed more than once. One of the others was Lexell’s comet,
which was removed from the range of observation by being thrown into
a new and larger orbit by Jupiter in 1779; and two are comparatively
recent discoveries, whose returns are soon expected. The other four have
failed to be observed at second and subsequent returns for unknown
reasons; quite possibly for the same reason, whatever that may be, that
has deprived us of Biela’s comet.

Similarly, Saturn is at present credited with two comets, one of
which is Tuttle’s comet, given in the catalogue of periodic comets.

Uranus stands sponsor for three,—one of them Tempel’s comet,
which is very interesting in its relation to the November meteors,
and is expected back in 1900. Finally, Neptune has a family of
six. Halley’s comet is one of them, and two of the others have been
observed for a second time since 1880; the other three are not due
on their second return for some years to come.

740. Origin of Comets:

the “Capture” Theory.—The
generally accepted theory as to the origin of these comet families
is that the comets which compose them have been captured by the
planets to which they now belong.

A comet entering the system from an infinite distance, and mov-
ing in a parabolic orbit, when it comes near a planet will be either
accelerated or retarded. If accelerated, its orbit becomes hyperbolic,
and that is the end of that comet so far as the solar system is con-
cerned; it never returns for a second observation. If, on the other
hand, it is retarded, the orbit becomes elliptical, and the comet will
return at regular intervals, moving in a path which, of course, always
passes through the point where the disturbance took place.

It is true, as Mr. Proctor has pointed out, that the attraction
of Jupiter, huge as is his mass, could not at one effort transform a
parabolic orbit into an orbit so small as that, say, of Biela’s comet.

But it is not necessary that the thing should be done at one effort.

The comet’s orbit lies near to Jupiter’s, and after a lapse of time,

Jupiter and the comet will be sure to come alongside again: the

COMETS.

429
comet may then be sent into a hyperbolic or parabolic orbit,—the
chances for such a result are nearly even;—but it may also have its
velocity a second time diminished, and its orbit made still smaller ;
and this may be done over and over again unlimitedly, until the
aphelion of the comet falls at such a distance within the orbit of

Jupiter that the planet is no longer able to disturb it seriously.

Given time enough, and comets enough, for Jupiter to work upon,
and the final result would necessarily be a comet-family such as
really exists, with the aphelia of their orbits near to the orbit of

Jupiter, and periods roughly half his own. But it must be frankly
admitted that the extent of time, and the quantity of cometary
material demanded, are enormous.

It may be added that the results of the recent investigations of Profes-
sor Newton of New Haven upon the nature and distribution of cometary
orbits are favorable to the hypothesis that comets come into the solar
system from outer space, and do not originate within it.

741. The “Ejection” Theory.—Mr. Proctor has suggested, and
vigorously defended, a very different theory,—that comets are masses of
matter which have been thrown off from the heavenly bodies by eruptions
of some sort ; that the comets of Jupiter’s family, for instance, once
formed a portion of its mass, and were at some time ejected with a ve-
locity sufficient to set them free in space; and that many of the parabolic
comets may have been similarly ejected from our own, or from other
suns. The main difficulty with this theory is that there is no evidence
of the necessary eruptive energy in Jupiter, or in any of the planets.

A body would have to leave the upper surface of Jupiter’s atmosphere
with a velocity exceeding thirty-five miles a second, in order to fulfil the
conditions of the problem, and become independent of the parent planet.

It cannot be said, however, that there is any special mechanical diffi-
culty in supposing that some of the parabolic comets may owe their origin
to eruptions from distant suns. Our own sun unquestionably sometimes
ejects clouds of matter (in the form of the solar prominences) with enor-
mous velocity, perhaps in some cases sufficient to send them off into
space. But so far as we can make out from the spectroscopic evidence,
the material of comets is not identical with that of the prominences.

742. Remarkable Comets.—(1) Halley’s Comet. This was
the first periodic comet whose return was predicted. Halley based
his prediction upon the fact that he found its orbit in 1682 to be
nearly identical with those of the comets of 1607 and 1531, which
had been carefully observed by Kepler and Apian; and he also found

COMETS.

430
records of the appearance of great comets in 1456, in 1301, in 1145
and 1066, which would correspond as regards the time-intervals
concerned, though data were wanting for an accurate calculation
of their orbits. He noticed, of course, that the two intervals be-
tween 1531 and 1607, and between 1607 and 1682 were not quite
equal; but he had sagacity enough to see that the differences were
no greater than might be accounted for by the attractions of Jupiter
and Saturn.

The theory of perturbation was not then sufficiently developed to
make it possible to compute with precision just what the effect would be
upon the next return of the comet, but he saw that the action of Jupiter
would retard it, and he accordingly fixed upon the early part of 1759
as the time at which the comet might be expected. Before that date,
however, mathematics had so advanced that the necessary calculations
could be made. Clairaut, as the result of a most laborious investigation,
fixed upon April 13 for the perihelion passage; but in publishing his
result, he remarked that it might easily be a month out of the way owing
to the uncertainty as to the masses of the planets, and the possible action
of undiscovered planets beyond Saturn (Uranus and Neptune were then
unknown). The comet actually came to perihelion on March 13. At
this return it was best seen in the southern hemisphere, and at one time
had a tail nearly 50◦ long. At its next return, in 1835, it came to the
predicted time within two days. It did not appear on this occasion as an
extremely brilliant comet, but was reasonably conspicuous, with a tail
of the first type (hydrogen) about 15◦ in length.

Its next return will occur in or about 1911, but the necessary calcu-
lations have not yet been made to determine the date with accuracy.

The most remarkable of its earlier appearances were in 1066 and 1456.

The comet of 1066 figures on the Bayeux tapestry as a propitious omen
for William the Conqueror (of England). In 1455 the comet, according
to popular belief, was formally excommunicated by Pope Calixtus III.

in a bull directed mainly against the Turks, who were then threatening
eastern Europe. It is doubtful, however, whether such a formal bull was
ever really promulgated.

743. (2) Encke’s Comet. This is interesting us the first of the
short-period comets, and also as the comet having the shortest
known time of revolution,—only about three years and a half. Encke
first detected its periodicity in 1819, but it had been frequently ob-
served during the preceding fifty years, and has been observed at
almost every return since then. It is usually visible only in the tele-

COMETS.

431
scope, though sometimes, under very favorable circumstances, it can
be seen by the naked eye, with a tail a degree or two long. It is often
irregular in form, and “lumpy,” and seldom shows a well-defined nu-
cleus; nor does it exhibit very much that is interesting in the way of
jets, envelopes, and other cometary freaks. We have already men-
tioned its remarkable contraction in volume on approaching the sun

(Art. 715), and the progressive shortening of its period, which has
been ascribed to a resisting medium (Art. 710).

744. (3) Biella’s Comet. This is also, or rather was, a small
comet with a period of 6.6 years,—the second comet of short period
in order of discovery. Its history is very interesting. It was discov-
ered in 1826 by Biela, an Austrian officer, and its periodic character
was soon detected by Gambart, whose name is connected with it by
many French authorities. Its orbit comes within a very few thou-
sand miles of the earth’s orbit, the nearness varying, of course, from
time to time, on account of perturbations. The approach is often so
close, however, that if the comet and the earth were to arrive at the
nearest point at the same time there would be a collision, and the
earth would pass through the outer portions of the comet’s head.

At the return of the comet in 1832, some one started the report
that such an encounter would occur, and in consequence there was
something hardly short of a panic in southern France, the first of the
since numerous “comet-scares.” At this time the comet passed the
critical point about a month before the earth reached it, so that the
two bodies were never really within 15,000000 miles of each other.

745. At the comet’s next return in 1839 it failed to be observed
on account of its unfavorable position in the sky; but in 1846 it duly
reappeared, and did a very strange thing, so far unprecedented. It
divided into two! When first seen on November 28, it presented the
ordinary appearance of any newly discovered comet. On Dec. 19 it
had become rather pear-shaped, and ten days later it had divided,
the duplication being first seen in New Haven, and soon after at

Washington, some days before any European astronomer had no-
ticed it.

The twin comets travelled along side by side for more than four
months, at an almost unvarying distance of about 160,000 miles, without
showing the least sign of mutual attraction or disturbance; but internally
both comets were intensely active, each developing a nucleus very bright
for a telescopic comet, with a tail some half a degree in length, and

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432
showing curious fluctuations of light, which seemed as a general rule to
alternate; i.e., whenever comet A brightened up, comet B grew fainter,
and vice versa. During a part of the time the two comets were connected
by a faint arc of light.

When next the comet returned in August, 1852, it was under rather
unfavorable circumstances for observation, but the twins were both seen,
now separated by about 1,500000 miles, and travelling quietly in their
appointed orbits. Neither of them has ever been seen again, although
they ought to have returned five times, and more than once under favor-
able conditions for visibility.

746. But the story is not yet ended, though the remainder per-
haps belongs more properly in the next chapter of our book.

On the night of Nov. 27, 1872, just as the earth was passing the
old track of the lost comet, she encountered a wonderful meteoric
shower. As Miss Clerke expresses it, perhaps a little too positively,

“It became evident that Biela’s comet was shedding over us the
pulverized products of its disintegration.”1

The same thing happened again in November, 1886, when the
earth once more passed the comet’s path.

The meteors of this so-called Bielid swarm, in their motion
through the sky, all appear to come from a point in the constel-
lation of Andromeda, and are therefore sometimes called the “An-
dromedes,” and their motion is parallel to the comet’s orbit, at the
point where it intersects our own.

747. (4) Donati’s Comet of 1858. This, on the whole, was per-
haps the finest (though not the largest or the most extraordinary)

of the comets of the present century, having been very favorably
situated for observation in the October sky.

It was discovered at Florence as a telescopic object on June 2. It
did not, however, become visible to the naked eye until near the end of

August, when it began to exhibit the beautiful phenomena which have
made it, so to speak, the normal and typical comet. The comet had
an apparently well-defined nucleus, which varied in diameter at differ-
ent times from 500 miles to 3000. For several weeks the coma exhib-

1“Hist. Ast. in 19th Century,” p. 384. It is probable enough that the meteors
were really the product of the comet’s “disintegration”; still it is by no means
certain. It is, of course, beyond question that they bear some relation to the
lost comet.

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433
ited in unrivalled perfection the development and structure of concentric
envelopes. Its tail was of the second or hydrocarbon type, with faint
tangential streamers which belong to the first or hydrogen type; it had
a maximum apparent length of about 50°, and was some 5° or 6° wide
at the extremity, and its real length was about 45,000000 miles, with a
width of 10,000000. The object was kept under accurate observation for
fully nine months, so that its orbit is unusually well determined as a very
long ellipse, with a periodic time of nearly 2000 years. Figs. 197 and 199
show its principal features.

Our space permits us to cite in detail only one other comet:—

748. (5) The Great Comet of 1882, which will always be remem-
bered, not only for its beauty, but for the great variety of unusual
phenomena it presented.

Discovery and Brightness. The comet seems to have been first
seen as a naked-eye object by some one whose name is not given, at

Auckland, New Zealand, on Sept. 3. By the 7th or 8th it had become
somewhat conspicuous, and was observed both at Cordova (South

America) and at the Cape of Good Hope. It was telegraphed to the
northern hemisphere by Cruls, of Rio Janeiro, on Sept. 11, but was
not seen in the north until the day when it passed its perihelion,

Sept. 17. It was then independently discovered by Common, in

England (who had not heard of Cruls’s telegram), in broad daylight,
within 2◦ of the sun; and the next day it was similarly discovered by
a number of observers, especially by Thollon, at Nice, who observed
its spectrum in full sunlight, and measured the displacement of the
sodium lines produced by its motion. It was so bright that there
was not the slightest difficulty in seeing it by simply shutting off
the sun with the hand held at arm’s length.

749. Transit across the Sun’s Disc.—On the afternoon of
the 17th, its approach to the sun’s disc and actual contact with it was
observed by two persons at the Cape of Good Hope, with the same
telescope and dark glasses ordinarily used for observing sun spots. The
comet seemed to be as bright as the sun’s surface itself, and was followed
right up to the sun’s limb, where it vanished so utterly that the observers
supposed that it had gone behind the sun. On the contrary, however, it
passed directly across the sun’s disc, but absolutely invisible; there was
not the least trace of it upon the sun’s surface, so that we must suppose it
to have been sensibly transparent. It must have traversed the disc in less
than fifteen minutes, though unfriendly clouds prevented the observation
of its exit. For four days after the perihelion passage it remained still

COMETS.

434
visible to the eye by daylight. On Sept. 22d, a French observer in Paris
ascended in a balloon to observe it, and succeeded in seeing it, but not
in getting any valuable results. A few days more carried it so far from
the sun that it became a magnificent object for the hours before sunrise.

750. Member of a Comet Group.—As has been stated be-
fore (Art. 705), its orbit—at least, that portion of it within the
earth’s orbit—coincides almost exactly with the orbits of three other
comets belonging to the same group; viz., the comets of 1668, 1843,
and 1880. The salient peculiarity of these orbits lies in the closeness
of their approach to the sun, the perihelion distance of each of them
being less than 750,000 miles, so that they all passed within 300,000
miles of the sun’s surface, and with a velocity which at perihelion
exceeded 300 miles per second, and carried them through 180◦ of
their orbit in less than three hours. And yet, this passage through
the sun’s coronal regions did not disturb their motion in the least,
as is shown by the fact that the orbit of the comet of 1882, deduced
from the observations made before the perihelion passage, agrees
exactly with that deduced from those made after it. The inference
as to the extreme rarity of the sun’s corona is obvious. Only one
other comet—Newton’s comet of 1680—has ever approached even
nearly as close to the sun as the four comets of this group.

Fig. 206.—Orbit of the Great Comet of 1882.

The comet continued visible until March, and this long period of
observation enabled the computers to determine the orbit with a greater

COMETS.

435
degree of accuracy than is usual. They all agree in making it a very
elongated ellipse, with a period ranging from 652 years (according to

Morrison) to 843 years (according to Kreutz). Fig. 206 represents the
form of the orbit and the way in which its plane is related to the orbit
of the earth.

751. Telescopic Features.—When the comet first became
telescopically observable in the morning sky it presented very nearly
the normal appearance.

The nucleus was sensibly circular, and
there were a number of clearly developed, concentric envelopes in
the head; the dark, shadow-like stripe behind the nucleus was also
well marked. In a few days the nucleus became elongated, and fi-
nally stretched out into a lengthened, luminous streak some 50,000
miles in extent, upon which there were six or eight star-like knots
of condensation. The largest and brightest of these knots was the
third from the forward end of the line, and was some 5000 miles
in diameter. This “string of pearls” continued to lengthen as long
as the comet was visible, until at last the length exceeded 100,000
miles. This fact, of course, made it difficult to fix upon the precise
point to be observed in determining the comet’s position.

Fig. 207.—The Head of the Great Comet of 1882.

When the nucleus first broke up in this way, the dark stripe
behind it was still conspicuous, making, however, a slight angle
with the line of nuclei. A bright streak followed the line of nuclei,
and in a few days this seemed to encroach on and to obliterate the
dark stripe, so that after that time the backbone of the comet’s

COMETS.

436
tail became bright instead of dark, as it had been previously. The
engraving (Fig. 207) represents the telescopic appearance at Prince-
ton on Oct. 9 and 15. The comet continued to be visible with the
telescope until it was more than 470,000000 miles from the earth,
a distance to which no other such object has ever been followed,
except the single comet of 1729,—the one whose perihelion distance
exceeded four times the earth’s distance from the sun.

752. Tail.—The comet was so situated that its tail was not
seen to the best advantage, being directed nearly away from the
earth, and never having an apparent length much exceeding 35◦. In
this respect it has, therefore, been often surpassed by much inferior
comets. The actual length of the tail, however, at one time exceeded

100,000000 miles,—more than the distance of the earth from the
sun. It was of the second or hydrocarbon type.

Fig. 208.—The “Sheath,” and the Attendants of the Comet of 1882.

A unique and so far utterly unexplained phenomenon was a faint,
straight-edged beam of light, or “sheath,” that accompanied the
comet, enveloping the head and projecting three or four degrees in
front of it, as shown in the figure (Fig. 208). Besides this, at dif-
ferent times, three or four irregular shreds of cometary matter were
detected by Schmidt, of Athens, and other observers, accompanying
the comet at a distance of three or four degrees when first seen, but
gradually receding from it, and at the same time growing fainter.

COMETS.

437

Possibly they may have been fragments of the tail which belonged
to the comet before passing the perihelion, or of the matter repelled
from the comet when near perihelion. Since the comet, in passing the
perihelion, changed the direction of its motion by nearly 180◦ in less than
three hours, it was, of course, physically impossible that the tail it had
before the perihelion passage could have made the circuit of the sun in
that time. Its tail after the perihelion passage was in no sense the same
that it had before; the comet ran away from the old one, and developed
a new one. Visible or invisible, the particles of the old train must have
kept on their way under the combined action of the sun’s gravitation
and repulsion, and calculation can show whether any of its fragments
could have occupied the position and had the motion possessed by these
companion-comets.

We close the chapter with a few remarks upon a subject which
has been much discussed.

753. The Earth’s Danger from Comets.—It has been sup-
posed that comets might do us harm in two ways,—either by actu-
ally striking the earth, or by falling into the sun, and thus producing
such an increase of solar heat as to burn us up.

As regards the possibility of a collision with a comet, it is to be
admitted that such an event is possible. In fact, if the earth lasts
long enough, it is practically sure to happen; for there are several
comets’ orbits which pass nearer to the earth’s orbit than the semi-
diameter of the comet’s head, and at some time the earth and comet
will certainly come together. Such encounters will, however, be very
rare. If we accept the estimate of Babinet, they will occur about
once in 15,000000 years in the long run.

As to the consequence of such a collision it is impossible to speak
with confidence, for want of sure knowledge of the state of aggre-
gation of the matter composing a comet. If the theory presented in
this chapter is true, everything depends on the size of the separate
solid particles which form the main portion of the comet’s mass. If
they weigh tons, the bombardment experienced by the earth when
struck by a comet would be a very serious matter: if, as seems more
likely, they are for the most part smaller than pin-heads, the result
would be simply a meteoric shower.

754. Effect of the Fall of a Comet into the Sun.—As
regards the effect of the fall of a comet into the sun, it may be stated
that, except in the case of Encke’s comet, there is no evidence of

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438
any action going on that would cause a now existing periodic comet
to strike the sun’s surface; it is, however, undoubtedly possible that
a comet may enter the system from without, so accurately aimed
that it will hit the sun.

But, if a comet actually strikes the sun, it is not likely that
the least harm will be done. If a comet, having a mass equal to

1
of the earth’s mass, were to strike the sun’s surface with the

100 000
parabolic velocity of nearly 400 miles a second, it would generate
about as much heat as the sun radiates in eight or nine hours.

If this were all instantly effective in producing increased radiation
at the sun’s surface (increasing it, say eightfold, for even a single
hour), mischief would follow, of course. But it is almost certain that
nothing of the sort would happen. The cometary particles would
pierce the photosphere, and liberate their heat mostly below the
solar surface, simply expanding, by some slight amount, the sun’s
diameter, and so adding to its store of potential energy about as
much as it ordinarily expends in a few hours. There might, and
very likely would be, a flash of some kind at the solar surface, as
the shower of cometary particles struck it, but probably nothing
that the astronomer would not take delight in watching.

METEORS.

439

CHAPTER XVIII.

755. Meteors.—Occasionally bodies fall on the earth from the
sky,—masses of stone or iron which sometimes weigh several tons.

During its flight through the sky such a body is called a meteor, and
the pieces which fall from it are called meteorites, or aerolites (air-
stones), or uranoliths (heaven-stones), or simply meteoric stones.

756. Circumstances of their Fall.—The circumstances
which attend the fall of a meteorite are in most cases substantially
as follows. If it occurs at night a ball of fire is seen, which moves
with an apparent speed depending both on its real velocity and
on the observer’s position. If the body is coming “head on,” so to
speak, the motion will be comparatively slow; so also if it is very dis-
tant. The fire-ball is generally followed by a luminous train, which
marks out the path of the body, and often continues visible for a
long time after the meteor itself has disappeared. The motion is sel-
dom exactly straight, but is more or less irregular, and every here
and there along its path the meteor seems to throw off fragments,
and to change its course more or less abruptly. If the observer is
near enough, the flight is accompanied by a heavy, continuous roar,
accentuated by sharp detonations which accompany the visible ex-
plosions by which fragments are burst off from the principal body.

The noise is sometimes tremendous, and heard for distances of forty
or fifty miles.

If the fall occurs by day the luminous appearances are, of course,
principally wanting, and white clouds take the place of the fire-ball
and the train.

757. The Aerolites themselves.—The mass that falls is
sometimes a single piece, but more usually there are many separate
fragments, sometimes numbering many thousands. In such cases
the stones are mostly small, and sometimes they are mere grains of
sand. Nearly all the aerolites that are actually seen to fall, and are
found at the time, are masses of stone; but a very few, perhaps three
or four per cent of the whole number, consist of nearly pure iron,

METEORS.

440
more or less alloyed with nickel. There are also a good many cases of
uranoliths, which are mainly stony, but have a considerable portion
of iron disseminated through the mass in grains and globules; and
nearly all the stony uranoliths contain as much as twenty or thirty
per cent of iron in the form of sulphides or analogous compounds.

758.

The only iron meteors which have been actually seen to
fall so far, and are represented by specimens in our museums, are the
following:—

(1)

Agram, Bohemia, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1751.

(2)

Dickson Co., Tennessee,. . . . . .. . . . . . . . . . . . . . . . . .1835.

(3)

Braunau, Bohemia, . . . . . . . . . . . . . . . . . . . . . . . . . . . .1847.

(4)

Tabarz, Saxony, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1854.

(5)

Nejed, Arabia, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1865.

(6)

Nedagollah, India, . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1870.

(7)

Maysville, California, . . . . . . . . . . . . . . . . . . . . . . . . . .1873.

(8)

Rowton, Shropshire, England,. . . . . . . . . . . . . . . . . .1876.

(9)

Emmett Co., Iowa, . . . . . . . . . . . . . . . . . . . . . . . . . . . .1879.

(10)

Mazapil, Mexico, . . . . . . . . . . . . . . . . . . . . . Nov. 27, 1885.

(11)

Johnson Co., Arkansas, . . . . . . . . . . . . . . . . . . . . . . . .1886.

The Emmett County iron was mostly in small fragments, and along with
them there were many large stones with quantities of iron included. The
separate fragments of pure iron which reached the earth probably came
by the breaking up of the stony masses.

Besides these iron meteors which have been seen to fall, our cabi-
nets contain a very large number of so-called meteoric irons; i.e., masses
of iron found under such circumstances that they cannot easily be ac-
counted for in any way except by supposing them to be of meteoric
origin.

759. The number of meteorites which have fallen since 1800 and
been gathered into our cabinets1 is about 250. The most remarkable falls
in the United States have been the six following: namely, Weston, Con-
necticut, 1807; Bishopsville, So. Carolina, 1843; Cabarrus Co., No. Car-
olina, 1849; New Concord, Ohio, 1860; Amana, Iowa, 1875; and Emmett

Co., Iowa, 1879. In the first case and the three last, several hundred frag-

1In this country the cabinets of Amherst College and Harvard and Yale

Universities are especially rich in meteorites. The finest collection in the world,
however, is that at Vienna. The collection of the British Museum is also note-
worthy, as well as that at Paris.

METEORS.

441
ments fell at the same time, ranging in size from five hundred pounds to
half an ounce.

760. Twenty-four of the sixty-
seven known chemical elements have
been found in meteors, and not a single
new one. The minerals of which me-
teorites are composed present a great
resemblance to terrestrial minerals of
volcanic origin, but many of them are
peculiar, and found in meteors only.

(The study of these meteoric minerals
is a very curious and important branch
of mineralogy, though naturally it has
not many votaries.)

The occasional
presence of carbon is to be specially
noted, and in a meteor which recently
fell in Russia the carbon appeared to
be in a crystalline form, identical with
the black diamond, though in exceed-
ingly minute particles.

Fig. 209 is

Fig. 209.

Fragment of one of the Amana
from a photograph of a fragment of

Meteoric Stones.

one of the meteoric stones which fell
at Amana, Iowa, in 1875. The picture is taken by the permission of the
publishers from Professor Langley’s “New Astronomy,” where the body
is designated as “part of a comet.”

761. The Crust.—The most characteristic external feature of
an aerolite is the thin, black crust that covers it, usually, but not
always, glossy like a varnish. It is formed by the fusion of the surface
in the meteor’s swift motion through the air, and in some cases
penetrates deeply into the mass of the meteor through fissures and
veins. It is largely composed of oxide of iron, and is always strongly
magnetic. The crusted surface usually exhibits pits and hollows,
such as would be produced by thrusting the thumb into a mass of
putty. These cavities are explained by the burning out of certain
more fusible substances during the meteor’s flight.

762. Magnitude.—Of the meteors actually seen to fall the
largest pieces found thus far weigh about 500 pounds, though the
whole mass of the body when it first entered the atmosphere has
sometimes been much larger, perhaps, in a few cases, amounting to

METEORS.

442
two or three tons.1

As seen from a distance of many miles, the meteoric fire-ball
sometimes appears to have a diameter as large as the moon, which
would indicate a real diameter of several hundred feet. The great
apparent size, however, is an illusion, partly due to irradiation, and
partly, undoubtedly, to the fact that the meteor itself is surrounded
by an extensive envelope of heated air and smoke which becomes
luminous throughout. No meteor ever yet investigated would make
a mass as large as ten feet in diameter.

763. Path.—When a meteor has been observed by a number
of persons at different points, who have noted any data which will
give its altitude and bearing at identified moments, the path can
be computed. Observations from a single point are worthless for
the purpose, since they can give no information as to the meteor’s
distance.

The meteor is generally first seen at an altitude of between eighty
and 100 miles, and disappears at an altitude of between five and ten
miles. The length of the path may be anywhere from 50 miles to

500, according to its inclination to the earth’s surface. The veloc-
ity is rather difficult to ascertain, but is found to range from ten
to forty miles per second at the moment when the meteor first be-
comes visible, and diminishes to one or two miles per second, at the
time when it disappears. The average velocity with which meteors
enter the atmosphere appears not to vary much from the “parabolic
velocity” of twenty-six miles per second, due to the sun’s attraction
at the earth’s distance—a fact which, of course, indicates that these
bodies, whatever their origin may be, are now moving in space, like
the comets, under the sun’s attraction.

With possibly a very few exceptions in cases where the meteor
glances, so to speak, on the earth’s atmosphere, like a skipping-stone
on water, a body which has once entered the air is sure to be brought
to the ground: it is hardly possible that one meteor in a million should
escape after becoming involved in the atmosphere. We mention this es-

1Some of the masses of iron supposed to be of meteoric origin, but not
actually seen to fall, are very much larger. The iron mass from Otumpa in

Mexico is said to weigh fully sixteen tons. As regards some of these hypothetical
meteorites, however, their meteoric origin is more than questionable; such, for
instance, is the case with the Ovifak iron found by Nordenskiold on the coast
of Greenland, and exhibited at the Philadelphia Centennial Exhibition.

METEORS.

443
pecially, because some authorities erroneously speak of it as a usual thing
for the meteor to keep on its course, and leave the earth, after throwing
off a few fragments.

764. Observation of Meteors.—The object of the observa-
tion should be to obtain accurate estimates of the altitude and az-
imuth of the body at moments which can be identified. At night
this is best done by noting the position of the meteor with reference
to neighboring stars at the moments of its appearance and disap-
pearance, or of the intervening explosions. In the daytime it can
often be done by noting the position of the object with reference to
trees or buildings. The observer should then mark the exact posi-
tion where he is standing, so that by going there afterwards with
proper instruments he can determine the data desired.

Of course, all such measurements must be given in angular units.

To speak of a meteor as having an altitude of twenty feet, and pursuing
a path 100 feet long, is meaningless, unless the size of the “foot” is
somehow defined.

The determination of the meteor’s velocity is more difficult, as it
is seldom possible to look at a watch-face quickly enough, even in the
daytime. The usual course is for the observer to repeat some familiar
piece of doggerel as rapidly as possible, beginning when the object
first becomes visible and stopping when it explodes or disappears,
noting also the precise syllable where he stops. By repeating the
same sentence over again before a clock it is possible to determine
within a few tenths of a second the time occupied by the meteor’s
flight.

765. Explanation of the Heat and Light of a Meteor.

—These are due simply to the destruction of the body’s velocity;
its kinetic mass-energy of motion is transformed into heat by the
friction of the air. If a moving body whose mass is M kilograms,
and its velocity V metres per second, is stopped, the number of
calories of heat developed is given by the equation

M V 2

Q =

(Art. 354).

8339

The quantity of heat evolved in bringing to rest a body which
has a velocity of forty-two kilometres, or twenty-six miles a second,
is enormous, vastly more than sufficient to fuse it even if it were
made of the most refractory material, and hundreds of times more

METEORS.

444
than would be produced by its combustion in oxygen if it were a
mass of coal.

This heat is developed all along the meteor’s course, and mostly
just upon its surface. As Sir William Thomson has shown, the
thermal effect of the rush through the air is the same as if the
meteor were immersed in a blow-pipe flame having a temperature of
many thousand degrees; and it is to be noted that this temperature
is independent of the density of the air through which the meteor
may be passing. The quantity of heat developed in a given time
is greater, of course, where the air is dense; but the temperature
produced in the air itself, at the surface where it rubs against the
moving body, is the same whether the gas be dense or rare.

When a moving body has a velocity of about 1500 metres per second,
the virtual temperature of the surrounding air is about that of red heat;
i.e., the body becomes heated as fast as it would if it were at rest and the
air about it were heated to that temperature. When the velocity reaches
twenty or thirty miles per second, it is acted upon as if the surrounding
gas were heated to the liveliest incandescence at a temperature of several
thousand degrees.

The surface is fused, and the liquefied portion is
continually swept off by the rush of the air, condensing as it cools into
the luminous powder that forms the train. The fused surface itself is
continually renewed until the velocity falls below two miles a second or
thereabouts, when it solidifies and forms the characteristic crust. As a
general rule, therefore, the fragments are hot if found soon after their fall;
but if the stone is a large one and falls nearly vertically, so as to have but
a short path through the air, the heating effect will be mainly confined
to its surface; and owing to the low conducting power of stone, the
centre may still remain intensely cold, retaining nearly the temperature
which it had in interplanetary space. It is recorded that one of the large
fragments of the Dhurmsala (India) meteorite, which fell in 1860, was
found in moist earth half an hour or so after the fall, coated with ice.

766. Train.—One unexplained feature of meteoric trains de-
serves notice. They often remain luminous for a long time, some-
times as much as half an hour, and are carried by the wind like
clouds. It is impossible to suppose that such a cloud of dust re-
mains incandescent from heat for so long a time in the cold upper
regions of the atmosphere; and the question of its enduring lumi-
nosity or phosphorescence is an interesting and puzzling one.

METEORS.

445

767. Origin.—We may at once dismiss the theories which
make meteors to be the immediate product of volcanic eruption
on the earth or on the moon. They come to us for the most part, as
has been said, from the depths of space, with the velocity of planets
and comets, and there is no certain reason for assuming that they
originated in any manner different from the larger heavenly bodies.

At the same time, many of them so closely resemble each other as
almost to compel the idea of some common source; and though lunar
volcanoes are now extinct, and no terrestrial volcano, not even Krakat˜
ao,
is now competent to send off its ejected missiles through the earth’s
atmosphere into space, it is not certain that this was always so. Some
still maintain that these bodies may be fragments which were shot off
millions of years ago when the moon’s volcanoes were in full vigor and
the earth was young. Since then, according to this view, these masses
have been travelling around the sun in long ellipses which intersect the
orbit of the earth, until at last they happen to come along at the right
time and encounter her surface, and so return to the old home.

As to the iron meteors, some maintain that they are masses which
have been ejected from our sun, or from some other star; and they fortify
their opinion by the remarkable but unquestionable fact that these mete-
oric irons are full of “occluded ” hydrogen and carbon oxides which can be
extracted from them by heating them in a proper receptacle connected
with a mercurial pump. They argue that these gases could have been
absorbed by the iron only when it was in the liquid state and overlaid
by a dense, hot atmosphere containing them; and that, they say, is just
the condition of the minute drops of iron which are supposed to form
part of the photosphere of the sun: upon our sun or in some other sun
only could such conditions be found. There is no question of the sun’s
ability to project masses to planetary distances, as shown by the chro-
mospheric eruptions which have been repeatedly observed by students of
solar physics. And if our sun can do this, it is natural to suppose that
other suns can do it also.

However these bodies originated, it is quite certain that before
they reach the earth they have been moving independently in space
for a long time, just as planets and comets do. But a recent im-
portant research by Professor Newton has shown that more than 90
per cent of some 200 aerolites, for the approximate determination
of whose paths we have the data, were moving before their fall in
orbits, not parabolic, but analogous to those of the short-period
comets; and direct, not retrograde.

METEORS.

446

768. Detonating Meteors, or “Bolides,” of which Frag-
ments are not known to reach the Earth.—Some writers dis-
criminate between these meteors and aerolites, but the distinction
does not seem to be well founded. The phenomena appear to be
precisely the same, except that in the one case the fragments are
actually found, and in the other they fall into the sea, the forest,
or the desert; or sometimes when the path is nearly horizontal, and
therefore long, they may be consumed and dissipated in the dust
and vapor of the train, without reaching the earth’s surface at all,
except ultimately as impalpable dust.

769. Number.—As to the number of aerolites which strike
the earth, it is difficult to make a trustworthy estimate. Since the
beginning of the century, at least two or three have been seen to fall
every year, and have been added to our cabinets (see Art. 759),—in
some years as many as half a dozen. This, of course, implies a vastly
greater number which are not seen, or are not found. Schreibers,
some years ago, estimated the number as high as 700 a year, and

Reichenbach sets it still higher—not less than 3000 or 4000.

SHOOTING STARS.

770. A few minutes’ watching on any clear, moonless night will
be sure to reveal one or more of the swiftly moving, evanescent
points of light that are known as “shooting stars.” No sound is ever
heard from them, nor (with a single exception to be mentioned fur-
ther on) has anything ever been known to reach the earth’s surface
from them, not even when the sky was “as full of them as of snow-
flakes,” as sometimes has happened in a great meteoric shower. For
this reason it is perhaps justifiable to allow the old distinction to
remain between them and the bodies we have been discussing, at
least provisionally. The difference may be, and according to opinion
at present prevalent very probably is, merely one of size, like that
between boulders and grains of sand. Still there are some reasons
for supposing that there is also a difference of constitution,—that
while the aerolite is a solid, compact mass, the shooting star is a
little cloud of dust and intermingled gas, like a puff of smoke.

771. Numbers.—The number of these bodies is very great.

A single watcher sees on the average from four to eight hourly.

If observers enough are employed to guard the whole sky, so that

METEORS.

447
none can escape unnoticed, the visible number becomes from thirty
to sixty an hour. Since ordinarily only these are seen which are
within two or three hundred miles of the observer, the estimated
total daily number of these which enter the earth’s atmosphere,
and are large enough to be visible to the naked eye, rises into the
millions. Professor Newton sets it at from 10,000000 to 15,000000,
the average distance between them being about 200 miles.

There is a still larger number too small to be seen with the naked
eye. One hardly ever works many hours with a telescope carrying a low
power, and having a field of view as large as 15 in diameter, without
seeing several of them flash across the field. In a few instances observers
have reported dark meteors crossing the moon’s disc while they were
watching it. There may be some question, however, as to the real nature
of the objects seen in such a case. Birds (?).

772. Comparative Number in Morning and Evening.—

The hourly number about six o’clock in the morning is fully double
the hourly number in the evening. The obvious reason is simply
that in the morning we are on the front of the earth as regards
its orbital motion, while in the evening we are in the rear; in the
evening we only see such meteors as overtake us; in the morning we
see all that we either meet or overtake. If they are really moving in
all directions alike, with the parabolic velocity corresponding to the
earth’s distance from the sun (twenty-six miles per second), theory
indicates that the relative hourly numbers for morning and evening
ought to be in just the observed proportion.

773. Brightness.—For the most part these bodies are much
like the stars in brightness,—a few are as brilliant as Venus or

Jupiter; more are like stars of the first magnitude; and the ma-
jority are like the smaller stars. The bright ones not unfrequently
show trains which sometimes last from five to ten minutes, when
they are folded up and wafted away by the winds of the upper air.1

774. Elevation,

Path,
and Velocity.—By observations
made by two or more observers thirty or forty miles apart, it is
possible to determine the height, path, and velocity of these bod-
ies. It is found as the result of a great number of such observations

1These air currents, at an elevation of forty miles above the earth’s surface,
are thus observed to have, ordinarily, velocities from fifty to seventy five miles
per hour.

METEORS.

