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Title: A Text-Book of Astronomy
Author: Comstock, George C.
Language: English
As this book started as an ASCII text book there are no pictures available.

*** Start of this LibraryBlog Digital Book "A Text-Book of Astronomy" ***

                      TWENTIETH CENTURY TEXT-BOOKS

                                EDITED BY
                     A. F. NIGHTINGALE, PH.D., LL.D.

                 [Illustration: A TOTAL SOLAR ECLIPSE.
    After Burckhalter's photographs of the eclipse of May 28, 1900.]

                      TWENTIETH CENTURY TEXT-BOOKS

                             A TEXT-BOOK OF

                           GEORGE C. COMSTOCK

                     PROFESSOR OF ASTRONOMY IN THE
                        UNIVERSITY OF WISCONSIN


                                NEW YORK
                        D. APPLETON AND COMPANY

                            COPYRIGHT, 1901
                       BY D. APPLETON AND COMPANY


The present work is not a compendium of astronomy or an outline course
of popular reading in that science. It has been prepared as a text-book,
and the author has purposely omitted from it much matter interesting as
well as important to a complete view of the science, and has endeavored
to concentrate attention upon those parts of the subject that possess
special educational value. From this point of view matter which permits
of experimental treatment with simple apparatus is of peculiar value and
is given a prominence in the text beyond its just due in a well-balanced
exposition of the elements of astronomy, while topics, such as the
results of spectrum analysis, which depend upon elaborate apparatus, are
in the experimental part of the work accorded much less space than their
intrinsic importance would justify.

Teacher and student are alike urged to magnify the observational side of
the subject and to strive to obtain in their work the maximum degree of
precision of which their apparatus is capable. The instruments required
are few and easily obtained. With exception of a watch and a protractor,
all of the apparatus needed may be built by any one of fair mechanical
talent who will follow the illustrations and descriptions of the text.
In order that proper opportunity for observations may be had, the study
should be pursued during the milder portion of the year, between April
and November in northern latitudes, using clear weather for a direct
study of the sky and cloudy days for book work.

The illustrations contained in the present work are worthy of as careful
study as is the text, and many of them are intended as an aid to
experimental work and accurate measurement, e. g., the star maps, the
diagrams of the planetary orbits, pictures of the moon, sun, etc. If the
school possesses a projection lantern, a set of astronomical slides to
be used in connection with it may be made of great advantage, if the
pictures are studied as an auxiliary to Nature. Mere display and scenic
effect are of little value.

A brief bibliography of popular literature upon astronomy may be found
at the end of this book, and it will be well if at least a part of these
works can be placed in the school library and systematically used for
supplementary reading. An added interest may be given to the study if
one or more of the popular periodicals which deal with astronomy are
taken regularly by the school and kept within easy reach of the
students. From time to time the teacher may well assign topics treated
in these periodicals to be read by individual students and presented to
the class in the form of an essay.

The author is under obligations to many of his professional friends who
have contributed illustrative matter for his text, and his thanks are in
an especial manner due to the editors of the Astrophysical Journal,
Astronomy and Astrophysics, and Popular Astronomy for permission to
reproduce here plates which have appeared in those periodicals, and to
Dr. Charles Boynton, who has kindly read and criticised the proofs.

                                                  GEORGE C. COMSTOCK.

  UNIVERSITY OF WISCONSIN, _February, 1901_.


 CHAPTER                                                        PAGE
    I.--DIFFERENT KINDS OF MEASUREMENT                             1
          The measurement of angles and time.

   II.--THE STARS AND THEIR DIURNAL MOTION                        10
          Finding the stars--Their apparent motion--
          Latitude--Direction of the meridian--Sidereal

  III.--FIXED AND WANDERING STARS                                 29
          Apparent motion of the sun, moon, and
          planets--Orbits of the planets--How to find
          the planets.

   IV.--CELESTIAL MECHANICS                                       46
          Kepler's laws--Newton's laws of motion--The law
          of gravitation--Orbital motion--Perturbations--
          Masses of the planets--Discovery of Neptune--
          The tides.

    V.--THE EARTH AS A PLANET                                     70
          Size--Mass--Precession--The warming of the
          earth--The atmosphere--Twilight.

   VI.--THE MEASUREMENT OF TIME                                   86
          Solar and sidereal time--Longitude--The

  VII.--ECLIPSES                                                 101
          Their cause and nature--Eclipse limits--Eclipse
          maps--Recurrence and prediction of eclipses.

          The clock--Radiant energy--Mirrors and lenses--
          The telescope--Camera--Spectroscope--Principles
          of spectrum analysis.

   IX.--THE MOON                                                 150
          Numerical data--Phases--Motion--Librations--Lunar
          topography--Physical condition.

    X.--THE SUN                                                  178
          Numerical data--Chemical nature--Temperature--
          Visible and invisible parts--Photosphere--Spots--
          sun-spot period--The sun's rotation--Mechanical
          theory of the sun.

   XI.--THE PLANETS                                              212
          Arrangement of the solar system--Bode's law--
          Physical condition of the planets--Jupiter--
          Saturn--Uranus and Neptune--Venus--Mercury--
          Mars--The asteroids.

  XII.--COMETS AND METEORS                                       251
          Motion, size, and mass of comets--Meteors--Their
          number and distribution--Meteor showers--Relation
          of comets and meteors--Periodic comets--Comet
          families and groups--Comet tails--Physical nature
          of comets--Collisions.

 XIII.--THE FIXED STARS                                          291
          Number of the stars--Brightness--Distance--Proper
          motion--Motion in line of sight--Double stars--
          Variable stars--New stars.

  XIV.--STARS AND NEBULÆ                                         330
          Stellar colors and spectra--Classes of stars--
          Clusters--Nebulæ--Their spectra and physical
          condition--The Milky Way--Construction of the
          heavens--Extent of the stellar system.

   XV.--GROWTH AND DECAY                                         358
          Logical bases and limitations--Development of the
          sun--The nebular hypothesis--Tidal friction--Roche's
          limit--Development of the moon--Development of stars
          and nebulæ--The future.

        APPENDIX                                                 383

        INDEX                                                    387


                                                         FACING PAGE
    I.--Northern Constellations                                  124
   II.--Equatorial Constellations                                190
  III.--Map of Mars                                              246
   IV.--The Pleiades                                             344
        Protractor                       _In pocket at back of book_


                                                         FACING PAGE
  A Total Solar Eclipse                               _Frontispiece_
  The Harvard College Observatory, Cambridge, Mass.               24
  Isaac Newton                                                    46
  Galileo Galilei                                                 52
  The Lick Observatory, Mount Hamilton, Cal.                      60
  The Yerkes Observatory, Williams Bay, Wis.                     100
  The Moon, one day after First Quarter                          150
  William Herschel                                               234
  Pierre Simon Laplace                                           364




1. ACCURATE MEASUREMENT.--Accurate measurement is the foundation of
exact science, and at the very beginning of his study in astronomy the
student should learn something of the astronomer's kind of measurement.
He should practice measuring the stars with all possible care, and
should seek to attain the most accurate results of which his instruments
and apparatus are capable. The ordinary affairs of life furnish abundant
illustration of some of these measurements, such as finding the length
of a board in inches or the weight of a load of coal in pounds and
measurements of both length and weight are of importance in astronomy,
but of far greater astronomical importance than these are the
measurement of angles and the measurement of time. A kitchen clock or a
cheap watch is usually thought of as a machine to tell the "time of
day," but it may be used to time a horse or a bicycler upon a race
course, and then it becomes an instrument to measure the amount of time
required for covering the length of the course. Astronomers use a clock
in both of these ways--to tell the time at which something happens or is
done, and to measure the amount of time required for something; and in
using a clock for either purpose the student should learn to take the
time from it to the nearest second or better, if it has a seconds hand,
or to a small fraction of a minute, by estimating the position of the
minute hand between the minute marks on the dial. Estimate the fraction
in tenths of a minute, not in halves or quarters.

EXERCISE 1.--If several watches are available, let one person tap
sharply upon a desk with a pencil and let each of the others note the
time by the minute hand to the nearest tenth of a minute and record the
observations as follows:

    2h. 44.5m.  First tap.   2h. 46.4m.  1.9m.
    2h. 44.9m.  Second tap.  2h. 46.7m.  1.8m.
    2h. 46.6m.  Third tap.   2h. 48.6m.  2.0m.

The letters h and m are used as abbreviations for hour and minute. The
first and second columns of the table are the record made by one
student, and second and third the record made by another. After all the
observations have been made and recorded they should be brought together
and compared by taking the differences between the times recorded for
each tap, as is shown in the last column. This difference shows how much
faster one watch is than the other, and the agreement or disagreement of
these differences shows the degree of accuracy of the observations. Keep
up this practice until tenths of a minute can be estimated with fair

2. ANGLES AND THEIR USE.--An angle is the amount of opening or
difference of direction between two lines that cross each other. At
twelve o'clock the hour and minute hand of a watch point in the same
direction and the angle between them is zero. At one o'clock the minute
hand is again at XII, but the hour hand has moved to I, one twelfth part
of the circumference of the dial, and the angle between the hands is one
twelfth of a circumference. It is customary to imagine the circumference
of a dial to be cut up into 360 equal parts--i. e., each minute space of
an ordinary dial to be subdivided into six equal parts, each of which
is called a degree, and the measurement of an angle consists in finding
how many of these degrees are included in the opening between its sides.
At one o'clock the angle between the hands of a watch is thirty degrees,
which is usually written 30°, at three o'clock it is 90°, at six o'clock
180°, etc.