448
that they first appear at an average elevation of about seventy-four
miles, and disappear at an average height of about fifty miles, after
traversing a distance of forty or fifty miles, with an average velocity
of about twenty-five miles per second. They do not begin to be
visible at so great an elevation as the aerolitic meteors, and they
vanish before they penetrate so deeply into the atmosphere.

775. Materials.—Occasionally it has been possible to catch a
glimpse of the spectrum of one of the brighter shooting stars, and
the lines of sodium and magnesium (probably) are quite conspicu-
ous, along with some other lines which cannot be securely identified
by such a hasty glance.

As these bodies are completely burned up before they reach the
earth, all we can ever hope to get of their material is the product
of the combustion. In most places the collection and identification
of this meteoric ashes is, of course, hopeless: but Nordenskiold has
thought he might find it in polar snows, and others have thought
it might be found in the material dredged up from the bottom of
the ocean. In fact, the Swedish naturalist, by melting several tons
of Spitzbergen snow and filtering the water, did find in it a sedi-
ment containing minute globules of oxide and sulphide of iron: sim-
ilar globules are also found in the products of deep sea dredging.

These may be meteoric ashes; but what we have lately learned from

Krakat˜
ao of the distance to which smoke and fine volcanic dust is
carried by the winds, makes it quite possible that the suspected
material is purely terrestrial in its origin.

776. Probable Mass of Shooting-Stars.—We have no very
certain means of getting at this. We can, however, fix a provisional
value by means of the amount of light they give. Photometric com-
parisons between a standard star and a meteor, when we know the
meteor’s distance and the duration of its flight, enable us to ascer-
tain how the total amount of light emitted by it compares with that
given by a standard candle shining for one minute. Now, accord-
ing to determinations made some thirty years ago by Thomson at

Copenhagen, (which ought to be repeated,) the light given by a stan-
dard candle in a minute is equivalent to about twelve foot-pounds of
energy. This excludes all the energy of the dark, invisible radiation
of the candle. Our observations of the meteor give us, therefore,
its total luminous energy in foot-pounds; and if the whole of the
meteor’s energy appeared as light, then, since Energy = 1 M V 2,

2

METEORS.

449
we could at once get its mass by dividing twice this luminous en-
ergy by the square of the meteor’s velocity. Since, however, only
a small portion of the meteor’s whole energy is transformed into
light, the mass obtained in this way would be too small, and must
be multiplied by a factor which expresses the ratio between the to-
tal energy and that which is purely luminous. It is not likely that
this factor exceeds one hundred, or is less than ten, though on this
point we very much need information. Assuming the largest value,
however, for this factor, the photometric observations made in 1866
and 1867 by Professors Newcomb and Harkness (stationed respec-
tively at Washington and Richmond), showed that the majority of
the meteors of those star-showers weighed less than a single grain.

The largest of them did not reach 100 grains, or about a quarter of
an ounce.

777. Growth of the Earth.—Since the earth (in fact, every
planet) is thus continually receiving meteoric matter, and sending noth-
ing away from it, it must be constantly growing larger: but this growth is
extremely insignificant. The meteoric matter received daily by the earth,
if we accept one grain as the average weight of a shooting star, would
be only about a ton, after making a reasonable addition for occasional
aerolites, if we multiply this estimate by one hundred, it certainly will be
exceedingly liberal, and at that rate the amount received by the earth in
a year would amount to the very respectable figure of 36,500 tons; and
yet, even at this rate, assuming the specific gravity of the average meteor
as three times that of water, it would take, about 1000,000000 years to
accumulate a layer one inch thick over the earth’s surface.

778. Effect on the Earth’s Orbit.—Theoretically, the en-
counter of the earth with meteors must shorten the year in three distinct
ways:—

First. By acting as a resisting medium, and so diminishing the size
of the earth’s orbit, and indirectly accelerating its motion, in the same
manner as is supposed to happen with Encke’s comet.

Second. By increasing the attraction between the earth and the sun
through the increase of their masses.

Third. By lengthening the day—the earth’s rotation being made
slower by the increase of its diameter, so that the year will contain a
smaller number of days.

The whole effect, however, of the three causes combined, does not
amount to

1
second in a million of years. The diminution of the earth’s

1000
distance from the sun, assuming that one hundred tons of meteoric mat-

METEORS.

450
ter fall daily, and also assuming that the meteors are moving equally in
all directions with the parabolic velocity of twenty-six miles per second,
comes out about

1
of an inch per annum.

20 000

Theoretically, also, the same meteoric action should produce a short-
ening of the month, and Oppolzer investigated the subject a few years
ago, to see what amount of meteoric matter would account for the ob-
served lunar acceleration (Art. 459). He found that it would require an
amount immensely greater than really falls.

779. Meteoric Heat: Effect of Meteors on the Trans-
parency of Space.—Of course each meteor brings to the earth a cer-
tain amount of heat developed in the destruction of its motion; and at
one time if was thought that a very considerable percentage of the to-
tal heat: received by the earth might be derived from this source (See

Art. 355, (2)). Assuming, however, as before, the fall of one hundred tons
of meteoric matter daily with an average velocity of twenty miles per sec-
ond relative to the earth, the whole amount of heat comes out about 1

20
calorie per annum for each square metre of the earth’s surface—as much
in a year as the sun imparts to the same surface in about one-tenth of a
second.

One other effect of meteoric matter in space should be alluded to. It
must necessarily render space imperfectly transparent, like a thin haze.

Less light reaches us from a remote star than if the meteors were absent.

780. Meteoric Showers.—At certain times the shooting
stars, instead of appearing here and there in the sky at intervals
of several minutes, and moving in all directions, appear by thou-
sands, and even hundreds of thousands, for a few hours.

The Radiant.—At such times they do not move without system;
but they all appear to diverge or “radiate” from one point in the sky;
that is, their paths produced backward all intersect at a common
point (or nearly so), which is called “the radiant.” As an old lady
expressed it, in speaking of the meteoric shower of 1833, “The sky
looked like a great umbrella.” The meteors which appear near the
radiant are stationary, or have paths extremely short, while those
which appear at a distance from it have long courses. The radiant
keeps its place among the stars unchanged during the whole con-
tinuance of the shower, and the shower is named accordingly. Thus
we have the meteor shower of the “Leonids,” whose radiant is in the
constellation of Leo; similarly the “Andromedes” (or Bielids), the

“Perseids,” the “Geminids,” the “Lyrids,” etc. Fig. 210. is a chart
of the tracks of meteors observed on the night of Nov. 13, 1866,
showing the radiant near ζ Leonis.

METEORS.

451

Fig. 210.—The Meteoric Radiant in Leo, Nov. 13, 1866.

The simple explanation is that the radiant is purely an effect
of perspective. The meteors are really moving, relatively to the
observer, in lines which are sensibly straight and parallel, as are also
the tracks of light which they leave in the air. Hence they all seem
to diverge from one and the same perspective “vanishing point.”

The position of the radiant depends entirely upon the direction of
the meteor’s motion relative to the earth.

On account of the irregular form of the meteoric particles, they
are deflected a little one way or the other by the air, so that their
paths do not intersect at an absolute point; neither is it likely that
before they enter the air their paths are exactly parallel. The con-
sequence is that the radiant, instead of being a point, is an area of
some little size, usually less than 2◦ in diameter.

781. Probably the most remarkable of all meteoric showers that
ever occurred was that which appeared in the United States on Nov. 12,

1833, in the early morning—a shower of Leonids. The number that fell
in the five or six hours during which the shower lasted was estimated at

Boston as fully 250,000. A competent observer declared that “he never
saw snow-flakes thicker in a storm than were the meteors in the sky at
some moments.” No sound was heard, nor was any particle known to
reach the earth.

782. Dates of Showers.—Since the meteor-swarm pursues a
regular orbit around the sun, the earth can only encounter it when

METEORS.

452
she is at the point where her orbit cuts the path of the meteors;
and this, of course, must always be on the same day of the year,
except as, in the process of time, the meteors’ orbits slowly shift
their positions on account of perturbations. The Leonid showers,
therefore, always appear on the 18th of November (within a day or
two); the Andromedes on the 27th or 28th of the same month; and
the Perseids early in August.

783. Meteoric Rings and Swarms.—If the meteors are scat-
tered nearly uniformly around their whole orbit, so as to form a ring,
the shower will recur every year ; but if the flock is concentrated, it
will occur only when the meteor group is at the meeting-place at
the same time as the earth. The latter is the case with the Leonids
and Andromedes. The great star-showers from these groups occur
only rarely,—for the Leonids once in thirty-three years, and for the

Andromedes (otherwise known as the Bielids) about once in thir-
teen. The Perseids are much more equally and widely distributed,
so that they appear in considerable numbers every year, and are not
sharply limited to a particular date, but are more or less abundant
for a fortnight in the latter part of July and the first of August.

The meteors which belong to the same group all have a resemblance
to each other. The Perseids are yellowish, and move with medium veloc-
ity. The Leonids are very swift, for we meet them almost directly, and
they are characterized by a greenish or bluish tint, with vivid and persis-
tent trains. The Andromedes are sluggish in their movements, because
they simply overtake the earth, instead of meeting it. They are usually
decidedly red in color and have only small trains.

784. The Mazapil Meteorite.—As has been said, during these
showers no sound is heard, no sensible heat perceived, nor do any masses
reach the ground; with the one exception, however, that on Nov. 27, 1885,
a piece of meteoric iron mentioned in the list given in Article 758, fell
at Mazapil in Northern Mexico during the shower of Andromedes which
occurred that evening. Whether the coincidence is accidental or not, it
is interesting. Many high authorities speak confidently of this particular
iron meteor as being really a piece of Biela’s comet itself.

And now we come to one of the most remarkable discoveries of
modern astronomy,—the discovery of—

785. The Connection between Comets and Meteors.—

At the time of the great meteoric shower of 1833, Professors Olm-
sted and Twining, of New Haven, recognized the fact and meaning

METEORS.

453
of the radiant as pointing to the existence of swarms of meteoric
particles revolving in regular orbits around the sun; and Olmsted at
the time went so far as even to call the body or swarm a “comet.”

In some respects, however, his views were seriously wrong, and soon
received modification and correction from other astronomers. Er-
man especially pointed out that in some cases, at least, it would
be necessary to suppose that the meteors were distributed in rings,
and he also developed methods by which the meteoric orbits could
be computed if the necessary data could be secured. Olmsted and

Twining, however, were the first to show that the meteors are not
terrestrial and atmospheric, but bodies truly cosmical.

The subject was taken up later by Professor Newton, of New

Haven, who in 1864 showed by an examination of old records that
there had been a number of great autumnal meteoric star-showers
at intervals of just about thirty-three years, and he predicted confi-
dently a shower for Nov. 13–14, 1866. As to the orbit of the meteoric
body (or ring, according to Erman’s view), he found that it might,
consistently with what had been so far observed, have either of five
different orbits; one with a period of 33 1 years, two with periods of

4
one year ±11 days, and two with periods of half a year ±5 1 days.

2

He considered rather most probable the period of 354 days; but he
pointed out that the slow change that had taken place in the annual
date of the shower1 would furnish the means of determining which
of the orbits was the true one.

This change of date indicates a slow motion of the nodes of the
orbit of the meteoric body at the rate of about 52 a year. Adams,
of Neptunian fame, made the laborious calculation of the effect
of planetary perturbations upon each of the five different orbits
suggested by Professor Newton, and showed that the true orbit
must be the largest one which has a period of 33 1 years.

4

The meteoric shower occurred in 1866 as predicted, and was
repeated in 1867, the meteor-swarm being stretched out along its
orbit for such a distance that the procession is nearly three years in
passing any given point.

1In a.d. 902 (the “year of the stars” in the old Arab chronicles), the date
was what would be Oct. 19, in our “new style” reckoning. In 1202 the shower
occurred five days later, and in 1833 the date was Nov. 12.

METEORS.

454

Fig. 211.—Orbits of Meteoric Swarms which are known to be associated with

Comets.

786. Identification of Cometary and Meteoric Orbits.—

The researches of Newton and Adams had awakened lively interest
in the subject, and Schiaparelli, of Milan, a few weeks after the

Leonid shower, published a paper upon the Perseids, or August me-
teors, in which he brought out the remarkable fact that they were
moving in the same path as that of the bright comet of 1862, known
as Tuttle’s Comet. Shortly after this Leverrier published his orbit of
the Leonid meteors, derived from the observed position of the radi-
ant in connection with the periodic time assigned by Adams; and al-
most simultaneously, but without any idea of a connection between
them, Oppolzer published his orbit of Tempel’s comet of 1866; and
the two orbits were at once seen to be practically identical. Now a
single case of such a coincidence as that pointed out by Schiaparelli,
might possibly be accidental, but hardly two. Then five years later,
in 1872, came the meteorite shower of the Andromedes, following in
the track of Biela’s comet; and among the more than one hundred
distinct meteor-swarms now recognized, Professor Alexander Her-
schel finds four or five others which have a “comet annexed,” so to
speak. Fig. 211 represents the orbits of four of the meteoric swarms
which are known to be associated with comets.

787. In the cases of the Leonids and Andromedes the meteor-swarm
follows the comet. Many believe, however, that the comet itself is sim-

METEORS.

455
ply the thickest part of the swarm. Kirkwood and Schiaparelli have both
pointed out that a body constituted as a comet is supposed to be, must
almost necessarily break up in consequence of the “tide-producing” per-
turbations of the sun, independent of any repulsive action such as is
supposed to be the cause of a comet’s tail. They hold that these meteor-
swarms are therefore merely the product of a comet’s disintegration.

Fig. 212.—Transformation of the Orbit of the Leonids by the Encounter with

Uranus, a.d. 126.

The longer the comet has been in the system, the more widely scat-
tered will be its particles. The Perseids are supposed, therefore, to be
old inhabitants of the solar system, while the Leonids and Andromedes
are comparatively new-comers. Leverrier has shown that in the year
a.d. 126 Tempel’s comet must have been very near to Uranus, and a
natural inference is that it was introduced into the solar system at that
time. Fig. 212 illustrates his hypothesis. However these things may be,
it is now certain that the connection between comets and meteors is a
very close one, though it can hardly be considered certain as yet that
every scattered group of meteors is the result of cometary disintegration.

We are not sure that when a cometary mass first enters the solar system
from outer space, it comes in as a close-packed swarm.

THE STARS.

456

CHAPTER XIX.

THE STARS: THEIR NATURE AND NUMBER.—THE CON-

STELLATIONS.—STAR-CATALOGUES.—DESIGNATION AND

NOMENCLATURE.—PROPER MOTIONS AND THE MOTION

OF THE SUN IN SPACE.—STELLAR PARALLAX AND DIS-

TANCE.

788. We enter now upon a vaster subject. Leaving the confines
of the solar system we cross the void that makes an island1 of the
sun’s domains, and enter the universe of the stars. The nearest star,
so far as we have yet been able to ascertain, is one whose distance
is more than 200,000 times the radius of the earth’s annual orbit; so
remote that, seen from that star, the sun itself would appear only
about as bright as the pole star, and from it no telescope ever yet
constructed could render visible a single one of all the retinue of
planets and comets that make up the solar system.

789. Nature of the Stars.—As shown by their spectra the
stars are suns; that is, they are bodies comparable in magnitude
and in physical condition with our own sun, shining by their own
light as the sun does, and emitting a radiance which in many cases
could not be distinguished from sunlight by any of its spectroscopic
characteristics. Some of them are vastly larger and hotter than
our sun, others smaller and cooler, for, as we shall see, they differ
enormously among themselves.

790. Number of the Stars.—The impression on a dark night
is of absolute countlessness; but, in fact, the number visible to the
naked eye is very limited, as one can easily discover by taking some
definite area in the sky, say the “bowl of the dipper,” and counting

1That the solar system is thus isolated by a surrounding void is proved by
the almost undisturbed movements of Uranus and Neptune; for their perturba-
tions would betray the presence of any body, at all comparable with the sun
in magnitude, within a distance a thousand times as great as that between the
earth and sun.

THE STARS.

457
the stars which he can see within it. He will find that the number
which he can fairly count is surprisingly small, though by averted
vision he will get uncertain glimpses of many more. In the whole
celestial sphere the number bright enough to be visible to the naked
eye is only from 6000 to 7000 in a clear, moonless sky. A little haze
or moonlight cuts out fully half of them, and of course there is
a great difference in eyes. But the sharpest eyes could probably
never fairly see more than 2000 or 3000 at one time, since near the
horizon the smaller stars are invisible, and they are immensely the
most numerous, fully half of the whole number being those which are
just on the verge of visibility. The total number that can be seen well
enough for observation with such instruments as were used before
the invention of the telescope is not quite 1100.

With even a small telescope the number is enormously increased.

A mere opera-glass an inch a half in diameter brings out at least

100,000. The telescope with which Argelander made his Durch-
musterung of more than 300,000 stars—all north of the celestial
equator—had a diameter of only two inches and a half. The num-
ber visible in the great Lick1 telescope of three feet diameter is
probably nearly 100,000000.

791. Constellations.—In
ancient
times
the
stars
were
grouped by “constellations,” or “asterisms,” partly as a matter of
convenient reference and partly as superstition. Many of the con-
stellations now recognized,—all of those in the zodiac and those
about the northern pole,—are of prehistoric antiquity. To these
groups were given fanciful names, mostly of persons or objects con-
spicuous in the mythological records of antiquity; a great number
of them are connected in some way or other with the Argonautic
expedition.

In some cases the eye can trace in the arrangement of the stars a
vague resemblance to the object which gives name to the constellation;
but generally no reason can be assigned why the constellation should be
so named or so bounded. Of the sixty-seven constellations now usually
recognized on celestial globes, forty-eight have come down from Ptolemy.

The others have been formed by Hevelius, Bayer, Royer, and one or two

1Neglecting the loss of light in the lenses, the Lick telescope ought, theo-
retically, to show stars so faint that it would take more than 30,000 of them
to make a star equal to the faintest that can be seen with the naked eye. (See

Art. 38.)

THE STARS.

458
other astronomers, to embrace stars not included in Ptolemy’s constel-
lations, and especially to furnish a nomenclature for the stars never seen
by Ptolemy on account of their nearness to the southern pole. A con-
siderable number of other constellations, which have been tentatively
established at various times, and are sometimes found on globes and
star-maps, have been given up as useless and impertinent.

792. We present a list of the constellations, omitting, however, some
of the modern ones which are now not usually recognized by astronomers.

The constellations are arranged both vertically and horizontally. The or-
der in the vertical columns is determined by right ascension, as indicated
by the Roman numbers at the left. Horizontally the arrangement is ac-
cording to distance from the north pole, as shown by the headings of the
columns. The number appended to each constellation gives the num-
ber of stars it contains, down to and including the 6th magnitude. The
zodiacal constellations are italicized, and the modern constellations are
marked by an asterisk.

The different groups of constellations are found near the meridian at
half-past eight o’clock, p.m., on the dates indicated below.

Group (I., II.), Dec. 1. These constellations contain no first-magnitude
stars, but Cassiopeia, Andromeda, Aries, and Cetus include
enough stars of the second and third magnitude to be fairly
conspicuous.

Group (III., IV.), Jan. 1. Perseus north of the zenith, and the Pleiades
and Aldebaran in Taurus, are characteristic.

Group (V., VI.), Feb. 1. On the whole this is the most brilliant region
of the sky and Orion the finest constellation.

Group (VII., VIII.), March 1. Characterized by Procyon and Sirius,
the latter incomparably the brightest of all the fixed stars.

Group (IX., X.), April 1. Leo is the only conspicuous constellation.

Group (XI., XII.), May 1. A barren region, except for Ursa Major
north of the zenith.

Group (XIII., XIV.), June 1. Marked by Arcturus, the brightest of
the northern stars, with the paler Spica south of the equator.

Group (XV., XVI.), July 1. The Northern Crown and Hercules are
the most characteristic configurations.

Group (XVII., XVIII.), Aug. 1. Vega is nearly overhead, and the red

Antares low down in the south, with Altair near the equator,
just east of Ophiuchus.

Group (XIX., XX.), Sept. 1. Cygnus is in the zenith, and Sagittarius
low down, while the brightest part of the Milky Way lies athwart
the meridian.

THE STARS.

459

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enat.

Bor.

to

V
us,

◦

Minor,
us,
rse
otes,
riangulum,
e

—

—

+50

Andromeda,

T

P

Auriga,

Lynx,

Leo

Canes

Bo¨

Corona

Hercules,

Lyra,

Cygn

Lacerta,

CONSTELLA

*

*

*

*

OF

.

46

36

53

23

80

44

◦

—

—

—

—

—

—

LIST

+50

,a
jor,
to
ei

◦

Ma

Minor,

90

—

—

—

—

—

—

−

Cassiop

Camelopardus,

Ursa

Ursa

Draco,

Cepheus,

*
h.

II,

I,

I,

IV,

VI,

III,

X,

II,

V

XI

XIV,

XVI,

XX,

Decl.

XVI

XXI

XXIV,
h.

I,

I,

I,

A.

III,

V,

II,

II,

VI

IX,

XI,

XI

XV,

XIX,

XXI,

R.

XVI

XXI

THE STARS.

460

Group (XXI., XXII.), Oct. 1. A barren region, relieved only by the
bright star Fomalhaut of the Southern Fish near the southern
horizon.

Group (XXIII., XXIV.), Nov. 1. This region also is rather barren,
though the “great square” of Pegasus is a notable configuration
of stars.

793. A thorough knowledge of these artificial groups, and of
the names and locations of the stars in them, is not at all essential,
even to an accomplished astronomer; but it is a matter of very
great convenience to know the principal constellations, and perhaps
a hundred of the brightest stars, well enough to be able to recognize
them readily and to use them as points of reference. This amount of
knowledge is easily acquired by three or four evenings’ study of the
sky in connection with a good star-map or celestial globe, taking
care to observe on evenings at different seasons of the year, so as to
command the whole sky.

At present the best star-atlas for reference is probably that of Mr.

Proctor. The maps of Argelander’s “Uranometria Nova” and Heis’s atlas

(both in German) are handsomer, and for some purposes more conve-
nient. There are many others, also, which are excellent. The smaller
maps which are found in the text-books on astronomy are not on a scale
sufficiently large to be of much scientific use (as, for instance, in the ob-
servation of meteors), though they answer well enough the purpose for
which they were designed, of introducing the student to the principal
star-groups.

794. Designation of Bright Stars.—(a) Names. Some fifty
or sixty of the brighter stars have names of their own in common use.

A majority of the names belonging to stars of the first magnitude are
of Greek or Latin origin, and significant, as, for instance, Arcturus,

Sirius, Procyon, Regulus, etc. Some of the brightest stars, however,
have Arabic names, as Aldebaran, Vega, and Betelgueze, and the
names of most of the smaller stars are Arabic, when they have names
at all.

(b) Place in Constellation. Spica is the star in the handful of
wheat carried by Virgo; Cynosure signifies the star at the end of the

Dog’s Tail (in ancient times the constellation we now call Ursa Mi-
nor seems to have been a dog); Capella is the goat which Auriga, the
charioteer, carries in his arms. Hipparchus, Ptolemy, and, in fact,
all the older astronomers, including Tycho Brahe, used this clumsy

THE STARS.

461
method almost entirely in designating particular stars; speaking, for
instance, of the star in the “head of Hercules,” or in the “right knee
of Boötes,” and so on.

(c) Constellation and Letters. In 1603 Bayer, in publishing a
new star-map, adopted the excellent plan, ever since in vogue, of
designating the stars in the different constellations by the letters of
the Greek alphabet, assigned usually in order of brightness. Thus

Aldebaran is α Tauri, the next brightest star in the constellation is
β Tauri, and so on, as long as the Greek letters hold out; then the

Roman letters are used as long as they last; and finally, whenever
it is found necessary, we use the numbers which Flamsteed assigned
a century later. At present every naked-eye star can be referred
to and identified by some letter or number in the constellation to
which it belongs.

(d ) Current Number in a Star-Catalogue. Of course all the above
methods fail for the hundreds of thousands of smaller stars. In
their case it is usual to refer to them as number so-and-so of some
well-known star-catalogue; as, for instance, 22,500 Ll. (Lalande),
or 2573 B.A.C. (British Association Catalogue). At present our
various star-catalogues contain from 600,000 to 800,000 stars, so
that, except in the Milky Way, almost any star visible in a telescope
of two or three inches’ aperture can be identified and referred to by
means of some star-catalogue or other.

Synonyms. Of course all the brighter stars which have names
have also letters, and are sure to be included in every star-catalogue
which covers their part of the sky. A given star, therefore, has often
a large number of aliases, and in dealing with the smaller stars great
pains must be taken to avoid mistakes arising from this cause.

STAR-CATALOGUES.

795. These are lists of stars arranged in regular order (at
present usually in order of right ascension), and giving the places
of the stars at some given epoch, either by means of their right as-
censions and declinations, or by their (celestial) latitudes and longi-
tudes. The so-called “magnitude,” or brightness of the star, is also
ordinarily indicated. The first of these star-catalogues was that of

Hipparchus, containing 1080 stars (all that are easily visible and
measurable by naked-eye instruments), and giving their longitudes
and latitudes for the epoch of 125 b.c.

THE STARS.

462

This catalogue has been preserved for us by Ptolemy in the Almagest,
and from it he formed his own catalogue, reducing the positions of the
stars (i.e., correcting for precession the positions given by Hipparchus)

to his own epoch, about 150 a.d. The next of the old catalogues of
any value is that of Ulugh Beigh made at Samarcand about 1450 a.d.

This appears to have been formed from independent observations. It was
followed in 1580 by the catalogue of Tycho Brahe containing 1005 stars,
the last which was constructed before the invention of the telescope.

The modern catalogues are numerous. Some give the places of a
great number of stars rather roughly, merely as a means of identifying
them when used for cometary observations or other similar purposes. To
this class belongs Argelander’s Durchmusterung of the northern heav-
ens, which contains over 324,000 stars,—the largest number in any one
catalogue thus far published. Then there are the “catalogues of preci-
sion,” like the Pulkowa and Greenwich catalogues, which give the places
of a few hundred stars as accurately as possible in order to furnish “fun-
damental stars,” or reference points in the sky. The so-called “Zones”

of Bessel, Argelander, and many others, are catalogues covering limited
portions of the heavens, containing stars arranged in zones about a de-
gree wide in declination, and running some hours in right ascension. To
the practical astronomer the most useful catalogue is likely to be the
one which is now in process of formation by the co-operation of various
observatories under the auspices of the Astronomische Gesellschaft (an

International Astronomical Society, with its headquarters in Germany).

This catalogue will contain accurate places of all stars above the ninth
magnitude in the northern sky. Most of the necessary observations have
already been made.

796. Determination
of

Star-Places.—The observations
from which a star-catalogue is constructed are usually made with
the meridian circle (Art. 63). For the catalogues of precision, com-
paratively few stars are observed, but all with the utmost care and
during several years, taking all possible means to eliminate instru-
mental and observational errors of every sort.

In the more extensive catalogues most of the stars are observed
only once or twice, and everything is made to depend upon the
accuracy of the places of the fundamental stars, which are assumed
as correct. The instrument in this case is used only “differentially”

to measure the comparatively small differences between the right
ascension and declination of the fundamental stars and those of the
stars to be catalogued.

THE STARS.

463

Fig. 213.—The Photographic Telescope of the Henry Brothers, Paris.

THE STARS.

464

797. Method of using a Catalogue.—The catalogue con-
tains the mean right ascension and declination of its stars for the
beginning of some given year; i.e., the right ascension and declina-
tion the star would have at that time if there were no aberration of
light and no irregular motion in the celestial pole to affect the posi-
tion of the equator and equinox. To determine the actual apparent
right ascension and declination of a star for a given date (which is
what we want in practice), the catalogue place must be “reduced ”

to the date in question; i.e., it must be corrected for precession,
nutation, and aberration.

The operation with modern tables and formulæ is not a very tedious
one, involving perhaps five minutes’ work, but without the catalogue
places are useless for most purposes. Vice versa, the observations of a
fixed star with the meridian circle do not give its mean right ascension
and declination ready to go into the catalogue, but the observations must
be reduced from apparent place to mean before they can be tabulated.

798. Star-Charts.—For many purposes charts of the stars are
more convenient than a catalogue, as, for instance, in searching for
new planets. The old-fashioned way of making such charts was by
plotting the results of zone observations. The modern way, intro-
duced within the last few years, is to do it by photography. The plan
decided upon at the Paris Astronomical Congress in 1887 contem-
plates the photographing of the whole sky upon glass plates about
six inches square, each covering an area of 2◦ square (four square
degrees), showing all stars down to the fourteenth magnitude,—a
project which is entirely feasible, and can be accomplished in five
or six years by the co-operation of about a dozen different observa-
tories in the northern and southern hemispheres. The instruments
are now (1888) in process of construction.

The figure (Fig. 213) is a representation of the Paris instrument of
the Henry Brothers, which was adopted as the typical instrument for
the operation. It has an aperture of about fourteen inches, and a length
of about eleven feet, the object-glass being specially corrected for the
photographic rays. A 9-inch visual telescope is enclosed in the same tube
so that the observer can watch the position of the instrument during the
whole operation.

It was originally planned to give each plate 20 minutes’ exposure,
but improvements in the photographic plates since the meeting of the

Congress now make it possible to cut down the time very materially. It
will require about 11,000 plates of the size named to cover the whole sky,

THE STARS.

465
and as each star is to appear on two plates at least, the whole number
of plates, allowing for overlaps, will be about 22,000. As every plate
will contain upon it a number of well-determined catalogue stars, it will
furnish the means of determining accurately, whenever needed, the place
of any other star which appears upon the same plate.

STAR MOTIONS.

799. The stars are ordinarily called “fixed,” in distinction from
the planets or “wanderers,” because as compared with the sun and
moon and planets they have no evident motion, but keep their rela-
tive relations and configurations unchanged. Observations made at
sufficiently wide intervals of time, and observations with the spec-
troscope, show, however, that they are really moving, and that with
velocities which are comparable to the motion of the earth in her
orbit.

If we compare the right ascension and declination of a star deter-
mined to-day with that determined a hundred years ago, it will be
found different. The difference is mainly due to precession and nu-
tation, which are not motions of the stars at all, but simply changes
in the position of the reference circles used, and due to alterations
in the direction of the earth’s axis (Arts. 205 and 214). Aberration also comes in, and this also is not a real motion of the stars, but
only an apparent one.

800. Proper Motions.—But after allowing for all these ap-
parent and common motions, which depend upon the stars’ places
in the sky, and are sensibly the same for all stars in the same tele-
scopic field of view, whatever may be their real distance from us, we
find that most of the larger stars have a “proper motion” of their
own, (“proper” as opposed to “common,”) which displaces them
slightly with reference to the stars about them. There are only a
few stars for which this proper motion amounts to as much as 1
a year; perhaps 150 such stars are now known, but the number is
constantly increasing, as more and more of the smaller stars come
to be accurately observed.

The maximum proper motion known is that of the seventh mag-
nitude star 1830 Groombridge (i.e., No. 1830 in Groombridge’s
catalogue of circumpolar stars), which has an apparent drift of

7
annually,—enough to carry it completely around the heavens
in 185,000 years.

The largest known proper motions are the
following:—

THE STARS.

466

1830, Groombridge,

7th mag., 7 .0

Indi,

5th mag., 4 .5

9352, Lacaille,

7th

“

6 .9

Lalande 21,258

8th

“

4 .4

32,416, Gould,

9th

“

6 .2
o2 Eridani

6th

“

4 .1

61 Cygni,

6th

“

5 .2
µ Cassiopeiæ,

5th

“

3 .8

Lalande 21,185,

7th

“

4 .7
α Centauri,

1st

“

3 .7

The proper motions of Arcturus (2 .1), and of Sirius (1 .2), are
considered “large,” but are exceeded by a considerable number of
stars besides these given above. Since the time of Ptolemy, Arcturus
has moved more than a degree, and Sirius about half as much. Those
motions were first detected by Halley in 1718.

It is found, as might be expected, that the brighter stars, which
as a class are presumably nearer than the fainter ones, have on the
average a greater proper motion; on the average only, however, as
is evident from the list given above. Many smaller stars have larger
proper motions than any bright one, for there are more of them.

801. Real Motions of Stars.—The average proper motion
of the first-magnitude stars appears to be about 1
annually, and

4
that of a sixth-magnitude star,—the smallest visible to the eye,—is
about 1 .

25

The proper motion of a star gives
comparatively little information as to
its real motion until we know the dis-
tance of the star and the true direction
of the motion, since the proper motion

Fig. 214.

as determined from the star-catalogues

Components of a Star’s
is only the angular value of that part or

Proper Motion.

component of the star’s whole motion
which is perpendicular to the line of sight, as is clear from the fig-
ure. When the star really moves from A to B (Fig. 214), it will appear, as seen from the earth, to have moved from A to b. The
angular value of Ab as seen from the earth is the proper motion

(usually denoted by µ), as determined from the comparison of star-
catalogues. Expressed in seconds of arc, we have

Ab
µ = 206 265

.

distance

A body moving directly towards or from the earth has, therefore,
no (angular) proper motion at all,—none that can be obtained from
the comparison of star-catalogues.

THE STARS.

467

Since Ab in miles
µ × distance

=

,

206 265
these motions cannot be translated into miles without a knowledge
of the star’s distance; and this knowledge, as we shall see, is at
present exceedingly limited; nor can the true motion AB be found
until we also know either the angle BAE or else the line Aa.

But since AB is necessarily greater than Ab, it is possible in some
cases to determine a minor limit of velocity, which must certainly be
exceeded by the star. In the case of 1830, Groombridge, for instance,
we have certain knowledge that its distance is not less than 2,000000
times the earth’s distance from the sun. It may be vastly greater; but it
cannot be less. Now at that distance the observed proper motion of 7
a year would correspond to an actual velocity along the line Ab of more
than 200 miles a second, and the star may be moving many times more
swiftly. This star has sometimes been called the “runaway star.”

802. Motion in the Line of Sight.—Although the compari-
son of star-catalogues gives us no information of the body’s motion
towards or from us along the line of sight, the velocity Aa in the
figure, yet it is possible, when the star is reasonably bright, to deter-
mine somewhat roughly its rate of approach or recession by means
of the spectroscope. If the star is approaching us, the lines in the
spectroscope, according to Doppler’s principle (Art. 821, note), will
be shifted towards the blue; and vice versa, towards the red, if it
is receding from us. Dr. Huggins was the first actually to use this
method of investigating the star movements in 1868, and by means
of it he arrived at some very interesting results, which, however,
must be admitted to be somewhat uncertain as regards their quan-
titative value. He found, for instance, that Sirius was receding from
us at the rate of nineteen miles a second, and that Arcturus was
rushing towards us at the rate of nearly sixty miles a second; and
results of a similar character were found for a considerable number
of other stars.

Of late the investigations of this class have been carried on
mainly at the Greenwich Observatory, usually by comparing the
stellar spectra with hydrogen and sodium. The observations, how-
ever, are extremely difficult to make, for the displacements of the
lines are very small, and in most star spectra the lines are broad

THE STARS.

468
and hazy and not well adapted for accurate measurements. The re-
sults of different days’ observations, therefore, for a single star are
sometimes mournfully discrepant.

Vogel has recently taken up the work pho-
tographically at Potsdam, with very encourag-
ing results. Fig. 215 is from one of his plates

(a negative), showing the displacement towards
the red of the H

Fig. 215.

γ line in the spectrum of β Ori-

Displacement of Hγ Line
onis, or Rigel, and indicating a recession of the
in the Spectrum
star at a very rapid rate. Much is hoped in
β Orionis.

this line from the photographic spectrum-work
of the Draper Memorial at Cambridge, of which more will be said a few
pages farther on.

803. Star-Groups.—Star-atlases have been constructed by

Proctor and Flammarion, which show by arrows the direction and
rate of the angular proper motion of the stars as far as now known.

A moment’s inspection shows that in many cases stars in the same
neighborhood have a proper motion nearly the same in direction
and in amount.

Thus, Flammarion has pointed out that the stars in the “dipper” of

Ursa Major have such a community of motion, except α and η,—the
brighter of the pointers and the star in the end of the handle,—which
are moving in entirely different directions, and refuse to be counted as
belonging to the same group. Fig. 216 shows the proper motions of the
stars which compose this group. The same thing appears when their
motion is tested by the spectroscope. Huggins found that the five asso-
ciated stars are rapidly receding from the earth, while α is approaching
us, and η, though receding, has a widely different rate of motion from
the others.

Fig. 216.—Common Proper Motions of Stars in the “Dipper” of Ursa Major.

The brighter stars of the Pleiades are found in the same way to have
a common motion.

THE STARS.

469

In fact, it appears to be the rule rather than the exception that
stars apparently near each other are really connected as comrades,
travelling together in groups of twos and threes, dozens or hundreds.

They show, as Miss Clerke graphically expresses it, a distinctly

“gregarious tendency.”

804. The “Sun’s Way.”—The proper motions of the stars
are due partly to their own real motion, and partly also to the
motion of our sun, which is moving swiftly through space, taking
with it the earth and the planets. Sir William Herschel was the
first to investigate and determine the direction of this motion a
little more than 100 years ago. The principle involved is this: that
the apparent motion of each star is made up of its own motion
combined with the motion of the sun reversed (Art. 492).

The
effect must be that on the whole, the stars in that part of the sky
towards which the sun is moving are separating from each other,—
the intervals between them widening out,—while in the opposite part
of the heavens they are closing up; and in the intermediate part of
the sky the general drift must be backward with reference to the
sun’s (and earth’s) real motion. Just as one walking in a park filled
with people moving indiscriminately in different directions, would,
on the whole, find that those in front of him appeared to grow
larger,1 and the spaces between them to open out, while at the
sides they would drift backwards, and in the rear close up.

The spectroscope, moreover, ought to indicate this motion and un-
doubtedly will do so when the apparent motion in the line of sight has
been accurately determined for a considerable number of stars. In fact
one or two attempts have already been made to determine the solar
motion in this way; in the quarter of the sky towards which the sun
is moving, the star spectra should, on the whole, show displacement of
their lines indicating approach, and vice versa in the opposite quarter;
and these observations will have the advantage of showing directly the
sun’s rate of motion in miles, a result which is not given by investigations
founded upon the angular proper motions of the stars. As yet, however,
this spectroscopic method has not furnished results of any great weight.

1Theoretically, of course, the stars towards which we are moving must ap-
pear to grow brighter as well as to drift apart; but this change of brightness,
though real, is entirely imperceptible within a human lifetime.

THE STARS.

470

805. About twenty different determinations of the point in the
sky towards which this motion of the sun is directed have been
worked out by various astronomers, using in their discussions the
angular proper motions of from twenty to twenty-five hundred stars.

All the investigations present a reasonable accordance of results,
differing from each other only by a few degrees, and show that the
sun is now moving towards a point in the constellation of Hercules,
having a right ascension of about 267 ◦ and a declination of about

+31 ◦. This point is known as the “apex of the sun’s way.”

806. Velocity of the Sun’s Move-
ment.—This also is determined by the
discussion, and comes out to be such
as would carry the sun and its system
about 5 in 100 years as seen from the
average sixth magnitude star (the sixth
magnitude is the smallest easily visible
to the naked eye).

If we knew with
any certainty the distance of this average
sixth magnitude star we could translate
this motion into miles; but at present
this indispensable datum can be little
more than guessed at. On the reasonable

Fig. 217.

assumption adopted by Ludwig Struve

The Earth’s Motion in

(who has made the most recent and ex-

Space an affected by the
tensive of all the investigations upon the

Sun’s Drift.

motion of the solar system), that this dis-
tance is about 20,000000 times the astronomical unit, the velocity
of the sun’s motion in space comes out about five units per year,
that is, about five-sixths of the earth’s orbital velocity, or nearly
sixteen miles per second; but this result must be considered as still
very uncertain.

It is to be noted that this swift motion of the solar system, while of
course it affects the real motion of the planets in space, converting them
into a sort of corkscrew spiral like the figure (Fig. 217), does not in the
least affect the relative motion of sun and planets, as some paradoxers
have supposed it must.

807. The Central Sun.—We mention this subject simply to
say that there is no real foundation for the belief in the existence of
such a body. The idea that the motion of our sun and of the other

THE STARS.

471
stars is a revolution around some great central sun is a very fasci-
nating one to certain minds, and one that has been frequently sug-
gested. It was seriously advocated some fifty years ago by Mädler,
who placed this centre of the stellar universe at Alcyone, the prin-
cipal star in the Pleiades.

It is certainly within bounds to deny that any such motion has
been demonstrated, and it is still less probable that the star Alcyone
is the centre of such a motion, if the motion exists. So far as we can
judge at present it is most likely that the stars are moving, not in
regular closed orbits around any centre whatever, but rather as bees
do in a swarm, each for itself, under the action of the predominant
attraction of its nearest neighbors. The solar system is an absolute
monarchy with the sun supreme. The great stellar system appears
to be a republic, without any such central, unique, and dominant
authority.

THE PARALLAX AND DISTANCE OF THE STARS.

808. When we speak of the “parallax” of the moon, the sun,
or a planet, we always mean the diurnal parallax, i.e., the angular
semi-diameter of the earth as seen from the body in question. In
the case of the stars, this kind of parallax is hopelessly insensible,
never reaching an amount of

1
of a second of arc.

20 000

The expression “parallax of a star ” always means its annual
parallax, that is, the semi-diameter of the earth’s orbit as seen from
the star. Even this in the case of all stars but a very few is a mere
fraction of a second of arc, too small to be measured. In a few
instances it rises to about half a second, and in the one case of our
nearest neighbor (so far as known at present), the star α Centauri,
it appears to be about 0 .9, according to the earlier observers, or
about 0 .75, according to the latest determination of Gill and Elkin.

In Fig. 218 the angle at the star is the star’s parallax.

Fig. 218.—The Annual Parallax of a Star.

In accordance with the principle of relative motion (Art. 492),
every star has, superposed upon its own motion and combined with

THE STARS.

472
it, an apparent motion equal to that of the earth but reversed. If
the star is really at rest it must seem to travel around each year in
a little orbit 180,000000 miles in diameter, the precise counterpart
of the earth’s orbit in size and form, and having its plane parallel
to the ecliptic.

If the star is near the pole of the ecliptic this apparent “parallactic”

orbit will be viewed perpendicularly and appear as a circle; if the star is
on the ecliptic it will be seen edgewise as a short, straight line, while in
intermediate latitudes the parallactic orbit will appear as an ellipse. In
this respect it is just like the “aberrational” orbit of a star (Art. 220);
but while the aberrational orbit is of the same size for every star, having
always a semi-major axis of 20 .492, the size of the parallactic orbit
depends upon the distance of the star. Moreover, in the parallactic orbit
the star is always opposite to the earth, while in the aberrational orbit
it keeps just 90◦ ahead of her.

809. If we can find a way of measuring this parallactic orbit,
the star’s distance is at once determined. It equals

206 265 × R ,
p
in which p is the parallax in seconds of arc (the apparent semi-
major axis of the parallactic orbit), and R is the earth’s distance
from the sun.

The determination of stellar parallax had been attempted over
and over again from the days of Tycho down, but without success
until Bessel, in 1838, succeeded in demonstrating and measuring
the parallax of the star 61 Cygni; and the next year Henderson,
of the Cape of Good Hope, brought out that of α Centauri. It
will be remembered that it was mainly on account of his failure to
detect stellar parallax that Tycho rejected the Copernican theory
and substituted his own (Art. 504).

Roemer of Copenhagen, in 1690, thought that he had detected the
effect of stellar parallax in his observations of the difference of right as-
cension between Sirius and Vega at different times of the year. A few
years later, Horrebow, his successor, from his own discussion of Roemer’s
observations, made out the amount to be nearly four seconds of time or

1 , and published his premature exultation in a book entitled “Coper-
nicus Triumphans.” The discovery of aberration by Bradley explained
many abnormal results of the early astronomers which had been thought

THE STARS.

473
to arise from stellar parallax, and proved that the parallax must be ex-
tremely small. About the beginning of the present century, Brinkley of

Dublin and Pond, the Astronomer Royal, had a lively controversy over
their observations of α Lyræ (Vega). Brinkley considered that his obser-
vations indicated a parallax of nearly 3 . Pond, on the other hand, from
his observations deduced a minute negative parallax, which, as some one
has expressed it, would put the star “somewhere on the other side of
nowhere.” In fact, as it turns out, Pond was nearer right than Brink-
ley, the actual parallax as deduced from the latest observations being
only about 0 .2. The negative parallax, like the much too large result
of Brinkley, simply indicates the uncertainties and errors incident to
the instruments and methods of observation then used. The periodical
changes of temperature and air pressure continually lead to fallacious
results, except under the most extreme precautions.

810. Methods of determining Parallax.—The operation of
measuring a stellar parallax is, on the whole, the most delicate in
the whole range of practical astronomy. Two methods have been
successfully employed so far—the absolute and the differential.

(a) The first method consists in making meridian observations
of the right ascension and declination of the star in question at dif-
ferent seasons of the year, applying till known corrections for preces-
sion, nutation, aberration, and proper motion, and then studying
the resulting star-places. If the star is without parallax, the places
should be identical after the corrections have been duly applied. If
it has parallax, the star will be found to change its right ascension
and declination systematically, though slightly, through the year.

But the changes of the seasons so disturb the constants of the in-
strument that the method is treacherous and uncertain. There is no
possibility of getting rid of these temperature effects (in producing
changes of refraction and varying expansions of the instrument it-
self) by merely multiplying observations and taking averages, since
the changes of temperature are themselves annually periodic, just
as is the parallax itself.

Still, in a few cases the method has proved successful. Different
observers at different places with different instruments have found
for a few stars fairly accordant results; as, for instance, in the case
of α Centauri, already mentioned as our nearest neighbor.

811. (b) The Differential Method.—This consists in mea-
suring the change of position of the star whose parallax we are

THE STARS.

474
seeking (which is supposed to be comparatively near to us), with
reference to other small stars, which are in the same telescopic field
of view, but are supposed to be so far beyond the principal star as to
have no sensible parallax of their own. If the comparison stars are
near the large one (say within two or three minutes of arc), the ordi-
nary wire micrometer answers very well for the necessary measures;
but if they are farther away, the heliometer (Art. 677) represents
special and very great advantages. It was with this instrument that

Bessel, in the case of 61 Cygni, obtained the first success in this line
of research.

The great advantage of the differential method is that it avoids
entirely the difficulties which arise from the uncertainties as to the
exact amount of precession, etc.; and in great measure, though not
entirely, those arising from the effect of the seasons upon refraction
and the condition of the instruments. On the other hand, however,
it gives as the final result, not the absolute parallax of the star, but
only the difference between its parallax and that of the comparison
star. If the work is accurate the parallax deduced cannot be too
great; but it may be sensibly too small, and so may make the star
apparently too remote. This is because the parallax of the compar-
ison star can never be quite zero: if the comparison star happens
to have a parallax of its own as large as that of the principal star,
there will be no relative parallax at all; if larger, the parallax sought
will come out negative.

812. Determination
of

Parallax
by

Photography.—

Recently it has been attempted to press photography into the ser-
vice, and Professor Pritchard has obtained apparently excellent re-
sults from an extensive series of photographs of 61 Cygni and its
neighboring stars, made at Oxford during 1886.

Now and then a plate was found in which the sensitive film appeared
to have slipped a little during the development of the picture, but the
measurements at once showed up the faulty plates, so that they could
be confidently rejected. The measurements made upon the other plates
appeared to be just as trustworthy as measurements made upon the ac-
tual objects in the sky; and of course the measurement of a photographic
plate at leisure in one’s laboratory is a vastly more comfortable operation
than that of making micrometric settings by night.

813. Selection of Stars.—It is important to select for investi-
gations of this kind these stars which may reasonably be supposed to

THE STARS.

475
be near, and, therefore, to have a sensible parallax. The most impor-
tant indication of proximity is a large proper motion, and brightness is,
of course, confirmatory. At the same time, while it is probable that a
bright star with large proper motion is comparatively near, it is not cer-
tain. The small stars are so much more numerous than the large ones
that it will be nothing surprising if we should find among them one or
more neighbors nearer than α Centauri itself.

814. Unit of Stellar Distance.—The Light-Year. The ordi-
nary “astronomical unit,” or distance of the sun from the earth, is
not sufficiently large to be convenient in expressing the distances of
the stars. It is found more satisfactory to take us a unit the dis-
tance that light travels in a year, which is about 63,000 times the
distance of the earth from the sun. A star with a parallax of 1 is
at a distance of 3.262 “light-years,” so that the distance of any star
in “light-years” is expressed by the formula

3.262

Dy =

.

p

815. Table IV. in the Appendix, based upon that given in

Houzeau’s “Vade Mecum of Astronomy,” but brought down to date
by some additions and changes, gives the parallaxes, and the dis-
tances in light-years, of these stars whose parallaxes may be con-
sidered as now fairly determined.

The student will, of course, see that the tabulated distance in the case
of a remote star is liable to an enormous percentage of error. Considering
the amount of discordance between the results of different observers, it
is extremely charitable to assume that any of the parallaxes are certain
within 1 of a second of arc: but in the case of a star like the pole-star,

50
which appears to have a parallax of less than 0 .08, this 1 of a second

50
is 1 of the whole amount; so that the distance of that star is uncertain

4
by at least twenty-five per cent. ( 1 of a second is the angle subtended

50
by 1 of an inch at the distance of ten miles.)

16

A vigorous campaign has lately been organized for the purpose of ob-
taining within a reasonable time the parallaxes of a considerable number
of stars (perhaps one or two hundred),—enough to enable us to deduce
some general laws by statistical methods. The instruments to be used
are heliometers of six or seven inches’ aperture, one of which is at the

Cape of Good Hope, another has just been erected at Bamberg in Ger-
many, and a third is at New Haven in this country, under the charge of

Dr. Elkin.

THE STARS.

476

As regards the distance of stars, the parallax of which has not yet
been measured, very little can be said with certainty. It is probable that
the remoter ones are so far away that light in making its journey occupies
a thousand and perhaps many thousand years.

815*. Since the above was written Dr. Elkin has published the
result of his observations upon ten first-magnitude stars, as follows:

α Tauri (Aldebaran), 0 .116 ± .029; α Aurigæ (Capella), 0 .107 ± .047;
α Orionis (Betelgueze), −0 .009 ± .049; α Canis Minoris (Procyon),

0 .266 ± .047; β Geminorum (Pollux ), 0 .068 ± .047; α Leonis (Regu-
lus), 0 .093 ± .048; α Bootis (Arcturus), 0 .018 ± .022; α Lyræ (Vega),

0 .034 ± .045; α Aquillæ (Altair ), 0 .199 ± .047; α Cygni (Deneb),

−0 .042 ± .047.

Of course the two negative results simply indicate that the parallax
of the large star was less than that of the comparison stars employed.

The very small results for Vega and Arcturus are also rather surprising.

See Table IV. of Appendix.

THE LIGHT OF THE STARS.

477

CHAPTER XX.

THE

LIGHT

OF

THE

STARS.—STAR

MAGNITUDES

AND

PHOTOMETRY.—VARIABLE STARS.—STELLAR SPECTRA.

—SCINTILLATION OF STARS.

816. Star Magnitudes.—The term “magnitude,” as applied
to a star, refers simply to its brightness. It has nothing to do with
its apparent angular diameter. Hipparchus and Ptolemy arbitrar-
ily graded the visible stars, according to their brightness, into six
classes, the stars of the sixth magnitude being the smallest visible to
the eye, while the first class comprises about twenty of the bright-
est. There is no assignable reason why six classes should have been
constituted, rather than eight or ten.

After the invention of the telescope the same system was ex-
tended to the smaller stars, but without any general agreement or
concert, so that the magnitudes assigned by different observers to
telescopic stars vary enormously. Sir William Herschel, especially,
used very high numbers: his twentieth magnitude being about the
same as the fourteenth on the scale now generally used, which more
nearly corresponds with that of the elder Struve.

817. Fractional Magnitudes.—Of course, the stars classed
together under one magnitude are not exactly alike in brightness,
but shade from the brighter to the fainter, so that exactness requires
the use of fractional magnitudes. It is now usual to employ decimals
giving the brightness of a star to the nearest tenth of a magnitude.

Thus, a star of 4.3 magnitude is a shade brighter than one of 4.4,
and so on.

A peculiar notation was employed by Ptolemy, and used by Arge-
lander in his “Uranometria1 Nova.” It recognizes thirds of a magnitude
as the smallest subdivision. Thus, 2, 2,3, 3,2, and 3 express the gra-

1The term “Uranometria” has come to mean a catalogue of naked-eye stars;
like the catalogues of Hipparchus, Ptolemy, and Ulugh Beigh.

THE LIGHT OF THE STARS.

478
dations between second and third magnitude, 2,3 being applied to a star
whose brightness is a little inferior to the second, and 3,2 to one a little
brighter than the third magnitude.

818. Stars Visible to the Naked Eye.—Heis enumerates the
stars clearly visible to the naked eye in the part of the sky north of 35◦
south declination, as follows:—

1st magnitude, . . . . . . . . . . . .

14

4th magnitude, . . . . . . . . . . . .

313

2d

“

. . . . . . . . . . . .

48

5th

“

. . . . . . . . . . . .

854

3d

“

. . . . . . . . . . . . 152

6th

“

. . . . . . . . . . . . 2010

Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3391

According to Newcomb, the number of stars of each magnitude is
such that united they would give, roughly speaking, somewhere nearly
the same amount of light as that received from the aggregate of those
of the next brighter magnitude. But the relation is very far from exact,
and seems to fail entirely for the fainter magnitudes below the tenth or
eleventh, the smaller stars being less numerous than they should be. In
fact, if the law held out perfectly, and if light was transmitted through
space without loss, the whole sky would be a blaze of light like the surface
of the sun.

819. Light-Ratio and Absolute Scale of Star Magni-
tudes.—It was found by Sir John Herschel, about fifty years ago,
that the light given by the average star of the first magnitude is
just about one hundred times as great as that received from one
of the sixth, and that a corresponding ratio has been pretty nearly
maintained throughout the scale of magnitudes, the star’s of each

√
magnitude being about 2 1 times ( 5 100) brighter than those of the

2
next inferior magnitude. The number which expresses the ratio of
the light of a star to that of another one magnitude fainter is called
the light-ratio.

In the star magnitudes of the maps by Argelander, Heis, and
others, which are most used at present, the divergence from a strict
uniformity of light-ratio is, however, sometimes serious. Some forty
years ago it was proposed by Pogson to reform the system, by adopt-

√
ing a scale with the uniform light-ratio of 5 100, adjusting the first
six magnitudes to correspond as nearly as possible with the mag-
nitudes hitherto assigned by leading authorities, and then carrying
forward the scale indefinitely among the telescopic stars. Until re-
cently this “absolute scale of magnitude,” as it has been called, has

THE LIGHT OF THE STARS.

479
not been much used; but in the New Uranometrias lately made at

Cambridge and Oxford it has been adopted, and astronomers gen-
erally now endeavor to conform to it.

820. Relative Brightness of Different Star Magnitudes.

—In this scale the light-ratio between successive magnitudes is made

√
exactly 5 100, or the number whose logarithm is 0.4000, viz., 2.512.

Its reciprocal is the number whose logarithm is 9.6000, viz., 0.3981.

If b1 is the brightness of a standard first-magnitude star, expressed
either in candle-power or other convenient unit, and bn be the
brightness of a star of the nth magnitude on this scale, we shall
therefore have

4
log bn = log b1 −

(n − 1);

10

(n − 1) in this equation being the number of magnitudes between
the star of the first magnitude and the star of the nth magnitude:

i.e., for a star of the sixth magnitude (n − 1) is 5; so that for a star
of the sixth magnitude, the equation reads,

4
log b6 = log b1 −

× 5 = log b1 − 2.

10

With this light-ratio, every difference of five magnitudes corresponds
to a multiplication or division of the star’s light by 100; i.e., to
make one star as bright as the standard star of the first magnitude
it would require 100 of the sixth, 10,000 of the eleventh, 1,000000
of the sixteenth, and 100,000000 of the twenty-first magnitude.

As nearly standard stars of the first magnitude on this scale we
have α Aquilæ and Aldebaran (α Tauri). The other stars usually
counted as of first magnitude are some of them sensibly brighter,
and others fainter than these. The pole-star and the two “pointers”

are very nearly standard stars of the second magnitude.

821. Negative Magnitudes.—According to this scale, stars
that are one magnitude brighter than those of the standard first would
be of the zero magnitude, (as is the case with Arcturus), and those that
are brighter yet would be of a negative magnitude; e.g., the magnitude of

Sirius is −1.13; and Jupiter at opposition, in conformity to this system,
is described as a star of nearly −2d magnitude, which means that it is
nearly 2.513, or about 16 times brighter than a star of the +1st magni-
tude like Aldebaran. According to Seidel, Jupiter at opposition is about

8 1 times as bright as Vega, which would make its “magnitude” −2.09,

4

Vega being of magnitude 0.2.

THE LIGHT OF THE STARS.

480

822. Relation of Size of Telescope to the Magnitude of
the Smallest Star Visible with it.—If a telescope just shows a
star of a given magnitude, then to show stars one magnitude smaller
we require an instrument having its aperture larger in the ratio of

√

√

2.512 (or 10 100) to 1: i.e. as 1.59:1. Every tenfold increase in the
diameter of the object-glass will therefore carry the power of vision
just five magnitudes lower.

Assuming what seems to be very nearly true for normal eyes and good
telescopes, that the minimum visible for a one-inch aperture is a star of
the ninth magnitude, we obtain the following little table of apertures
required to show stars of a given magnitude.

Star Magnitude . . .

7

8

9

10

11

12

Aperture . . . . . . . . .

0in.40

0in.63

1in.00

1in.59

2in.51

3in.98

Star Magnitude . . .

13

14

15

16

17

18

Aperture . . . . . . . . .

6in.31

10in.00

15in.90

25in.10

39in.80

63in.10

But on account of the increased thickness necessary in the lenses
of large telescopes, they never quite equal their theoretical capacity as
compared with smaller ones.

The smallest stars visible by the Lick telescope (thirty-six inches
aperture), after allowing for all the advantage of the site, are not quite
a whole magnitude below the smallest visible with the Washington tele-
scope; but the number visible will be at least double; since the smaller
stars are vastly the more numerous.

823. Measurement of Star Magnitudes and Brightness.

—Until recently all such measurements were mere eye-estimates,
and even yet all photometric measurements depend ultimately on
the judgment of the eye. But it is possible by the help of instruments
to aid this judgment very much by limiting the point to be decided,
to the question whether two lights as seen are, or are not, exactly
equal, or else making the decision depend on the visibility or non-
visibility of some appearance.

824. Method of Sequences.—For some purposes the unas-
sisted eye is quite as good as any photometric instrument. It judges
directly with great precision of the order of brightness in which a
number of objects stand. In the method of “sequences,” as it is
called, the observer merely arranges the stars he is comparing, say

THE LIGHT OF THE STARS.

481
to the number of fifty or so, in the order of their brightness, tak-
ing care that the stars in each sequence list are nearly at the same
altitude, and seen under equally favorable circumstances. Then he
makes a second sequence, taking care to include in it some of the
stars that were in the first: and so on. Finally, from the whole set
of sequences, a list can be formed, including all the stars contained
in any of them, arranged in the order of brightness. This process
gives, however, no determination of the light-ratio, nor of the num-
ber of times by which the light of the brightest exceeds that of the
faintest.

Variable stars are still often observed in this way, the stars with
which they are compared, being such as have their magnitudes already
well determined.

825. Instrumental Methods.—These are based on two dif-
ferent principles:—
a. The measurement is made by causing the star to disappear
by diminishing its light in some measurable way. This is usually
referred to as the “method of extinctions.”

b. The measurement is effected by causing the light of the star
to appear just equal to some other standard light, by decreasing the
brightness of the star or of the standard in some known ratio until
they are perfectly equalized.

Under the first head come the photometers which act upon the prin-
ciple of “limiting apertures.” The telescope is fitted with some arrange-
ment, often a so-called “cat’s-eye,” by which the available aperture of
the object-glass can be diminished at will, and the observation consists
in determining with what area of object-glass the star is just visible. The
method is embarrassed by constant errors from the fact that the greater
thickness of the glass in the middle of the lens comes into account, and,
still worse, from the fact that the image of the star becomes large and
diffuse on account of diffraction when the aperture is very much reduced.

826. The Wedge Photometer.—The method of producing
the “extinction” by a “wedge” of dark, neutral-tinted glass is much
better. The wedge is usually five or six inches long, by perhaps a
quarter of an inch wide, and at the thick end cuts off light enough
to extinguish the brightest stars that are to be observed. In the

Pritchard form of the instrument this wedge is placed close to the
eye at the eye-hole of the eye-piece; in some other forms it is placed
at the principal focus of the object-glass, where micrometer wires

THE LIGHT OF THE STARS.

482
would be put.

In observation the wedge is pushed along promptly until the star
just disappears, and a graduation on the edge of the slider is read.

The great simplicity of the instrument commends it, and if the wedge
is a good one of really neutral glass (which is not easy to get), the results
are remarkably accurate. But the observations are very trying to the eyes
on account of the straining to keep in sight an object just as it is becoming
invisible. The constant of the wedge must be carefully determined in the
laboratory, i.e., what length of the wedge corresponds to a diminution of
the light of a star by just one magnitude (cutting off 0.602 of its light).

It is convenient to have the slider graduated into inches or millimeters
on the one edge and magnitudes on the other. The “Uranometria Nova

Oxoniensis” is a catalogue of the magnitudes of the naked-eye stars to the
number of 2784, between the pole and 10◦ south declination, observed
with an instrument of this kind by Professor Pritchard, and published
in 1885.

827. Polarization Photometers.—The instruments, how-
ever, with which most of the accurate photometric work upon the
stars has been done, are such as compare the light of the star with
some standard by means of an “equalizing apparatus” based on the
application of the principles of double refraction and polarization.

The light of either the observed star or the comparison star (real
or artificial) is polarized by transmission through a Nicol prism, or else
both pencils are sent through a double refracting prism. The images are
viewed with a Nicol prism in the eye-piece; and by turning this the po-
larized image or images can be varied in brightness at pleasure, and the
amount of variation determined by reading a small circle attached to it.

In the photometers of Seidel and Zöllner, who observed comparatively
few objects, but very accurately, the artificial star with which the real
stars were compared was formed by light from a petroleum lamp, shining
through a small aperture, and reflected to the eye by a plate of glass in
the telescope tube. Professor Pickering, in his extensive work embodied
in the “Harvard Photometry” (published in 1884, and giving the mag-
nitudes of 4260 stars) used the pole-star as the standard, bringing it by
an ingenious arrangement into the same field with the star observed.

Photometric observations in many cases require large and some-
what uncertain corrections, especially for the absorption of light
by the atmosphere at different altitudes, and the final results of
different observers naturally fail of absolute accordance. Still the
agreement between the two catalogues of Pickering and Pritchard

THE LIGHT OF THE STARS.

483
is remarkably close, generally within one or two tenths of a magni-
tude.

Fig. 219.—Pickering’s Meridian Photometer.

828. The Meridian Photometer.—This instrument, contrived
and used by Professor Pickering in the observations of the Harvard Pho-
tometry, consists of a telescope with two object-glasses side by side. The
telescope is pointed nearly east and west, and in front of each object-
glass is placed a silvered glass mirror (M1 and M2, Fig. 219) at an angle
of 45◦. One of the mirrors is so set as to bring the rays of the pole-star to
one object-glass; the other mirror is capable of being turned around the
optical axis of the telescope, in such a way as to command a star at any
part of the meridian, and bring its light into the other object-glass. At
the eye-end is placed, first (i.e., next the object-glass), a double-image
prism D, which separates any pencil of light falling upon it into two, po-
larized at right angles to each other. The “ordinary” rays come through
nearly undeflected, but the “extraordinary” are bent out of their course,
as indicated in the figure, where the pencil A, coming from object-glass

No. 1, is divided into two pencils ao and ae, and in the same way the
pencil B, from the second objective, is divided into bo and be. The angle
of the double-image prism D is so chosen that ao and be will be nearly
parallel to each other, and a suitable diaphragm1 E cuts off the two
other pencils ae and b0. A Nicol prism N receives the two pencils that
come through the diaphragm, and the eye views the two images through
the eye-piece at I. These pencils, being polarized at right angles to each
other, will vary in their brightness when the Nicol is turned, one of them
becoming brighter and the other fainter; and four positions of the Nicol
can be found at which the images will appear equal in brightness, what-
ever may be the original ratio of brightness between the pole-star and the
object observed. On looking into the instrument the observer sees two
stars, the pole-star at rest, the other moving along as in a transit instru-
ment. He simply turns the Nicol until the images are equalized, setting
the Nicol at all the four different positions which will produce the effect,

1The diaphragm E may be replaced by an eye-stop at I.

THE LIGHT OF THE STARS.

484
and reading the graduated circle C. The whole operation consumes not
more than a minute, with the help of an assistant to record the numbers
as read off. The “Harvard Photometry” (usually referred to simply as

“H. P.”) was made by means of an instrument with object-glasses only
two inches and a half in diameter. An instrument with four-inch lenses is
now at work in Cambridge, measuring the magnitudes of all the nearly

80,000 stars of Argelander’s Durchmusterung, which are of the eighth
magnitude or brighter.

829. Photometry by means of Photography.—It has been
found that, excepting a few strongly colored stars, the intensity, or more
simply the size, of the image of a star formed upon a photographic plate
may be used as a measure of its brightness as compared with other
stars taken on the same plate, or on similar plates similarly exposed.

The comparison becomes easier and more accurate if the photographic
telescope is not made to follow the stars exactly, but is allowed to lag
a little so that the star forms a “trail.” It will, therefore, be possible
to use the plates of the great photographic star campaign to determine
star magnitudes as well as positions. But, as has been intimated, there
are some anomalies; certain stars, for instance, that are hardly visible
to the naked eye, photograph as bright stars, and there are others—
red stars—that are abnormally faint on the plate. The exceptions are
numerous enough to make it necessary to use photographic magnitudes
with caution.

830. Star Colors and their Effects on Photometry.—The
stars differ considerably in color. The majority are of a very pure
white, like Sirius and the sun, but there are not a few of a yellowish
hue, like Capella, or reddish, like Arcturus and Antares; and there
are some, mostly small stars, which are as red as garnets and rubies.

We also have, associated with larger ones in double-star systems,
numerous small stars which are strongly green or blue; and there
are a few large isolated stars, which, like Vega, are of a decidedly
bluish tinge.

These differences of color embarrass photometric measurements made
by either of the methods described, because it is impossible to make a red
star look identical with a blue one by any mere increase or diminution
of brightness, and because different observers will differ in setting the
wedge of an extinction photometer according to the color of the star.

Some eyes are abnormally sensitive to blue light, some to red. To the
writer, for instance, Vega is decidedly superior to Arcturus, while the
majority of observers see the difference as decidedly the other way.

THE LIGHT OF THE STARS.

485

831. Spectrum Photometry.—The only completely satisfac-
tory and scientific method would be to compare the spectra of the stars
with some standard spectrum, say that of the pole-star, dividing the
spectrum into a considerable number of portions, and determining and
recording the amount of light in each portion of the spectrum as com-
pared with homologous parts of the standard spectrum. This, of course,
would immensely increase the work of comparing the brightness of the
stars; but it is quite feasible to do it for a few hundred of the brighter
ones, and it would be well worth accomplishment. If we ever succeed in
getting photographic plates equally sensitive to rays of all wave length,
photography would answer the purpose well.

832. Starlight compared with Sunlight.—The light re-
ceived from a first-magnitude star like Vega is about

1

(one

40000 000000
forty thousand millionth) of that from the sun, according to the de-
terminations of Zöllner and others. The measurement is not easy,
and must be taken as having a very considerable margin of error.

Sirius is nearly equivalent to six of Vega, its light being about

1
of the sun’s.

7000 000000

Since the light of a sixth-magnitude star is only

1
of that

100
of a standard first-magnitude, it follows that it would require

4,000000000000 of stars of the 6th magnitude to give us sunlight.

833. Total Light of the Stars.—Assuming what is roughly,
though not exactly, true, that Argelander’s magnitudes follow the
standard scale, it appears that the 324,000 stars north of the equator
enumerated in his Durchmusterung give a light about equal to that
of 240 first-magnitude stars; but it is noticeable that the aggregate
amount of light given by the stars in each of the fainter magnitudes
increases rapidly.

The following is the estimate, substantially according to Newcomb:—

10 stars

(above the 2d magnitude) =

6.0 first-magnitude stars.

37

“
from 2d to 3d

“

=

7.3

“

“

122

“

“

3d to 4th

“

=

9.6

“

“

310

“

“

4th to 5th

“

=

9.8

“

“

1016

“

“

5th to 6th

“

= 12.7

“

“

4322

“

“

6th to 7th

“

= 21.6

“

“

13593

“

“

7th to 8th

“

= 27.1

“

“

57960

“

“

8th to 9th

“

= 46.

“

“

247544

“

“

9th to 9 1

“

= 100.

“

“

2

Total . . . . . . . . . . . . . . . . . . . . . . . . . = 240.

THE LIGHT OF THE STARS.

486

How much to add for the still smaller magnitudes is very uncertain.

Beyond the tenth magnitude the number of small stars does not increase
proportionately fast, so that if we could carry on the account of stars
to the twentieth magnitude, it is practically certain that we should not
find the total light of the aggregate stars of each succeeding magnitude
increasing at any such rate as from the seventh to the tenth. Perhaps
it would be a not unreasonable estimate to put the total starlight of the
northern hemisphere as equivalent to about 1500 first-magnitude stars,
or that of the whole sphere at 3000. This would make the total starlight
on a clear night about 1 of the light of the full moon, and about

1

60

27 000000
that of the sun. The light from the stars which are visible to the naked
eye would not be as much as 1 of the whole. But the above estimate of

25
the light received from the extremely small stars is hardly more than a
mere guess, and may hereafter receive important corrections.

834. Heat from the Stars.—Attempts have been made to
measure by a sensitive thermopile the heat received from certain
stars.

Both Huggins and Stone (about 1870) thought they had
obtained sensible indications of heat from Arcturus and Vega; but
their results have not since been confirmed; and unless the radiation
of invisible energy by these stars is much greater in comparison with
their light than is the case with the sun, it is almost certain that
there must be some illusion in the matter.

1
of the sun’s

40000 000000
heat could hardly be shown by any instrument known to science,
and there is no present reason to suppose that the total heat received
from the stars bears a larger ratio to that received from the sun than
starlight does to sunlight.

835. Amount of Light emitted by Certain Stars.—This,
of course, is something vastly different from that received by us.

A star may emit hundreds of times as much light as the sun, and
yet, if the star is remote enough, the amount that reaches the earth
will be only an excessively minute fraction of sunlight. If l be the
amount of light that we receive from a star, expressed in terms of
sunlight at the earth, then the total amount of light emitted, L, is
given by the simple equation,

L = l × D2,

D being the distance of the star in astronomical units, while L is
expressed in terms of the sun’s light emission.

Turning to the table of stellar parallax (Appendix, Table IV.),

THE LIGHT OF THE STARS.

487
we find that, according to Gill & Elkin, D for Sirius equals 542,000;

5420002
whence, for Sirius,

L =

= 42.0;

7000 000000
that is, the light emitted by Sirius is more than forty times as much
as that emitted by the sun.

If we use the value of the parallax of this star as determined by Abbe,
namely, p = 0 .273, we shall get L = 68, while Gyldén’s value of the
parallax (0 .193) gives L = 155. In either case, however, it is clear that

Sirius emits vastly more light than the sun.

Similarly for the pole-star (p = 0 .06), L = 93; for Vega (p = 0 .14),

L = 69; α Centauri (p = 0 .75), L = 1.9; 70 Ophiuchi (p = 0 .16),

L = 1; 61 Cygni (p = 0 .43), L = 1 ; 24,258 Ll. (p = 0 .26), L = 1 .1

15

113

The companion of Sirius is a little star of the ninth magnitude,
which forms a double-star system with Sirius itself. The light emit-
ted by this companion does not exceed

1
that of Sirius.

12 000

836. Causes of Differences of Brightness in Stars.—It
used to be thought that the stars were all very much alike in mag-
nitude and constitution; not, indeed, without considerable differ-
ences, but us much resembling each other as do individuals of the
same race. It is now quite certain that this is not the case, as is
obvious from the short list of actual light emissions just given.

If the stars were all alike, all the differences of apparent bright-
ness would be traceable simply to differences of distance; but as
the facts are, we have to admit other causes to be equally effec-
tive. The differences of brightness are due, first, to difference of
distance; second, to difference of dimensions, or of light-giving area;
third, to difference in the brilliance of the light-giving surface, de-
pending upon difference of temperature and constitution. There are
stars near and remote, large and small, intensely incandescent and
barely glowing with incipient or failing light.

As Bessel puts it, there is no reason why there may not be

“as many dark stars as bright ones.” As we shall soon see, the
companion of Sirius, though only giving about

1
part as much

12 000
light as Sirius itself, is at least 1 part as heavy; so that, mass for

10
mass, it cannot be

1
part as luminous.

1000

1In making this calculation the magnitudes of the Harvard Photometry were
used.

THE LIGHT OF THE STARS.

488

When we compare stars by the thousand, we can say of the
tenth-magnitude stars, for instance, as compared with the fifth, that
as a class they are more remote; and also, just as truly, that their
average diameters are smaller, and also that their surfaces are less
brilliant ; but we must be careful not to make any assertions of this
sort regarding any one star of the tenth magnitude compared with
a particular individual of the fifth, unless we have some absolute
knowledge of their relative distances. The faint star may be the
larger of the two, or the hotter, or the nearer.

We must know
something beyond their relative “magnitudes” before it is possible
to settle such questions.