A watch may be used to measure angles. How? But a more convenient
instrument is the protractor, which is shown in Fig. 1, applied to the
angle _A B C_ and showing that _A B C_ = 85° as nearly as the protractor
scale can be read.

The student should have and use a protractor, such as is furnished with
this book, for the numerous exercises which are to follow.

[Illustration: FIG. 1.--A protractor.]

EXERCISE 2.--Draw neatly a triangle with sides about 100 millimeters
long, measure each of its angles and take their sum. No matter what may
be the shape of the triangle, this sum should be very nearly
180°--exactly 180° if the work were perfect--but perfection can seldom
be attained and one of the first lessons to be learned in any science
which deals with measurement is, that however careful we may be in our
work some minute error will cling to it and our results can be only
approximately correct. This, however, should not be taken as an excuse
for careless work, but rather as a stimulus to extra effort in order
that the unavoidable errors may be made as small as possible. In the
present case the measured angles may be improved a little by adding
(algebraically) to each of them one third of the amount by which their
sum falls short of 180°, as in the following example:

            Measured angles.    Correction.  Corrected angles.
                 °                   °              °
  A            73.4               + 0.1           73.5
  B            49.3               + 0.1           49.4
  C            57.0               + 0.1           57.1
              -----                              -----
  Sum         179.7                              180.0
  Defect      + 0.3

This process is in very common use among astronomers, and is called
"adjusting" the observations.

[Illustration: FIG. 2.--Triangulation.]

3. TRIANGLES.--The instruments used by astronomers for the measurement
of angles are usually provided with a telescope, which may be pointed at
different objects, and with a scale, like that of the protractor, to
measure the angle through which the telescope is turned in passing from
one object to another. In this way it is possible to measure the angle
between lines drawn from the instrument to two distant objects, such as
two church steeples or the sun and moon, and this is usually called the
angle between the objects. By measuring angles in this way it is
possible to determine the distance to an inaccessible point, as shown in
Fig. 2. A surveyor at _A_ desires to know the distance to _C_, on the
opposite side of a river which he can not cross. He measures with a tape
line along his own side of the stream the distance _A B_ = 100 yards and
then, with a suitable instrument, measures the angle at _A_ between the
points _C_ and _B_, and the angle at _B_ between _C_ and _A_, finding _B
A C_ = 73.4°, _A B C_ = 49.3°. To determine the distance _A C_ he draws
upon paper a line 100 millimeters long, and marks the ends _a_ and _b_;
with a protractor he constructs at _a_ the angle _b a c_ = 73.4°, and at
_b_ the angle _a b c_ = 49.3°, and marks by _c_ the point where the two
lines thus drawn meet. With the millimeter scale he now measures the
distance _a c_ = 90.2 millimeters, which determines the distance _A C_
across the river to be 90.2 yards, since the triangle on paper has been
made similar to the one across the river, and millimeters on the one
correspond to yards on the other. What is the proposition of geometry
upon which this depends? The measured distance _A B_ in the surveyor's
problem is called a base line.

EXERCISE 3.--With a foot rule and a protractor measure a base line and
the angles necessary to determine the length of the schoolroom. After
the length has been thus found, measure it directly with the foot rule
and compare the measured length with the one found from the angles. If
any part of the work has been carelessly done, the student need not
expect the results to agree.

[Illustration: FIG. 3.--Finding the moon's distance from the earth.]

In the same manner, by sighting at the moon from widely different parts
of the earth, as in Fig. 3, the moon's distance from us is found to be
about a quarter of a million miles. What is the base line in this case?

4. THE HORIZON--ALTITUDES.--In their observations astronomers and
sailors make much use of the _plane of the horizon_, and practically any
flat and level surface, such as that of a smooth pond, may be regarded
as a part of this plane and used as such. A very common observation
relating to the plane of the horizon is called "taking the sun's
altitude," and consists in measuring the angle between the sun's rays
and the plane of the horizon upon which they fall. This angle between a
line and a plane appears slightly different from the angle between two
lines, but is really the same thing, since it means the angle between
the sun's rays and a line drawn in the plane of the horizon toward the
point directly under the sun. Compare this with the definition given in
the geographies, "The latitude of a point on the earth's surface is its
angular distance north or south of the equator," and note that the
latitude is the angle between the plane of the equator and a line drawn
from the earth's center to the given point on its surface.

A convenient method of obtaining a part of the plane of the horizon for
use in observation is as follows: Place a slate or a pane of glass upon
a table in the sunshine. Slightly moisten its whole surface and then
pour a little more water upon it near the center. If the water runs
toward one side, thrust the edge of a thin wooden wedge under this side
and block it up until the water shows no tendency to run one way rather
than another; it is then level and a part of the plane of the horizon.
Get several wedges ready before commencing the experiment. After they
have been properly placed, drive a pin or tack behind each one so that
it may not slip.

5. TAKING THE SUN'S ALTITUDE. EXERCISE 4.--Prepare a piece of board 20
centimeters, or more, square, planed smooth on one face and one edge.
Drive a pin perpendicularly into the face of the board, near the middle
of the planed edge. Set the board on edge on the horizon plane and turn
it edgewise toward the sun so that a shadow of the pin is cast on the
plane. Stick another pin into the board, near its upper edge, so that
its shadow shall fall exactly upon the shadow of the first pin, and with
a watch or clock observe the time at which the two shadows coincide.
Without lifting the board from the plane, turn it around so that the
opposite edge is directed toward the sun and set a third pin just as the
second one was placed, and again take the time. Remove the pins and draw
fine pencil lines, connecting the holes, as shown in Fig. 4, and with
the protractor measure the angle thus marked. The student who has
studied elementary geometry should be able to demonstrate that at the
mean of the two recorded times the sun's altitude was equal to one half
of the angle measured in the figure.

[Illustration: FIG. 4.--Taking the sun's altitude.]

When the board is turned edgewise toward the sun so that its shadow is
as thin as possible, rule a pencil line alongside it on the horizon
plane. The angle which this line makes with a line pointing due south is
called the sun's _azimuth_. When the sun is south, its azimuth is zero;
when west, it is 90°; when east, 270°, etc.

EXERCISE 5.--Let a number of different students take the sun's altitude
during both the morning and afternoon session and note the time of each
observation, to the nearest minute. Verify the setting of the plane of
the horizon from time to time, to make sure that no change has occurred
in it.

6. GRAPHICAL REPRESENTATIONS.--Make a graph (drawing) of all the
observations, similar to Fig. 5, and find by bisecting a set of chords
_g_ to _g_, _e_ to _e_, _d_ to _d_, drawn parallel to _B B_, the time at
which the sun's altitude was greatest. In Fig. 5 we see from the
intersection of _M M_ with _B B_ that this time was 11h. 50m.

The method of graphs which is here introduced is of great importance in
physical science, and the student should carefully observe in Fig. 5
that the line _B B_ is a scale of times, which may be made long or
short, provided only the intervals between consecutive hours 9 to 10, 10
to 11, 11 to 12, etc., are equal. The distance of each little circle
from _B B_ is taken proportional to the sun's altitude, and may be upon
any desired scale--e. g., a millimeter to a degree--provided the same
scale is used for all observations. Each circle is placed accurately
over that part of the base line which corresponds to the time at which
the altitude was taken. Square ruled paper is very convenient, although
not necessary, for such diagrams. It is especially to be noted that from
the few observations which are represented in the figure a smooth curve
has been drawn through the circles which represent the sun's altitude,
and this curve shows the altitude of the sun at every moment between 9
A. M. and 3 P. M. In Fig. 5 the sun's altitude at noon was 57°. What was
it at half past two?

[Illustration: FIG. 5.--A graph of the sun's altitude.]

7. DIAMETER OF A DISTANT OBJECT.--By sighting over a protractor, measure
the angle between imaginary lines drawn from it to the opposite sides of
a window. Carry the protractor farther away from the window and repeat
the experiment, to see how much the angle changes. The angle thus
measured is called "the angle subtended" by the window at the place
where the measurement was made. If this place was squarely in front of
the window we may draw upon paper an angle equal to the measured one and
lay off from the vertex along its sides a distance proportional to the
distance of the window--e. g., a millimeter for each centimeter of real
distance. If a cross line be now drawn connecting the points thus found,
its length will be proportional to the width of the window, and the
width may be read off to scale, a centimeter for every millimeter in the
length of the cross line.

The astronomer who measures with an appropriate instrument the angle
subtended by the moon may in an entirely similar manner find the moon's
diameter and has, in fact, found it to be 2,163 miles. Can the same
method be used to find the diameter of the sun? A planet? The earth?



8. THE STARS.--From the very beginning of his study in astronomy, and as
frequently as possible, the student should practice watching the stars
by night, to become acquainted with the constellations and their
movements. As an introduction to this study he may face toward the
north, and compare the stars which he sees in that part of the sky with
the map of the northern heavens, given on Plate I, opposite page 124.
Turn the map around, upside down if necessary, until the stars upon it
match the brighter ones in the sky. Note how the stars are grouped in
such conspicuous constellations as the Big Dipper (Ursa Major), the
Little Dipper (Ursa Minor), and Cassiopeia. These three constellations
should be learned so that they can be recognized at any time.