837. Real Diameter of Stars.—We have no knowledge what-
ever as to the real diameter of any star. As to the apparent angular
diameter, we can only say negatively that it is insensible, in no case
being known to reach 0 .01. If there be a star of the same diameter
as our sun, at such a distance that its parallax equals one second,

1924
its apparent diameter must be

. (The sun’s mean angular

206265
diameter is 1924 (Art. 276).) This equals 0 .0093—a quantity far
too small to be reached by any direct measurement, especially since,
even in the Lick telescope, the “spurious” disc of a star has a diam-
eter of nearly 1 , and in smaller telescopes is much larger (about

7

0 .4 in a ten-inch telescope).

There is a theoretical connection between the diameter of the diffrac-
tion rings seen around the image of a star in the telescope, and the real

(as opposed to the spurious) diameter of the image; by comparing, there-
fore, the actual size of the rings with the size they should have if the star
were an absolute optical point, we might hope to get a determination
of the star’s diameter. But no such attempts have succeeded, and at
present no way seems open for obtaining the desired measurement. In
some cases, as we shall see later, the mass of a star as compared with
that of the sun can be found; but not its volume or its density, since
these require a knowledge of its diameter.

VARIABLE STARS.

838. A close examination shows that many stars change their
brightness. As a general rule, these which do this, and are called

“variable,” are reddish in their color; while comparatively few of the

THE LIGHT OF THE STARS.

489
white stars belong to this class. The variable stars may be classified1
as follows:—

I. Cases of slow continuous change.

II. Irregular fluctuations of light: alternately brightening and
darkening without any apparent law.

III. Temporary stars, which blaze out suddenly and then disap-
pear.

IV. Periodic stars of the type of o Ceti, usually of long period.

V. Periodic stars of short period, of the type of β Lyræ.

VI. Periodic stars in which the variation is like that which would
be the result of an eclipse by some intervening body—the

Algol type.

839. I. Gradual Changes. On the whole, the changes in the
brightness of the stars since the time of Hipparchus and Ptolemy
have been surprisingly small. There has been no general increase or
decrease in the brightness of the stars as a whole; and there are few
cases where any individual star has altered its brightness by a half
or even a quarter of a magnitude. The general appearance of the
sky is the same as it was 2000 years ago; so that notwithstanding all
the effect of proper motions in the meantime and the whole amount
of the variation that has taken place in the brightness of the stars,
there is no doubt that if either of these old astronomers were to come
to life he would immediately recognize the familiar constellations.

There are a few instances, however, in which it is almost cer-
tain that change has taken place and is going on. In the time of

Eratosthenes the star in the “claw of the Scorpion” (now β Libræ)

was reckoned the brightest in the constellation. At present, it is
a whole magnitude below Antares, which is now much superior to
any star in the vicinity. So when the two stars Castor and Pollux
in the constellation Gemini were lettered by Bayer, the former, α,
was brighter than Pollux, which was lettered β; but β is now no-
tably the brighter of the two. Taking the whole heavens, we find a
considerable number of such cases; perhaps a dozen or more.

840. Missing Stars.—It is a common belief that since accu-
rate star-catalogues began to be made, many stars have disappeared

1This classification is substantially that of Professor Pickering, slightly mod-
ified, however, by Houzeau.

THE LIGHT OF THE STARS.

490
and not a few new ones have come into existence. While it would not
do to deny absolutely that anything of the kind has ever happened,
it is certainly unsafe to assert that it has.

There are a considerable number of cases where stars are now missing
from the older catalogues as published,—nearly, if not quite, a hundred,—
but in almost every instance examination of the original observations
shows that the place as printed was a mistake of some sort which can
now be traced,—sometimes a mistake of the recorder, sometimes in the
reduction of the observation, and sometimes of the press. In a few cases
the star observed was a planet (Uranus, Neptune, or an asteroid); and in
some cases the missing star may have been a “temporary star,” as, for
instance, 55 Herculis, which was observed by the elder Herschel. So many
of the missing stars are now satisfactorily explained that it is natural to
suppose that the few cases remaining are of the same sort.

There, is no known instance of a new star appearing and remaining
permanently visible.

841. II. Stars that exhibit Irregular Fluctuations
in Brightness. The most conspicuous example of this class is the
strange star η Argus (not visible in the United States). This star
ranges all the way from the zero magnitude (in 1843, when, accord-
ing to Sir John Herschel, it was brighter than every star but Sirius)

down to the seventh magnitude, which is its present brightness and
has been ever since 1865.

Fig. 220.—Light Curve of η Argus according to Loomis.

Between 1677 (when it was observed by Halley as of the fourth mag-
nitude) and 1800, it oscillated in brightness, so far as we can judge from
the few observations extant, between the fourth and second magnitudes.

About 1810, it rose rapidly in brightness, and between 1826 and 1850
it was never below the standard first magnitude. When brightest, in

THE LIGHT OF THE STARS.

491

1843, it was giving more than 25,000 times as much light as in 1865. A
singular fact is that the star is in the midst of a nebula which apparently
sympathizes with it to some extent in its fluctuations. (There are other
instances of connection of nebulæ with variable or temporary stars, as
will appear later on.) Fig. 220 represents the “light-curve” of this object
from 1800 to 1870, according to Loomis; who, however, imagines the star
to be periodic, with a period of about seventy years. If so, it ought to be
again increasing in brightness by this time.

α Orionis and α Cassiopeiæ behave in a somewhat similar man-
ner, only the whole range of variation in their brightness is less
than a single magnitude. The catalogue of variable stars shows a
considerable number of other similar cases.

842. III. Temporary Stars. There are eleven well authenti-
cated cases in which new stars have appeared temporarily,—that is,
for a few weeks or months,—blazing up suddenly and then gradually
fading away. The list is as follows:—

1.

134 b.c. The star of Hipparchus.

2.

329 a.d. A star in Aquilla.

3. 1572 a.d. Tycho’s star in Cassiopeiæ.

4. 1600 a.d. P Cygni, 3d magnitude, observed by Jansen.

5. 1604 a.d. Kepler’s star in Ophiuchus.

6. 1670 a.d. 11 Vulpeculæ, 3d magnitude, observed by Anthelum.

7. 1848 a.d. A star of the 5th magnitude, observed by Hind—also
in Ophiuchus.

8. 1860 a.d. T Scorpii, 7th magnitude, in star cluster M 80, ob-
served by Auwers.

9. 1866 a.d. T Coronæ–Borealis, 2d magnitude.

10. 1876 a.d. Nova Cygni, 2d magnitude.

11. 1885 a.d. A star in the nebula of Andromeda, 6th magnitude.

As regards the first of these stars, we know almost nothing. Hip-
parchus has left no record of its position or brightness; but the Chinese
annals mention a star as appearing in Scorpio at just that date, and
probably the same object; though the Chinese observations may refer to
a comet. The appearance of this new star led Hipparchus to form his
catalogue of stars.

As to the second on the list, we know hardly more; the records do
not even make it absolutely certain that the object was not a comet, not
being explicit on the point of its motion.

THE LIGHT OF THE STARS.

492

843. The third is justly famous. When it was first seen by Ty-
cho in November, 1572, it was already brighter than Jupiter, having
probably appeared a few days previously. It very soon became as
bright as Venus herself, being even visible by day. Within a week
or two it began to fail, but continued visible to the naked eye for
fully sixteen months before it finally disappeared. It is not certain
whether it still exists or not as a telescopic star: Tycho determined
its position with as much accuracy as was possible to his instru-
ments; and there are a number of small stars, any one of which is
so near to Tycho’s place that it might be the real object.

There has been an entirely unfounded notion that this star may have
been identical with the “Star of Bethlehem,” it being imagined that the
star is periodically variable, with a period of 314 years. If so, it might
have been expected to reappear in 1886, and it was so expected by certain
persons “as a sign of the second coming of the Lord.” It is difficult to see
how the idea came to be so generally prevalent, as it certainly has been.

Probably every astronomer of any note has received hundreds of letters
on the subject. At Greenwich a printed circular was prepared and sent
out as a reply to such inquiries.

The fifth star, observed by Kepler, was nearly, though not quite,
as bright as that of Tycho, and lasted longer—fully two years. It
also has disappeared so that it cannot now be identified.

844. The ninth star excited much interest. It blazed out be-
tween the 10th and 12th of May as a star of the second magnitude,
remained at its maximum for three or four days, and then, in five or
six weeks, faded away to its original faintness, for it now is, and was
before the outburst, a nine and one-half magnitude star on Arge-
lander’s catalogue, with nothing noticeable to distinguish it from
its neighbors. While at the maximum its spectrum was carefully
studied by Huggins, and exhibited brightly and strongly the bright
lines of hydrogen, just as if it were a sun like our own, but entirely
covered with outbursting “prominences” of incandescent hydrogen.

The tenth star had a very similar history. It also rose to its full
brightness (second magnitude) on November 24, within four hours
according to Schmidt, remained at a maximum for only a day or
two, and faded away to invisibility within a month. But it still
exists as a very minute telescopic star of the fifteenth magnitude.

It was also spectroscopically studied by several observers (by Vogel
especially) with the strange result that the spectrum, which at the

THE LIGHT OF THE STARS.

493
maximum was nearly continuous, though marked by the bright lines
of hydrogen and by bands of other unknown substances, lost more
and more of this continuous character, until at last it became a
simple spectrum of three bright lines like that of a nebula.

845. The eleventh and last of these temporary stars was very
peculiar in one respect; not in its brightness, for it never exceeded
the six and one-half magnitude, but because it appeared right in
the midst of the great nebula of Andromeda, only 12 or 13 dis-
tant from the nucleus. It came out suddenly like all the others, and
faded away gradually in about six months, to absolute extinction
so far as any existing telescope can show. It showed under photo-
metric measurements many fluctuations in brightness, not losing its
light smoothly and regularly but in a rather paroxysmal manner.

Its spectrum, even when brightest, was simply continuous without
anything more than the merest trace of bright lines in it. The eighth
star on the list resembled it in the fact that it appeared in the midst
of a star cluster.

846. IV. Variables of the o Ceti Type. These objects
almost without exception behave like the temporary stars in re-
maining generally faint, suddenly brightening up for a short time,
and then fading back to the original condition; but they do it pe-
riodically. The periods are generally of considerable length, from
six months to two years; but they are very apt to be considerably
irregular, not unfrequently to the extent of several weeks.

The star o Ceti (often called Mira, that is, “the Wonderful”) may
be taken as the type of this class. Its variability was discovered by

Fabricius in 1596. During most of the time it remains simply a faint
twelfth-magnitude star, but once in about eleven months it runs up
to the fourth, third, or even the second magnitude, and then back
again, occupying usually about 100 days in the rise and fall. Its
brightness increases more rapidly than it fails, and it remains at its
maximum for a week or ten days. At the maximum its spectrum is
very beautiful, containing a large number of intensely bright lines
which, however, are not yet certainly identified. Its light-curve is A,
in Fig. 221. A large proportion of the known variables belong to this
class, and many suppose that the temporary stars also belong to it,
differing from their classmates only in the length of their periods.

THE LIGHT OF THE STARS.

494

847. V. Variables of the Type β Lyræ. In these the main
characteristic is that there are two or more maxima and minima in
each period, as if we were dealing with several superposed causes
of variation. The light-curve of β Lyræ is given by B, Fig. 221. Its
period is about thirteen days.

848. VI. Variables of the Algol Type. The sixth and
last class consists of stars which seem to suffer a partial eclipse
at short intervals, their light-curves (C, Fig. 221) being the re-
verse of the o Ceti type. Of this type of stars, Algol, or β Persei,
may be taken as the most conspicuous representative. Its period
is 2d 20h 48m 55s.4, which is subject to almost no variation, except
certain slow changes that appear to be the result of some unknown
disturbance. During most of the time the star remains of the sec-
ond magnitude. At the time of obscuration it loses about five-sixths
of its light, falling to the fourth magnitude in about four and one-
half hours, remaining at the minimum for about twenty minutes,
and then in three and one-half hours recovering its original condi-
tion. During the minimum the spectrum undergoes no considerable
change, though there are suspicions of some slight variations in its
lines.

Fig. 221.—Light Curves of Variable Stars.

Only nine stars are so far known belonging to this class, and
among them are these of the shortest period. One in Cepheus,
discovered by Ceraski in 1880, has a period of two days and eleven
hours; another in Ophiuchus of only twenty hours.

THE LIGHT OF THE STARS.

495

849. Explanation of Variability.—Evidently no simple ex-
planation will hold for all the different classes. For the gradually
progressive changes no explanation need be looked for; on the con-
trary, it is surprising that such changes are no greater than they
are, for the stars are all growing older.

As for the irregular changes, no sure account can yet be given of
them. Where the range of variation is small, as it is in most cases,
one thinks of spots on the surface like these of our own sun, (but
much more extensive and numerous) and running through a period
just as our sun spots do. Let a star with such spots upon it revolve
on its own axis, as of course it must do, and in the combination we
have at least a possible explanation of a great proportion of all the
known cases, both the irregular variables and the regularly periodic.

850. Collision Theory.

—For the temporary stars,
and those of the o Ceti type,

Mr. Lockyer has recently (in
connection with a much more
extended subject) suggested
a collision theory, illustrated
by Fig. 222.

The funda-
mental idea that the phenom-
ena of the temporary stars
may be due to collisions is
not new.

Newton long ago
brought it out, and to some ex-
tent discussed it; but consider-

Fig. 222.—The Collision Theory of

Variable Stars.

ing the probable diameters of
the stars as compared with the distances between them, it seems im-
possible that collisions could have been frequent enough to account
for the number of temporary stars actually observed. Mr. Lockyer,
however, imagines that the temporary stars, and also variable stars
of the o Ceti class, are, in their present stage of development, not
compact bodies, but only pretty dense swarm of meteorites of con-
siderable extent, each such swarm being accompanied by another
smaller one revolving around it in an eccentric orbit, just as comets
revolve around the sun, or as the components of double stars re-
volve around each other. He supposes that the perihelion distance
is so small that the swarms interpenetrate and pass through each

THE LIGHT OF THE STARS.

496
other at the perihelion, which could happen without disturbing the
general motion of either of the two meteoric flocks; but while they
are thus passing, the collisions are immensely increased in number
and violence, with a corresponding increase in the evolution of light.

There are many good points about this ingenious theory, but also
serious objections to it—as, for instance, the great irregularity of
the periods of stars of this class, an irregularity which seems hardly
consistent with such an orbital revolution.

851. Stellar Eclipses.—As to the Algol type, the natural ex-
planation is by means of an eclipse of some sort. The interposi-
tion of a more or less opaque object between the observer and the
star,—a dark companion revolving around it,—would produce just
the effect observed. If, however, this is really the case, the mass
of Algol must be absolutely enormous compared with that of our
sun in order to produce so swift a revolution in the eclipsing body.

But Ceraski’s variable of this type in Cepheus is very refractory and
exhibits changes of period and other phenomena that are extremely
difficult to reconcile with the idea that its obscuration is due to an
eclipse.

852. Number and Designation of Variables.—The “Cat-
alogue of Variable Stars,” of Mr. S. C. Chandler, published in 1888,
contains 225 objects. One hundred and sixty are distinctly periodic
stars: in 12 cases the periodicity is perhaps uncertain, while 14 are
certainly irregular.

The remainder includes the temporary stars and some twenty-five or
thirty stars in respect to the variations of which very little is yet known.

Table VI. in the Appendix presents the principal data for the naked-eye
variables which are visible in the United States.

When a star is discovered to be variable which previously had no
special appellation of its own, it is customary to designate it by one of
the last letters in the alphabet, beginning with R. Thus R Sagittarii is
the first discovered variable in Sagittarius; S Sagittarii is the second;

T Sagittarii, the third, and so on.

853. Range of Variation.—In many cases the whole range is
only a fraction of a magnitude (especially among the more newly
discovered variables), but in a great number it extends from four to
eight magnitudes, the maximum brightness exceeding the minimum
by from fifty to a thousand times; and in a few cases the range is

THE LIGHT OF THE STARS.

497
greater yet. Not unfrequently considerable changes of color accom-
pany the changes of brightness; the star as a rule being whiter at
its maximum, and frequently showing bright lines in its spectrum.

854. Method of Observation.—There is no better way than
that of comparing the star by the eye, or with the help of an opera-
glass, with surrounding stars of about the same brightness at the
time when its light is near the maximum or minimum; noting to
which of them it is just equal at the moment, and also these which
are a shade brighter or fainter.

It is possible for an amateur to do really valuable work in this way,
by putting himself in relation with some observatory which is interested
in the subject. The observations themselves require so much time that it
is difficult for the working force in a regular observatory to attend to the
matter properly, and outside assistance is heartily welcomed in gathering
the needed facts. The observations themselves are not specially difficult,
require no very great labor or mathematical skill in their reduction, and,
as has been said, can be made without instruments; but they require
patience, assiduity, and a keen eye.

STAR SPECTRA.

855. In 1824 Fraunhofer, in connection with his study of the
lines of the solar spectrum, investigated also the spectra of certain
stars, using an apparatus essentially similar to that which is now
employed at Cambridge. He placed a prism in front of the object-
glass of a small telescope and looked at the stars through this, using
a cylindrical lens in the eye-piece to widen the spectrum, which
otherwise would be a mere line.

He found that Sirius, Castor, and many other stars show very few
dark lines in their spectrum, but strong ones; while, on the other hand,
the spectra of Pollux (β Geminorum) and Capella resemble closely the
spectrum of the sun. In all the spectra he recognized the D line, although
it was not then known that it had anything to do with sodium.

856. Observations of Huggins and Secchi.—Almost as
soon as the spectroscope had taken its place as a recognized in-
strument of science it was applied by Huggins to the study of the
stars, and Secchi followed hard in his footsteps. The former studied
the spectra of comparatively few stars, but with all the dispersive
power he could obtain, and in detail; while Secchi, using a much
less powerful instrument, examined several thousand star spectra,

THE LIGHT OF THE STARS.

498
in a more general way, for purposes of classification.

Huggins identified with considerable certainty in the spectra of
α Orionis (Betelgueze) and α Tauri (Aldebaran) a number of ele-
ments that are familiar on the earth, and are most of them promi-
nent in the solar spectrum. He found in the former sodium, mag-
nesium, calcium, iron, bismuth, and hydrogen; and in α Tauri he
reported in addition, tellurium, antimony, and mercury; but these
latter metals have not yet been verified.

857. Classification of Stellar Spectra.—Secchi, in his spec-
troscopic survey, found that the 4000 stars which he observed could
all be reduced to four classes.

Fig. 223.—Secchi’s Types of Stellar Spectra.

The first class comprises the white or blue stars. To it belong

Sirius and Vega, and, in fact, considerably more than half of all the
stars examined. The spectrum is characterized by the great strength
of the hydrogen lines, which are wide, hazy bands, much like the

H and K lines in the solar spectrum. Other lines are extremely
faint or entirely absent; the K line especially, which in the solar
spectrum is especially prominent, in the spectra of most of these
stars is hardly visible.

The second class is also numerous, and is composed of stars with
a spectrum substantially like that of our sun. The H and K lines
are both strong. Capella and Pollux (β Geminorum) are prominent

THE LIGHT OF THE STARS.

499
examples of this class. There are certain stars which form a con-
necting link between these two first classes, stars like Procyon and
α Aquilæ, which, while they show the hydrogen lines very strongly,
also exhibit a great number of other lines between them. The first
and second classes together embrace fully seven-eighths of all the
stars he observed.

The third class includes most of the red and variable stars, some

500 in number, and the spectrum is characterized by dark bands
instead of lines (though lines are generally present also). These
bands, which are probably due to carbon, shade from the blue to-
wards the red; that is, they are sharply defined and darkest at the
more refrangible edge. Occasionally in spectra of this type some
of the hydrogen lines are bright. α Herculis, α Orionis, and Mira

(o Ceti) are fine examples of this third class.

The fourth class is composed of a very small number of stars,
less than sixty so far as now known, mostly small red stars. This
spectrum is also a banded one; but compared with the third class the
bands are reversed, that is, faced the other way, and shaded towards
the blue. These generally show also a number of bright lines. None
of the conspicuous stars belong to this class—none above the fifth
magnitude.

The sixth magnitude star, 152 Schjellerup, may be
taken as its finest example (α, 12h39m; δ, +40◦9 , in the constellation
of Canes Venatici). Fig. 223 exhibits the light-curves of these four
types of spectrum.1

858. Vogel has revised Secchi’s classification of spectra as fol-
lows, making only three main classes, but with subdivisions:

I. (a) Same as Secchi’s I. The white stars.

(b) Nearly continuous; all lines wanting or very faint. β Ori-
onis is the type.

(c) Showing the lines of hydrogen bright, and also the helium
line D3 (Art. 323).

1It is difficult to represent spectra accurately by any process of engraving
that can be readily reproduced in a book like the present. The curve, on the
other hand, is easily managed, and, though it does not please the eye like the
spectrum itself, it is capable of conveying all the information that could be
obtained from the most finished engraving. Dark lines are represented by lines
running downward from the upper boundary line of the curve, and bright lines
by lines running upward, while the bands and their shading are represented by
variations in the contour of the curve.

THE LIGHT OF THE STARS.

500

II. (a) Same as Secchi’s II.

(b) Like II. (a), but showing in addition bright lines which
are not the lines of hydrogen or helium. (This is rare.)

III. (a) Same as Secchi’s III.

(b) Same as Secchi’s IV.

Vogel’s classification is based in part on the very doubtful assumption
that stars of Class I. are hottest and also youngest, while the other classes
belong to stars which are either beginning to fail or are already far gone
in decrepitude. But it is very far from certain that a red star is not just
as likely to be younger than a white one, as to be older. It probably is
now at a lower temperature, and possesses a more extensive envelope of
gases; but it may be increasing in temperature as well as decreasing. At
any rate we have no certain knowledge about its age.

859. Photography of Stellar Spectra.—As early as 1863

Huggins attempted to photograph the spectrum of Vega, and succeeded
in getting an impression of the spectrum, but without any of the lines. In

1872 Dr. Henry Draper of New York, working with the reflector which
he had himself constructed, succeeded in getting an impression of the
spectrum of the same star, showing for the first time four of its hydrogen
lines. The introduction of the more sensitive dry plates in 1876, induced

Mr. Huggins to resume the subject (as did Dr. Draper soon after), and
they soon succeeded in getting pictures showing many lines. The spec-
tra were about half an inch long by 1 or 1 of an inch wide. After the

12

16
lamented death of Dr. Draper in 1882, Professor Pickering took up the
work at Cambridge (U. S.); and with such success that Mrs. Draper, who
had intended to establish and to endow her husband’s observatory as an
establishment for astro-physical research, and a most fitting monument
to his memory, concluded to transfer the instruments to Cambridge, and
there establish the “Draper Memorial.”

860. The Slitless Spectroscope.—Professor Pickering has
attained his remarkable success by reverting to the “slitless spec-
troscope,” arranged in the manner first used by Fraunhofer, and
later revived by Secchi. The instrument consists of a telescope with
the objective corrected not for the visual, but for the photographic
rays, equatorially mounted and carrying in front of the object-glass
one or more prisms, with a refracting angle of nearly 10◦, and large
enough to cover the whole lens.

The refracting edge of the prism is placed east and west, so that the
linear spectrum of a star formed on a plate at the focus of the object-
glass runs north and south. If, now, the clock-work of the instrument is

THE LIGHT OF THE STARS.

501
adjusted to follow the star exactly, the image (i.e., the spectrum) will be
a mere line, broken here and there where the dark lines of the spectrum
should appear. By merely retarding or accelerating the clock a trifle,
the linear spectrum will drift a little sidewise upon the plate, and so will
form a spectrum having a width depending on the amount of this drift
during the time of exposure. If the air is calm the lines of the spectrum
thus formed are as clean and sharp as if a slit were used; otherwise not.

861. The most powerful in-
strument
used
in
this
work
at Cambridge is Dr. Draper’s
eleven-inch photographic refrac-
tor, with four huge glass prisms
in a box in front of the object-
glass, arranged as indicated in

Fig. 224.

With this appara-
tus, photographic spectra of the
brighter stars are now obtained

Fig. 224.

having, before enlargement, a

Arrangement of the Prisms in the
length of fully three inches from

Slitless Spectroscope.

F in the blue of the spectrum to
the extremity of the ultra violet. It is a pity, of course, that the
lower portions of the spectrum below F cannot be reached in the
same way; but no plates sufficiently sensitive to green, yellow, and
red rays have yet been found. The exposure necessary to obtain
the impression of even the most powerful photographic rays is from
half an hour to an hour. Fig. 225 is enlarged about one-third from
one of these photographs of the spectrum of Vega, which extends
far into the ultra violet.

Fig. 225.—Photographic Spectrum of Vega. Cambridge, 1887.

These spectra bear tenfold enlargement perfectly, making them
more than two feet long by two inches in width, and then in the
spectrum of such a star as Capella they show hundreds of lines. It
is simply amazing that the feeble, twinkling light of a star can be

THE LIGHT OF THE STARS.

502
made to produce such an autographic record of the substance and
condition of the inconceivably distant luminary.

862. Peculiar Advantages of the Slitless Spectroscope.

—The slitless spectroscope has three great advantages. First, that
it utilizes all the light that comes from the star to the object-glass,
much of which in the usual form of the instrument is lost in the
jaws of the slit: Secondly, that by taking advantage of the length
of a large telescope, it produces a very high dispersion with even
a single prism: Thirdly, and most important, it gives on the same
plate and with a single exposure the spectra of all the many stars
whose images fall upon it. With the smaller eight-inch instrument
made at Cambridge, and one prism, as many as 100 or 150 spec-
tra are sometimes taken together; as, for instance, in a spectrum
photograph of the Pleiades.

863. Disadvantages of the Slitless Spectroscope.—Per
contra, the giving up of the slit precludes all the usual methods
of identifying the lines by actually confronting them with compari-
son spectra; the comparison prism (Art. 315) cannot be used. This
makes it extremely difficult to utilize these magnificent pictures for
purposes of scientific measurement.

If it turns out that any of the lines photographed in the spectrum
are of atmospheric origin, the difficulty will be largely removed, as these
atmospheric lines could be used as reference points. Professor Pickering
has tried, by the interposition of various vapors in the path of the light
within the telescope tube, to get lines that will answer the purpose, but
thus far without any really satisfactory results. Until this difficulty is
overcome it will be impossible to make these spectra yield us all of the
information which they undoubtedly contain with respect to the motions
of the stars in the line of sight.

Vogel in his photographic work mentioned in connection with this
subject (Art. 802) used an ordinary spectroscope with a slit. This ar-
rangement required a very long exposure, and limited the dispersion it
was possible to employ; but it permitted him to use a hydrogen Geissler
tube placed within the telescope itself, to furnish a comparison spectrum.

864. Twinkling or Scintillation of the Stars.—This is a
purely atmospheric effect, usually violent near the horizon and al-
most null at the zenith. It differs greatly on different nights accord-
ing to the steadiness of the air.

If the spectrum of a star near the eastern horizon be examined

THE LIGHT OF THE STARS.

503
with a spectroscope so held as to make the spectrum vertical, it will
appear to be continually traversed by dark bands running through
the spectrum from the blue end towards the red. At the western
horizon the bands move in the opposite direction, from red to blue;
on the meridian they merely oscillate back and forth.

Cause of Scintillation.—Authorities differ as to the exact explana-
tion of scintillation, but probably it is mainly due to two causes (optically
speaking), both depending on the fact that the air is full of streaks of
unequal density that are carried by the wind.

(1) In the first place, light transmitted through such a medium is
concentrated in some places and turned away from others by simple re-
fraction: so that, if the light of a star were strong enough, a white surface
illuminated by it would look like the sandy bottom of a shallow, rippling
pool of water illuminated by sunlight, with light and dark mottlings
which move with the ripples on the surface. So, as we look towards the
star, and the mottlings due to the irregularities of the air move by us,
we see the star alternately bright and faint; in other words, it twinkles;
and if we look at it in a telescope we shall see that it not only twinkles,
but dances, i.e., it is slightly displaced back and forth by the refraction.

(2) The other cause of twinkling is “interference.” Pencils of light
coming from the star (which optically is a mere point), and feebly re-
fracted by the air in the way above explained, reach the observer by
slightly different routes, and are just in a condition to interfere. The
result of the interference is the temporary destruction of rays of certain
wave-lengths, and the reinforcement of others. At a given moment the
green rays, for instance, will be destroyed, while the red and blue will
be abnormally intense; hence the quivering dark bands in the spectrum.

If the star is very near the horizon, the effects are often sufficient to
produce marked changes of color.

865. Why Planets Twinkle Less than Stars.—This is
mainly because they have discs of sensible diameter, so that there
is a general unchanging average of brightness for the sum total of
all the luminous points of which the disc is composed. When, for
instance, point A of the disc becomes dark for a moment, point

B, very near it, is just as likely to become bright; the interference
conditions being different for the two points. The different points
of the disc do not keep step, so to speak, in their twinkling.

AGGREGATIONS OF STARS.

504

CHAPTER XXI.

DOUBLE

AND

MULTIPLE

STARS.—ORBITS

AND

MASSES

OF DOUBLE STARS.—CLUSTERS.—NEBULÆ.—THE MILKY

WAY.—DISTRIBUTION

OF

STARS.—CONSTITUTION

OF

THE STELLAR UNIVERSE.—COSMOGONY AND THE NEB-

ULAR HYPOTHESIS.

Fig. 226.—Double and Multiple Stars.

866. Double and Multiple Stars.—The telescope shows nu-
merous instances in which two stars lie very near each other, in
many cases so near that they can be seen separate only under a
high magnifying power. These are called “double stars.” At present
something over 10,000 such couples are known, and the number is
continually increasing. In not a few instances we have three stars
together, two of which are usually very close and the third farther
away; and there are several cases of quadruple stars, where there

AGGREGATIONS OF STARS.

505
are two pairs of stars lying close together (as in

Lyræ), or a pair
of stars with two single stars close by; and there are some cases
where more than four form a “multiple star.” Fig. 226 represents a
number of such double and multiple stars.

867. Distance, Magnitudes, and Colors.—The apparent
distances usually range from 30 to 1 , few telescopes being able to

4
separate double stars closer than 1 .

4

In a very large proportion of cases (perhaps about one-third of
all) the two stars are nearly equal; in many others they are extremely
unequal, a minute star near a large one being usually known as its

“companion.”

Not infrequently the components of a double star present a fine
contrast of color; never, however, in cases where they are nearly
equal in magnitude. It is a remarkable fact, as yet wholly unex-
plained, that when we have such a contrast of color the tint of the
smaller star always lies higher in the spectrum than that of the
larger one. The larger one is reddish or yellowish, and the smaller
one green or blue, without a single exception among the many hun-
dreds of such tinted couples now known, γ Andromedæ and β Cygni
are fine examples for a small telescope.

868.

Measurement of

Double

Stars.—Such mea-
sures are generally made with
a filar position-micrometer, es-
sentially such as shown in

Fig. 28 and 29 (Art. 73). The
quantities to be determined are
the distance and position-angle
of the couple. By “distance”

we mean simply the apparent
distance in seconds of arc be-
tween the centres of the two
star discs. The position-angle
of a double star is the angle

Fig. 227.—Measurement of Distance
made with the hour-circle by
and Position-Angle of a Double Star.

the line drawn from the larger
star to the smaller, reckoning around from the north through the
east, as shown in Fig. 227.

AGGREGATIONS OF STARS.

506

Photography may also be used, and promises to become a favorite
method. The first photograph of such an object was by Bond in 1851.

869. Stars Optically and Physically Double.—Stars may
be double in two different ways. They may be merely optically
double,—that is, simply in line with each other, but one far beyond
the other; or they may be really very near together, in which case
they are said to be “physically connected,” because they are then
under the influence of their mutual attraction, and move accord-
ingly.

870. Criterion for distinguishing between Physically
and Optically Double Stars.—This cannot be done off-hand.

It requires a series of measurements long enough continued to de-
termine whether the relative movement of the stars is in a curve or
a straight line. If the stars are really close together their attraction
will force them to describe curves around each other. If they are
really at a great distance and only accidentally in line, then their
proper motions, being sensibly uniform and rectilinear, will produce
a relative motion of the same kind. Taking either star as fixed, the
other star will appear to pass it in a straight line, and with a steady,
uniform drift.

871. Relative Number of Stars Optically Double, and

Physically connected.—Double-star observations practically be-
gan with Sir William Herschel only a little more than a hundred
years ago. When he took up the subject less than 100 such pairs
had been recognized, such as had been accidentally encountered in
making observations of various kinds. The great majority of dou-
ble stars have been discovered so recently that sufficient time has
not yet elapsed to make the criterion above given effective with
more than a small proportion of them. But it is already perfectly
clear that the optically double stars are, as the theory of probability
shows they ought to be, very few in number, while several hundred
pairs have shown themselves to be physically connected, i.e., to be
what are known as “binary” stars, or couples which revolve around
their common centre of gravity.

872. Binary Stars.—Sir W. Herschel began his observations
of double stars in the hope of ascertaining stellar parallax.

He
had supposed in the case of couples where one was large and the
other small that the smaller one was usually a long way beyond the

AGGREGATIONS OF STARS.

507
other (as sometimes is really the fact). In this case there should
be perceptible variations in the distance and position of the two
stars during the course of the year; precisely such variations as
those by which, fifty years later, Bessel succeeded in getting the
parallax of 61 Cygni (Art. 811). But Herschel, instead of finding
the yearly oscillation of distance and position which he expected,
found quite a different, and, at the time, a surprising thing,—a
regular, progressive change, which showed that one of the stars was
slowly describing a regular orbit around the other. To use his own
expression, he “went out like Saul to seek his father’s asses, and
found a kingdom,”—the dominion of gravitation1 extended to the
stars, unlimited by the bounds of the solar system. γ Virginis,
ξ Ursæ Majoris, and ζ Herculis were among the most prominent of
the systems which he pointed out.

At present the number of pairs known to be binary is at least

200, and as many more begin to show signs of movement. (Up to
the present time of course only the quicker moving ones are obvi-
ous.) About fifty have progressed so far,—having made at least one
entire revolution or a great part of one,—that their orbits have been
computed more or less satisfactorily.

873. Orbits of Binary Stars.—The real orbit described by
each of such a pair of stars is always found to be an ellipse, and
assuming the applicability of the law of gravitation, the common
centre of gravity must be at the focus. The two ellipses are precisely
similar, the one described by the smaller star being larger than the
other in inverse proportion to the star’s mass.

So far as the relative motion of the two bodies goes, we may
regard either of them (usually the larger is preferred) as being at
rest, and the other as moving around it in a relative orbit of precisely
the same shape as either of the two actual orbits which are described
around the centre of gravity. But the relative orbit is larger, having
for its semi-major axis the sum of the two semi-axes of the real
orbits (Art. 427).

Usually the relative orbit is all that we can ascertain, as this

1It is not yet fully demonstrated that the motions of binary stars are due to
gravitation, though it is extremely probable, and the burden of proof seems to
be shifted upon those who are disposed to doubt it. See, however, the footnote
to Article 901.

AGGREGATIONS OF STARS.

508
alone can be deduced from the micrometer measures when they
consist only of position-angles and distances measured between the
two stars.

In a few cases where such measures have been made from small stars
in the same field of view with the couple, but not belonging to the
system, or when the couple has long been observed with the meridian
circle, it becomes possible to work out separately the orbit of each star
of the pair with reference to their common centre of gravity; then we can
deduce their relative masses, as for instance, in the case of Sirius and its
companion.

874. Calculation of the Orbit of a Binary Star.—If the
observer is so placed as to view the orbit perpendicularly, he will see
it in its true form and having the larger star in its focus, while the
smaller moves around it, describing “equal areas in equal times.”

But if the observer is anywhere else, the orbit will be apparently
more or less distorted. It will still be an ellipse (since every pro-
jection of a conic is also a conic), but the large star will no longer
occupy its focus, nor will the major and minor axes be apparently
at right angles to each other; nor will they even coincide with the
longest and shortest diameters of the ellipse. In this distorted el-
lipse the smaller star will, however, still describe equal areas in equal
times around the larger one.

Theoretically five absolutely accurate observations of the posi-
tion and distance are sufficient to determine the elements of the
relative orbit, if we assume that the orbital motion is described un-
der the law of gravitation. Practically a greater number are needed
in most cases, because the motions are so slow and the stars so
near each other that observation-errors of 0 .1 (which in most cal-
culations are of small account) here become important. The work
requires not only labor, but judgment and skill, and unless the pair
has completed or nearly completed an entire revolution the result is
apt to be seriously uncertain. So far, as has been said, about fifty
such orbits are fairly well determined. Catalogues, more or less com-
plete, will be found in Flammarion’s book on “Double Stars,” also
in Gledhill’s “Handbook of Double Stars,” and Houzeau’s “Vade

Mecum.” See Table V. in the Appendix.

875. Sirius and Procyon.—The cases of these two stars are
remarkable.

In both instances the large stars have been found from
meridian-circle observations to be slowly moving in little ellipses, al-

AGGREGATIONS OF STARS.

509
though when this discovery was first made neither of them was known
to be double. In 1862 the minute companion of Sirius was discovered by

Clark with the object-glass of the Chicago telescope, then just finished,
and at that time the largest object-glass in the world. And this little
companion was found to be precisely the object needed to account for
the peculiar motion of Sirius itself.

In the case of Procyon, the companion, if it exists, is yet to be dis-
covered; not, however, because it has not been carefully looked for with
the most powerful instruments.

876. Periods.—The
periods
of binary stars, so far as at present
known, vary from twelve years (the
period of δ Equulei) to nearly 1600
years, as ζ Aquarii.

It is possible that one or two oth-
ers may be found with periods even
shorter than fourteen years, and it is
practically certain that as time goes
on, pairs of longer period than 1500
years will present themselves. Com-
puted periods much exceeding 200
years must, however, be received at
present with much reserve.

Fig. 228 shows the apparent orbits

Fig. 228.—Orbits of Binary Stars.

of several of the most interesting binaries. The figure for ζ Cancri (copied
from Gledhill) is incorrect; 1878 and 1827 should be interchanged, and
the arrow reversed.

877. Size of the Orbits.—The angular semi-major axes of
the orbits thus far computed range from about 0 .3 for δ Equulei,
to 18 for α Centauri. The real dimensions are, of course, only to
be obtained when we know the star’s parallax and distance.1 For-
tunately several of the stars whose parallaxes have been determined
are also binary stars. Assuming the data as to parallax and orbits
given in the tables in the Appendix (mainly taken from Houzeau’s

“Vade Mecum”) we find the following short table of results:—

1The real semi-axis of the orbit in astronomical units is simply the angular
semi-axis divided by the parallax.

AGGREGATIONS OF STARS.

510

Assumed

Apparent

Real

Mass.

Name.

Period.

Parallax.

Semi-axis.

Semi-axis.

= 1.

η Cassiopeiæ . .

0 .15

8 .64

57.6

195y.2

5.0

0.38

8.53

22.4

44.0

5.8

Sirius . . . . . . . . .

(0.19)

—

(44.8)1

—

(46.4)

α Geminorum .

0.20

7.54

37.7

997.0

0.54
α Centauri . . . .

0.75

17.50

23.3

77.0

2.14

70 Ophiuchi . . .

0.16

4.79

29.9

94.5

3.0

61 Cygni . . . . . .

0.43

15.40?

35.8?

450.0?

0.23

It is obvious from the table that the double-star orbits are com-
parable with the larger orbits of the solar system; that of α Centauri
being just about equal in size to that of Uranus, and that of η Cas-
siopeiæ not quite double the size of Neptune’s orbit. Of course the
many binary stars whose distance is so great as to make their paral-
lax insensible while their apparent orbits are as large as those given
in the list must have real orbits of still vaster dimensions.

878. Masses of Binary Stars.—When we know both the size
of the orbit of a binary and its period, the mass, according to the
law of gravitation, follows at once from the equation of Article 536,
a3

M + m = 4π2

.

t2

If t and a are given respectively in years and astronomical units
of distance, then, by omitting the factor 4π2, M + m comes out in
terms of the sun’s mass. The final column of the little table above
gives the masses of the six pairs of stars as compared with the mass
of the sun. But the student must bear in mind that the parallaxes
of stars are so uncertain, that these results are to be accepted with
a very large margin of error.

879. Relation between the “Mass-Brightness” of Binary

Stars.—Monck of Dublin has recently called attention to a curious re-
lation between the apparent brightness of a binary, its period and (angu-
lar) distance on the one hand, and its “mass-brightness,” or candle-power
per ton, so to speak, on the other,—a relation which does not involve a
knowledge of the parallax of the stars. He shows that if we let l be the
apparent brightness of any double star, photometrically determined, a
the semi-major axis of its orbit, (in seconds of arc), and t its period in

1The distance 44.8 is computed with Gyldén’s parallax, 0 .193.

AGGREGATIONS OF STARS.

511
years, while k is its “mass-brightness,” or candle-power per ton,—then
we have1

4

2

4

4
k1
l1
t1

3
a1
k

3

3

1 : k2 = l1t a2
a2

=

.

1

1 : l2t2

2
or
k2
l2
t2
a2

(The Observatory, February, 1887.)

He takes ξ Ursæ Majoris as a standard (i.e., its k = 1), and finds for
γ Leonis k = 93; for α Geminorum, 38; for Sirius, 7; for ζ Herculis, 4;
for η Cor. Bor., 1.5; for 70 Ophiuchi, 0.36; for η Cassiopeiæ, 0.22; and
for 61 Cygni, 0.08. In other words, γ Leonis is enormously brilliant in
proportion to its mass, Sirius is medium, while 61 Cygni is extremely
faint. Upon this scale the sun’s mass-brightness would be about 2, judg-
ing from its mass and brightness compared with that of Sirius (Arts. 832
and 877). (We have given above only a few of Mr. Monck’s numbers.)

880. Have the Stars Planets attending them?—It is a
very natural supposition that the minute companions which attend
some of the larger stars may be really planetary in their nature,
shining more or less by reflected light. As to this we can only say,
that while it is quite possible that other stars besides our sun may
have their retinues of planets, it is quite certain that such planets
could not be seen by us with any existing telescope. If our sun were
viewed from α Centauri, Jupiter would be a star of less than the
twenty-first magnitude, at a distance of only 5 from the sun which
itself would be a smallish first-magnitude star.

881. The statement can be verified as follows: Jupiter at opposition
is certainly not equivalent in brightness to twenty stars like Vega (most
photometric measurements make it from eight to fourteen). Assuming,
however, that it is equal to twenty Vegas, its light received by the earth
would be about

1
of the sun’s. At opposition our distance from

2000 000000

Jupiter is about four astronomical units, so that seen from the same
distance as the sun, its light would be sixteen times that quantity, or

(nearly)

1
of the sun’s.

125 000000

1This is strictly true only on the assumption that either the two components
of the double star have the same mass-brightness, or else that the smaller one
is so much smaller that its motion is practically the same as if it were a mere
particle. If the stars of the pair are both about alike, or if they differ greatly
in mass, the equation is practically correct in either case. It breaks down when
the two stars of a pair do not differ much in mass, but do differ greatly in
brightness—probably an unusual case.

AGGREGATIONS OF STARS.

512

Now a ratio of 125,000000 between the light of two stars corresponds
to a difference of 20+ magnitudes

8.0969
log 125,000000 = 8.0969; but

= 20.24 magnitudes. (Art. 820.)

0.4000

Accordingly, if the observer were removed to such a distance that the
sun would appear like a first-magnitude star (as would be the case from
α Centauri), Jupiter would be a star of the twenty-first magnitude. Ac-
cording to Art. 822, it would require a 25-inch telescope to show a star
of the sixteenth magnitude; it would therefore require an instrument
with an aperture of 250 inches, or nearly 21 feet, to show a star five
magnitudes fainter, even if there were no large star near to add to the
difficulty.

882. Triple and Multiple Stars.—There are a considerable
number of objects of this kind, and some of them constitute physical
systems. In the case of ζ Cancri the two larger stars revolve around
their common centre in a nearly circular orbit less than 2 in diameter,
and with a period of about sixty years; while the third star, smaller
and more distant, moves around the closed pair in an orbit not yet
well determined, but with a period that must be several hundred years;
and in its motion there is evidence of a peculiar perturbation, as yet
unexplained. In

Lyræ we have two pairs, each making a very slow
revolution, of periods not yet determined, but probably ranging from 300
to 500 years. And since the pairs have also a common proper motion it is
practically certain that they also are physically connected, and revolve
around their common centre of gravity in a period to be reckoned by
millenniums—the motion during the last hundred years being barely
perceptible. In other cases, as for instance, in the multiple star θ Orionis,
we have a number of stars not organized in pairs, but at more or less
equal distances from each other: we are confronted by the problem of n
bodies in its most general and unmanageable form. Nature challenges
the mathematicians.

883. Clusters.—There are in the sky numerous groups of stars
containing from one hundred to many thousand members. Some of
them are made up of stars visible separately to the naked eye, as the

Pleiades; some of them require a small telescope to resolve them,
as, for instance, the Præsepe in Cancer, and the group of stars in
the sword-handle of Perseus; while others yet, even in telescopes of
some size, look simply like wisps or balls of shining cloud, and break
up into stars only in the most powerful instruments.

AGGREGATIONS OF STARS.

513

In a large instrument some of the telescopic clusters are magnificent
objects, composed of thousands of stellar sparks compressed into a ball
which is dazzlingly bright at the centre and thinning out towards the
edge. In some of them vividly colored stars add to the beauty of the
group. In the northern hemisphere the finest cluster is that known as

13 M Hercules (α, 16h 37m and δ, 36◦ 40 ) not very far from the “apex
of the sun’s way.”

Fig. 229.—The Pleiades.

884. The Pleiades.—Of the naked-eye clusters the Pleiades
is the most interesting and important. To an ordinary eye six stars
are easily visible in it, the six largest ones indicated in the figure

(Fig. 229). Eyes a little better see easily five more—these next in
size in the figure (the two stars of Asterope being seen as one).

A very small telescope (a mere opera-glass) increases the number
to nearly a hundred; and with large instruments more than 400
are catalogued in the group. A few of the stars, apparently in the
cluster, are really only accidentally on the same line of vision, and
are distinguished by proper motions different from those of the rest

AGGREGATIONS OF STARS.

514
of the group; but the great majority have proper motions nearly the
same in amount and direction; they have also identical spectra, and
therefore undoubtedly constitute a single system.

The distances and positions of the principal stars with respect to the
central star Alcyone have been carefully measured three or four times
during the last fifty years. The relative motions during the period have
not proved large enough to admit of satisfactory determination, but it
is clear that such motions exist. A curious and interesting fact is the
presence of nebulous matter in considerable quantity. A portion of this
nebulosity hanging around Merope (the northeast star of the dipper-
bowl in the figure) was discovered many years ago; but it was reserved to
photography to detect very recently other wisps of nebulosity attached
to other stars, especially to Maia, and to show that the whole space
is covered with streaks and streamers of it, emitting light of such a
character as to impress the photographic plate much more strongly than
the eye. The eye cannot see the nebula with the same telescope which is
able to photograph it strongly. The figure shows roughly the outlines of
some of the principal nebulous filaments.

885. Distance of Star-Clusters and Size of the Compo-
nent Stars.—The question at once arises whether clusters, such as
the one mentioned in Hercules, are composed of stars each comparable
with our sun, and separated by distances corresponding to the distance
between the sun and its neighboring stars, or whether the bodies which
compose the swarm are really very small,—mere sparks of stellar matter:

whether the distance of the mass from us is about the same as that of
the stars among which it seems to be set, or whether it is far beyond
them. Forty years ago the accepted view was that the stars composing
the clusters are no smaller than ordinary stars, and that the distance
of the star-clusters is immensely greater than that of the isolated stars.

There are many eloquent passages in the writings of that period based
upon the belief that these star-clusters are stellar universes,—“galaxies,”

like the group of stars to which the writers supposed the sun to belong,
but so inconceivably remote that in appearance they shrank to these
mere balls of shining dust.

It is now, however, quite certain that the other view is correct,—that
star-clusters are among our stars and form part of our universe. Large
and small stars are so associated in the same group in many cases, as
to leave us no choice of belief in the matter. It is true that as yet no
parallax has been detected in any star-cluster; but that is not strange,
since a cluster is not a convenient object for observations of the kind
necessary to the detection of parallax.

AGGREGATIONS OF STARS.

515

886. The Nebulæ.—There are also in the sky a multitude of
faintly shining bodies,—shreds and balls of cloudy stuff that are known
as “nebulæ” (the word meaning strictly a “little cloud”). Some 10,000
or 11,000 of these objects are already catalogued.

Two or three of them are visible to the naked eye. The nebula in
the girdle of Andromeda is the brightest of them, in which, it will be
remembered, the temporary star of 1885 appeared.

The next brightest is the wonderful nebula of Orion, which, in the
beauty and variety of its details, in the interesting relations of the in-
cluded stars, the delicate tint of the filmy light, and in its spectroscopic
interest, far exceeds the other,—indeed, all others.

It is so difficult to represent these delicate objects by any process
available to text-books, that we limit ourselves to cuts of two only, the
great nebula of Andromeda and the “ring-nebula” in Lyra (both from
original drawings), referring the reader to the elaborate engravings that
can be found in the “Philosophical Transactions,” the “Memoirs of the

Royal Astronomical Society,” and other similar works, for adequate rep-
resentations of other interesting objects of this kind.

With a small telescope a nebula cannot be distinguished from a
close star-cluster, and it is quite likely that the clusters and nebulæ
shade into each other by insensible gradations. Forty years ago it
was supposed that there was no distinction between them except
that of mere remoteness,—that all nebulæ could be resolved into
stars by sufficient increase of telescopic power. When Lord Rosse’s
great telescope was first erected, it was for a time reported (and the
statement is still often met with) that it had “resolved” the Orion
nebula. This was a mistake however. No telescope ever has resolved
that nebula into stars nor ever will, for we now know that it is not
composed of stars.

887. Forms and Magnitudes of Nebulæ.—The larger and
brighter nebulæ are, many of them, very irregular in form, stretch-
ing out sprays and streamers in all directions, and containing dark
openings or “lanes.” The so-called “fish-mouth” in the nebula of

Orion, and the dark streaks in the nebula of Andromeda, are strik-
ing examples. Some of these bodies are of enormous volume. The
nebula of Orion, with its outlying streamers, extends over several
square degrees, and the nebula of Andromeda covers more than one.

Now, as seen from even the nearest star, the apparent distance of

Neptune from the sun is only 30 , and the diameter of its orbit

1 . It is perfectly certain that neither of these nebulæ is as near as

AGGREGATIONS OF STARS.

516
α Centauri, and therefore the cross-section of the Orion nebula, as
seen from the earth, must be at least many thousand times the area
of Neptune’s orbit.

Fig. 230.—The Great Nebula in Andromeda.

We do not know what is the real shape of either of these nebulæ,
whether it is a thin, flat sheet, or a voluminous bulk; but some things
about these two nebulæ and several others favor strongly the idea
that their thickness does not correspond to their apparent area.

888. The Smaller Nebulæ.—

The smaller nebulæ are for the most
part elliptical in outline, some nearly
circular, others more elongated, and
some narrow, slender streaks of light.

Generally they are brighter at the
centre, and in many cases the centre
is occupied by a star. Indeed, there
is a considerable number of so-called

“nebulous stars,” that is, stars with
a hazy envelope around them.

There are some nebulæ which
present nearly a uniform disc of light,
and are known as “planetary” neb-

Fig. 231.—The Annular Nebula
ulæ, and there are some which are
in Lyra.

dark in the centre and are known as

“annular ” or ring nebulæ. The finest of these annular nebulæ is the
one in the constellation of Lyra, about half-way between the stars
β and γ; it is shown in Fig. 231.

AGGREGATIONS OF STARS.

517

There are also a number of double nebulæ, and some of these
exhibit a remarkable spiral structure when examined by telescopes
of the largest aperture like the great reflector of Lord Rosse. The
so-called “whirlpool ” nebula in the constellation “Canes Venatici”

is the most striking specimen. This spiral structure, however, is to
be made out only in large telescopes; in fact, very little of the real
beauty of most of these objects is accessible to instruments of less
than 12 inches aperture.

889. Variable Nebulæ.—There are several nebulæ which
vary in their brightness from time to time; one especially, near

Tauri, at times has been visible with a small telescope, while
at other times it is entirely invisible even with large ones. So far no
regular periodicity has been ascertained in such cases.

890. Their Spectra.—One of the earliest and most remark-
able achievements of the spectroscope was its demonstration of the
fact that the light of many of the nebulæ proceeds mainly from
luminous gas. They give a spectrum of six or seven bright lines1

(Fig. 232), three of which are fairly conspicuous; while the rest are
very faint.

Fig. 232.—Spectrum of the Gaseous Nebulæ.

Three of the lines, F , Hγ, and h, are due to Hydrogen. One, the
faintest of all, recently discovered by Copeland, seems to be identical
with the D3 line of the solar chromosphere; but the origin of the rest
remains unknown. The brightest line of the whole number is in the
green, λ, 500.4, and was for a while referred to nitrogen; but under
closer examination the identification breaks down, though the student
will find it still called “the nitrogen line” in many astronomical works.

Mr. Lockyer identifies it with a “fluting” in the low-temperature
spectrum of magnesium, which he has found in the spectrum of me-

1The wave-lengths of the lines are the following, in the order of brightness:

(1) 500.4 ? (2) 495.8 ? (3) 486.1 Hydrogen (F ); (4) 434.0 Hydrogenγ; (5) 410.1

Hydrogen (h); (6) 587.4 Helium (?); (7) 448?

AGGREGATIONS OF STARS.

518
teorites; but this seems rather doubtful, as the nebula line is fine and
sharp, and does not look at all like the relic of a band ; nor has the coinci-
dence been satisfactorily established by actually confronting the nebula
spectrum with that of magnesium.

All the nebulæ which give a gaseous spectrum at all, present this
same spectrum entire or in part. If the nebula is faint only the brightest
line appears, while the Hγ line and the other fainter lines are seen only
in the brightest nebulæ and under favorable circumstances.

891. But not all the nebulæ by any means give a gaseous spec-
trum: those which do so—about half the whole number—are of a
more or less distinct greenish tint, which is at once recognizable in
the telescope. The white nebulæ, the nebula of Andromeda at their
head, give only a continuous and perfectly expressionless spectrum,
unmarked by any lines or bands, either bright or dark. This must
not be interpreted as showing that these nebulæ cannot be gaseous;
for a gas under pressure gives just such a spectrum; but so also
do masses of solid or liquid when heated to incandescence. The
spectroscope simply declines to testify in this case. The telescopic
evidence as to the nature of the white nebulæ is the same as for the
green. They withstand all attempts at resolution, none more firmly
than the Andromeda nebula itself, the brightest of them all.

892. Changes in the Nebulæ.—The question has been raised
whether some of the nebulæ have not sensibly changed, even within
the few years since it has become possible to observe them in detail.

It is quite certain that in important respects the early drawings dif-
fer seriously from these of recent observers; but the appearance of a
nebula depends so much upon the telescope and the circumstances
under which it is used, the features are so delicate and indefinite,
and the difficulty of representing them on paper is such, that very
little reliance can be placed on discrepancies between drawings, un-
less supported by the evidence of measures of some kind.

Thus far, the best authenticated instance of such a change, according
to Professor Holden, is in the so-called “trifid” nebula, in Sagittarius. In
this object there is a peculiar three-legged area of darkness which divides
the nebula into three lobes. A bright triple star, which in the early part
of the century was described and figured by Herschel and other observers
as in the middle of one of these dark lanes, is now certainly in the edge of
the nebula itself. The star does not seem to have moved with reference
to the neighboring stars, and it seems therefore necessary to suppose

AGGREGATIONS OF STARS.

519
that the nebula itself has drifted and changed its form.

As to the nebula of Orion, Professor Holden’s conclusion is, that while
the outlines of the different features have probably undergone but little
change, their relative brightness and prominence have been continually
fluctuating. This, however, can hardly be considered certain; to settle
the question will probably require another fifty years or so, and the
comparison, not of drawings but of photographs—for it is now possible
to photograph the brighter nebulæ.

893. Photographs of Nebulæ.—The first success in pho-
tographing an object of this sort was obtained in 1880 by Dr. Henry

Draper in experiments upon the nebula of Orion; in 1881 and 1882
he greatly improved upon his first essays; and not long before his
death in 1882 he obtained a really fine photograph of the beautiful
object. Mr. Common, of England, with his three-foot silver-on-glass
reflector, and exposures ranging from half an hour upward, has since
then obtained pictures of the same object, still finer, and that seem
to leave little to be desired. They fail of perfection only because the
stars which jewel the nebula so brilliantly as seen by the eye, be-
come in a photograph mere blotches, encroaching upon and cutting
out large patches of the most interesting portions of the nebula.

A number of nebulæ have also been photographed by others as well
as by Common; but he still maintains the lead, and it is hoped that when
his new five-foot reflector is finished, he will procure for us an authentic
record of the present appearance of all the most important objects of
this class.

894. Nature of the Nebulæ.—As to the constitution of these
clouds we can only speculate. In the green nebulæ we can say with
confidence that hydrogen and some other gas or gases are certainly
present, and that the gases emit most of the light that reaches us
from such objects. But how much solid or liquid matter in the form
of grains and drops may be included within the gaseous cloud we
have no means of knowing.

The idea of Mr. Lockyer (a part of his wide induction as to what
we may call the “meteoritic constitution of the universe”) is that they
are clouds of “sparse meteorites, the collisions of which bring about
a rise of temperature sufficient to render luminous one of their chief
constituents,—magnesium.”

How far this theory will stand the test of time and future investiga-
tions remains to be seen. At first view it seems very doubtful whether

AGGREGATIONS OF STARS.

520
the collisions in such a body could be frequent or violent enough to ac-
count for its luminosity, and one is tempted to look to other causes for
the source of light.

895. Number and Distribution of Nebulæ.—Sir William

Herschel was the first extensive investigator of these interesting ob-
jects, and left his unfinished work as a legacy to his son, Sir John

Herschel, who completed the survey of the heavens by a residence of
several years at the Cape of Good Hope. His “General Catalogue”

is the standard of reference for objects of this kind, and contains
about 5000 of them. Between 1000 and 2000 more have been added
since its publication, most of them extremely faint. It is hardly
possible that any important ones remain to be found.

As to their distribution, it is a curious and important fact that
it is in contrast to the distribution of the stars. The stars, as we
shall soon see, gather especially in and about the Milky Way, as do
also the star-clusters; but the nebulæ specially crowd together in
regions as far from the Milky Way as it is possible to get. As has
been pointed out by more than one, this shows, however, not a want
of relation between the stars and the nebulæ, but some “relation of
contrariety.” Precisely what this is, and why the nebulæ avoid the
regions thickly starred, is not yet clear. Possibly the stars devour
them, that is, gather in and appropriate surrounding nebulosity so
that it disappears from their neighborhood.

896. Distance of the Nebulæ.—On this point we have very
little absolute knowledge. Attempts have been made to measure the
parallax of one or two, but so far unsuccessfully. Still it is probable,
indeed almost certain, that they are at the same order of distance
as the stars. The wisps of nebulosity which photography shows
attached to the stars in the Pleiades (and a number of similar cases
appear elsewhere), the nebulous stars of Herschel, and numerous
nebulæ which have a star exactly in the centre,—these compel us to
believe that in such cases the nebulosity is really at the star. Then in
the southern hemisphere there are two remarkable luminous clouds
which look like detached portions of the Milky Way (though they
are not near it), and are known as the Nubeculæ (major and minor).

These are made up of stars and star-clusters, and of nebulæ also,
all swarming together, and so associated that it is not possible to
question their real proximity to each other.

AGGREGATIONS OF STARS.

521

897. Fifty years ago a very different view prevailed. As has been
said already, astronomers at that time very generally believed that there
was no distinction between nebulæ and star-clusters except in regard to
distance, the nebulæ being only clusters too remote to show the separate
stars. They considered a nebula, therefore, as a “universe of stars,” like
our own “galactic cluster” to which the sun belongs, but as far beyond
the “star-clusters” as these were believed to be beyond the isolated stars.

In some respects this old belief strikes one as grander than the truth even.

It made our vision penetrate more deeply into space than we now dare
think it can.

898. The Galaxy, or Milky Way.—This is a luminous belt
which surrounds the heavens nearly in a great circle. It varies much
in width and brightness, and for about a third of its extent, from

Cygnus to Scorpio, is divided into two nearly parallel streams. In
several constellations, as in Cygnus, Sagittarius, and Argo Navis,
it is crossed by dark straight-edged bars that look as if some light
cloud lay athwart it, and in the constellation of Centaurus there is
a dark pear-shaped orifice,—the “coal sack,” as it is called.

The galaxy intersects the ecliptic at two opposite points near
the solstices, making with it an angle of about 60◦. The northern

“galactic pole,” as it is called, lies, according to Sir John Herschel, in
declination +37◦, and right ascension 12h47m; the southern “galactic
pole” is of course at the opposite point in the southern hemisphere.

As Herschel remarks, the “galactic plane” “is to sidereal what the
ecliptic is to planetary astronomy, a plane of ultimate reference, the
ground plan of the sidereal system.”

The Milky Way is made up almost wholly of small stars from
the eighth magnitude down. It contains also a large number of
star-clusters, but (as has been already mentioned) very few true
nebulæ. In some places the stars are too thickly packed for counting,
especially in the bright knots which abound here and there.

(An excellent detailed description of its appearance and course may
be found in Herschel’s “Outlines of Astronomy.”)

899. Distribution of Stars in the Sky: Star-Gauges.—

It is obvious that the stars are not uniformly scattered over the
heavens. They show a decided tendency to collect in groups here
and there, and to form connected streams; but besides this, an
enumeration of the stars in the great star-catalogues shows that
the number increases with considerable regularity from the galactic

AGGREGATIONS OF STARS.

522
poles, where they are most sparse, towards the galactic circle, where
they are most crowded. The “star-gauges” of the Herschels make
this fact still more obvious.

These gauges consisted merely in the counting of the number of stars
visible in the field of view (15 in diameter) of the twenty-foot reflector.

Sir William Herschel made 3400 of these gauges, directing the telescope
to different parts of the sky; and his son followed up the work at the

Cape of Good Hope. Struve’s discussion of these gauges in their relation
to the galactic circle gives the following result:—

Distance from Galaxy.

Number of Stars in Field.

90◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.15

75◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.68

60◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.52

45◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.36

30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.68

15◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.30

0◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.00

900. Structure of the Heavens.—Our space does not permit
a discussion of the untenable conclusions reached by Herschel and
others by combining the unquestionable data derived from observa-
tion, with the unfounded and untrue assumptions that the stars are
substantially of a size and spaced at approximately equal distances.

Many of these conclusions relating to the form and dimensions of
the Milky Way, and of the stellar universe to which our sun belongs,
have become almost classical; but they are none the less incorrect.

It is certain, however, that the faint stars as a class are smaller
and darker and more remote than are the bright ones as a class:

and accepting this, we can safely draw from the star-gauges a few
general conclusions, as follows:—

We present them substantially as given by Newcomb in his “Popular

Astronomy,” p. 491.

1. “The great mass of the stars which compose this (stellar)

system are spread out on all sides in or near a widely extended plane,
passing through the Milky Way. In other words, the large majority
of the stars which we can see with the telescope are contained in a
space having the form of a round, flat disc, the diameter of which
is eight or ten times its thickness.”

2. “Within this space the stars are not scattered uniformly, but
are for the most part collected into irregular clusters or masses,

AGGREGATIONS OF STARS.

523
with comparatively vacant spaces between them.” They are “gre-
garious,” to use Miss Clerke’s expression.

3. Our sun is near the centre of this disc-like space.

4. The naked-eye stars “are scattered in this space with a near
approach to uniformity,” the exceptions being a few star-clusters
and star-groups like the Pleiades and Coma Berenices.

5. “The disc described above does not represent the form of
the stellar system, but only the limits within which it is mostly
contained.” The circumstances are such as to “prevent our assigning
any more definite form to the system than we could assign to a cloud
of dust.”

6. “On each side of the galactic region the stars are more evenly
and thinly scattered, but probably do not extend out to a distance
at all approaching the extent of the galactic region,” or if they do
they are very few in number; but it is impossible to set any definite
boundaries.

7. On each side of the galactic and stellar region we have a neb-
ular region, comparatively starless, but occupied by great numbers
of nebulæ.

As to the Milky Way itself, it is not yet certain whether the stars
which compose it are distributed pretty equally near the galactic cir-
cle, or whether they form something like a ring with a comparatively
vacant space in the middle. The ring theory seems at present rather
the more probable of the two.

As to the distance of the remotest stars in the stellar system, it is
impossible to say anything very definite, but it seems quite certain
that it must be at least so great that light would occupy from 10,000
to 20,000 years in traversing it. If one asks what is beyond the stellar
system, whether the star-filled space extends indefinitely or not, no
certain answer can be given.

901. Do the Stars Form a System?—That is, do they form
an organized unit, in which, as in the solar system, each of the dif-
ferent members has its own function and permanently maintains its
relation to the rest? Gravitation certainly operates, as the binary
stars demonstrate,1 and the stars are moving swiftly in various di-

1This is perhaps rather too strong an expression. It would be truer to
say that none of the observed phenomena of the binary stars contravene the
universality of gravitation, and for the most part they are just what gravitation

AGGREGATIONS OF STARS.

524
rections with enormous velocities, as shown by their proper motions,
and by the spectroscope. The question is whether these motions are
controlled by gravitation, and whether they carry the stars in orbits
that can be known and predicted.

That the stars are organized into a system or systems of some
sort can hardly be doubted, for this seems to be a necessary con-
sequence of their mutual attraction. But that the system is one
at all after the pattern of the solar system, in which the different
members move in closed orbits,—orbits that are permanent except
for the slow changes produced by perturbation,—this is almost cer-
tainly impossible, as was said a few pages back.

902. Is there a Revolution of the Whole Mass of Stars?

—A favorite idea has been that the mass of stars which constitutes our
system has a slow rotation like that of a body on its axis, the plane of
this general revolution coinciding with the plane of the galaxy. Such
a general motion is not in any way inconsistent with the independent
motions of the individual stars, and there is perhaps a slight inherent
probability in favor of such a movement; but thus far we have no evidence
that it really exists—indeed, there hardly could be any such evidence at
present, because exact Astronomy is not yet old enough to have gathered
the necessary data.

903. Central Suns.—A number of speculative astronomers,

Mädler perhaps most prominently, have held the belief that there is
a “central sun,” standing in some such relation to the stellar system as
our sun does to the solar system. It is hardly necessary to say that the
notion has not the slightest foundation, or even probability.

Lambert supposed many such suns as the centres of subordinate
stellar systems, and because we cannot see them, he imagined them to
be dark.

If we conceive of boundaries drawn around our stellar system, and
count all the stars within the limits as members of it, leaving out of the
account all that fall outside, then, of course, our system so limited has
at any moment a perfectly definite centre of gravity. There is no reason
easily accounts for. But they do not demonstrate that the central force varies
inversely with the square of the distance, because we do not know the angle
made by the orbit-plane with the line of sight, except from calculations based
on the assumption that the central force really varies in that way. The orbits,
as directly observed, are consistent with several other laws of central force than
the law of inverse squares. (See “Astron. Journal,” Vol. VIII., article by Prof.

A. Hall.)

AGGREGATIONS OF STARS.

525
why some particular star may not be very near that centre, and in that
sense a “central sun” is possible; but its central position would not give it
any preëminence or rule over its neighbors, or put it in any such relation
to the rest of the stars as the sun bears to the planets.

904. Orbits of Sun and Stars.—It is practically certain that
the motions of the stars are not orbital in any strict sense. Excepting
stars which are in clusters, all other stars are simultaneously acted
upon by many forces drawing in various and opposite directions;
and these forces must in most cases be so nearly balanced that
the resultant cannot be very large. The motions of the stars must
consequently, as a rule, be nearly rectilinear.

Still the balancing of the forces will seldom be exact, and ac-
cordingly the path of a star will almost always be slightly curved;
and since the amount and direction of the resultant force which acts
on the star is continually changing, the curvature of its motion will
alter correspondingly, and the result will be a path which does not
lie in any one plane, but is bent about in all ways like a piece of
crooked wire. It is hardly likely, however, that the curvature of a
star’s path would, in any ordinary case, be such as could be detected
by the observations of a single century, or even of a thousand years.

As has been said before, in connection with the proper motions
of the stars, the probability is that the separate stars move nearly
independently, “like bees in a swarm.”

In the solar system the
central power is supreme, and perturbations or deviations from the
path which the central power prescribes are small and transient. In
the stellar system, on the other hand, the central force, if it exists
at all (as an attraction towards the centre of gravity of the whole
mass of stars) is trifling. Perturbation prevails over regularity, and

“individualism” is the method of the greater system of the stars, as
solar despotism is that of the smaller system of the planets.

905. Cosmogony.—Unquestionably one of the most interest-
ing, and also most baffling, topics of speculation is the problem of
the way in which the present condition of the universe came about.

By what processes have moons and earths and Jupiters and Sat-
urns, come to their present state and into their relation to the sun?

What has been their past history, and what has the future in store
for them? How has the sun come to his present glory and domin-
ion? and in the stellar universe, what is the meaning and mutual
relation of the various orders of bodies we see,—of the nebulæ, the

AGGREGATIONS OF STARS.

526
star-clusters, and the stars themselves?

In a forest, to use a comparison long ago employed by the elder

Herschel, we see around us trees in all stages of their life history.

There are the seedlings just sprouting from the acorn, the slender
saplings, the sturdy oaks in their full vigor, those also that are old
and near decay, and the prostrate trunks of the dead. Can we apply
the analogy to the heavens, and if we can, which of the objects before
us are to be regarded as in their infancy, and which of them as old
and near dissolution?

906. Fundamental Principles of a Rational Cosmogony.

—In the present state of science many of the questions thus sug-
gested seem to be hopelessly beyond the reach of investigation, while
others appear like problems which time and patient work will solve,
and others yet have already received clear and decided answers. In
a general way it may be said that the condensation and aggregation
of rarefied masses of matter under the force of gravitation; the con-
version into heat of the (potential) “energy of position” destroyed
by the process of condensation; the effect of this heat upon the con-
tracting mass itself, and the radiation of energy into space and to
surrounding bodies as waves of light and heat,—these principles con-
tain nearly all the explanations that can thus far be given of the
present state of the heavenly bodies.

907. The Planetary System.—We see that our planetary
system is not a mere accidental aggregation of bodies. Masses of
matter coming hap-hazard towards the sun would move, as comets
do, in orbits, always conic sections to be sure, but of every degree of
eccentricity and inclination. There are a multitude of relations actu-
ally observed in the planetary system which are wholly independent
of gravitation and demand an explanation.

1. The orbits are all nearly circular.

2. They are all nearly in one plane (excepting the cases of some
of the little asteroids).

3. The revolution of all is in the same direction.

4.

There is a curiously regular progression of distances (ex-
pressed by Bode’s law, which, however, breaks down at Neptune).

5. There is a roughly regular progression of density, increasing
both ways from Saturn, the least dense of all the planets in the
system.

AGGREGATIONS OF STARS.

527

As regards the planets themselves, we have

6. The plane of the planets’ rotation nearly coinciding with that
of the orbit (probably excepting Uranus).

7. The direction of the rotation the same as that of the orbital
revolution (excepting probably Uranus and Neptune).

8. The plane of orbital revolution of the satellites coinciding
nearly with that of the planet’s rotation.

9. The direction of the satellites’ revolution also coinciding with
that of the planet’s rotation.

10. The largest planets rotate most swiftly.

908. Origin of the Nebular Hypothesis.—Now this is evi-
dently a good arrangement for a planetary system, and therefore some
have inferred that the Deity made it so, perfect from the first. But to
one who considers the way in which other perfect works of nature usually
come to their perfection—their processes of growth and development—
this explanation seems improbable. It appears far more likely that the
planetary system grew than that it was built outright.

Three different philosophers in the last, century, Swedenborg, Kant,
and La Place (only one of them an astronomer), independently proposed
essentially the same hypothesis to account for the system as we now know
it. La Place’s theory, as might have been expected from his mathematical
and scientific attainments, was the most carefully and reasonably worked
out in detail. It was formulated before the discovery of the great principle
of the “conservation of energy,” and before the mechanical equivalence
of heat with other forms of energy was known, so that in some respects
it is defective, and even certainly wrong. In its main idea, however,
that the solar system once existed as a nebulous mass and has reached
its present state as the result of a series of purely physical processes, it
seems certain to prove correct, and it forms the foundation of all the
current speculations upon the subject.

909. La Place’s Theory.—(a) He supposed that at some past
time, which may be taken as the starting-point of our system’s
history (though it is not to be considered as the beginning of the
existence of the substance of which our system is composed), the
matter now collected in the sun and planets was in the form of a
nebula.

(b) This nebula was a cloud of intensely heated gas, perhaps
hotter, as he supposed, than the sun is now.

(c) This nebula under the action of its own gravitation, assumed

AGGREGATIONS OF STARS.

528
an approximately globular form with a rotation around an axis. As
to this movement of rotation, it appears to be necessary to account
for it by supposing that the different portions of the nebula, before
the time which has been taken as the starting-point, had motions
of their own. Then, unless these motions happened to be balanced
in the most perfect and improbable manner, a motion of rotation
would set in of itself as the nebula contracted, just as water whirls
in a basin when drawn off by an orifice in the bottom. The velocity
of this rotation would become continually swifter as the volume
of the nebula diminished, the so-called “moment of momentum”

remaining necessarily unchanged.