_The names of the stars._--Facing the star map is a key which contains
the names of the more important constellations and the names of the
brighter stars in their constellations. These names are for the most
part a Greek letter prefixed to the genitive case of the Latin name of
the constellation. (See the Greek alphabet printed at the end of the

9. MAGNITUDES OF THE STARS.--Nearly nineteen centuries ago St. Paul
noted that "one star differeth from another star in glory," and no more
apt words can be found to mark the difference of brightness which the
stars present. Even prior to St. Paul's day the ancient Greek
astronomers had divided the stars in respect of brightness into six
groups, which the modern astronomers still use, calling each group a
_magnitude_. Thus a few of the brightest stars are said to be of the
first magnitude, the great mass of faint ones which are just visible to
the unaided eye are said to be of the sixth magnitude, and intermediate
degrees of brilliancy are represented by the intermediate magnitudes,
second, third, fourth, and fifth. The student must not be misled by the
word magnitude. It has no reference to the size of the stars, but only
to their brightness, and on the star maps of this book the larger and
smaller circles by which the stars are represented indicate only the
brightness of the stars according to the system of magnitudes. Following
the indications of these maps, the student should, in learning the
principal stars and constellations, learn also to recognize how bright
is a star of the second, fourth, or other magnitude.

10. OBSERVING THE STARS.--Find on the map and in the sky the stars
α Ursæ Minoris, α Ursæ Majoris, β Ursæ Majoris. What geometrical
figure will fit on to these stars? In addition to its regular name,
α Ursæ Minoris is frequently called by the special name Polaris, or
the pole star. Why are the other two stars called "the Pointers"? What
letter of the alphabet do the five bright stars in Cassiopeia suggest?

EXERCISE 6.--Stand in such a position that Polaris is just hidden behind
the corner of a building or some other vertical line, and mark upon the
key map as accurately as possible the position of this line with respect
to the other stars, showing which stars are to the right and which are
to the left of it. Record the time (date, hour, and minute) at which
this observation was made. An hour or two later repeat the observation
at the same place, draw the line and note the time, and you will find
that the line last drawn upon the map does not agree with the first one.
The stars have changed their positions, and with respect to the vertical
line the Pointers are now in a different direction from Polaris.
Measure with a protractor the angle between the two lines drawn in the
map, and use this angle and the recorded times of the observation to
find how many degrees per hour this direction is changing. It should be
about 15° per hour. If the observation were repeated 12 hours after the
first recorded time, what would be the position of the vertical line
among the stars? What would it be 24 hours later? A week later? Repeat
the observation on the next clear night, and allowing for the number of
whole revolutions made by the stars between the two dates, again
determine from the time interval a more accurate value of the rate at
which the stars move.

The motion of the stars which the student has here detected is called
their "diurnal" motion. What is the significance of the word diurnal?

In the preceding paragraph there is introduced a method of great
importance in astronomical practice--i. e., determining something--in
this case the rate per hour, from observations separated by a long
interval of time, in order to get a more accurate value than could be
found from a short interval. Why is it more accurate? To determine the
rate at which the planet Mars rotates about its axis, astronomers use
observations separated by an interval of more than 200 years, during
which the planet made more than 75,000 revolutions upon its axis. If we
were to write out in algebraic form an equation for determining the
length of one revolution of Mars about its axis, the large number,
75,000, would appear in the equation as a divisor, and in the final
result would greatly reduce whatever errors existed in the observations

Repeat Exercise 6 night after night, and note whether the stars come
back to the same position at the same hour and minute every night.

[Illustration: FIG. 6. The plumb-line apparatus.]

[Illustration: FIG. 7. The plumb-line apparatus.]

11. THE PLUMB-LINE APPARATUS.--This experiment, and many others, may be
conveniently and accurately made with no other apparatus than a plumb
line, and a device for sighting past it. In Figs. 6 and 7 there is
shown a simple form of such apparatus, consisting essentially of a board
which rests in a horizontal position upon the points of three screws
that pass through it. This board carries a small box, to one side of
which is nailed in vertical position another board 5 or 6 feet long to
carry the plumb line. This consists of a wire or fish line with any
heavy weight--e. g., a brick or flatiron--tied to its lower end and
immersed in a vessel of water placed inside the box, so as to check any
swinging motion of the weight. In the cover of the box is a small hole
through which the wire passes, and by turning the screws in the
baseboard the apparatus may be readily leveled, so that the wire shall
swing freely in the center of the hole without touching the cover of the
box. Guy wires, shown in the figure, are applied so as to stiffen the
whole apparatus. A board with a screw eye at each end may be pivoted to
the upright, as in Fig. 6, for measuring altitudes; or to the box, as in
Fig. 7, for observing the time at which a star in its diurnal motion
passes through the plane determined by the plumb line and the center of
the screw eye through which the observer looks.

The whole apparatus may be constructed by any person of ordinary
mechanical skill at a very small cost, and it or something equivalent
should be provided for every class beginning observational astronomy. To
use the apparatus for the experiment of § 10, it should be leveled, and
the board with the screw eyes, attached as in Fig. 7, should be turned
until the observer, looking through the screw eye, sees Polaris exactly
behind the wire. Use a bicycle lamp to illumine the wire by night. The
apparatus is now adjusted, and the observer has only to wait for the
stars which he desires to observe, and to note by his watch the time at
which they pass behind the wire. It will be seen that the wire takes the
place of the vertical edge of the building, and that the board with the
screw eyes is introduced solely to keep the observer in the right place
relative to the wire.

12. A SIDEREAL CLOCK.--Clocks are sometimes so made and regulated that
they show always the same hour and minute when the stars come back to
the same place, and such a timepiece is called a sidereal clock--i. e.,
a star-time clock. Would such a clock gain or lose in comparison with an
ordinary watch? Could an ordinary watch be turned into a sidereal watch
by moving the regulator?

[Illustration: FIG. 8.--Photographing the circumpolar stars.--BARNARD.]

13. PHOTOGRAPHING THE STARS.--EXERCISE 7.--For any student who uses a
camera. Upon some clear and moonless night point the camera, properly
focused, at Polaris, and expose a plate for three or four hours. Upon
developing the plate you should find a series of circular trails such as
are shown in Fig. 8, only longer. Each one of these is produced by a
star moving slowly over the plate, in consequence of its changing
position in the sky. The center indicated by these curved trails is
called the pole of the heavens. It is that part of the sky toward which
is pointed the axis about which the earth rotates, and the motion of the
stars around the center is only an apparent motion due to the rotation
of the earth which daily carries the observer and his camera around this
axis while the stars stand still, just as trees and fences and telegraph
poles stand still, although to the passenger upon a railway train they
appear to be in rapid motion. So far as simple observations are
concerned, there is no method by which the pupil can tell for himself
that the motion of the stars is an apparent rather than a real one, and,
following the custom of astronomers, we shall habitually speak as if it
were a real movement of the stars. How long was the plate exposed in
photographing Fig. 8?

14. FINDING THE STARS.--On Plate I, opposite page 124, the pole of the
heavens is at the center of the map, near Polaris, and the heavy trail
near the center of Fig. 8 is made by Polaris. See if you can identify
from the map any of the stars whose trails show in the photograph. The
brighter the star the bolder and heavier its trail.

Find from the map and locate in the sky the two bright stars Capella and
Vega, which are on opposite sides of Polaris and nearly equidistant from
it. Do these stars share in the motion around the pole? Are they visible
on every clear night, and all night?

Observe other bright stars farther from Polaris than are Vega and
Capella and note their movement. Do they move like the sun and moon? Do
they rise and set?

In what part of the sky do the stars move most rapidly, near the pole or
far from it?

How long does it take the fastest moving stars to make the circuit of
the sky and come back to the same place? How long does it take the slow

15. RISING AND SETTING OF THE STARS.--A study of the sky along the lines
indicated in these questions will show that there is a considerable part
of it surrounding the pole whose stars are visible on every clear night.
The same star is sometimes high in the sky, sometimes low, sometimes to
the east of the pole and at other times west of it, but is always above
the horizon. Such stars are said to be circumpolar. A little farther
from the pole each star, when at the lowest point of its circular path,
dips for a time below the horizon and is lost to view, and the farther
it is away from the pole the longer does it remain invisible, until, in
the case of stars 90° away from the pole, we find them hidden below the
horizon for twelve hours out of every twenty-four (see Fig. 9). The sun
is such a star, and in its rising and setting acts precisely as does
every other star at a similar distance from the pole--only, as we shall
find later, each star keeps always at (nearly) the same distance from
the pole, while the sun in the course of a year changes its distance
from the pole very greatly, and thus changes the amount of time it
spends above and below the horizon, producing in this way the long days
of summer and the short ones of winter.

[Illustration: FIG. 9.--Diurnal motion of the northern constellations.]

How much time do stars which are more than 90° from the pole spend above
the horizon?

We say in common speech that the sun rises in the east, but this is
strictly true only at the time when it is 90° distant from the
pole--i. e., in March and September. At other seasons it rises north or
south of east according as its distance from the pole is less or greater
than 90°, and the same is true for the stars.

16. THE GEOGRAPHY OF THE SKY.--Find from a map the latitude and
longitude of your schoolhouse. Find on the map the place whose latitude
is 39° and longitude 77° west of the meridian of Greenwich. Is there any
other place in the world which has the same latitude and longitude as
your schoolhouse?

The places of the stars in the sky are located in exactly the manner
which is illustrated by these geographical questions, only different
names are used. Instead of latitude the astronomer says _declination_,
in place of longitude he says _right ascension_, in place of meridian he
says _hour circle_, but he means by these new names the same ideas that
the geographer expresses by the old ones.