910. (d ) In consequence of this rotation, the mass, instead of
remaining spherical, would become much flattened at the poles,
and as the rotation went on and the motion became accelerated,
the time would come when the centrifugal force at the equator of
the nebula would become equal to gravity, and “rings of nebulous
matter” would be abandoned (not thrown off), resembling the rings
of Saturn, which, indeed, suggested this feature of the theory.

(e) A ring would revolve for a while as a whole, but in time would
break, and the material would collect into a single globe. La Place
supposed that the ring would revolve as if it were solid, the outer
edge, therefore, moving more swiftly than the inner. If this were so,
the mass formed from the collection of the matter of the ruptured
ring would necessarily rotate in the same direction as the ring had
revolved.

(f ) The planet thus formed would continue to revolve around the
central mass, and might itself in turn abandon rings which might
break, and so furnish it with a retinue of satellites.

911. It is obvious that this theory meets completely most of
the conditions of the problem. It explains every one of the facts
just mentioned as demanding explanation in the solar system. In-
deed, it explains them almost too well; for as the theory stands it
meets a most serious difficulty in the exceptional cases of the plan-
etary system, such as the anomalous and retrograde revolutions of
the satellites of Uranus and Neptune. Another difficulty lies in the
swift revolution of Phobos (Art. 589), the inner satellite of Mars.

According to the unmodified nebular hypothesis, no planet or satel-
lite could have a time of revolution less than the time of rotation
which the central body would have, if expanded until its radius be-

AGGREGATIONS OF STARS.

529
comes equal to the radius of the satellite’s orbit; still less could it
have a period shorter than the central body now has.

912. Necessary Modifications.—The principal modifica-
tions which seem essential to the theory in the light of our present
knowledge, are the following. (The small letters indicate the articles
of the original theory to which reference is made.)

(b) It is not probable that the original nebula could have been at
a temperature even nearly as high as the present temperature of the
sun. The process of condensation of a gaseous cloud from loss of heat
by radiation, would cause the temperature to rise, according to the
remarkable and almost paradoxical law of Lane. (Art. 357), until
the mass had begun to liquefy or solidify. And it appears probable
that the original nebula, instead of being purely gaseous, was rather
a cloud of dust than a “fire-mist ”; i.e., that it was made up of finely
divided particles of solid or liquid matter, each particle enveloped
in a mantle of permanent gas. Such a nebula in condensing would
rise in temperature at first as if purely gaseous, so that its central
mass after a time would reach the solar stage of temperature, the
solid and liquid particles melting and vaporizing as the mass grew
hotter. At a subsequent stage, when yet more of the original energy
of the mass had been dissipated by radiation, the temperature of
the bodies which were formed from and within the nebula would
fall again.

And yet La Place may have been right in ascribing a high tempera-
ture to the original nebula. If that were really the case, the only difference
would be that the nebula would be longer in reaching the condition of a
solar system; but it is not necessary, as he supposed, to assume that the
original temperature was high, and that the matter was originally in a
purely gaseous condition, in order to account for the present existence
of such a group of bodies as the incandescent sun and its cool attendant
planets.

913. (d ) As regards the manner in which the planetary bodies
were probably liberated from the parent mass, it seem to be very
doubtful whether the matter accumulated at the equator of the ro-
tating mass would usually separate itself as a ring. If a plastic mass
in swift rotation is not absolutely homogeneous and symmetrical,
it is more likely to become distorted by a lump formed somewhere
on its equator, which lump may be finally detached and circulate
around its primary. The formation of a ring, though possible, would

AGGREGATIONS OF STARS.

530
seem likely to be only a rare occurrence.

La Place seems to have believed also that the outer rings must
necessarily have been abandoned first, and the others in regular
succession, so that the outer planets are much the older. It seems,
however, quite possible, and even probable, that several of the plan-
ets may be of about the same age, more than one ring having been
liberated at the same time; or several planets having been formed
from different zones of the same ring.

(e) In the case where a ring was formed, it is practically certain
that it could not have revolved as a solid sheet; i.e., with the same
angular velocity for all the particles, and with the outer portions,
therefore, moving more swiftly than the inner. If, for instance, the
matter which now constitutes the earth were ever distributed to
form a ring occupying anything like half the distance from Venus
to Mars, it must have been of a tenuity comparable only to that of
a comet. The separate particles of such a ring could have had very
little control over each other, and must have moved substantially
as independent bodies; the outer ones, like remoter planets, making
their circuits in longer periods and moving more slowly than those
near the inner edge of the ring.

914. Explanation of the Anomalous Rotation of Uranus
and Neptune.—When the matter of such a ring concentrates into
a single mass, the direction of the rotation of the resultant planet
depends upon the manner in which the matter was originally dis-
tributed in the ring.

If the ring be nearly of the same density
throughout, the resulting planet (which would be formed at about
the middle of the ring’s width) must have a retrograde rotation like

Uranus and Neptune. But if the particles of the ring are more closely
packed near its inner edge, so that the resultant planet would be
formed much within the middle of its width, its axial rotation must
be direct. In the first case, illustrated in Fig. 233 (a), the particles
near the inner edge of the ring would control the rotation, having a
greater moment of rotation with respect to M , where the planet is
supposed to be formed, than those at the outer edge. The rotation,
therefore, will be retrograde, on account of their greater velocity.

In the other case, Fig. 233 (b), where the inner edge of the ring
is densest, and the planet is formed as at N , much nearer the inner
than the outer edge of the ring, the aggregate moment of rotation
with respect to N is greater for the particles beyond N (because

AGGREGATIONS OF STARS.

531
of their greater distance from it) than that of the swifter moving
particles within, and this determines a direct rotation.

Fig. 233.—Rotation of Planets formed from Rings according to the Nebular

Hypothesis.

The fact that the satellites of Uranus and Neptune revolve back-
wards is not, therefore, at all a bar to the acceptance of the nebular
hypothesis, as sometimes represented. If a new planet should ever
be discovered outside of Neptune it is altogether probable that its
satellites would be found to retrograde.

This is not the only way in which the retrograde rotation of the
outer planets may be accounted for. There are a number of other possible
assumptions as to the constitution of the mass thrown off from the parent
nebula, and the manner in which its particles ultimately coalesce to form
a planet, which lead to a similar result.

915. Faye has recently propounded a modification of the nebular
hypothesis which makes the planets of the “terrestrial group” (Mercury,

Venus, the Earth, and Mars) older than the outer ones. He supposes that
the planets were formed by local condensations (not by the formation
of rings) within the revolving nebula. At first, before the nebula was
much condensed at the centre, the inward attraction would be at any
point directly proportional to the distance of that point from the centre
of gravity of the nebula; i.e., the force could be expressed by the equation

F = ar. After the condensation has gone so far that practically almost
the whole of the matter is collected at the centre of the nebula, the force
is inversely proportional to the square of the distance,—the ordinary law

AGGREGATIONS OF STARS.

532
of gravitation,
b
i.e.,

F =

.

r2

At any intermediate time, during the gradual condensation of the nebula,
the intensity of the central force will, therefore, be given by an expression
having the form
b

F = ar +

,
r2
r being the distance of the body acted upon from the centre of gravity of
the nebula, while a and b are coefficients which depend upon its age; a
continually decreasing as the nebula grows older, while b increases. The
planets formed within the nebula when it was young, i.e., when a was
large and b was small, would have direct rotation upon their axes, while
those formed after a had sensibly vanished would have a retrograde rota-
tion; and this he supposes to be the case with Uranus and Neptune, which
he considers younger than the inner planets. Faye’s work “L’Origine du

Monde,” 1885, contains an excellent summary of the views and theories
of the different astronomers who have speculated upon the cosmogony.

916. Tidal Evolution.—Within a few years Prof. G. H. Dar-
win (son of the great naturalist) has made some important investi-
gations upon the effect of tidal reaction between a central mass and
a body revolving about it, both of them being supposed to be of
such a nature (i.e., not absolutely rigid ), that tides can be raised
upon them by their mutual attraction. We have already alluded
to the subject in connection with the tides (Art. 484). He finds
in this reaction an explanation of many puzzling facts. It appears,
for instance, that if a planet and its satellite have ever had their
times of rotation of the same length as the time of their orbital rev-
olution around their common centre of gravity, then, starting from
that time, either of two things might happen,—the satellite might
begin to recede from the planet, or it might fall back to the central
mass. The condition is one of unstable equilibrium, and the slight-
est cause might determine the subsequent course of things in either
of the two opposite directions. Whenever the time of rotation of the
planet is shorter than the orbital period of the satellite (as it would
naturally become by condensation continuing after the separation
of the satellite), the tendency would be, as explained in Article 484,
slightly to accelerate the satellite, and so to cause it continually to
recede by an action the reverse of that produced by the hypothetical

AGGREGATIONS OF STARS.

533
resisting medium which is supposed to disturb Encke’s comet. This,
it will be remembered, is thought to be the case with our moon.

917. But if by any means the rotation of the planet were re-
tarded, so that its day should become longer than the period of the
satellite, the tides produced by the satellite upon the planet will
then retard the motion of the satellite like a resisting medium, and
so will cause a continual shortening of its period, precisely as in
the case of Encke’s comet. If nothing intervenes, this action will
in time bring down the satellite upon the planet’s surface. Now
in the case of Mars there is a known cause operating to retard its
rotation (namely, the tides which are raised by the sun upon the
planet), and those who accept the theory of tidal evolution suggest
that this was the cause which first made the length of the planet’s
day to exceed the period of the satellite, and so enabled the planet
to establish upon the satellite that retardation which has shortened
its little month, and must ultimately bring it down upon the planet.

Processes such as these of tidal evolution must necessarily be
extremely slow. How long are the periods involved, no one can yet
estimate with any precision, but it is certain that the years are to
be counted by the million.

(We have already referred the reader (Art. 484) to the last chapter
of Ball’s “Story of the Heavens” as containing an excellent and easily
understood explanation of this subject.)

918. Conclusions derived from the Theory of Heat.—

As Professor Newcomb has said, “Kant and La Place seem to have
arrived at the nebular hypothesis by reasoning forwards. Modern
science obtains a similar result by reasoning backwards from actions
which we now see going on before our eyes.”

We have abundant evidence that the earth was once at a much
higher temperature than now. As we penetrate below the surface
we find the temperature continually rising at a rate of about 1° F.

for every fifty or sixty feet, thus indicating that at the depth of a
few miles the temperature must be far above incandescence. Now,
since the surface temperature is so much lower, this implies one (or
both) of two things,—either that heat-making processes are going
on within the earth (which may be true to some extent), or else
that the earth has been much hotter than it now is, and is cooling
off,—and this seems to be a most probable supposition. It is just as

AGGREGATIONS OF STARS.

534
reasonable, as Sir W. Thomson puts it, to suppose that the earth
has lately been intensely heated as to suppose that a warm stone
that one picks up in the field has been lately somewhere in the fire.

919. Evidence derived from the Condition of the Moon
and Planets.—In the case of the moon we find a body bearing
upon its surface all the marks of past igneous action, but now in
appearance intensely cold.

The planets, so far as we can judge
from what we can see through the telescope, corroborate the same
conclusion. Their testimony is not very strong, but it is at least true
that nothing in the aspect of any of them militates against the view
that they also are bodies cooling like the earth; and in the cases of

Jupiter and Saturn many phenomena go to show that they are still

(or at least now ) at a high temperature,—as might be expected of
bodies of such an enormous mass, which, necessarily, other things
being equal, would cool much more slowly than smaller globes like
the earth.

The ratio of surface to mass is smaller as the diameter of a globe
grows larger, and upon this ratio the rate of cooling of a body largely
depends. In short, everything we can ascertain from the observation of
the planets agrees completely with the idea that they have come to their
present condition by cooling down from a molten or even gaseous state.

920. The Sun’s Testimony.—In the sun we have a body
steadily pouring forth an absolutely inconceivable amount of heat,
without any visible source of supply. Thus far the only reason-
able hypothesis to account for this, and for a multitude of other
phenomena which it shows us, is the one which makes it a great
cloud-mantled ball of incandescent gases, slowly shrinking under its
own central gravity, converting continually a portion of its “poten-
tial energy of position”1 into the kinetic-energy of heat, which at
present is mainly radiated off into space.

We say mainly, because it is not impossible that the sun’s tempera-
ture is even yet slowly rising, and that the maximum has not yet been
reached. We are not sure whether all the heat produced by the sun’s
annual shrinkage is radiated into space, or whether a portion is retained

1By “potential energy of position” is meant the energy due to the separated
condition of its particles from each other. As they fall together and towards the
centre in the shrinkage of the sun, they “do work” in precisely the same way as
any falling weight.

AGGREGATIONS OF STARS.

535
within its mass, thus raising its temperature; or whether, again, it ra-
diates more than the amount thus generated, so that its temperature is
slowly diminishing.

921. —That the sun is really shrinking is admittedly only an in-
ference, for the shrinkage must be far too slow for direct observation.

Our case is like that of a man who, to use one of Professor Newcomb’s
illustrations, when he comes into a room and finds a clock in motion,
concludes that the clock-weight is descending, even though its motion
is too slow to be observed. Knowing the construction of the clock and
the arrangement of its gearing, and the number of teeth in each of its
different wheels, he states confidently just how many thousandths of an
inch the weight sinks at each vibration of the pendulum; and looking
into the clock-case and measuring the length of the space in which the
weight can move, and noting its present place, he proceeds to state how
long ago the clock was wound up, and how long it has yet to run. We
must not push the analogy too far, but it is in some such way that we
conclude from our measurements of the sun’s annual output of energy in
the form of heat, how fast it is shrinking, and we find that its diameter
must diminish not far from 250 feet in a year; at least, the loss of po-
tential energy corresponding to that amount of shrinkage would account
for one year’s running of the solar mechanism.

922. Age of the Solar System.—Looking backward, then,
in imagination we see the sun growing continually larger through
the reversed course of time, expanding and becoming ever less and
less dense, until at some epoch in the past it filled all the space now
included within the largest orbit of the solar system.

How long ago that was no one can say with certainty. If we could
assume that the amount of potential energy lost by contraction,
converted into the actual energy of heat and radiated into space,
has been the same each year through all the intervening ages, and
moreover, that all the heat radiated has come from this source
only, without subsidy from any original store of heat contained in
an original “fire mist,” or from energy derived from outside sources,
then it is not difficult to conclude that the sun’s past history must
cover some 15,000000 or 20,000000 years.

But the assumption that the loss of heat has been even nearly
uniform is extremely improbable, considering how high the present
temperature of the sun must be as compared with that of the orig-
inal nebula, and how the ratio of surface to solid content has in-
creased with the lessening diameter.

AGGREGATIONS OF STARS.

536

Nor is it unlikely that the sun may have received energy from
other sources than its own contraction. Altogether it would seem
that we must consider the 15,000000 years to be the least possible
value of a duration which may have been many times more extended.

If the nebular hypothesis and the theory of the solar contraction be
true, the sun must be as old as that,—how much older no one can
tell.

923. Future Prospects.—Looking forward toward the future,
it is easy to conclude also that at its present rate of radiation and
contraction the sun must, within 5,000000 or 10,000000 years, be-
come so dense that the conditions of its constitution will be radically
changed, and to such an extent that life on the earth, as we now
know life, would probably be impossible. If nothing intervenes to
reverse the course of things, the sun must at last solidify and be-
come a dark, rigid globe, frozen and lifeless among its lifeless family
of planets. At least, this is the necessary consequence of what now
seems to science to be the true account of its present activity and
the story of its life.

924. Stars, Star-Clusters, and the Nebulæ.—It is obvious
that the same nebular hypothesis applies satisfactorily to the ex-
planation of the relation of these different classes of bodies to each
other. In fact, Herschel, appealing only to the law of continuity,
had concluded before La Place formulated his theory, that nebulæ
develop sometimes into clusters, sometimes into double or multiple
stars, and sometimes into single ones. He showed the existence in
the sky of all the intermediate forms between the nebula and the
finished star. For a time, about the middle of our century, while
it was generally supposed that all nebulæ were nothing but star-
clusters, too remote to be resolved by existing telescopes, his views
fell rather into abeyance; but when the spectroscope demonstrated
the substantial differences between the gaseous nebulæ and the star-
clusters, they regained acceptance in their essential features; with
perhaps the reservation, that many are disposed to believe that the
rarest even of nebulous matter, instead of being purely gaseous, is
full of solid and liquid particles like a cloud of fog or smoke.

925. The Present System not Eternal.—One lesson seems
to stand out clearly,—that the present system of stars and worlds
is not an eternal one. We have before us irrefragable evidence of

AGGREGATIONS OF STARS.

537
continuous, uncompensated progress, inexorable in one direction.

The hot bodies are losing their heat, and distributing it to the cold
ones, so that there is a steady, unremitting tendency towards a
uniform (and therefore useless) temperature throughout the uni-
verse: for heat does work, and is available as energy only when it
can pass from hotter to cooler bodies, so that this warming up of
cooler bodies at the expense of hotter ones always involves a loss,
not of energy (for that is indestructible), but of available energy.

To use the technical language now usually employed, energy is un-
ceasingly “dissipated ” by the processes which maintain the present
life of the universe; and this dissipation of energy can have but one
ultimate result,—that of absolute stagnation when a uniform tem-
perature has been everywhere attained. If we carry our imagination
backwards we reach at last a “beginning of things,” which has no
intelligible antecedent: if forwards, an end of things in stagnation.

That by some process or other this end of things will result in “new
heavens and a new earth” we can hardly doubt, but science has as
yet no word of explanation.

926. Mr. Lockyer’s Meteoric Hypothesis.—The idea that
the heavenly bodies in their present state may have been formed by
the aggregation of meteoric matter, rather than by the condensation
of a gaseous mass, is not new, and not original with Mr. Lockyer, as
he himself points out. But his adoption and advocacy of the theory,
and the support he brings to it from spectroscopic experiments on
the light emitted by fragments of meteoric stones under different
conditions, has given it such currency within the last two years that
his name will always be justly associated with it. We have already
referred to it in several places (Arts. 850 and 894 especially).

He believes that he finds in the spectra of meteorites, under var-
ious conditions, an explanation1 of the spectra of comets, nebulæ,
and all the different types of stars, as well as the spectra of the

Aurora Borealis and the Zodiacal Light.

Assuming this, he considers that nebulæ are meteoric swarms
in the initial stages of condensation, the separate individuals being

1In some cases the explanation appears to be at least doubtful, especially
in instances where the presence of a sharply defined line in the spectrum of a
heavenly body is referred to the degradation of a band observed in the meteoric
spectrum.

AGGREGATIONS OF STARS.

538
still widely separated, and collisions comparatively infrequent.

As aggregation goes on, the nebulæ become stars, which run
through a long life-history, the temperature first increasing slowly
to a maximum, and then falling to non-luminosity. During this life-
history the stars pass through successive stages, each stage charac-
terized by its own typical spectrum; and according to these views
he divides the stars into six spectroscopic classes, as follows:—

I.

Those which, like the nebulæ, show bright lines or flut-
ings in the spectrum, without dark lines or bands. Vogel’s II. (b)

and I. (c) (Art. 858) are included in this class. γ Cassiopeiæ and

152 Schjellerup are types.

II. Those which show both dark lines and dark flutings in the
spectrum. Vogel’s III. (a): (α Orionis).

III. Those which show the fine dark lines of metals. Vogel’s

II. (a) is included here in part. α Aurigæ and the sun are typical.

IV. Those whose spectra are characterized by the conspicuous
hydrogen lines, all other lines and markings being faint. These stars
are at the summit of the temperature curve. Vogel’s I. (a): Sirius
and Vega.

V. This class, on the descending branch of the temperature
curve (stars past middle life), should have sensibly the same spec-
trum as these of Class III. We cannot be sure in which of the two
classes our sun, for instance, should be counted.

VI. The red stars, their spectrum characterized by heavy ab-
sorption bands. These are stars verging to extinction according to

Mr. Lockyer’s view. Vogel’s III. (b).

It is impossible to go into detail and give here Mr. Lockyer’s
ingenious applications of the theory to explain the phenomena of
the different classes of variable stars. (See Art. 850, however.)

This hypothesis has recently been much strengthened by a most
interesting mathematical investigation of Prof. G. Darwin, who
shows that, if we assume a meteoric swarm comparable in dimen-
sions with our solar system, composed of individual masses such as
fall on the earth, and endowed with such velocities as meteors are
known to have, such a swarm, seen from the distance of the stars,
would behave like a mass composed of a continuous gas. This is not
strange since, according to the kinetic theory of gases, a gas is sim-
ply a swarm of molecules, behaving in just the way the meteorites
are supposed to act. But it follows that the meteoric theory of a

AGGREGATIONS OF STARS.

539
nebula does not in the least invalidate, or even to any great extent
modify, the reasoning of La Place in respect to the development of
suns and systems from a gaseous nebula. The old hypothesis has
no quarrel with the new.

While it would be premature to indorse this speculation of

Mr. Lockyer’s as an established discovery (since there remain in it
many obscure and doubtful points), there can be little doubt that
it marks an epoch in the history of the science.

APPENDIX.

THE GREEK ALPHABET.

Letters.

Name.

Letters.

Name.

Letters.

Name.

A, α,

Alpha.

I, ι,

Iota.

P, ρ, ,

Rho.

B, β,

Beta.

K, κ,

Kappa.

Σ, σ, ς,

Sigma.

Γ, γ,

Gamma.

Λ, λ,

Lambda.

T, τ ,

Tau.

∆, δ,

Delta.

M, µ,

Mu.

Υ, υ,

Upsilon.

E, ,

Epsilon.

N, ν,

Nu.

Φ, φ,

Phi.

Z, ζ,

Zeta.

Ξ, ξ,

Xi.

X, χ,

Chi.

H, η,

Eta.

O, o,

Omicron.

Ψ, ψ,

Psi.

Θ, θ, ϑ,

Theta.

Π, π,

,

Pi.

Ω, ω,

Omega.

MISCELLANEOUS SYMBOLS.

, Conjunction.

A.R., or α, Right Ascension.

, Quadrature.

Decl., or δ, Declination.

, Opposition.

λ, Longitude (Celestial).

, Ascending Node.

β, Latitude (Celestial).

, Descending Node.

φ, Latitude (Terrestrial).

ω, Angle between line of nodes and line of apsides.

Also obliquity of the ecliptic.

DIMENSIONS OF THE TERRESTRIAL SPHEROID.

(According to Clarke’s Spheroid of 1878.

For the spheroid of 1866, see Art. 145.)

Equatorial semi-diameter,—

20 926 202 feet = 3963.296 miles = 6 378 190 metres.

Polar semi-diameter,—

20 854 895 feet = 3949.790 miles = 6 356 456 metres.

Ellipticity, or Polar Compression,

1

.

293.46

APPENDIX.

541

Length (in metres) of 1◦ of meridian in lat. φ =

111 132.09 − 556.05 cos 2φ + 1.20 cos 4φ.

Length (in metres) of 1◦ of parallel, in lat. φ =

111 415.10 cos φ − 94.54 cos 3φ.

1◦ of lat. at
pole = 111 699.3 metres = 69.407 miles.

1◦ of lat. at equator = 110 567.2 metres = 68.704 miles.

These formulæ correspond to the Clarke Spheroid of 1866, used by
the U.S. Coast and Geodetic Survey.

TIME CONSTANTS.

The sidereal day

= 23h 56m 4s.090 of mean solar time.

The mean solar day = 24h 3m 56s.556 of sidereal time.

To reduce a time-interval expressed in units of mean solar time to
units of sidereal time, multiply by 1.00273791; Log. of 0.00273791 =

[7.4374191].

To reduce a time-interval expressed in units of sidereal time to units
of mean solar time, multiply by 0.99726957 = (1 − 0.00273043); Log.

0.00273043 = [7.4362316].

Tropical year (Leverrier, reduced to 1900), 365d 5h 48m 45s.51.

Sidereal year

“

“

“

365 6

9

8.97.

Anomalistic year “

“

“

365 6 13

48.09.

Mean synodical month (new moon to new),

29d 12h 44m 2s.684.

Sidereal month, . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

7 43

11.545.

Tropical month (equinox to equinox), . . . . . .

27

7 43

4.68.

Anomalistic month (perigee to perigee), . . .

27 13 18

37.44.

Nodical month (node to node), . . . . . . . . . . .

27

5

5

35.81.

Obliquity of the ecliptic (Leverrier),

23◦ 27 08 .0 − 0 .4757(t − 1900).

Constant of precession (Struve),

50 .264 + 0 .000227 (t − 1900).

Constant of notation (Peters),

9 .223.

Constant of aberration (Nyrén),

20 .492.

APPENDIX.

542
h,

3
o.

c

20

15

44

31

19

28

51

29
o

1,
d

1
e

33

46

25

40

52

17

33

Ep

°15

0.17

0.13

0.50

0.20?

0.26

0.62

0.52

0.64

0.46

Jan.

1850.

Alb

83

14

29

Longitude
at

327

245

100

103

160

334
yitv at

1

6

?

7

0.85

0.83

1.00

0.38

2.65

1.18

0.91

0.88

14

15

22

54

49

58

38

43

Gra

Surface.

27.65

7

6
of

27

21

17

37

54

50

59
ihelion.

°
er

75

11

90

45

Longitude
or

P

129

100

333

149

170

.y

?

?

?

?

1

1

?

1

1

?

it

203

220

10

92

11

1710

Oblate-
ness,

Elliptic-

9

52

00

53

39

17

53

54

25
de.

6
of
to

°33 19 00 23

46

56

20

13

12

No

0

46

75

48

30

98

73

?

?

?

?

?

Longitude

Ascending

112

130

°15

33

°27

50

°05

49

Orbit.

7

6

3

Inclination

23

24

26

Equator

8

2

2

35

00

10

41

40

20
s
s
m

.09

.67

SYSTEM.

23

±

24

°00 23 00 51

37

18

29

46

47
of

4

48

43

Ecliptic.

22

7

3

0

1

1

2

0

1
m
m
m

10
h

?

?

?

?

Inclination
to

Axial

7

7

21

56

37

55

14

Time

Rotation.

d
h
h
h

9

25

27

23

23

24

10

.

SOLAR
y

1.

=

Eccen-
tricit

.20560

.00684

.01677

.09326

.07631

.04825

.05607

.04634

.00896

?

1.39

3.40

4.85

5.58

4.01

1.33

0.72

1.22

1.11

THE
ater

12.44?

W

OF
y
er

Density.

p

35

1.

cit

=

?

to

8.1

6.0

4.2

3.4

0.25

0.61

2.23?

0.86

1.00

0.72

0.24

0.13

0.22

0.20

21.9

18.5

15.0

11.1

⊕

Orbit-
eloV
second).

23

(miles

?

1.

000

65

85

=

1

760
d
olume.

1309

⊕

310

0.020

0.056

0.920

1.000

0.152

50000

.24

.62

.00

.88

.60

.86

.46

.02

.78

V

1
erio
in
ears.

0

0

1

1

4

ELEMENTS

11

29

84

P

Y

164

?

1.

?

=

1

31

81

1

8

1

.

1

AL

0.78

9

94.9

14.7

17.1

⊕

331100

1.000

70000

316.0
d
solar

?

ys).

.96926

.7008

.2564

.9505

.414

.580

.22

.82

.11

Mass.

1.

?

erio

.

da

87

=

1

1

1

1

1

.

1

1

1

1

Sidereal

P

224

365

686

1681

4332

.

1048

3490

22600

19380

(mean

10759

30686

60181

1.00000

425000

331100

26821000

2668700

3093500

1.

I.—PRINCIP

.4

.273

.382

.972

.000

.534

..

.92

.17

.03

.39

.0

.2

.9

.5

.1

.3

.0

.9

.6

=

0

0

0

1

0

.

9

4

4

10

⊕

Miles.

36

67

92

109

Mean

Dist.

141

257

483

886

Millions

1781

2791
of

Diameter

ABLET

,400

,163

,030

,700

,918

,230

,500

,000

,900

,800

Miles.

2

3

7

7

4

Mean

100?

86

73

31

34
jor

866
in
of

.5
t

.1

.9

Axis

Orbit.

.387099

.723332

.000000

.523691

.767265

.202800

.538861

.18329

.05508

“

0

0

1

1

2

5

9

Semi-Ma

19

30
mean

67

24

50

20

4

2

13

.

.

.

.

to

.

to

.

to
to
to
to

Angular

04

07
to

.6

.8

.7

Symbol.

Â

Ã

Apparen

⊕

Ä

1○

Å

Æ

Ç

È

Diameter.

32

31

5

11

3

32

14

3

2

Symbol.

Â

Ã

⊕

Ä

1○

Å

Æ

Ç

È

Earth
us
us
en

Earth
us

Mercury

V

The

Mars

Ceres

Jupiter

Saturn

Uran

Neptune
on
us

Name.

en

Sun

Mo

Mercury

V

The

Mars

Ceres

Jupiter

Saturn

Uran

Neptune

Planets.

Planets.

Name.

Planets.

Planets.

errestrial

T
jor

Ma
errestrial

T
jor

Ma

APPENDIX.

543
y
vit
of
the
tly
the
with
gra
of
t
of
ered
sensibly
equator.

from
with
v
fic
enden
diameters
sensibly
are
planes
inner

Remarks.

ci
rest

Lassell.

e

Englemann’s.

5
plane

Disco
y

Sp

Orbits

The

The
indep
b

Retrograde.

Retrograde.

3.44.

coinciden
planet’s
are

The
data

Damoiseau.

the
orbits
coincide
the
ring.

(

.

in
of

1

81

?

?

?

?

?

?

?

1

?

?

?

?

?

?

?

erms

4600

Mass

T

Primary

.00001688

.00002323

.00008844

.00004248

Miles
in

7?

5?

Diam’r

600?

800?

500?

500?

400?

800?

2162

2500

2100

3550

2960

1100?

1200?

1500?

3500?

2000?

1000?

2000?

.y

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.0013

.0072

.0299

.1189

.0296

Eccen-
tricit

.05491

0

0

0

0
the

=

18

—
of
celes-

±

—

—

—

—

.

(1900)

—

—

◦

Plane

◦

◦

◦
of

(1900)

satellites

75
to

28

28

27
celestial

Planet’s

43

Orbit.

to

05

SYSTEM.

of

—

—

—

—

—
equator

—

—

◦

Inc.

out
plane

57
inner

◦

Ab

Inclination

5
to
tial

6

Inc.

equator

(1900).

120

3

40

57

53

00

10

49

09

48
of
to

.2

8

◦

◦

SOLAR

“

“

“

“

4.

“

“

“

08

17

47.2

08

38

59

57

10

38

31.5

51 82

12

34

Inc.

− “

“

“

−

Orbit

Ecliptic.

◦

◦

◦

◦

◦

◦

5

2

1

1

1

=

=

26

25

28

27

27

18

97

THE

145

.7

.9
d.

TURN
s
s
on
on
de,

2

MARS

6.9

OF

JUPITER

35

53.7

35.9

SA

.

URANUS

No
erio

—

—

1900*,

(1900).

NEPTUNE

P
m

OF
m

5

OF

OF

(1900).

44

OF

28

17

59

Ascend.

orbits

35

Ascend.

for
satellites
orbits
ecliptic

32

OF
dic
h
h
of
of
of

Asc.

3
of

10
of

◦

25

12

—

—

18

13

18
ring.)

de

◦
inner
de

◦

165

Syno
d
d 1

3

7

29

16

Long.

No
ecliptic

168

(5
and

Long.

No
plane

=

Long.

184

TELLITES

.5

.1

.5

.7

.1

.2

TELLITES

TELLITES

TELLITES
d.

s

TELLITES

TELLITE

SA
s
s
s
s
s

5

6.9

9.3

6.4

11

15

54.0

SA

33

42.1

33.4

11.2

SA

25.6

11.6

23.2

27.0

17.1

SA

21

37.2

29.5

SA

44

SA
erioP
m
m
m
m
m
m

7

2

43

39

17

27

13

42

32

37

53

18

41

25

41

39

54

20

27

56
h
h
h
h
h
h

7

7

6

3

8

6

7

3

18

13

16

22

21

17

12

22

12

16

11

21
d
d
d
d
d
d

Sidereal

1

1

3

7

1

1

2

4

2

4

8

5

27

16

13

21

79

13

II.—THE

840

850

650

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

5

Miles.

14

Mean

238

261

415

664

167

117

157

186

238

332

771

934

225

120

167

273

365

225

Distance
in

1

2

ABLET
anet
in

Pl

.27035

.771

.921

.933

.430

.057

.486

.11

.98

.95

.34

.86

.48

.07

.58

.52

.46

.12

.90

.93
of

2

6

5

9

3

3

4

6

8

7

60

15

26

20

25

59

10

17

22

12

Dist.

Equatorial

Radii

..

“

“

“

“

“

“

“

“

.

.

.

1877

1610

1789

1684

1672

1655

1848

1671

1851

1787

1846

..

ery

.

v

..

hel,
hel,

.

“

“

“

“

..

Cassini,

Bond,

Cassini,

Disco

..

.

.

Hersc

P

Hersc

.

“

“

“

“

“

D.

“

“

D.

“

“

.

Hall,

Galileo,

W.

J.

Huyghens,

G.

J.

Lassell,

W.

Lassell,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

os

..

ymede
ys
erion
on

.

etus
briel

Name
eron
eth

Mo

Phob

Deimos

Io

Europa

Gan

Callisto

Mimas

Enceladus

T

Dione

Rhea

Titan

Hyp

Iap

Ariel

Um

Titania

Ob

Nameless

1

2

1

2

3

4

1

2

3

4

5

6

7

8

1

2

3

4

1

APPENDIX.

544

GE.

7

00

17

37

10

27

50

47

40

23

20

53

3

54

45

23

23

27

50

33

41

19

14

33

15

ASSAP

◦

5

Inclination.

12

12

29

14

10

12

15

11

55

74

44

162
of

8

9

1

55

39

16

29

21

25

15

41

15
de.

2

0

7

6

36

19

56

24

58

35

42

29

10

No

◦

PERIHELION

72

84

55

Longitude

334

121

297

101

101

245

146

209

269

254
of

ONE

4

3

1888.

45

54

50

17

11

47

59

48

47

48

7

9

4

32

15

21

58

11

48

28

20

45

48

◦

THAN
erihelion.

43

50

93

Longitude

P

158

306

116

276

241

108

319

116

149

165

.y

MORE

T

LONGITUDES,
tricit

A

0.845781

0.552541

0.655951

0.809797

0.726775

0.405128

0.755119

0.626277

0.549017

0.821544

0.954996

0.931088

0.967281

DES

Eccen

VED

U

OBSER

BUREA

Aphelion

Distance.

4.096935

4.665563

5.162744

5.612808

5.582030

4.897332

6.196874

5.771986

5.970090

10.459624

33.67129

33.61592

35.41121

DU

BEEN

VE
erihelionP Distance.

0.342309

1.344665

1.072638

0.589892

0.883240

2.073322

0.860592

1.326420

1.738140

1.024728

0.77511

1.19961

0.58895

HA
d.

ANNUAIRE
erio
ears.)

P

3.307

5.209

5.505

5.462

5.812

6.507

6.629

6.686

7.566

WHICH

(Y

13.760

71.48

72.63

76.37

THE

7

9

4

8

20

30

25

23

13

22

11

25

15

OM
v.

y
v.

FR

Mar.

No

Ma

Mar.

Sept.

Sept.

Sept.

Jan.

Jan.

Sept.

Jan.

Oct.

No

COMETS
erihelion
assage.

P

P

DIC

1885,

1883,

1886,

1879,

1886,

1885,

1852,

1884,

1881,

1885,

1884,

1887,

1835,

O

RI

....

.....

.

ift

.......

ucleus,n ........

..........

w

.........

e

..........

oks

...........

..........

l
l-S
k
l

...........

...........

III.—PE

2d

Name

............

e
e
e
e
k
e
ers
emp
emp
emp
ya
uttle
ons-Bro

Enc

T

T

Brorsen

Winnec

T

Biela,

D’Arrest

F

T

P

Olb

Halley

ABLE

.

No

T

1

2

3

4

5

6

7

8

9

10

11

12

13

APPENDIX.

545

TABLE IV.—STELLAR PARALLAX.

The star-places are only approximate,—sufficient merely for identification.

The parallaxes are mostly from Houzeau’s “Vade Mecum,” but a few are in-
cluded from later authorities.

ears.

Name of Star.

Approx. Place.

Authority.

Date.

Method.

t-Y

(1900)

arallax.P

Distance.

Magnitude.

α
δ

Ligh

1. β Cassiopeiæ . . .

2

0h 4m +58◦ 36

Pritchard

1887

Photography

0 .187

17.4

2. Groombridge 34 . 8

0 12

+43 26

Auwers

1867

Mer. Circle

0.307

10.6

3. η Cassiopeiæ . . . . 4

0 43

+57 17

O. Struve

1856

Micrometer

0.154

21.3

4. µ Cassiopeiæ . . .

5.5

1

1

+54 26

O. Struve

1856

Micrometer

0.342

9.5ff

Pritchard

1887

Photography

0.036

90.6

5. Polaris . . . . . . . . . . 2

1 22

+88 46

Peters

1846–47 Mer. Circle

0.073

44.7ff

Pritchard

1887

Photography

0.052

62.7

6.

Eridani . . . . . . .

4.4

3 16

−43 27

Elkin

1882

Heliometer

0.143

22.8

7. o2 Eridani . . . . . . 4.4

4 11

− 7 46

Gill

1882

Heliometer

0.166

19.6ff

Hall

1884

Micrometer

0.223

14.6

8. α Aurigæ . . . . . . . 1

5

9

+45 54

Peters

1846

Mer. Circle

0.046

70.9ff

(Capella)

O. Struve

1856

Micrometer

0.305

10.7

9. α Canis Maj. . . . 1

6 41

−16 35

Gyldén

1864

Mer. Circle

0.193

16.99

=

(Sirius)

Abbe

1868

Mer. Circle

0.273

11.9

;

Gill & Elkin

1882

Heliometer

0.380

8.6

10. α Geminorum . . . 1.5

7 28

+32

6

Johnson

1856

Heliometer

0.198

16.5

(Castor)

11. α Canis Min. . . . 1

7 34

+ 5 29

Auwers

1873

Micrometer

0.123

26.5

(Procyon)

12. ι Ursæ Maj. . . . . 3.5

8 52

+48 26

Peters

1846

Mer. Circle

0.133

24.5

1858ff

13. Lalande 21 185 . . 7

10 56

+36 42

Winnecke

Micrometer

0.506

6.6

1872

14. Lalande 21 258 . . 8.5

11

1

+44 02

Auwers

1863

Micrometer

0.262

12.4ff

Krüger

1864

Micrometer

0.260

12.6

15. Groombr’ge 1830

6.5

11

7

+38 32

Peters

1846

Micrometer

0.226

14.49

>

>

Wichmann

1848

Micrometer

0.182

17.9>

=

O. Struve

1850

Micrometer

0.034

95.9

>

Johnson

1854

Heliometer

0.033

98.8>

>

;

Brünnow

1873

Micrometer

0.095

34.3

16. Oeltzen 11 677 . . 9.5

11 15

+66 23

Geelmuyden

1880

Mer. Circle

0.265

12.3

17. α Bootis . . . . . . . . 1

14 11

+19 42

Peters

1846

Mer. Circle

0.127

25.7ff

(Arcturus)

Johnson

1856

Heliometer

0.138

23.6

18. α Centauri . . . . . . 1

14 33

−60 25

Henderson

1842

Mer. Circle

0.913

3.69

=

(Probably best.)

Gill & Elkin

1882

Heliometer

0.750

4.35

;

19. Oeltzen 17 415 . . 9

17 37

+68 28

Krüger

1863

Micrometer

0.243

13.4

Pond

1817ff

20. γ Draconis . . . . . . 2

17 54

+51 30

Mural Circle

0.127

25.7

Gyldén

1877

21. 70, p, Ophiuchi . 4.5

18 00

+ 2 33

Krüger

1859–63 Micrometer

0.162

20.1

22. α Lyræ . . . . . . . . . 1

18 34

+38 41

W. Struve

1840

Micrometer

0.262

12.49

>

>

(Vega)

Peters

1846

Mer. Circle

0.103

32.4>

>

>

=

Johnson

1856

Heliometer

0.140

23.3

O. Struve

1859

Micrometer

0.140

23.3>

>

>

Brünnow

1873

Micrometer

0.188

17.3>

>

;

Hall

1881

Micrometer

0.134

24.3

23. α Draconis . . . . . . 5

19 33

+69 30

Brünnow

1870–73 Micrometer

0.234

14.0

24. 61 Cygni . . . . . . . . 5.5

21

2

+38 15

Bessel

1838–40 Heliometer

0.348

9.49

>

>

Pogson

1853

Heliometer

0.384

8.5>

>

>

>

Johnson

1854

Heliometer

0.397

8.2>

>

>

=

O. Struve

1852–53 Micrometer

0.505

6.5

Auwers

1863

Micrometer

0.564

5.8>

>

>

Ball

1878

Micrometer

0.465

7.0>

>

>

>

Hall

1880–86 Micrometer

0.270

12.1>

>

;

Pritchard

1887

Photography

0.432

7.55

25.

Indi . . . . . . . . . . . 5.5

21 56

−57 69

Gill & Elkin

1881–82 Heliometer

0.222

14.6

26. Lacaille 9352 . . . . 7.5

22 59

−36 26

Gill

1881

Heliometer

0.285

11.4

27. Bradley 3077 . . .

6

23

8

+56 37

Brünnow

1873

Micrometer

0.069

47.3

28. 85 Pegasi . . . . . . . 6

23 57

+26 35

Brünnow

1873

Micrometer

0.054

60.4

APPENDIX.

546
it.

orbit

Orb
of
computed.

k.

k.

k.

k.

k.

k.

k.

k.

k.

hard.

hard.

hard.

er.

satisfactory
et
erc
erc
erc
erc
erc
erc
erc
erc
erc
y

Calculator

No

Dubiago.

Dob

Duner.

Pritc

Dob

Seeliger.

Pritc

Elkin.

Pritc

Dob

Dob

Dob

Dob

Thiele.

Grub

Dob

Dob

Dob

Thiele.

5

6

5.5

6.5

9

6.2

5

4

6

7

7

6.5

7

3.2

7.3

7

3.5

6.5

3

MECUM.”

10.5

4.5,

6,

3,

6,

1,

9.5,

5.5,

4,

1,

4.5,

4,

6,

4.5,

6,

3,

4,

6,

2,

5.5,

2.5,

ADE

Magnitudes.

−

“V

U’S

?

eriastron.

1869.92

1864.79

1850.26

1889.44

1877.13

1866.02

1875.26

1875.97

1808.90

1843.70

1841.81

1770.69

1751.96

1836.68

1901.95

1798.80

1741.00

1826.93

1750.33

P

HOUZEA
c.

?

OM

0.480

0.463

0.263

0.591

0.302

0.391

0.416

0.526

0.467

0.350

0.536

0.708

0.461

0.896

0.624

0.634

0.733

0.752

0.344

FR
a.

.30

0.657

1.284

0.827

8.53

1.46

0.853

2.580

4.790

0.70

0.890

4.860

0.717

3.97

8.639

1.54

1.98

5.88

7.538

ARS.

0

17.50

ST
y

Y
d.

12
to
erioP
y

25.71

34.411

41.576

44.0

54.25

60.327

60.80

77.42

94.44

95.50

BINAR

10

110.82

127.35

129.84

185.01

195.24

349.10

407.04

845.86

996.85

OF

5

2

32

10

49

43

32

48

01

13

22

32

40

35

36

47

11

53

27

10

◦ 9

2

9

6

0

(1880).

16

60
δ

+

+18

+31

+30

−

+27

+18

+32

−

+

+26

+

+19

−

−

+57

+23

+20

+34

+32

.6

V.—ORBITS
m

4.2

5.3

8

38.8

18.2

39.7

41.8

11.8

32.0

59.4

37.7

22.0

45.8

45.0

35.6

41.8

48.0

31.3

10.2

27.0

(1880).

h

6

8

9

0

0

7
α

21

13

16

15

17

11

14

17

15

14

20

12

10

16

ABLET

.

..

..

ar.

....

....

chi

.....

.....

Bor.

j.

Bor.

.....

.....

Bor.

St

Ber.

......

......

.......

eiæ

......

.........

Ma
of
tauri

Ophiu
otis

Comæ

Herculis
p,

Andromedæ

Equulei

Herculis

Coronæ

Cancri

Cen

Coronæ

Leonis

Virginis

Cassiop

Leonis

Coronæ

Geminorum

2

Ursæ

Bo

Aquarii
δ

42
ζ
η

Sirius
µ
ζ
ξ
α

70,
γ
ω
ξ

4
γ
η

36
γ
σ
α

Name

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

APPENDIX.

547

TABLE VI.—THE PRINCIPAL VARIABLE STARS.

A selection from S. C. Chandler’s catalogue of 225 variables (Astro-
nomical Journal, Sept. 1888), containing such as are visible to the naked
eye, have a range of variation exceeding half a magnitude, and can be
seen in the United States.

Place, 1900.

.

Range of

Name.

Period (days).

Remarks.

Variation.

No
α
δ

1

T Ceti . . . . . . .

0h 16m.7

−20° 37

5.1 to 7

65 ±

Irregular.

2

R Andromedæ

0

18.8

+38

1

5.6

13

411.2

3

R Sculptoris . .

1

22.4

−33

4

5.8

7.8

207

(Mira.

Varia-

4
o Ceti . . . . . . . .

2

14.3

− 3 26

1.7

9.5

331.3363
tions in length
of period.

5
ρ Persei . . . . . .

2

58.7

+38 27

3.4

4.2

33
nAlgol.

Period

6
β Persei . . . . . .

3

1.6

+40 34

2.3

3.5

2d 23h 48m 55s.43
now shortening.

7
λ Tauri . . . . . .

3

55.1

+12 12

3.4

4.2

3d 22h 52m 12s
nAlgol type, but

8

Aurigæ . . . . .

4

54.8

+43 41

3

4.5

Irregular
irregular.

9
α Orionis . . . .

5

49.7

+ 7 23

1

1.6

196 ?

Irregular.

10
η Geminorum

6

8.8

+22 32

3.2

4.2

229.1

11
ζ Geminorum

6

58.2

+20 43

3.7

4.5

10d

3h 41m 30s

12

R Canis Maj.

7

14.9

−16 12

5.9

6.7

1d

3h 15m 55s

Algol type.

13

R Leonis Min.

9

39.6

+34 58

6

13

373.5

Period short’ing.

14

R Leonis . . . . .

9

42.2

+11 54

5.2

10

312.87

15

U Hydræ . . . . .

10

32.6

−12 52

4.5

6.3

194.65

16

R Ursæ Maj. .

10

37.6

+69 18

6.0

13.2

305.4

Period short’ing.

17

R Hydræ . . . . .

13

24.2

−22 46

3.5

9.7

496.91

Period short’ing.

18

S Virginis . . . .

13

27.8

− 6 41

5.7

12.5

376.0

19

R Bootis . . . . .

14

32.8

+27 10

5.9

12.2

223.9

20
δ Libræ . . . . . .

14

55.6

− 8

7

5.0

6.2

2d

7h 51m 22s.8

Algol type.

21

S Coronæ . . . .

15

17.3

+31 44

6.0

12.5

360.57

22

R Coronæ . . . .

15

44.4

+28 28

5.8

13

Irregular

23

R Serpentis . .

15

46.1

+15 26

5.6

13

357.6

24
α Herculis . . . .

17

10.1

+14 30

3.1

3.9

Two or three months, but very irreg.

nShortest period

25

U Ophiuchi . .

17

11.5

+ 1 19

6.0

6.7

20h

7m 41s.6
known.

26

X Sagittarii . .

17

41.3

−27 48

4

6

7.01185

27

W Sagittarii . .

17

58.6

−29 35

5

6.5

7.59445

28

Y Sagittarii . .

18

15.5

−18 54

5.8

6.6

5.76900

29

R Scuti . . . . . .

18

42.1

− 5 49

4.7

9

71.10

(Secondary min-

30
β Lyræ . . . . . . .

18

46.4

+33 15

3.4

4.5

12d 21h 46m 58s.3
imum
about
midway.

31

R Cygni . . . . . .

19

34.1

+49 58

5.9

13

425.7

32
χ Cygni . . . . . .

19

46.7

+32 40

4.0

13.5

406.045

Period length’ing

33
η Aquilæ . . . . .

19

47.4

+ 0 45

3.5

4.7

7d

4h 14m

0s.0

34

S Sagittæ . . . .

19

51.4

+16 22

5.6

6.4

8d

9h 11m

35

X Cygni . . . . . .

20

39.5

+35 13

6.4

7.7

15d 14h 24m
nMinimum
not
constant.

36

T Vulpeculæ .

20

47.2

+27 52

5.5

6.5

4d 10h 29m

37

T Cephei . . . . .

21

8.2

+68

5

5.6

9.9

383.20

38
µ Cephei . . . . .

21

40.4

+58 19

4

5

432 ?

39
δ Cephei . . . . .

22

25.4

+57 54

3.7

4.9

5d

8h 47m 39s.97

40
β Pegasi . . . . .

22

58.9

+27 32

2.2

2.7

Irregular

41

R Aquarii . . . .

23

38.6

−15 50

5.8

11

387.16

42

R Cassiopeiæ

23

53.3

+50 50

4.8

12

429.00

INDEX.

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

A.

Algol, or β Persei, 848.

Aberration of light, annual, 99, 224–

Almagest of Ptolemy, 500, 700, 795.

226; used to determine the solar

Almucantar defined, 12.

parallax, 692; diurnal, 226*; spher-

Altitude defined, 21; parallels of, 12;
ical and chromatic, 39.

of pole equals latitude, 30; of sun,

Aboul Wefa, discoverer of the lunar
how measured with sextant, 77.

variation, 457.

Altitude and azimuth instrument,

Absolute scale of stellar magnitude,

71.

819.

Amplitude defined, 22.

Acceleration of Encke’s comet, 710;

Andromeda, the nebula in, 886;
of Winnecke’s comet, 711; of the
temporary star in the nebula of,
sun’s equator, 283–285; secular, of

845.

moon’s mean motion, 459–461; sec-

Andromedes, the, 780, 784, 786.

ular, of moon’s mean motion as af-

Angle, position, of a double star, 868;
fected by meteors, 778.

of the vertical, 156.

Achromatic object-glasses for tele-

Angular and linear dimensions, 5;
scopes, 41.

velocity under central force, its law,

Actinometer of Violle, 341.

408, 409.

Adams, J. C., the discovery of Nep-

Annual equation of the moon’s mo-
tune, 654; investigation of the orbit
tion, 458; motion of the sun, 172,
of the Leonids, 785.

173.

Adjustments of the transit instru-

Annular eclipse, 382; nebula in Lyra,
ment, 60.

888.

Aerolites. See Meteorites.

Anomalistic month, the, 397, note;

Age, relative, of the planets, 913, 915;
revolution of the moon, 250; year
of the solar system, 922; of the sun,
defined, 216.

359.

Anomaly defined, mean and true,

Air-currents at high elevations, 773,

189.

note.

Apertures, limiting, in photometry,

Airy, G. B., density of the earth,

825.

169.

Apex of the sun’s way, 805.

Albedo defined and determined, 546;

Apparition, perpetual, circle of, 33.

of Jupiter, 644; of Mars, 583; of

Apsides, line of, defined, 183; its rev-

Mercury, 558; of the Moon, 259;
olution in case of the earth’s orbit,
of Neptune, 660; of Saturn, 636; of

199; its revolution in case of the

Uranus, 648; of Venus, 572.

moon’s orbit, 454; its revolution in

INDEX.

549

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

case of the planets’ orbits, 527.

Barometer, changes of, affecting at-

Arc of meridian, how measured, 147.

mospheric refraction, 91; effect on

Areal or areolar velocity, law of, un-
height of the tides, 480.

der central force, 402–406.

Barometric error of a clock and its

Areas, equable description of, in
compensation, 52.

earth’s orbit, 186, 187.

Beginning of the day, 123; of the

Argelander, his Durchmusterung
year, 222.

and zones, 795, 833; his star mag-

Benzenberg, experiments on the de-
nitudes, 817, 833.

viation of falling bodies, 138.

Argus, η, 841.

Bessel, the parallax of 61 Cygni, 809,

Ariel, the inner satellite of Uranus,

811; formation of comets’ tails, 728;

650.

his “zones,” 795.

Aries, first of, 17.

Biela’s comet, 744–746.

Aristarchus, method of determin-

Bielids, the, 746, 780, 784, 786.

ing the sun’s distance, 666, 670.

Bielid meteorite, Mazapil, 784.

Artificial horizon, the, 78.

Binary
stars,

872–875;
number

Ashes of meteors, 775.

known
at
present,

872;
their

Aspects of planets defined by dia-
masses, 877, 878;
their “mass-
gram, 494.

brightness,” 879; their orbits, 875–

Asteroids, the, or minor planets,

877.

592–601; theories as to their origin,

Bissextile year, explanation of term,

600.

219.

Astræa, the fifth asteroid, discovered

Black Drop, the, at a transit of
by Heneke, 593.

Venus, 681.

Astro-Physics defined, 2.

Bode’s law, 488, 489.

Atlases of the stars, 793.

Bolides, or detonating meteors, 768.

Atmosphere of the moon, 255–257;

Bolometer, the, of Langley, 343.

of Venus, 573; height of the earth’s,

Bond, G. P., first photograph of a

98.

double star, 868.

Attraction, intensity of the solar,

Bond, W. C., discovery of Hyperion,
on the earth, 436; within a hollow

643; of Saturn’s dusky ring, 637.

sphere,169; of universal gravitation,

Boyle, law of, 360, note.

161, 162.

Brahe, Tycho. See Tycho.

Axis of the earth, its direction, 14; of

Bredichin, his theory of comets’

the earth, disturbed by precession,
tails, 731, 732.

206; of the sun, its direction, 282.

Brightness of comets, 699; of planets

Azimuth defined, 22; determination
in various positions, Mercury, 551;
of, 127; method of reckoning, 22; of

Venus, 563, 568; Mars, 579; Aster-transit instrument, its adjustment,
oids, 596, 599; Jupiter, 610; Saturn,

60.

632; Uranus, 647; Neptune, 660; of an object in the telescope, 38; of

B.

shooting stars, 773; of stars, causes

Baily, determination of the density
of the difference in this respect, 836;
of the earth, 166.

of stars, its measurement, 823–831.

Balance, common, used in determin-
ing the density of the earth, 170;

C.

torsion, used in determining the

Calendar, the, 217–223.

density of the earth, 165.

Callisto,
the
outer
satellite
of

INDEX.

550

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Jupiter, 621, 627.

occultation, 33.

Calories of different magnitude, 338,

Circular motion, central force in,
note.

411.

Candle power, its mechanical equiv-

Clairaut’s equation concerning the
alent, 776; power of sunlight, 332,
ellipticity of the earth, 155.

333.

Clarke, Col., dimensions of the

Candle standard, 333, note.

earth, 145 and Appendix.

Capture theory of comets, 740.

Classification of stellar spectra, 857,

Cardinal points defined, 20.

858.

Carlini, earth’s density, 168.

Clerke, Miss A. M., her history of

Carrington, law of the sun’s rota-
astronomy, Preface, 570, 626, note,
tion, 283, 284.

723, 746, 900.

Cassegrainian telescope, 48.

Clocks, general remarks on, 50.

Cassini, J. D., discovery of the divi-

Clock-breaks (electric), 57.

sion in Saturn’s ring, 637; discovery

Clock-error, or correction, and rate,
of four satellites of Saturn, 643.

53; or correction determined by

Catalogues of stars, 795.

transit instrument, 59.

Cavendish, the torsion balance, 165.

Clusters of stars, 883–885.

Celestial latitude and longitude, 178,

Coggia’s Comet, 730.

179; sphere, conceptions of it, 4.

Collimating eye-piece, 67.

Cenis, Mt., determination of the

Collimation of transit instrument,
earth’s density, 168.

60.

Central force, motion under it, 400–

Collimator, the, used with transit

410; force, its measure in case of cir-
instrument, 60; of a spectroscope,
cular motion, 411.

311.

Central suns, 807, 903.

Collision theory of variable stars,

Centrifugal force of the earth’s rota-

850.

tion, 154.

Colors of stars in photometry, 830; of

Ceres, discovery of, 592.

double stars, 867.

Chandler, S. C., catalogue of vari-

Colures defined, 25.

able stars, 852, Appendix, Table

Comet, Biela’s, 744; Donati’s, 727,

VI.

730, 747; Encke’s, 710, 743; great, Changes on the moon’s surface, 268;
of 1882, 748–752; Halley’s, 742;
in the nebulæ, 892.

Winnecke’s, 711.

Characteristics of different meteoric

Comets, acceleration of Encke’s and
swarms, 783.

Winnecke’s, 710, 711; brightness

Charts of the stars, 798.

of, 699, 723; capture theory of,

Chemical elements recognized in

740; chemical elements in, 724, 725;
comets, 724, 725; elements recog-
constituent parts, 713; contraction
nized in stars, 856; elements recog-
of head when near the sun, 715;
nized in the sun, 315–317.

danger from, 753, 754; density of,

Chromatic aberration of a lens, 39.

720; designation of, 697; dimen-

Chromosphere, the, 291, 322, 363.

sions of, 714, 717; ejection theory,

Chronograph, the, 56.

741; fall upon earth or sun, proba-

Chronometer, the, 54; longitude by,
ble effect, 754; groups of, with sim-

121 [A].

ilar orbits, 705; their light, 721;

Circle, the meridian, 63.

their masses, 718, 719; and mete-

Circles of perpetual apparition and
ors, their connection, 785–787; na-

INDEX.

551

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

ture of, 737; their orbits, 700–709;

Crust of meteorites, 761.

origin of, 738–741; perihelia, dis-

Curvature of comet’s tails, 729.

tribution of, 706; physical charac-

Curvilinear motion the effect of
teristics, 712; planetary families of,
force, 401.

739; their spectra, 724, 726; super-Cycle, the metonic, 218.

stitions regarding them, 695; their

Cyclones as proofs of the earth’s ro-
tails or trains, 713, 717, 728–736;
tation, 143.

variations in brightness, 723; visi-

D.

tors in the solar system, 709.

Comparison of starlight with sun-

Dalton, his law of gaseous mixtures,
light, 334, 832.

360, note.

Compensation pendulums, 51.

Danger from comets, 753, 754.

Compensation
of
pendulum
for

Darkening of the sun’s limb, 337.

barometric changes, 52.

Darwin, G. H., rigidity of the earth,

Components of the disturbing force,

171; tidal evolution, 484, 916.

445.

Dawes, diameter of the spurious discs

Co-ordinates, astronomical, 20.

of stars, 43; nucleoli in sun spots,

Common, A. A., photographs of

293.

nebulæ, 893.

Day, the civil and the astronomical,

Conics, the. 422, 423.

117; effect of tidal friction upon its

Connection between comets and
length, 461; changes in its length,
meteors, 785–787.

144; where it begins, 123.

Constant of aberration, the, 225; the

Declination defined, 23; parallels of,
solar, 338–340.

23; determined with the meridian

Constancy, secular, of the mean dis-
circle, 128.

tances and periods of the planets,

Degree of the meridian, how mea-

526.

sured, 135, 147.

Constellations, list of, 792; their ori-

Deimos, the outer satellite of Mars,
gin, 791.

590, 591.

Contact
observations,
transit
of

Delisle, method of determining the

Venus, 679–682.

solar parallax, 682.

Contraction theory of solar heat,

Denning, drawings of Jupiter’s red

356.

spot, 618.

Conversion of R. A. and Decl. to lat-

Density of comets, 720; of the earth,
itude and longitude, 180.

determinations of it, 164–170; of

Copernicus, his system, 503; “Tri-
the moon, 246; of a planet, how de-
umphans,” 809.

termined, 540; of the sun, 279.

Cornu, determination of the earth’s

Detonating meteors, or “Bolides,”

density, 166; photometric observa-

768.

tion of eclipses of Jupiter’s satel-

Development of sun spots, 297.

lites, 630.

Dhurmsala meteorite,
ice-coated,

Corona, the solar, 291, 327–331, 364.

765.

Cosmogony, 905–917.

Diameter (apparent) as related to

Cotidal lines, 475.

distance, 6; of a planet, how mea-

Craters on the moon, 265–267.

sured, 534.

Crew,

H., spectroscopic observa-

Differential method of determining
tions of the sun’s rotation, 285,
a body’s place, 129; method of de-
note.

termining stellar parallax, 811.

INDEX.

552

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Diffraction of an object-glass, 43.

how measured, 134, 135; constitu-

Dione, fourth satellite of Saturn, 643,
tion of its interior, 171; its dimen-
note.

sions, Appendix and 145; its dimen-

Dip of the horizon, 81.

sions determined geodetically, 147–

Disc, spurious, of stars in telescope,

149; form of, from pendulum ex-

43.

periments, 152–155; growth of, by

Discovery of comets, 698.

accession of meteoric matter, 777;

Dissipation of energy, 925.

mass compared with that of the

Distance of the moon, 239; of the
sun, 278; its mass and density, 159–
nebulæ, 896; and parallax, rela-

170; its orbit, form of, determined,
tion between, 84; of a planet in as-

182; principal facts relating to it,
tronomical units, how determined,

132; proofs of its rotation, 138–143.

515–518; of the stars, 808–815; of

Earth-shine on the moon, 254.

the sun, 274, 275, also Chap. XVI.

Eccentricity of the earth’s orbit, dis-

Distinctness of telescopic image, its
covered by Hipparchus, 184; of the
conditions, 39.

earth’s orbit, how determined, 185;

Distribution of the nebulæ, 895; of
of the earth’s orbit, secular change
the stars, 899; of the sun spots, 301.

of, 198; of an ellipse defined, 183,

Disturbing
force,
the,

439–444;

506.

force, diagram of, 441; force, its res-

Eclipses, duration of lunar, 373; du-
olution into components, 445.

ration of solar, 385; number in a

Diurnal aberration, 226*; inequal-
year, 391–393; recurrence of, the
ity of the tides, 471; parallax, 82,
saros, 395; of Jupiter’s satellites,

86; phenomena in various latitudes,

627–630; of the moon, 370–378; of

191.

the sun, 379–390; total, of the sun,

Divisions of astronomy, 2.

as showing the solar atmosphere
and corona, 319, 323.

Doerfel proves that a comet moves
in a parabola, 700.

Ecliptic, the, defined, 175; obliquity
of, 176; limits, lunar, 374, 375; lim-

Donati’s comet, 727, 730, 747.

its, solar, 386.

Doppler’s principle, 321, note.

Double stars, 866–879; their colors,

Effective temperature of the sun,

867; criterion for distinguishing be-

351.

tween these optically and physically

Ejection theory of comets and mete-
double, 870; method of measuring
ors, 741.

them, 868; optically and physically

Elbowed equatorial, the, 74.

double, 869; having orbital motion,

Electrical registration of observa-
see Binary Stars.

tions, 56.

Electro-dynamic theory of gravita-

Draper, H., oxygen in the sun, 316;
photograph of the nebula in Orion,
tion, 602.

893; photography of stellar spectra,

Elements, chemical, not truly ele-

859; memorial, the, 859.

mentary, 318; chemical, recognized

Duration, future, of the sun, 358; of
in comets, 724, 725; chemical, rec-
sun spots, 300.

ognized in stars, 856; chemical,
recognized in sun, 316, 317; of a

E.

planet’s orbit, 505–508.

Earth, the, her annual motion proved

Elkin, stellar parallaxes, 808, 814,
by aberration and stellar paral-

815, and Appendix, Table IV.

lax, 174; approximate dimensions,

Ellipse defined, 183; described as a

INDEX.

553

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

conic, 422, 423.

Escapement of clock, 50.

Elliptic comets, their number, 702;

Establishment of a port (harbor)

their orbits, 703; recognition of,
defined, 463.

704.

Europa,
the
second
satellite
of

Ellipticity or oblateness of a planet

Jupiter, 621.

defined, 150.

Evection, the, 456.

Elongation of moon or planet de-

Evolution, tidal, 484, 916.

fined, 230.

Eye-pieces, telescopic, 44.

Enceladus, the second satellite of

Extinctions, the method of, in pho-

Saturn, 643, note.

tometry, 825.

Encke’s comet, 710, 743.

F.

Encke, his reduction of the transits
of Venus, 667.

Faculæ, solar, 292.

Energy, the dissipation of, 925; and

Fall of a planet to the sun, time re-
work of solar radiation, 345.

quired, 413, 3; of a comet on the

Enlargement, apparent, of bodies
sun, probable effect, 754.

near horizon, 4, note, 88, 93.

Falling bodies, eastward deviation,

Envelopes in the head of a comet,

138.

713, 727.

Families (planetary) of comets, 739.

Epoch of a planet’s orbit defined,

Faye, H. A., his modification of the

508.

nebular hypothesis, 915; theory of

Epsilon Lyræ, 653, 866, 882.

sun spots, 304.

Equal altitudes,
determination of

Flattening, apparent, of the celestial
time, 115.

sphere, 4, note.

Equation, annual, of moon’s mo-

Force, evidenced not by motion, but
tion, 458; of the centre, 189; of the
by change of motion, 400; projec-
equinoxes, 213; of light, by means of
tile, term carelessly used, 401; cen-

Jupiter’s satellites, 628–630; of time
tral, motion under it, 400–410; re-
explained, 201–204; expressing the
pulsive, action on comets, 728–733.

relation between the light of differ-

Form of the earth, 145–155.

ent stellar magnitudes, 820.

Formation of comets’ tails, 728.

Equator, the celestial, 16.

Foucault, the gyroscope, showing

Equatorial acceleration of the sun’s
earth’s rotation, 142; his pendulum
rotation, 283–285; coudé, Paris, 74;
experiment, showing earth’s rota-
parallax, 85; telescope, 72; tele-
tion, 139–141; measures velocity of
scope used to determine the place
light, 690.

of a heavenly body, 129.

Fourteen hundred and seventy-four

Equinoctial, the, see Equator, ce-
line of the spectrum of the corona,
lestial; points, or equinoxes, 17.

329.

Equinoxes, the, equation of, 213;

Fraunhofer lines in the solar spec-
precession of, 205–212.

trum, 315, 855; observations on

Eratosthenes, his measure of the
stellar spectra, 855.

earth, 136.

Free wave, velocity of, 473.

Erecting eye-piece for telescope, 45.

Frequency, relative, of solar and lu-

Ericsson, his solar engine, 345; ex-
nar eclipses, 394.

periment upon radiation of molten
iron, 350.

G.

Eruptive prominences, 325.

Galaxy, the, 898.

INDEX.

554

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Galileo, discovery of Jupiter’s satel-

Gravity,
increase of,
below the
lites, 621; discovery of Saturn’s
earth’s surface, 169; variation of,
rings, 637; discovery of phases of
between equator and pole, 152.

Venus, 567; use of pendulum in

Gregorian calendar, the, and its
time-keeping, 50.

adoption in England, 220, 221; tele-

Galle, optical discovery of Neptune,
scope, 48.

654.

Groups, cometary, 705; of stars hav-

Ganymede, the third satellite of
ing common motion, 803.

Jupiter, 621.

Growth of the earth by meteoric

Gas contracting by loss of heat,
matter, 777.

Lane’s law, 357.

Gyroscope,

Foucault’s
proof
of

Gauss, computes the orbit of Ceres,
earth’s rotation, 142; illustrating

592; determination of the elements
the precession of the equinoxes,
of an orbit, 519; peculiar form of

210, 211.

achromatic object-glass, 41.

H.

Gay Lussac, law of gaseous expan-
sion, 360, note.

Hall, A., discovery of the satel-

Geocentric latitude, 156; place of a
lites of Mars, 590; on the ques-
heavenly body, 511.

tion whether it is certain that grav-

Geodetic
determination
of
the
itation extends through the stellar
earth’s dimensions, 147, 149.

universe, 901, note.

Genesis of the solar system, 908–915;

Halley, his comet, 742; his com-
of star clusters and nebulæ, 924.

putation of cometary orbits, 700;

Georgium Sidus, the original name
his method of determining the sun’s
for Uranus, 645.

parallax, 679, 680; the moon’s sec-

Gill, solar parallax from observa-
ular acceleration, 459; proper mo-
tions of Mars, 676; stellar paral-
tions of stars, 800.

laxes, 808. Appendix, Table IV.

Hansen, correction of the solar par-

Globe, celestial, rectification of, 33,
allax, 667; opinion on the form of
note.

the moon, 258.

Gnomon, determination of latitude

Harding discovers Juno, 593.

with it, 107; determination of the

Harkness, observations on the light
obliquity of the ecliptic, 176.

of meteors, 776; observation of the

Golden number, the, 218.

corona spectrum, 329.

Gradual changes in the light of the

Harmonic law, Kepler’s, 412–117.

stars, 839.

Harton colliery, density of the earth,

Graduation errors of a circle, 69.

169.

Grating diffraction, 311, note.

Harvard photometry, the, 827, 828.

Gravitation, electro-dynamic, the-

Harvest and hunter’s moons ex-
ory of, 602; law stated, 161; na-
plained, 237.

ture unknown, 161; law extending

Heat and light of meteors explained,
to the stars, 872, note, 873, 901,

765; of the moon, 260; of the sun,
note; Newton’s verification of the

338–358; received by the earth from
law by means of the moon’s motion,
meteors, 355, 779; from the stars,

419, 420.

834.

Gravitational astronomy defined, 2;

Height of lunar mountains, 270.

methods of determining the solar

Heis, enumeration of naked-eye stars,
parallax, 687–689.

818.

INDEX.

555

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Heliocentric place of a planet, 512.

Huggins, W., attempts to photo-

Heliometer, the, 677; used in de-
graph the solar corona without an
termining solar parallax, 676, 683;
eclipse, 328; attempted observation
used in determining stellar paral-
of stellar heat, 834; observations of
lax, 811, 815.

stellar spectra, 856; photography of

Helioscopes, or solar eye-pieces, 286,
stellar spectra, 859; spectroscopic

287.

observations of T coronæ, 844; star-

Helium, an unidentified metal in the
motions in line of sight, 802.

solar chromosphere, 323.

Humboldt, A. von, classification of

Helmholtz, contraction theory of
the planets, 549.

solar heat, 356.

Hunt,

Sterry,
carbonic
acid

Hencke, discovers Astræa, the fifth
brought to earth by comets, 735.

asteroid, 593.

Huyghens,
discovery of Saturn’s

Henderson, measures the parallax
rings, 637; discovery of Saturn’s
of α Centauri, 809, 810.

satellite, Titan, 643; invention of

Henry

Brothers,
astronomical
the pendulum clock, 50; his long
photography, 798.

telescope, 40.

Henry, Prof. J., heat of sun spots,

Hydrogen in the solar chromosphere

310; at sun’s limb, 348.

and prominences, 323–325; bright

Herschel, Sir John, astrometry,
lines of its spectrum in the nebulæ,

819; illustration of the planetary

890; bright lines of its spectrum in
system, 664.

temporary stars, 844; bright lines of

Herschel, Sir W., discovery of the
its spectrum in variable stars, 857.

sun’s motion in space, 804; dis-

Hyperbola,
the,
described as a
covery of two satellites of Saturn,
conic, 422.

643; discovery of Uranus, 645; dis-

Hyperbolic comets, 702.

covery of two satellites of Uranus,

Hyperion, the seventh and last dis-

650; star-gauges, 899; theory of sun
covered satellite of Saturn, 643,
spots, 302; his reflecting telescope,

644.

48.

Hypothesis, nebular. See Nebular

Hevelius, his view of cometary or-
hypothesis.

bits, 700.

I.

Hipparchus, discovers eccentricity of
earth’s orbit, 184; discovers preces-

Iapetus, the outermost satellite of
sion, 205; his value of the solar par-

Saturn, 643.

allax, 671; the first star-catalogue,

Ibn Jounis, use of pendulum in ob-

795.

servation, 50.

Holden, E. S., on changes in neb-

Ice, amount melted by solar radia-
ulæ, 892.

tion, 344–346, 364*.

Horizon, apparent enlargement of

Illumination of the moon’s disc dur-
bodies near it, 4, note, 88, 93; arti-ing a lunar eclipse, 376.

ficial, 78; rational and apparent de-

Image, telescopic, conditions of dis-
fined, 10; dip of, 81; visible, defined,
tinctness, 39.

11.

Inequality, diurnal, of the tides, 471.

Horizontal parallax, 83, 84; point of

Inferences deducible from Kepler’s
the meridian circle, 67.

laws, 418.

Hour-angle defined, 24.

Inferior planet, motion of, 497.

Hour-circle defined, 18.

Infinity, velocity from, 429.

INDEX.

556

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Influences of the moon on the earth,

Lane’s law, rise of temperature con-

262.

sequent on the contraction of a

Intra-Mercurial planets, 602–606;
gaseous mass, 357.

planets, supposed observations of,

La Place, his equations relating to
during solar eclipse, 605.

the eccentricities and inclinations of

Interior of the earth, its constitution,
the planetary orbits, 532; explana-

171.

tion of the moon’s secular accelera-

Invariable plane of the solar system,
tion, 459, 460; the invariable plane

531.

of the solar system, 531; the nebu-

Invariability of the earth’s rotation,
lar hypothesis, 901–911.

144.

Lassell, discovery of the two inner

Io, the first satellite of Jupiter, 621.

satellites of Uranus, 650; indepen-

Iron meteorites, 758; in the sun, 315.

dent discovery of Hyperion, 643.