Imagine the earth swollen up until it fills the whole sky; the earth's
equator would meet the sky along a line (a great circle) everywhere 90°
distant from the pole, and this line is called the _celestial equator_.
Trace its position along the middle of the map opposite page 190 and
notice near what stars it runs. Every meridian of the swollen earth
would touch the sky along an hour circle--i. e., a great circle passing
through the pole and therefore perpendicular to the equator. Note that
in the map one of these hour circles is marked 0. It plays the same part
in measuring right ascensions as does the meridian of Greenwich in
measuring longitudes; it is the beginning, from which they are reckoned.
Note also, at the extreme left end of the map, the four bright stars in
the form of a square, one side of which is parallel and close to the
hour circle, which is marked 0. This is familiarly called the Great
Square in Pegasus, and may be found high up in the southern sky whenever
the Big Dipper lies below the pole. Why can it not be seen when Ursa
Major is above the pole?

Astronomers use the right ascensions of the stars not only to tell in
what part of the sky the star is placed, but also in time reckonings, to
regulate their sidereal clocks, and with regard to this use they find
it convenient to express right ascension not in degrees but in hours,
24 of which fill up the circuit of the sky and each of which is equal
to 15° of arc, 24 × 15 = 360. The right ascension of Capella is
5h. 9m. = 77.2°, but the student should accustom himself to using it
in hours and minutes as given and not to change it into degrees. He
should also note that some stars lie on the side of the celestial
equator toward Polaris, and others are on the opposite side, so that the
astronomer has to distinguish between north declinations and south
declinations, just as the geographer distinguishes between north
latitudes and south latitudes. This is done by the use of the + and -
signs, a + denoting that the star lies north of the celestial equator,
i. e., toward Polaris.

[Illustration: FIG. 10.--From a photograph of the Pleiades.]

Find on Plate II, opposite page 190, the Pleiades (Plēadēs),
R. A. = 3h. 42m., Dec. = +23.8°. Why do they not show on Plate I,
opposite page 124? In what direction are they from Polaris? This is one
of the finest star clusters in the sky, but it needs a telescope to
bring out its richness. See how many stars you can count in it with the
naked eye, and afterward examine it with an opera glass. Compare what
you see with Fig. 10. Find Antares, R. A. = 16h. 23m. Dec. = -26.2°. How
far is it, in degrees, from the pole? Is it visible in your sky? If so,
what is its color?

Find the R. A. and Dec. of α Ursæ Majoris; of β Ursæ Majoris; of
Polaris. Find the Northern Crown, _Corona Borealis_, R. A. = 15h. 30m.,
Dec. = +27.0°; the Beehive, _Præsepe_, R. A. = 8h. 33m., Dec. = +20.4°.

These should be looked up, not only on the map, but also in the sky.

17. REFERENCE LINES AND CIRCLES.--As the stars move across the sky in
their diurnal motion, they carry the framework of hour circles and
equator with them, so that the right ascension and declination of each
star remain unchanged by this motion, just as longitudes and latitudes
remain unchanged by the earth's rotation. They are the same when a star
is rising and when it is setting; when it is above the pole and when it
is below it. During each day the hour circle of every star in the
heavens passes overhead, and at the moment when any particular hour
circle is exactly overhead all the stars which lie upon it are said to
be "on the meridian"--i. e., at that particular moment they stand
directly over the observer's geographical meridian and upon the
corresponding celestial meridian.

An eye placed at the center of the earth and capable of looking through
its solid substance would see your geographical meridian against the
background of the sky exactly covering your celestial meridian and
passing from one pole through your zenith to the other pole. In Fig. 11
the inner circle represents the terrestrial meridian of a certain
place, _O_, as seen from the center of the earth, _C_, and the outer
circle represents the celestial meridian of _O_ as seen from _C_, only
we must imagine, what can not be shown on the figure, that the outer
circle is so large that the inner one shrinks to a mere point in
comparison with it. If _C P_ represents the direction in which the
earth's axis passes through the center, then _C E_ at right angles to it
must be the direction of the equator which we suppose to be turned
edgewise toward us; and if _C O_ is the direction of some particular
point on the earth's surface, then _Z_ directly overhead is called the
_zenith_ of that point, upon the celestial sphere. The line _C H_
represents a direction parallel to the horizon plane at _O_, and _H C P_
is the angle which the axis of the earth makes with this horizon plane.
The arc _O E_ measures the latitude of _O_, and the arc _Z E_ measures
the declination of _Z_, and since by elementary geometry each of these
arcs contains the same number of degrees as the angle _E C Z_, we have

_Theorem._--The latitude of any place is equal to the declination of its

_Corollary._--Any star whose declination is equal to your latitude will
once in each day pass through your zenith.

[Illustration: FIG. 11.--Reference lines and circles.]

18. LATITUDE.--From the construction of the figure

    ∠ _E C Z_ + ∠ _Z C P_ = 90°
    ∠ _H C P_ + ∠ _Z C P_ = 90°

from which we find by subtraction and transposition

    ∠ _E C Z_ = ∠ _H C P_

and this gives the further

_Theorem._--The latitude of any place is equal to the elevation of the
pole above its horizon plane.

An observer who travels north or south over the earth changes his
latitude, and therefore changes the angle between his horizon plane and
the axis of the earth. What effect will this have upon the position of
stars in his sky? If you were to go to the earth's equator, in what part
of the sky would you look for Polaris? Can Polaris be seen from
Australia? From South America? If you were to go from Minnesota to
Texas, in what respect would the appearance of stars in the northern sky
be changed? How would the appearance of stars in the southern sky be

[Illustration: FIG. 12.--Diurnal path of Polaris.]

EXERCISE 8.--Determine your latitude by taking the altitude of Polaris
when it is at some one of the four points of its diurnal path, shown in
Fig. 12. When it is at _1_ it is said to be at upper culmination, and
the star ζ Ursæ Majoris in the handle of the Big Dipper will be
directly below it. When at _2_ it is at western elongation, and the star
Castor is near the meridian. When it is at _3_ it is at lower
culmination, and the star Spica is on the meridian. When it is at _4_ it
is at eastern elongation, and Altair is near the meridian. All of these
stars are conspicuous ones, which the student should find upon the map
and learn to recognize in the sky. The altitude observed at either _2_
or _4_ may be considered equal to the latitude of the place, but the
altitude observed when Polaris is at the positions marked _1_ and _3_
must be corrected for the star's distance from the pole, which may be
assumed equal to 1.3°.

The plumb-line apparatus described at page 12 is shown in Fig. 6
slightly modified, so as to adapt it to measuring the altitudes of
stars. Note that the board with the screw eye at one end has been
transferred from the box to the vertical standard, and has a screw eye
at each end. When the apparatus has been properly leveled, so that the
plumb line hangs at the middle of the hole in the box cover, the board
is to be pointed at the star by sighting through the centers of the two
screw eyes, and a pencil line is to be ruled along its edge upon the
face of the vertical standard. After this has been done turn the
apparatus halfway around so that what was the north side now points
south, level it again and revolve the board about the screw which holds
it to the vertical standard, until the screw eyes again point to the
star. Rule another line along the same edge of the board as before and
with a protractor measure the angle between these lines. Use a bicycle
lamp if you need artificial light for your work. The student who has
studied plane geometry should be able to prove that one half of the
angle between these lines is equal to the altitude of the star.

After you have determined your latitude from Polaris, compare the result
with your position as shown upon the best map available. With a little
practice and considerable care the latitude may be thus determined
within one tenth of a degree, which is equivalent to about 7 miles. If
you go 10 miles north or south from your first station you should find
the pole higher up or lower down in the sky by an amount which can be
measured with your apparatus.

19. THE MERIDIAN LINE.--To establish a true north and south line upon
the ground, use the apparatus as described at page 13, and when Polaris
is at upper or lower culmination drive into the ground two stakes in
line with the star and the plumb line. Such a meridian line is of great
convenience in observing the stars and should be laid out and
permanently marked in some convenient open space from which, if
possible, all parts of the sky are visible. June and November are
convenient months for this exercise, since Polaris then comes to
culmination early in the evening.

20. TIME.--What is _the time_ at which school begins in the morning?
What do you mean by "_the time_"?

The sidereal time at any moment is the right ascension of the hour
circle which at that moment coincides with the meridian. When the hour
circle passing through Sirius coincides with the meridian, the sidereal
time is 6h. 40m., since that is the right ascension of Sirius, and in
astronomical language Sirius is "_on the meridian_" at 6h. 40m. sidereal
time. As may be seen from the map, this 6h. 40m. is the right ascension
of Sirius, and if a clock be set to indicate 6h. 40m. when Sirius
crosses the meridian, it will show sidereal time. If the clock is
properly regulated, every other star in the heavens will come to the
meridian at the moment when the time shown by the clock is equal to the
right ascension of the star. A clock properly regulated for this purpose
will gain about four minutes per day in comparison with ordinary clocks,
and when so regulated it is called a sidereal clock. The student should
be provided with such a clock for his future work, but one such clock
will serve for several persons, and a nutmeg clock or a watch of the
cheapest kind is quite sufficient.


EXERCISE 9.--Set such a clock to sidereal time by means of the transit
of a star over your meridian. For this experiment it is presupposed that
a meridian line has been marked out on the ground as in § 19, and the
simplest mode of performing the experiment required is for the observer,
having chosen a suitable star in the southern part of the sky, to place
his eye accurately over the northern end of the meridian line and to
estimate as nearly as possible the beginning and end of the period
during which the star appears to stand exactly above the southern end of
the line. The middle of this period may be taken as the time at which
the star crossed the meridian and at this moment the sidereal time is
equal to the right ascension of the star. The difference between this
right ascension and the observed middle instant is the error of the
clock or the amount by which its hands must be set back or forward in
order to indicate true sidereal time.

A more accurate mode of performing the experiment consists in using the
plumb-line apparatus carefully adjusted, as in Fig. 7, so that the line
joining the wire to the center of the screw eye shall be parallel to the
meridian line. Observe the time by the clock at which the star
disappears behind the wire as seen through the center of the screw eye.
If the star is too high up in the sky for convenient observation, place
a mirror, face up, just north of the screw eye and observe star, wire
and screw eye by reflection in it.

The numerical right ascension of the observed star is needed for this
experiment, and it may be measured from the star map, but it will
usually be best to observe one of the stars of the table at the end of
the book, and to obtain its right ascension as follows: The table gives
the right ascension and declination of each star as they were at the
beginning of the year 1900, but on account of the precession (see
Chapter V), these numbers all change slowly with the lapse of time, and
on the average the right ascension of each star of the table must be
increased by one twentieth of a minute for each year after 1900--i. e.,
in 1910 the right ascension of the first star of the table will be
0h. 38.6m. + (10/20)m. = 0h. 39.1m. The declinations also change
slightly, but as they are only intended to help in finding the star on
the star maps, their change may be ignored.

Having set the clock approximately to sidereal time, observe one or two
more stars in the same way as above. The difference between the observed
time and the right ascension, if any is found, is the "correction" of
the clock. This correction ought not to exceed a minute if due care has
been taken in the several operations prescribed. The relation of the
clock to the right ascension of the stars is expressed in the following
equation, with which the student should become thoroughly familiar:

    A = T ± U

_T_ stands for the time by the clock at which the star crossed the
meridian. _A_ is the right ascension of the star, and _U_ is the
correction of the clock. Use the + sign in the equation whenever the
clock is too slow, and the - sign when it is too fast. _U_ may be found
from this equation when _A_ and _T_ are given, or _A_ may be found when
_T_ and _U_ are given. It is in this way that astronomers measure the
right ascensions of the stars and planets.

Determine _U_ from each star you have observed, and note how the several
results agree one with another.

21. DEFINITIONS.--To define a thing or an idea is to give a description
sufficient to identify it and distinguish it from every other possible
thing or idea. If a definition does not come up to this standard it is
insufficient. Anything beyond this requirement is certainly useless and
probably mischievous.

Let the student define the following geographical terms, and let him
also criticise the definitions offered by his fellow-students: Equator,
poles, meridian, latitude, longitude, north, south, east, west.

Compare the following astronomical definitions with your geographical
definitions, and criticise them in the same way. If you are not able to
improve upon them, commit them to memory:

_The Poles_ of the heavens are those points in the sky toward which the
earth's axis points. How many are there? The one near Polaris is called
the north pole.

_The Celestial Equator_ is a great circle of the sky distant 90° from
the poles.

_The Zenith_ is that point of the sky, overhead, toward which a plumb
line points. Why is the word overhead placed in the definition? Is there
more than one zenith?

_The Horizon_ is a great circle of the sky 90° distant from the zenith.

_An Hour Circle_ is any great circle of the sky which passes through the
poles. Every star has its own hour circle.

_The Meridian_ is that hour circle which passes through the zenith.

_A Vertical Circle_ is any great circle that passes through the zenith.
Is the meridian a vertical circle?

_The Declination_ of a star is its angular distance north or south of
the celestial equator.

_The Right Ascension_ of a star is the angle included between its hour
circle and the hour circle of a certain point on the equator which is
called the _Vernal Equinox_. From spherical geometry we learn that this
angle is to be measured either at the pole where the two hour circles
intersect, as is done in the star map opposite page 124, or along the
equator, as is done in the map opposite page 190. Right ascension is
always measured from the vernal equinox in the direction opposite to
that in which the stars appear to travel in their diurnal motion--i. e.,
from west toward east.

_The Altitude_ of a star is its angular distance above the horizon.

_The Azimuth_ of a star is the angle between the meridian and the
vertical circle passing through the star. A star due south has an
azimuth of 0°. Due west, 90°. Due north, 180°. Due east, 270°.

What is the azimuth of Polaris in degrees?

What is the azimuth of the sun at sunrise? At sunset? At noon? Are these
azimuths the same on different days?

_The Hour Angle_ of a star is the angle between its hour circle and the
meridian. It is measured from the meridian in the direction in which the
stars appear to travel in their diurnal motion--i. e., from east toward

What is the hour angle of the sun at noon? What is the hour angle of
Polaris when it is at the lowest point in its daily motion?

22. EXERCISES.--The student must not be satisfied with merely learning
these definitions. He must learn to see these points and lines in his
mind as if they were visibly painted upon the sky. To this end it will
help him to note that the poles, the zenith, the meridian, the horizon,
and the equator seem to stand still in the sky, always in the same place
with respect to the observer, while the hour circles and the vernal
equinox move with the stars and keep the same place among them. Does the
apparent motion of a star change its declination or right ascension?
What is the hour angle of the sun when it has the greatest altitude?
Will your answer to the preceding question be true for a star? What is
the altitude of the sun after sunset? In what direction is the north
pole from the zenith? From the vernal equinox? Where are the points in
which the meridian and equator respectively intersect the horizon?



23. STAR MAPS.--Select from the map some conspicuous constellation that
will be conveniently placed for observation in the evening, and make on
a large scale a copy of all the stars of the constellation that are
shown upon the map. At night compare this copy with the sky, and mark in
upon your paper all the stars of the constellation which are not already
there. Both the original drawing and the additions made to it by night
should be carefully done, and for the latter purpose what is called the
method of allineations may be used with advantage--i. e., the new star
is in line with two already on the drawing and is midway between them,
or it makes an equilateral triangle with two others, or a square with
three others, etc.

A series of maps of the more prominent constellations, such as Ursa
Major, Cassiopea, Pegasus, Taurus, Orion, Gemini, Canis Major, Leo,
Corvus, Bootes, Virgo, Hercules, Lyra, Aquila, Scorpius, should be
constructed in this manner upon a uniform scale and preserved as a part
of the student's work. Let the magnitude of the stars be represented on
the maps as accurately as may be, and note the peculiarity of color
which some stars present. For the most part their color is a very pale
yellow, but occasionally one may be found of a decidedly ruddy
hue--e. g., Aldebaran or Antares. Such a star map, not quite complete,
is shown in Fig. 13.

So, too, a sharp eye may detect that some stars do not remain always of
the same magnitude, but change their brightness from night to night,
and this not on account of cloud or mist in the atmosphere, but from
something in the star itself. Algol is one of the most conspicuous of
these _variable stars_, as they are called.

[Illustration: FIG. 13.--Star map of the region about Orion.]

24. THE MOON'S MOTION AMONG THE STARS.--Whenever the moon is visible
note its position among the stars by allineations, and plot it on the
key map opposite page 190. Keep a record of the day and hour
corresponding to each such observation. You will find, if the work is
correctly done, that the positions of the moon all fall near the curved
line shown on the map. This line is called the ecliptic.

After several such observations have been made and plotted, find by
measurement from the map how many degrees per day the moon moves. How
long would it require to make the circuit of the heavens and come back
to the starting point?

On each night when you observe the moon, make on a separate piece of
paper a drawing of it about 10 centimeters in diameter and show in the
drawing every feature of the moon's face which you can see--e. g., the
shape of the illuminated surface (phase); the direction among the stars
of the line joining the horns; any spots which you can see upon the
moon's face, etc. An opera glass will prove of great assistance in this

Use your drawings and the positions of the moon plotted upon the map to
answer the following questions: Does the direction of the line joining
the horns have any special relation to the ecliptic? Does the amount of
illuminated surface of the moon have any relation to the moon's angular
distance from the sun? Does it have any relation to the time at which
the moon sets? Do the spots on the moon when visible remain always in
the same place? Do they come and go? Do they change their position with
relation to each other? Can you determine from these spots that the moon
rotates about an axis, as the earth does? In what direction does its
axis point? How long does it take to make one revolution about the axis?
Is there any day and night upon the moon?

Each of these questions can be correctly answered from the student's own
observations without recourse to any book.

25. THE SUN AND ITS MOTION.--Examine the face of the sun through a
smoked glass to see if there is anything there that you can sketch.

By day as well as by night the sky is studded with stars, only they can
not be seen by day on account of the overwhelming glare of sunlight, but
the position of the sun among the stars may be found quite as
accurately as was that of the moon, by observing from day to day its
right ascension and declination, and this should be practiced at noon on
clear days by different members of the class.

EXERCISE 10.--The right ascension of the sun may be found by observing
with the sidereal clock the time of its transit over the meridian. Use
the equation in § 20, and substitute in place of _U_ the value of the
clock correction found from observations of stars on a preceding or
following night. If the clock gains or loses _with respect to sidereal
time_, take this into account in the value of _U_.

EXERCISE 11.--To determine the sun's declination, measure its altitude
at the time it crosses the meridian. Use either the method of Exercise
4, or that used with Polaris in Exercise 8. The student should be able
to show from Fig. 11 that the declination is equal to the sum of the
altitude and the latitude of the place diminished by 90°, or in an

    Declination = Altitude + Latitude - 90°.