Irradiation
in
micrometric
mea-

Latitude (astronomical) of a place on
sures, 256, 534.

the earth’s surface, 30, 100, 156; astronomical, geodetic, and geocen-

J.

tric, distinguished, 156; determina-

Janssen, discovery of the spectro-
tion of, methods used, 101–107; at
scopic method of observing the so-
sea, its determination, 103; possi-
lar prominences, 323; solar photog-
ble variations of it, 108; station er-
raphy, 289.

rors, 158; celestial, defined, 178,

Jets issuing from the nucleus of a

179; and longitude (celestial), con-
comet, 713, 727.

version into α and δ, 180.

Jolly, observations of the earth’s

Law of angular velocity under cen-
density, 170.

tral force, 408; Bode’s, 488, 489;

Julian calendar, 219.

of Boyle or Mariotte, density of a

Juno, discovered by Harding, 593.

gas, 360, note; of Dalton, mixture

Jupiter, the planet, 609–631; bright-
of gases, 360, note; of earth’s or-
ness as seen from α Centauri, 881;
bital motion, 186, 187; of equal ar-
a semi-sun, 619.

eas, 186, 402–406, 412; of Gay Lus-K.

sac, gaseous expansion, 360, note;
of gravitation, 161, 162, 419, 872;

Kant, proposes the nebular hypoth-

Lane’s, of temperature in gaseous
esis, 908.

contraction, 357; of linear velocity

Kepler, his belief as to cometary or-
in angular motion, 407.

bits, 700; his three laws of plan-

Laws of Kepler, 412–418; motion un-
etary motion, 412–418; his “prob-
der a central force, 400–411.

lem,” 188; his “regular solid” the-

Leap year, rule for, 220.

ory of the planetary distances, 592,
note.

Length, of the day, possible changes
in it, 144; of the year, its invariabil-

Kirchoff, his fundamental princi-
ity, 526, 778.

ples of spectrum analysis, 314.

Leonids, the, 780, 786.

L.

Lescarbault, supposed discovery of

Langley, S. P., his Bolometer, 343;

Vulcan, 603.

the color of the sun, 337; observa-

Level adjustment of the transit in-
tions on lunar heat, 260, 261; on
strument, 60.

solar heat, 348; sun-spot drawings,

Leverrier, discovery of Neptune,

292; heat in sun spots, 293.

653, 654; on an intra-Mercurial

INDEX.

557

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

planet, 603; motion of the per-

894; meteoric hypothesis, 926.

ihelion of Mercury’s orbit, 602;

Loewy, peculiar method of determin-
method of determining the solar
ing the refraction, 95.

parallax by planetary perturba-

Longitude, arcs of, to determine the
tions, 689.

earth’s dimensions, 151; (terres-

Lexell’s comet, approach to Jupiter,
trial), determination of, 118–121;

718; recognition of Uranus as a

(celestial), 178–180; of perihelion,
planet, 645.

505, 506.

Librations of the moon, 249, 250,

Luminosity of bodies at low temper-

251.

atures, 737, note.

Light of comets, 722; of the moon,

Lunar distances, 120, B; eclipses,

259; emitted by certain stars, 835;

370–378; influences on the earth,
received by the earth from certain

262; methods of determining the
stars, 832; of the sun, 332–337; to-
longitude, 120; perturbations, 448–
tal, of the stars, 833; equation of,

461; perturbations used to deter-
from Jupiter’s satellites, 628–630;
mine the solar parallax, 687.

mechanical equivalent of, Thomsen,

Lyræ, α, see Vega; β, variable star,

776; time occupied by, in coming

847;

, quadruple star, 653, 866,
from the sun, 275, 629; velocity of,

882.

225, note, 668, 690.

M.

Light-curves of variable stars, 848.

Light-gathering
power
of
tele-

Mädler, speculations as to a central
scopes, 38.

sun, 807.

Light-ratio, the, in scale of star-

Magnifying power of a telescope, 37;
magnitudes, 819.

power, highest available, 43.

Light-year, the, the unit of stellar

Magnitudes of stars, 816–822.

distance, 814.

Magnitude of smallest star visible in

Limb of the sun, darkening of, 337;
a given telescope, 822.

of the sun, diminution of heat, 348.

Magnesium in the nebulæ, 890, 894.

Limiting apertures in stellar pho-

Maintenance of the solar heat, 353–
tometry, 825.

356.

Linear and angular dimensions, their

Mars, the planet, 578–591; observed
relation, 5; velocity under central
for solar parallax, 673–677.

force, its law, 407, 409.

Maskelyne, his mountain method

Linné, lunar crater supposed to have
of determining the earth’s density,
changed, 269.

164.

Listing, dimensions of the earth,

Mass and weight, distinction between

145.

them, 159, 160; of comets, 718, 719;

Local and standard time, 122.

of the earth compared with the sun,

Lockyer, J. N., discovery of the

278; of the earth in terms of the sun
spectroscopic method of observing
as determining the solar parallax,
the solar prominences, 323;
his

689; of the moon, its determination,

“collision theory” of variable stars,

243; of a planet, how determined,

850; views as to the compound na-

536–539; of the sun, compared with
ture of the so-called chemical “el-
the earth, 278; probable, of shoot-
ements,” 318; origin of the Fraun-
ing stars, 776.

hofer lines, 320; theory of sun spots,

Mass-brightness of binary stars,

306; meteoric theory of nebulæ,

879.

INDEX.

558

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Masses of binary stars, 877, 878.

Microscope, the reading, 64.

Mayer, R., meteoric theory of the

Midnight sun, the, 191.

solar heat, 353.

Milky Way, or galaxy, 898.

Maxwell, Clerk, meteoric theory of

Mimas, the innermost satellite of

Saturn’s rings, 641.

Saturn, 643 and note.

Mazapil, the meteorite of, 784.

Minor planets, or asteroids, 592–601.

Mechanical equivalent of light, 776.

Mira, Omicron Ceti, 846.

Mercury, the planet, 551–562.

Missing stars, 840.

Mercury’s orbit, motion of its peri-

Mohammedan calendar, 217.

helion, 602.

Monck, the mass-brightness of bi-

Meridian, the celestial, defined, 19;
nary stars, 879.

arc of, how measured, 147.

Monocentric eye-piece for telescope,

Meridian-circle, the, 63; used to

45.

determine the place of a heavenly

Moon, the, 227–272; her atmosphere,
body, 128.

255–258; the, regarded as a clock,

Meridian photometer, the, 828.

120; distance of, etc., 240; her

Meteors and shooting stars, 755–
heat and temperature, 260, 261;

787; ashes of, 775; and comets, their
her light as compared with sunlight,
connection, 785–787; daily num-

259; influences on the earth, 262;
ber of, 771; detonating, 768; ef-
mass of, determined, 243, 244; her
fect on the earth’s orbital motion,
motion (apparent), 228; her motion

778; effect upon the moon’s mo-
relative to the sun, 241; her moun-
tion, 778; effect upon the trans-
tains, measurement of their eleva-
parency of space, 779; explanation
tion, 270; her orbit with reference
of their light and heat, 765; heat
to the earth, 238; her parallax de-
from them, 355, 779; magnitude
termined, 239; perturbations, 448–
of, 762; method of observing them,

461; her rotation and librations,

764; their trains, 766.

248–251; her phases, 253; her sur-

Meteoric growth of the earth, 777;
face character, 263–270; culmina-
showers, 780–786; shower of the
tions for longitude, 120, A.

Bielids, 1872, 1886, 746; shower of

Month, the anomalistic, 397, note;
the Leonids, 1833, 1866–67, 781;
the nodical, 397, note; the sidereal,
swarms and rings, 783; swarms,

229, 232; the synodic, 229, 232;
special characteristics, 783; theory
length of, increased by perturba-
of Saturn’s rings, 641; theory of the
tion, 453; slightly shortened by the
sun’s heat, 353–355; theory of sun
secular acceleration, 459.

spots, 306.

Motion, direct and retrograde, of the

Meteorites, or uranoliths, or aero-
planets, 494; of the solar system in
lites, 755–769; chemical elements
space, 804–807; in line of sight, ef-
in them, 760; crust, 761; fall of,
fect on spectrum, 321; of stars in

756; which have fallen in the United
line of sight, spectroscopically ob-

States, 759; iron, list of, 758; num-
served, 802.

ber of, 769; question of their origin,

Motions, proper, of the stars, 800–

767; their paths, 763.

803.

Metonic cycle, the, 218.

Mountains, lunar, their height, 270.

Michelson, determination of the ve-

Mountain method of determining
locity of light, 225, note, 668, 690.

the earth’s density, 164.

Micrometer, the filar, 73, 534, 867.

Multiple stars, 882.

INDEX.

559

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Mural circle, the, 70.

ber of meteors, 771; investigation of
meteoric orbits, 767, 785; theory of

N.

the constitution of a comet, 737.

Nadir, the, defined, 9;
point of

Newton, Sir Isaac, discovery of
meridian circle, 67.

gravitation, 161, 419; verification of

Names of the constellations, 792; of
the idea of gravitation by means of

Jupiter’s satellites, 621; of satel-
the moon’s motion, 419, 420; dis-
lites of Mars, 590; of the planets,
covery that planetary orbits must

487, 489; of Saturn’s satellites, 643,
be conics, 421; computation of a
note; of the satellites of Uranus,
cometary orbit, 700; his reflecting

650; of stars, 794.

telescope, 48.

Neap tide defined, 463.

Nitrogen, suspected in the nebulæ,

Nebula, the great, in Andromeda,

890.

886; the annular, in Lyra, 888; of

Node of an orbit defined, 233, 506.

Orion, the, 886, 892, 893.

Nodes of moon’s orbit, their regres-

Nebulæ, the, 886–897; changes in,
sion, 455; of the planetary orbits,

892; their distance, 896; Lockyer’s
their motion, 527.

meteoric theory, 894; their nature,

Nodical month, the, 249, 397, note.

894; their number and distribution,

Nordenskiold, meteoric ashes, 775.

895; photographs of, 893; plane-

Nucleus of a comet, 713, 716.

tary, 888; spiral, 888; their spec-

Number of eclipses in a year, 391; in
tra, and chemical elements in them,
a saros, 398; of meteors and mete-

890, 891.

orites, 759, 769, 771; and designa-

Nebular hypothesis, the, 908–915;
tion of variable stars, 852.

modifications of the original theory,

Nutation of the earth’s axis, 214,

912, 913.

215.

Negative eye-pieces for the tele-
scope, 44; shadow of the moon, 381;

O.

star magnitudes, 821.

Oberon,
the
outer
satellite
of

Neptune,
the
planet,

653–661;

Uranus, 650.

anomalous retrograde rotation, in

Object-glass, achromatic, 41; resid-
relation to the nebular hypothesis,
ual or secondary spectrum of, 42;

914; appearance of sun and solar
designed for photography, 42.

system
from
it,

658;

(actual)

Oblateness,
or
ellipticity
of
a
discovery by Galle, 654; theoretical
spheroid, 150.

discovery by Leverrier and Adams,

Oblique sphere, the, 33.

653, 654; its discovery “no acci-

Obliquity of the ecliptic defined and
dent,” 655; the computed elements
measured, 176; of the ecliptic, sec-
erroneous, 655; its satellite, 661;
ular change of, 197.

spectrum of, 660.

Occultation, circle of perpetual, 33;

Newcomb, S., conclusions as to sun’s
of stars, 399; of stars used for
age and duration, 358, 359; ob-
longitude determination, 120 C; of
servations on meteors, 776; on the
stars proving absence of lunar at-
moon’s secular acceleration, 461; on
mosphere, 256.

the structure of the heavens, 900;

Olbers, discovers Pallas and Vesta,
his value of the solar parallax, 667;

593.

velocity of light, 225, note, 668, 690.

Olmsted, D., his researches on me-

Newton, Prof.

H. A., daily num-
teors, 785.

INDEX.

560

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Oppolzer, effect of meteors on the
ative, Pond, 809; stellar, absolute
moon’s motion, 778; orbit of Tem-
method, 810; stellar, differential
pel’s Comet, 786; motion of Win-
method, 811; stellar, table of, Ap-
necke’s Comet, 711, note.

pendix, Table IV.

Orbit of the earth, its form deter-

Parallactic inequality of the moon,
mined, 182; of the earth, effect of

687; orbit of a star, 808.

meteors upon it, 778; of the earth,

Parallel sphere, the, 32.

perturbations, 197; of the moon,

Parallels of declination, 23.

238; of the moon, its perturbations,

Peirce, B., heat from meteors, 355;

448–461; of a planet, determined
on the mass of comets, 719; theory
graphically, 428, 431, 432; plane-of sun spots, 306.

tary, its elements, 505–510; plane-

Pendulum, compensation, 51; use
tary, its elements, determination of,
in clocks, 50; used in determining

519.

form of the earth, 152–155; free, of

Orbits of binary stars,

873;
of

Foucault, showing earth’s rotation,
comets, 700–709; of planets, dia-

139–141.

gram, 489; of sun and stars in the

Penumbra of the earth’s shadow,
stellar system, 904.

368; of the moon’s shadow, 383; of

Origin of comets, 738–741; of mete-
a sun spot, 295.

orites or aerolites, 767.

Perigee and apogee defined, 238.

Orthogonal component of the dis-

Perihelia of comets, their distribu-
turbing force, 445, 455.

tion, 706.

Perihelion of earth’s orbit defined,

P.

183; its motion, 199; of Mercury’s

Pallas, discovered by Olbers, 593.

orbit, its motion, 602.

Palisa, discoverer of sixty-five aster-

Period, sidereal and synodic, of the
oids, 593.

moon, 229–232; sidereal and syn-

Parabola, the, described as a conic,
odic, of a planet, defined, 490; sid-

422, 423.

ereal, of a planet, determined, 513,

Parabolic comets,
their number,

514.

702; velocity, the, 429.

Periodic comets, 703, 704, 738–740;

Parallax (diurnal), defined and dis-
table of comets of short period, Ap-
cussed, 82, 83; of the moon, de-
pendix, Table III.

termined, 239; of the sun, classi-

Periodicity of sun spots, 307–309.

fication of methods, 669; of the

Persei, β, or Algol, 848.

sun, gravitational methods, 687–

Perseids, the, meteoric swarm, 780,

689; of the sun, history of investiga-

782, 783.

tions, 666–668; of the sun, method

Personal equation,

114,

120 A,
of Aristarchus, 670; of the sun,

121 B.

method of Hipparchus, 671; of the

Perturbations,
lunar,

448–461;
sun by observations on Mars, 673,
planetary, 521–523; of Mars and

676; of the sun by transits of Venus,

Venus by the earth as determining

678, 686; of the sun by the veloc-
the sun’s parallax, 689.

ity of light, 690–692; of the sun,

Peters, C. H. F., discovers fifty-two

Ptolemy’s value, 671; of the stars
asteroids, 593.

(annual), 808–814; of α Centauri,

Piazzi, discovery of Ceres, 592.

Henderson, 809, 810; of 61 Cygni,

Picard, measure of earth’s diameter,

Bessel, 809–811; of α Lyræ, neg-

136.

INDEX.

561

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Pickering, E. C., his meridian pho-

Table I.; data, accuracy of, 653;
tometer, 828; photography of stel-
nebulæ, 888; system, facts suggest-
lar spectra, 860, 862; photomet-
ing the theory of its origin, 907; sys-
ric observations of the eclipses of
tem, Sir J. Herschel’s illustration of

Jupiter’s satellites, 630; the Har-
its dimensions, 664.

vard photometry, 827.

Pleiades, the, 884.

Phases of Mercury, Venus, and Mars,

Pogson, his absolute scale of star-

559, 567, 582; of the moon, 253.

magnitudes, 819.

Phobos, the inner satellite of Mars,

Pole of the earth, 28; (celestial), de-

589.

fined, 14; its altitude equal to the

Photographs of the moon,

272;
latitude, 30, 100; its place affected
of the nebulæ, 893; of the solar
by precession, 206, 207.

corona, 328; of the sun’s surface

Pole-star, ancient, γ Draconis, 207;
and spots, 289.

its position and recognition, 15.

Photographic object-glasses, 42; ob-

Polar distance, defined, 23; point of
servations of eclipses of Jupiter’s
meridian circle, 66.

satellites, 630, and note; observa-

Position-angle of a double star, 868.

tions of transit of Venus, 684–686.

Position of a heavenly body, how de-

Photography as a means of photom-
termined, 128, 129.

etry, 829; solar, 289; spectroscopic,

Positive eye-pieces for telescopes, 44.

motion in line of sight, 802; applied

Pouillet, his pyrheliometer, 340.

to star-charting, 798; in determina-

Power, magnifying, of telescope, 37.

tion of stellar parallax, 812; of stel-

Poynting,
determination
of
the
lar spectra, 859–863.

earth’s density, 170.

Photometer, the meridian, 828; po-

Practical astronomy defined, 2.

larization, 827; the wedge, 826.

Precession of the equinoxes, 205–

Photometry,

Harvard,
the,

827,

212.

828; by means of photography, 829;

Prime vertical defined, 19; vertical
by the spectroscope, 831; of sun-
instrument, 62, 106.

light, 332–335; stellar, 823–831.

Priming and lagging of the tides,

Photosphere of the sun, its nature,

470.

291, 292, 361.

Pritchard, Prof. C., determina-

Photo-tachymetrical
determi-
tion of stellar parallax by means of
nation
of
the
sun’s
parallax,
photography, 812; stellar photom-

690–692.

etry, 826; Uranometria Oxoniensis,

Physical characteristics of comets,

826.

712; method of determining sun’s

Pritchett, C. W., discovery of the
parallax, 690–692.

great red spot on Jupiter, 618.

Planet,
intra-Mercurial,

602–606;

Problem of three bodies, 437–461; of
trans-Neptunian, 662.

two bodies, 424–433.

Planets attending certain stars, 880,

Problems illustrating Kepler’s third

881; distances and periods, 489;
law, 413.

enumerated, 486, 487; relative age,

Proctor, R. A., on the origin of
according to nebular hypothesis,
comets, 741; determination of the

913, 915; orbits, diagram of, 489;
rotation period of Mars, 584.

orbits, elements of, 505, 510.

Projectiles,
deviation caused by

Planetoid. See Asteroid, 591.

earth’s rotation, 143; their path

Planetary data, tables of, Appendix,
near the earth, 435.

INDEX.

562

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Projectile force, careless use of the
upon time of sunrise and sunset, 92.

term, 401.

Refracting telescope (simple), 36;

Prominences, or protuberances, the
telescope, achromatic, 41.

solar, 291, 323–326, 363; quiescent Refractors and reflectors compared,
and eruptive, 325, 326.

49.

Proper motions of the stars, 800,

Reich, determination of the density

803.

of the earth, 166; experiments upon

Proximity of a star, indications of it,
falling bodies, 138.

813.

Relative motion, law of, 492; sizes of

Ptolemaic system, the, 500, 502.

the planets, diagram, 550.

Ptolemy, his almagest, 500, 700,

Repulsive force acting on comets,

795.

728, 731, 732, 734.

Pyrheliometer of Pouillet, 340.

Retardation of earth’s rotation by
the tides, 461, 483.

Q.

Reticle used in telescope for point-

Quantity of the solar radiation in
ing, 46.

calories, 338–340; of sunlight in

Retrograde revolution of the satel-
candle power, 332, 333.

lites of Uranus and Neptune, 652,

Quiescent prominences, 325.

661, 914.

R.

Reversing layer of the solar atmo-
sphere, 291, 319, 320, 362.

Radial compound of the disturbing

Reversal of the spectrum, 314.

force, 446.

Rhea, the fifth satellite of Saturn,

Radian, the, defined as angular unit,

643, note.

5, note.

Rigidity of the earth, 171, 482.

Radiant, the, in meteoric showers,

Right ascension defined, 25, 27; as-

780.

cension determined by transit in-

Radius of curvature of a meridian,
strument, 59, 128, 129; sphere, the,

149.

31.

Ranyard, peculiar theory of the re-

Rings of Saturn, 637–642.

pulsive force operative in comets’

Rosse, Lord, observations of lunar
tails, 733.

heat, 260, 261; his great telescope,

Rate of a clock defined, 53.

48; spiral nebulæ, 888.

Reading microscope, the, 64.

Rotation, distinguished from revolu-

Recognition of elliptic comets, diffi-
tion, 248, 248*; of the earth, af-
culties, 704.

fected by the tides, 461, 483; of the

Red spot of Jupiter, the, 618.

earth, proofs of, 138–143; of plan-

Reduction of mean star places to ap-
ets, how determined, 543; period of
parent and vice versa, 797.

Jupiter, 615; period of Mars, 584;

Reflecting telescope, various forms,
period of the moon, 248, 252; pe-

47, 48;
telescopes, large instru-
riod of Saturn, 635; of the sun, 281,
ments, 48.

283; period of Venus, 570; periods,

Refraction, atmospheric, its law, 89,
see also Appendix, Table I.

90; determination of its amount,

94, 95; effect of temperature and

S.

barometric pressure, 91; effect upon

Saros,
the,

395–398;
number of
form and size of discs of sun and
eclipses in a saros, 398.

moon near the horizon, 93; effect

Satellites of Jupiter, 621–631; of

INDEX.

563

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Mars, 590, 591; of Neptune, 661; of
termining time, 115, 116.

Saturn, 643, 644; of Uranus, 650–

Shadow of the earth, its dimensions,

652; general table of, Appendix, Ta-

367; of the moon, 379, 380; of the
ble II.

moon, its velocity over the earth,

Satellite orbits, generally circular,

384.

548.

“Sheath” of the comet of 1882, 752.

Saturn, the planet, 632–644.

Ship’s place at sea, determination of,

Schehallien, determination of the

103, 120 B, 121 A, 124–126.

earth’s mass, 164.

Shooting Stars, 770, 787; ashes

Schiaparelli, connection between
of, 775; brightness of, 773; com-
comets and meteors, 786; his map
parative numbers in morning and
of Mars, 588.

evening, 772; daily number of, 771;

Schröter, the rotation of Mercury,
elevation of, 774; mass of, 776; ma-

559; the rotation of Venus, 570.

terials of, 775; path of, 774; showers

Schwabe, discovery of the periodic-
of, 780–786; spectra of, 775; veloc-
ity of sun spots, 307.

ity of, 774.

Scintillation of the stars, 864.

Short-period comets, 703; comets,

Sea, ship’s place at, 103, 120 B,
table of, Appendix, Table III.

121 A, 124–126.

Showers, meteoric, 780–786.

Seasons, the, explained, 190, 192,

Sidereal day defined, 26, 110; month,

193; difference between northern

229; time, 26, 110; year, its length, and southern hemispheres,

194,

216.

195.

Signals used in determining differ-

Secchi, theories of sun spots, 303,
ence of longitude, 119.

305; observations on stellar spectra,

Signs of the zodiac, 177.

856, 857.

Single-altitude method of determin-

Seidel, his photometer, 827.

ing local time, 116.

Secular acceleration of the moon’s

Sirius and its companion, 875; its
mean motion, 459–461; changes in
light compared with the sun’s, 334,
the earth’s orbit, 196–200; pertur-

832, 835; its mass, 877.

bations in the planetary system,

Sky, apparent distance of, 6.

525–529.

Slitless spectroscope, the, 860–862.

Semi-diameter,
augmentation
of

Solar
constant,
the,

338–340;
the moon’s, 88; correction for, in
eclipses,

319,

323,

327,

329,
sextant observations, 88.

387–390, 393, 398; eclipses, their Semi-major axis of a planet’s or-rarity, 398; engine of Ericsson and
bit, defined and discussed, 505, 506;

Mouchot, 345; eye-pieces, 286, 287;
axis of the planets’ orbits, invari-
parallax,
see Parallax
of
the
able, 526; axis as depending on

Sun; system, age of, 922; time,
planet’s velocity, 428–430.

apparent and mean, defined, 111,

Separating power of a telescope,

112.

Dawes, 43.

Solstice defined, 176.

Sequences, method of, in stellar

Sosigenes, devises the Julian calen-
photometry, 824.

dar, 219.

Sextant, the, described, 76; the, used

Spectra of comets, 724–726; of me-
to determine latitude, 103; the,
teors, 775; of nebulæ, 890, 891; of used in finding a ship’s place at sea,
stars, 855–863.

103, 116, 125, 126; the, used in de-Spectroscope, principles of its con-

INDEX.

564

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

struction, 311–313; how it shows

Variable Stars.

the solar prominences, 324; slitless,

Star-atlases, 793.

860–862.

Star-catalogues, 795.

Spectroscopic measurement of mo-

Star-charts, 798.

tions in the line of vision, 321, 802.

Star-clusters, 883, 884.

Spectrum of the chromosphere and

Star-gauges, 899.

prominences, 323; of the corona,

Star-motions, 799–803.

329; solar (photosphere), 312; so-

Star-places, how affected by aberra-
lar, compared with iron, 315; of sun
tion, etc., 226; their determination,
spots, 321; analysis, fundamental

796; mean and apparent, 797.

principles, 314; photometry, 831.

Statical theory of the tides, 469.

Sphere, the celestial, conceptions of

Station errors, 158.

it, 4; the oblique, 33; the parallel,

Stellar spectra, 855, 856; classifica-

32; the right, 31.

tion of, 857, 858; photography of,

Spheres, attraction of, 162.

859–863; system, the hypothetical,

Spheroid, terrestrial, its dimensions,

901–904.

145, Appendix, page 540.

Stone, E. J., attempted observation

Spherical aberration of a lens, 39;
of stellar heat, 834.

astronomy, defined, 3; shell, its at-

Stripe, central, in comets’ tails, 730.

traction, 169.

Structure of the heavens, 900–904.

Spider lines in a reticle, 46.

Struve, von, F. G. W., on distri-

Spring tide defined, etc., 463.

bution of stars, 899.

Spurious disc of stars in the tele-

Struve, von, Ludwig, investiga-
scope, 43.

tion of sun’s motion in space, 806.

Stability of the planetary system,

Struve, von, Otto, Saturn’s rings,

530–533.

637, 642.

Standard and local time, 122.

Sumner, Capt., his method of de-

Stars, binary, see Binary Stars;
termining a ship’s place at sea, 125,
causes of the difference in their

126.

brightness, 836; colors of, 830; dark,

Sun, the, 273–364; age and dura-

836; designations and names, 794;
tion of, 359, 922; apparent annual
their real diameters, 837; distri-
motion of, 172, 173; attraction on
bution of, 899; double, see Dou-
the earth, its intensity, 436; candle
ble

Stars;
gradual changes in
power of sunlight, 332, 333; chemi-
their light, 839; heat from them,
cal elements in it, 315–317; diame-

834; light compared with sunlight,
ter, surface, and volume, 276, 277;

334, 832, 835; magnitudes of, 816–
distance and parallax, 274, 275,

822; missing, 840; nature, as being

665–693; gravity at its surface, 280;
suns, 789; number of, 790; paral-
heat emission, 338–357; light, 332–
lax and distance, 808–814; photog-

336; mass and density, 278, 279; its raphy of, 798; photometric observa-motion in space, 804–805; physical
tions of, 823–831; proper motions
constitution, 360–364; its tempera-
of, 800–803; proximity of, its indi-
ture, 349–351; the central, 807.

cations, 813; seen by day with tele-

Sun spots, their development and
scope, 38; shooting, see Shooting
changes, 297, 298; distribution on

Stars; temporary, 842–845; triple
sun’s surface, 301; general descrip-
and multiple, 882; twinkling of,
tion, 293, 294; influence on terres-
or scintillation, 864; variable, see
trial conditions, 309, 310; period-

INDEX.

565

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

icity of, 307–309; their spectrum,
ture required to show stars of given

321; theories as to their cause and
magnitude, 822; temporary stars,
nature, 302–306.

842; total light from stars of differ-

Sun’s way, apex of, 804–806.

ent magnitude, 833; velocity of free

Sunrise and sunset affected by re-
wave at various depths, 473.

fraction, 92.

Tails or trains of comets, 713, 717,

Superior planet, motions of, 496.

728–736.

Surface errors in lenses and mirrors,

Talcott, Capt., his zenith tele-

49; of the moon, 263–270.

scope, 105.

Swarms, meteoric, 783.

Tangential component of the dis-

Swedenborg, a proposer of the neb-
turbing force, 447.

ular hypothesis, 908.

Telegraph used in determination of

System, planetary, facts suggesting
longitude, 121 B.

a theory of its origin, 907; numer-

Telescope, the, achromatic, 41; dis-
ical data, Appendix, Table I.; stel-
tinctness of image, 39;
equato-
lar, 901–904.

rial, 72; eye-pieces, 44; the gen-

Synodic month, or revolution, of
eral theory, 35; invention of, 35;
moon, 229–231; period, general def-
light-gathering power, 38; long, of
inition of, 490.

Huyghens, 40; magnifying power,

Syzygy, defined, 230.

37; object-glass, various forms, 41,

42; reflecting, various forms, 47, 48;

T.

refracting, simple, 36; relation of its

Tables, Appendix.

Greek alphabet,
aperture to the “magnitude” of the
page 540; miscellaneous symbols,
smallest star visible with it, 822;
page 540; dimensions of the earth,
separating or dividing power, 43.

page 540; time constants, page 541;

Telespectroscope, 313.

Table I., elements of solar system,

Temperature, cause of the annual
page 542; Table II., satellites of the
change, 192, 193; of the moon, 261;
system, page 543; Table III., short-
of the sun, 349–351.

period comets, page 544; Table

Temporary stars, 842–845.

IV., parallaxes of stars, page 545;

“Terminator,” the, on the moon’s

Table V., orbits of binary stars,
surface, its form, 253.

page 546; Table VI., the variable

Tethys, the third satellite of Saturn,
stars, page 547.

643, note.

Tables, in body of the book. The con-

Thomsen of Copenhagen, the me-
stellations, 792; approximate dis-
chanical equivalent of light, 776.

tances and periods of the planets,

Thomson, Sir W., on the tempera-

489; distance of sun correspond-
ture of meteors, 765; rigidity of the
ing to certain values of the paral-
earth, 171, 482.

lax, 668; distribution of stars with

Three bodies, the problem of, 437,
reference to the galaxy, 899; iron

438.

meteors seen to fall, 758; naked-

Tidal evolution, 484, 916; friction, ef-
eye stars north of celestial equator,
fect on the earth’s rotation, 461,

818; orbits and masses of certain bi-

483; wave, its origin and course,
nary stars, 877; parallaxes of first-

476.

magnitude stars, Elkin, 815; proper

Tides, the, definition of terms relat-
motions of certain stars, 800; signs
ing to them, 463; priming and lag-
of the zodiac, 177; telescopic aper-
ging of, 470; statical theory of, 469;

INDEX.

566

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

wave theory of, 472.

in Cassiopeia, 843; his planetary

Tide-raising force, the, 464–467.

system, 504.

Time, defined as an hour-angle, 109;

U.

its determination by the sextant,

115, 116; its determination by the

Umbriel,
the second satellite of
transit instrument, 113; equation

Uranus, 650.

of, explained and discussed, 201–

Unit of stellar distances, the light-

204; sidereal, defined, 26, 110; so-year, 814.

lar, apparent, 111; solar, mean, 112;

Uranolith. See Meteorites.

standard and local, 122.

Uranometria Nova:

Argelander,

Tisserand on peculiarities of satel-

817; Oxoniensis, 826.

lite orbits, 548.

Uranus and Neptune, their anoma-

Titan, the sixth and great satellite of
lous rotation in relation to the neb-

Saturn, 643.

ular hypothesis, 914; the planet,

Titania, the third satellite of Uranus,

645–652.

650.

Utility of astronomy, 2.

Todd, Prof.

D. P., search for

V.

trans-Neptunian planet, 662.

Torsion balance, determination of

Van der Kolk’s theorem, 434.

the earth’s density, 165.

Vanishing point of a system of par-

Trade winds, proving rotation of the
allel lines, 7.

earth, 143.

Variable nebulæ, 889.

Trains of meteors, 766, 773.

Variable stars, 838–854; classifica-

Transits of moon across meridian,
tion of, 838; explanation of their
the interval between them, 235; of
variation, 849–851; methods of ob-

Mercury, 561, 562; of Venus, law of
servation, 854; their number and
recurrence, 575–577; of Venus, used
designation, 852; their range of
for determination of solar parallax,
variation, 853.

678–686.

Variation, the lunar, 457.

Transit circle, see Meridian Circle,

Vega, or α Lyræ, its light compared

63; instrument, 59–61; instrument
with the sun’s, 334, 832, 835; its used in determining time, 113.

spectrum, 859.

Trans-Neptunian planet, hypothet-

Velocity of air currents at high ele-
ical, 662.

vations, 773, note; areal, linear and

Transparency of space as affected
angular, law of, 407–409; of earth in
by meteors, 779.

her orbit, 225, note, 278; of light,

Triple and multiple stars, 882.

225, note, 690–692; of the moon’s

Tropics, defined, 176.

shadow, 384; parabolic, or velocity

Tropical year, its definition and its
from infinity, 429; of planet at any
length, 216.

point in its orbit, 434; of stellar mo-

Twilight, theory and duration of, 96,
tions, 801.

97, 130.

Venus, the planet, 563–577; her at-

Twinkling, or scintillation, of the
mosphere and its effect upon obser-
stars, 864.

vations of a transit, 681; transits

Two bodies, problem of, 424–433.

of, used to determine solar paral-

Tycho

Brahe discovers the lu-
lax, 678–686.

nar variation, 457; observations of

Vertical, angle of the, 156; circles de-
comet of 1577, 700; temporary star
fined, 12.

INDEX.

567

[All references, unless expressly stated to the contrary, are to articles and not to pages.]

Vogel, his classification of stellar

307.

spectra, 858; star motions in line of

Worms, formula for the eastward de-
sight, 802, 863.

viation of a falling body, 138.

Vulcan, hypothetical intra-Mercurial

Y.

planet, 603, 604.

Year, bissextile, or leap, 219; be-

W.

ginning of, 222; of confusion, 219;

Waste of solar energy, 347.

eclipse, 391; sidereal, tropical, and

Water, absence of, on the moon, 258;
anomalistic, 216, also Appendix,
presence of, in atmosphere of plan-
page 541; of light, unit of stellar dis-
ets, 560, 573, 589.

tance, 814.

Watson, J. C. discovers and endows

Z.

twenty-two asteroids, 593, 601.

Zenith, the astronomical and geocen-

Wave-length of a ray of light, af-
tric, 8.

fected by motion in the line of

Zenith distance defined, 21; tele-
vision—Doppler’s principle,

321,
scope, for determination of latitude,
note.

105.

Wave-theory of the tides, 472.

Zenker, theory of a comet’s consti-

Weather, moon’s influence on it,
tution, 733.

262.

Zero points of a meridian circle, 66,

Wedge photometer, the, 826.

67.

Weight, loss of, between equator and

Zodiac, the, and its signs, 177; signs
pole, 152–155; and mass, distinc-
of, as affected by precession, 208.

tion between them, 159, 160.

Zodiacal light, the, 607, 608.

Wilsing,
determination
of
the

Zöllner, albedo of the planets, 558,
earth’s density, 167.

572, 583, 614, 636, 648, 660, also Winnecke’s comet, acceleration of,

Appendix, page 542; his photome-

711.

ter, 827; on the repulsive force act-

Wolf, periodicity of the sun spots,
ing upon comets, 732.

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Document Outline

- Project Gutenberg Boilerplate

- A Text-book of General Astronomy

- Frontispiece
- Preface
- Table of Contents
- Introduction
- I. The Doctrine of the Sphere
- II. Astronomical Instruments

- III. Corrections to Astronomical Observations

- IV. Problems of Practical Astronomy
- V. The Earth
- VI. The Earth's Orbital Motion
- VII. The Moon
- VIII. The Sun
- IX. The Sun's Light and Heat
- X. Eclipses
- XI. Central Forces
- XII. The Problem of Three Bodies

- XIII. The Planets: their Motions, Apparent and Real

- XIV. The Planets themselves

- XV. The Major Planets

- XVI. Determination of Sun's Parallax and Distance

- XVII. Comets
- XVIII. Meteors
- XIX. The Stars
- XX. The Light of the Stars
- XXI. Aggregations of Stars
- Appendix. Tables of Astronomical Data
- Index

- Project Gutenberg License

Table of Contents

- 1–4
- 5–20
- 21–58
- 59–71
- 72–93
- 94–120
- 121–148
- 149–176
- 177–216
- 217–232
- 233–251
- 252–273
- 274–299
- 300–322
- 323–355
- 356–383
- 384–402
- 403–438
- 439–455
- 456–476
- 477–503
- 504–539
- 540–547
- 548–567
- Figure 1,
- (Fig. 2)
- Fig. 3.
- Art. 30)
- (Fig. 4).—The
- (Fig. 5).—The
- Art. 179.)
- Fig. 6.
- (Fig. 7).—A
- (Fig. 8),
- Fig. 9,
- Fig. 10.
- Fig. 12,
- (Fig. 13)
- figure
- figure)
- (Fig. 16)
- Fig. 20
- (Fig. 21).—This
- Fig. 22,
- Fig. 23.
- Fig. 26
- Fig. 27
- Fig. 28,
- Fig. 30,
- figure,