If the declination as found from this equation is a negative number it
indicates that the sun is on the south side of the equator.

The right ascension and declination of the sun as observed on each day
should be plotted on the map and the date, written opposite it. If the
work has been correctly done, the plotted points should fall upon the
curved line (ecliptic) which runs lengthwise of the map. This line, in
fact, represents the sun's path among the stars.

Note that the hours of right ascension increase from 0 up to 24, while
the numbers on the clock dial go only from 0 to 12, and then repeat 0 to
12 again during the same day. When the sidereal time is 13 hours, 14
hours, etc., the clock will indicate 1 hour, 2 hours, etc., and 12 hours
must then be added to the time shown on the dial.

If observations of the sun's right ascension and declination are made
in the latter part of either March or September the student will find
that the sun crosses the equator at these times, and he should determine
from his observations, as accurately as possible, the date and hour of
this crossing and the point on the equator at which the sun crosses it.
These points are called the equinoxes, Vernal Equinox and Autumnal
Equinox for the spring and autumn crossings respectively, and the
student will recall that the vernal equinox is the point from which
right ascensions are measured. Its position among the stars is found by
astronomers from observations like those above described, only made with
much more elaborate apparatus.

Similar observations made in June and December show that the sun's
midday altitude is about 47° greater in summer than in winter. They show
also that the sun is as far north of the equator in June as he is south
of it in December, from which it is easily inferred that his path, the
ecliptic, is inclined to the equator at an angle of 23°.5, one half of
47°. This angle is called the obliquity of the ecliptic. The student may
recall that in the geographies the torrid zone is said to extend 23°.5
on either side of the earth's equator. Is there any connection between
these limits and the obliquity of the ecliptic? Would it be correct to
define the torrid zone as that part of the earth's surface within which
the sun may at some season of the year pass through the zenith?

EXERCISE 12.--After a half dozen observations of the sun have been
plotted upon the map, find by measurement the rate, in degrees per day,
at which the sun moves along the ecliptic. How many days will be
required for it to move completely around the ecliptic from vernal
equinox back to vernal equinox again? Accurate observations with the
elaborate apparatus used by professional astronomers show that this
period, which is called a _tropical year_, is 365 days 5 hours 48
minutes 46 seconds. Is this the same as the ordinary year of our

26. THE PLANETS.--Any one who has watched the sky and who has made the
drawings prescribed in this chapter can hardly fail to have found in the
course of his observations some bright stars not set down on the printed
star maps, and to have found also that these stars do not remain fixed
in position among their fellows, but wander about from one constellation
to another. Observe the motion of one of these planets from night to
night and plot its positions on the star map, precisely as was done for
the moon. What kind of path does it follow?

Both the ancient Greeks and the modern Germans have called these bodies
wandering stars, and in English we name them planets, which is simply
the Greek word for wanderer, bent to our use. Besides the sun and moon
there are in the heavens five planets easily visible to the naked eye
and, as we shall see later, a great number of smaller ones visible only
in the telescope. More than 2,000 years ago astronomers began observing
the motion of sun, moon, and planets among the stars, and endeavored to
account for these motions by the theory that each wandering star moved
in an orbit about the earth. Classical and mediæval literature are
permeated with this idea, which was displaced only after a long struggle
begun by Copernicus (1543 A. D.), who taught that the moon alone of
these bodies revolves about the earth, while the earth and the other
planets revolve around the sun. The ecliptic is the intersection of the
plane of the earth's orbit with the sky, and the sun appears to move
along the ecliptic because, as the earth moves around its orbit, the sun
is always seen projected against the opposite side of it. The moon and
planets all appear to move near the ecliptic because the planes of their
orbits nearly coincide with the plane of the earth's orbit, and a narrow
strip on either side of the ecliptic, following its course completely
around the sky, is called the _zodiac_, a word which may be regarded as
the name of a narrow street (16° wide) within which all the wanderings
of the visible planets are confined and outside of which they never
venture. Indeed, Mars is the only planet which ever approaches the edge
of the street, the others traveling near the middle of the road.

[Illustration: FIG. 14.--The apparent motion of a planet.]

27. A TYPICAL CASE OF PLANETARY MOTION.--The Copernican theory,
enormously extended and developed through the Newtonian law of
gravitation (see Chapter IV), has completely supplanted the older
Ptolemaic doctrine, and an illustration of the simple manner in which it
accounts for the apparently complicated motions of a planet among the
stars is found in Figs. 14 and 15, the first of which represents the
apparent motion of the planet Mars through the constellations Aries and
Pisces during the latter part of the year 1894, while the second shows
the true motions of Mars and the earth in their orbits about the sun
during the same period. The straight line in Fig. 14, with cross ruling
upon it, is a part of the ecliptic, and the numbers placed opposite it
represent the distance, in degrees, from the vernal equinox. In Fig. 15
the straight line represents the direction from the sun toward the
vernal equinox, and the angle which this line makes with the line
joining earth and sun is called the earth's longitude. The imaginary
line joining the earth and sun is called the earth's radius vector, and
the pupil should note that the longitude and length of the radius vector
taken together show the direction and distance of the earth from the
sun--i. e., they fix the relative positions of the two bodies. The same
is nearly true for Mars and would be wholly true if the orbit of Mars
lay in the same plane with that of the earth. How does Fig. 14 show that
the orbit of Mars does not lie exactly in the same plane with the orbit
of the earth?

EXERCISE 13.--Find from Fig. 15 what ought to have been the apparent
course of Mars among the stars during the period shown in the two
figures, and compare what you find with Fig. 14. The apparent position
of Mars among the stars is merely its direction from the earth, and this
direction is represented in Fig. 14 by the distance of the planet from
the ecliptic and by its longitude.

[Illustration: FIG. 15.--The real motion of a planet.]

The longitude of Mars for each date can be found from Fig. 15 by
measuring the angle between the straight line _S V_ and the line drawn
from the earth to Mars. Thus for October 12th we may find with the
protractor that the angle between the line _S V_ and the line joining
the earth to Mars is a little more than 30°, and in Fig. 14 the position
of Mars for this date is shown nearly opposite the cross line
corresponding to 30° on the ecliptic. Just how far below the ecliptic
this position of Mars should fall can not be told from Fig. 15, which
from necessity is constructed as if the orbits of Mars and the earth lay
in the same plane, and Mars in this case would always appear to stand
exactly on the ecliptic and to oscillate back and forth as shown in Fig.
14, but without the up-and-down motion there shown. In this way plot in
Fig. 14 the longitudes of Mars as seen from the earth for other dates
and observe how the forward motion of the two planets in their orbits
accounts for the apparently capricious motion of Mars to and fro among
the stars.

[Illustration: FIG. 16.--The orbits of Jupiter and Saturn.]

28. THE ORBITS OF THE PLANETS.--Each planet, great or small, moves in
its own appropriate orbit about the sun, and the exact determination of
these orbits, their sizes, shapes, positions, etc., has been one of the
great problems of astronomy for more than 2,000 years, in which
successive generations of astronomers have striven to push to a still
higher degree of accuracy the knowledge attained by their predecessors.
Without attempting to enter into the details of this problem we may say,
generally, that every planet moves in a plane passing through the sun,
and for the six planets visible to the naked eye these planes nearly
coincide, so that the six orbits may all be shown without much error as
lying in the flat surface of one map. It is, however, more convenient to
use two maps, such as Figs. 16 and 17, one of which shows the group of
planets, Mercury, Venus, the earth, and Mars, which are near the sun,
and on this account are sometimes called the inner planets, while the
other shows the more distant planets, Jupiter and Saturn, together with
the earth, whose orbit is thus made to serve as a connecting link
between the two diagrams. These diagrams are accurately drawn to scale,
and are intended to be used by the student for accurate measurement in
connection with the exercises and problems which follow.

In addition to the six planets shown in the figures the solar system
contains two large planets and several hundred small ones, for the most
part invisible to the naked eye, which are omitted in order to avoid
confusing the diagrams.

29. JUPITER AND SATURN.--In Fig. 16 the sun at the center is encircled
by the orbits of the three planets, and inclosing all of these is a
circular border showing the directions from the sun of the
constellations which lie along the zodiac. The student must note
carefully that it is only the directions of these constellations that
are correctly shown, and that in order to show them at all they have
been placed very much too close to the sun. The cross lines extending
from the orbit of the earth toward the sun with Roman numerals opposite
them show the positions of the earth in its orbit on the first day of
January (_I_), first day of February (_II_), etc., and the similar lines
attached to the orbits of Jupiter and Saturn with Arabic numerals show
the positions of those planets on the first day of January of each year
indicated, so that the figure serves to show not only the orbits of the
planets, but their actual positions in their orbits for something more
than the first decade of the twentieth century.

The line drawn from the sun toward the right of the figure shows the
direction to the vernal equinox. It forms one side of the angle which
measures a planet's longitude.

[Illustration: FIG. 17.--The orbits of the inner planets.]

EXERCISE 14.--Measure with your protractor the longitude of the earth on
January 1st. Is this longitude the same in all years? Measure the
longitude of Jupiter on January 1, 1900; on July 1, 1900; on September
25, 1906.

Draw neatly on the map a pencil line connecting the position of the
earth for January 1, 1900, with the position of Jupiter for the same
date, and produce the line beyond Jupiter until it meets the circle of
the constellations. This line represents the direction of Jupiter from
the earth, and points toward the constellation in which the planet
appears at that date. But this representation of the place of Jupiter in
the sky is not a very accurate one, since on the scale of the diagram
the stars are in fact more than 100,000 times as far off as they are
shown in the figure, and the pencil mark does not meet the line of
constellations at the same intersection it would have if this line were
pushed back to its true position. To remedy this defect we must draw
another line from the sun parallel to the one first drawn, and its
intersection with the constellations will give very approximately the
true position of Jupiter in the sky.

EXERCISE 15.--Find the present positions of Jupiter and Saturn, and look
them up in the sky by means of your star maps. The planets will appear
in the indicated constellations as very bright stars not shown on the

Which of the planets, Jupiter and Saturn, changes its direction from the
sun more rapidly? Which travels the greater number of miles per day?
When will Jupiter and Saturn be in the same constellation? Does the
earth move faster or slower than Jupiter?

The distance of Jupiter or Saturn from the earth at any time may be
readily obtained from the figure. Thus, by direct measurement with the
millimeter scale we find for January 1, 1900, the distance of Jupiter
from the earth is 6.1 times the distance of the sun from the earth, and
this may be turned into miles by multiplying it by 93,000,000, which is
approximately the distance of the sun from the earth. For most purposes
it is quite as well to dispense with this multiplication and call the
distance 6.1 astronomical units, remembering that the astronomical unit
is the distance of the sun from the earth.

EXERCISE 16.--What is Jupiter's distance from the earth at its nearest
approach? What is the greatest distance it ever attains? Is Jupiter's
least distance from the earth greater or less than its least distance
from Saturn?

On what day in the year 1906 will the earth be on line between Jupiter
and the sun? On this day Jupiter is said to be in _opposition_--i. e.,
the planet and the sun are on opposite sides of the earth, and Jupiter
then comes to the meridian of any and every place at midnight. When the
sun is between the earth and Jupiter (at what date in 1906?) the planet
is said to be in _conjunction_ with the sun, and of course passes the
meridian with the sun at noon. Can you determine from the figure the
time at which Jupiter comes to the meridian at other dates than
opposition and conjunction? Can you determine when it is visible in the
evening hours? Tell from the figure what constellation is on the
meridian at midnight on January 1st. Will it be the same constellation
in every year?

30. MERCURY, VENUS, AND MARS.--Fig. 17, which represents the orbits of
the inner planets, differs from Fig. 16 only in the method of fixing the
positions of the planets in their orbits at any given date. The motion
of these planets is so rapid, on account of their proximity to the sun,
that it would not do to mark their positions as was done for Jupiter and
Saturn, and with the exception of the earth they do not always return to
the same place on the same day in each year. It is therefore necessary
to adopt a slightly different method, as follows: The straight line
extending from the sun toward the vernal equinox, _V_, is called the
prime radius, and we know from past observations that the earth in its
motion around the sun crosses this line on September 23d in each year,
and to fix the earth's position for September 23d in the diagram we have
only to take the point at which the prime radius intersects the earth's
orbit. A month later, on October 23d, the earth will no longer be at
this point, but will have moved on along its orbit to the point marked
30 (thirty days after September 23d). Sixty days after September 23d it
will be at the point marked 60, etc., and for any date we have only to
find the number of days intervening between it and the preceding
September 23d, and this number will show at once the position of the
earth in its orbit. Thus for the date July 4, 1900, we find

    1900, July 4 - 1899, September 23 = 284 days,

and the little circle marked upon the earth's orbit between the numbers
270 and 300 shows the position of the earth on that date.

In what constellation was the sun on July 4, 1900? What zodiacal
constellation came to the meridian at midnight on that date? What other
constellations came to the meridian at the same time?

The positions of the other planets in their orbits are found in the same
manner, save that they do not cross the prime radius on the same date in
each year, and the times at which they do cross it must be taken from
the following table:

                      TABLE OF EPOCHS

    A. D. | Mercury.   |  Venus.   |   Earth.   |  Mars.
  Period  | 88.0 days. |224.7 days.|365.25 days.| 687.1 days.
   1900   | Feb. 18th. | Jan. 11th.| Sept. 23d. | April 28th.
   1901   | Feb. 5th.  | April 5th.| Sept. 23d. |   ...
   1902   | Jan. 23d.  | June 29th.| Sept. 23d. | March 16th.
   1903   | April 8th. | Feb. 8th. | Sept. 23d. |   ...
   1904   | March 25th.| May 3d.   | Sept. 23d. | Feb. 1st.
   1905   | March 12th.| July 26th.| Sept. 23d. | Dec. 19th.
   1906   | Feb. 27th. | March 8th.| Sept. 23d. |   ...
   1907   | Feb. 14th. | May 31st. | Sept. 23d. | Nov. 6th.
   1908   | Feb. 1st.  | Jan. 11th.| Sept. 23d. |   ...
   1909   | Jan. 18th. | April 4th.| Sept. 23d. | Sept. 23d.
   1910   | Jan. 5th.  | June 28th.| Sept. 23d. |   ...

The first line of figures in this table shows the number of days that
each of these planets requires to make a complete revolution about the
sun, and it appears from these numbers that Mercury makes about four
revolutions in its orbit per year, and therefore crosses the prime
radius four times in each year, while the other planets are decidedly
slower in their movements. The following lines of the table show for
each year the date at which each planet first crossed the prime radius
in that year; the dates of subsequent crossings in any year can be found
by adding once, twice, or three times the period to the given date, and
the table may be extended to later years, if need be, by continuously
adding multiples of the period. In the case of Mars it appears that
there is only about one year out of two in which this planet crosses the
prime radius.

After the date at which the planet crosses the prime radius has been
determined its position for any required date is found exactly as in the
case of the earth, and the constellation in which the planet will appear
from the earth is found as explained above in connection with Jupiter
and Saturn.

The broken lines in the figure represent the construction for finding
the places in the sky occupied by Mercury, Venus, and Mars on July 4,
1900. Let the student make a similar construction and find the positions
of these planets at the present time. Look them up in the sky and see if
they are where your work puts them.

31. EXERCISES.--The "evening star" is a term loosely applied to any
planet which is visible in the western sky soon after sunset. It is easy
to see that such a planet must be farther toward the east in the sky
than is the sun, and in either Fig. 16 or Fig. 17 any planet which
viewed from the position of the earth lies to the left of the sun and
not more than 50° away from it will be an evening star. If to the right
of the sun it is a morning star, and may be seen in the eastern sky
shortly before sunrise.

What planet is the evening star _now_? Is there more than one evening
star at a time? What is the morning star now?

Do Mercury, Venus, or Mars ever appear in opposition? What is the
maximum angular distance from the sun at which Venus can ever be seen?
Why is Mercury a more difficult planet to see than Venus? In what month
of the year does Mars come nearest to the earth? Will it always be
brighter in this month than in any other? Which of all the planets comes
nearest to the earth?

The earth always comes to the same longitude on the same day of each
year. Why is not this true of the other planets?

The student should remember that in one respect Figs. 16 and 17 are not
altogether correct representations, since they show the orbits as all
lying in the same plane. If this were strictly true, every planet would
move, like the sun, always along the ecliptic; but in fact all of the
orbits are tilted a little out of the plane of the ecliptic and every
planet in its motion deviates a little from the ecliptic, first to one
side then to the other; but not even Mars, which is the most erratic in
this respect, ever gets more than eight degrees away from the ecliptic,
and for the most part all of them are much closer to the ecliptic than
this limit.



32. THE BEGINNINGS OF CELESTIAL MECHANICS.--From the earliest dawn of
civilization, long before the beginnings of written history, the motions
of sun and moon and planets among the stars from constellation to
constellation had commanded the attention of thinking men, particularly
of the class of priests. The religions of which they were the guardians
and teachers stood in closest relations with the movements of the stars,
and their own power and influence were increased by a knowledge of them.

[Illustration: ISAAC NEWTON (1643-1727).]

Out of these professional needs, as well as from a spirit of scientific
research, there grew up and flourished for many centuries a study of the
motions of the planets, simple and crude at first, because the
observations that could then be made were at best but rough ones, but
growing more accurate and more complex as the development of the
mechanic arts put better and more precise instruments into the hands of
astronomers and enabled them to observe with increasing accuracy the
movements of these bodies. It was early seen that while for the most
part the planets, including the sun and moon, traveled through the
constellations from west to east, some of them sometimes reversed their
motion and for a time traveled in the opposite way. This clearly can not
be explained by the simple theory which had early been adopted that a
planet moves always in the same direction around a circular orbit having
the earth at its center, and so it was said to move around in a small
circular orbit, called an epicycle, whose center was situated upon
and moved along a circular orbit, called the deferent, within which the
earth was placed, as is shown in Fig. 18, where the small circle is the
epicycle, the large circle is the deferent, _P_ is the planet, and _E_
the earth. When this proved inadequate to account for the really
complicated movements of the planets, another epicycle was put on top of
the first one, and then another and another, until the supposed system
became so complicated that Copernicus, a Polish astronomer, repudiated
its fundamental theorem and taught that the motions of the planets take
place in circles around the sun instead of about the earth, and that the
earth itself is only one of the planets moving around the sun in its own
appropriate orbit and itself largely responsible for the seemingly
erratic movements of the other planets, since from day to day we see
them and observe their positions from different points of view.

[Illustration: FIG. 18.--Epicycle and deferent.]

33. KEPLER'S LAWS.--Two generations later came Kepler with his three
famous laws of planetary motion:

I. Every planet moves in an ellipse which has the sun at one of its

II. The radius vector of each planet moves over equal areas in equal

III. The squares of the periodic times of the planets are proportional
to the cubes of their mean distances from the sun.

These laws are the crowning glory, not only of Kepler's career, but of
all astronomical discovery from the beginning up to his time, and they
well deserve careful study and explanation, although more modern
progress has shown that they are only approximately true.

EXERCISE 17.--Drive two pins into a smooth board an inch apart and
fasten to them the ends of a string a foot long. Take up the slack of
the string with the point of a lead pencil and, keeping the string drawn
taut, move the pencil point over the board into every possible position.
The curve thus traced will be an ellipse having the pins at the two
points which are called its foci.

In the case of the planetary orbits one focus of the ellipse is vacant,
and, in accordance with the first law, the center of the sun is at the
other focus. In Fig. 17 the dot, inside the orbit of Mercury, which is
marked _a_, shows the position of the vacant focus of the orbit of Mars,
and the dot _b_ is the vacant focus of Mercury's orbit. The orbits of
Venus and the earth are so nearly circular that their vacant foci lie
very close to the sun and are not marked in the figure. The line drawn
from the sun to any point of the orbit (the string from pin to pencil
point) is a _radius vector_. The point midway between the pins is the
_center_ of the ellipse, and the distance of either pin from the center
measures the _eccentricity_ of the ellipse.

Draw several ellipses with the same length of string, but with the pins
at different distances apart, and note that the greater the eccentricity
the flatter is the ellipse, but that all of them have the same length.

If both pins were driven into the same hole, what kind of an ellipse
would you get?

The Second Law was worked out by Kepler as his answer to a problem
suggested by the first law. In Fig. 17 it is apparent from a mere
inspection of the orbit of Mercury that this planet travels much faster
on one side of its orbit than on the other, the distance covered in ten
days between the numbers 10 and 20 being more than fifty per cent
greater than that between 50 and 60. The same difference is found,
though usually in less degree, for every other planet, and Kepler's
problem was to discover a means by which to mark upon the orbit the
figures showing the positions of the planet at the end of equal
intervals of time. His solution of this problem, contained in the second
law, asserts that if we draw radii vectors from the sun to each of the
marked points taken at equal time intervals around the orbit, then the
area of the sector formed by two adjacent radii vectores and the arc
included between them is equal to the area of each and every other such
sector, the short radii vectores being spread apart so as to include a
long arc between them while the long radii vectores have a short arc. In
Kepler's form of stating the law the radius vector is supposed to travel
with the planet and in each day to sweep over the same fractional part
of the total area of the orbit. The spacing of the numbers in Fig. 17
was done by means of this law.

For the proper understanding of Kepler's Third Law we must note that the
"mean distance" which appears in it is one half of the long diameter of
the orbit and that the "periodic time" means the number of days or years
required by the planet to make a complete circuit in its orbit.
Representing the first of these by _a_ and the second by _T_, we have,
as the mathematical equivalent of the law,

    a^{3} ÷ T^{2} = C

where the quotient, _C_, is a number which, as Kepler found, is the same
for every planet of the solar system. If we take the mean distance of
the earth from the sun as the unit of distance, and the year as the unit
of time, we shall find by applying the equation to the earth's motion,
_C_ = 1. Applying this value to any other planet we shall find in the
same units, _a_ = _T_^{2/3}, by means of which we may determine the
distance of any planet from the sun when its periodic time, _T_, has
been learned from observation.

EXERCISE 18.--Uranus requires 84 years to make a revolution in its
orbit. What is its mean distance from the sun? What are the mean
distances of Mercury, Venus, and Mars? (See Chapter III for their
periodic times.) Would it be possible for two planets at different
distances from the sun to move around their orbits in the same time?

A circle is an ellipse in which the two foci have been brought together.
Would Kepler's laws hold true for such an orbit?

34. NEWTON'S LAWS OF MOTION.--Kepler studied and described the motion of
the planets. Newton, three generations later (1727 A. D.), studied and
described the mechanism which controls that motion. To Kepler and his
age the heavens were supernatural, while to Newton and his successors
they are a part of Nature, governed by the same laws which obtain upon
the earth, and we turn to the ordinary things of everyday life as the
foundation of celestial mechanics.

Every one who has ridden a bicycle knows that he can coast farther upon
a level road if it is smooth than if it is rough; but however smooth and
hard the road may be and however fast the wheel may have been started,
it is sooner or later stopped by the resistance which the road and the
air offer to its motion, and when once stopped or checked it can be
started again only by applying fresh power. We have here a familiar
illustration of what is called

THE FIRST LAW OF MOTION.--"Every body continues in its state of rest or
of uniform motion in a straight line except in so far as it may be
compelled by force to change that state." A gust of wind, a stone, a
careless movement of the rider may turn the bicycle to the right or the
left, but unless some disturbing force is applied it will go straight
ahead, and if all resistance to its motion could be removed it would go
always at the speed given it by the last power applied, swerving neither
to the one hand nor the other.

When a slow rider increases his speed we recognize at once that he has
applied additional power to the wheel, and when this speed is slackened
it equally shows that force has been applied against the motion. It is
force alone which can produce a change in either velocity or direction
of motion; but simple as this law now appears it required the genius of
Galileo to discover it and of Newton to give it the form in which it is
stated above.

35. THE SECOND LAW OF MOTION, which is also due to Galileo and Newton,

"Change of motion is proportional to force applied and takes place in
the direction of the straight line in which the force acts." Suppose a
man to fall from a balloon at some great elevation in the air; his own
weight is the force which pulls him down, and that force operating at
every instant is sufficient to give him at the end of the first second
of his fall a downward velocity of 32 feet per second--i. e., it has
changed his state from rest, to motion at this rate, and the motion is
toward the earth because the force acts in that direction. During the
next second the ceaseless operation of this force will have the same
effect as in the first second and will add another 32 feet to his
velocity, so that two seconds from the time he commenced to fall he will
be moving at the rate of 64 feet per second, etc. The column of figures
marked _v_ in the table below shows what his velocity will be at the end
of subsequent seconds. The changing velocity here shown is the change of
motion to which the law refers, and the velocity is proportional to the
time shown in the first column of the table, because the amount of force
exerted in this case is proportional to the time during which it
operated. The distance through which the man will fall in each second is
shown in the column marked _d_, and is found by taking the average of
his velocity at the beginning and end of this second, and the total
distance through which he has fallen at the end of each second, marked
_s_ in the table, is found by taking the sum of all the preceding values
of _d_. The velocity, 32 feet per second, which measures the change of
motion in each second, also measures the _accelerating force_ which
produces this motion, and it is usually represented in formulæ by the
letter _g_. Let the student show from the numbers in the table that the
accelerating force, the time, _t_, during which it operates, and the
space, _s_, fallen through, satisfy the relation

    s = 1/2 gt^{2},

which is usually called the law of falling bodies. How does the table
show that _g_ is equal to 32?


    _t_   _v_   _d_   _s_

     0      0     0     0
     1     32    16    16
     2     64    48    64
     3     96    80   144
     4    128   112   256
     5    160   144   400
    etc.  etc.  etc.  etc.

If the balloon were half a mile high how long would it take to fall to
the ground? What would be the velocity just before reaching the ground?

[Illustration: GALILEO GALILEI (1564-1642).]

Fig. 19 shows the path through the air of a ball which has been struck
by a bat at the point _A_, and started off in the direction _A B_ with a
velocity of 200 feet per second. In accordance with the first law of
motion, if it were acted upon by no other force than the impulse given
by the bat, it should travel along the straight line _A B_ at the
uniform rate of 200 feet per second, and at the end of the fourth second
it should be 800 feet from _A_, at the point marked 4, but during these
four seconds its weight has caused it to fall 256 feet, and its actual
position, 4', is 256 feet below the point 4. In this way we find its
position at the end of each second, 1', 2', 3', 4', etc., and drawing a
line through these points we shall find the actual path of the ball
under the influence of the two forces to be the curved line _A C_. No
matter how far the ball may go before striking the ground, it can not
get back to the point _A_, and the curve _A C_ therefore can not be a
part of a circle, since that curve returns into itself. It is, in fact,
a part of a _parabola_, which, as we shall see later, is a kind of orbit
in which comets and some other heavenly bodies move. A skyrocket moves
in the same kind of a path, and so does a stone, a bullet, or any other
object hurled through the air.

[Illustration: FIG. 19.--The path of a ball.]

36. THE THIRD LAW OF MOTION.--"To every action there is always an equal
and contrary reaction; or the mutual actions of any two bodies are
always equal and oppositely directed." This is well illustrated in the
case of a man climbing a rope hand over hand. The direct force or action
which he exerts is a downward pull upon the rope, and it is the reaction
of the rope to this pull which lifts him along it. We shall find in a
later chapter a curious application of this law to the history of the
earth and moon.

It is the great glory of Sir Isaac Newton that he first of all men
recognized that these simple laws of motion hold true in the heavens as
well as upon the earth; that the complicated motion of a planet, a
comet, or a star is determined in accordance with these laws by the
forces which act upon the bodies, and that these forces are essentially
the same as that which we call weight. The formal statement of the
principle last named is included in--

37. NEWTON'S LAW OF GRAVITATION.--"Every particle of matter in the
universe attracts every other particle with a force whose direction is
that of a line joining the two, and whose magnitude is directly as the
product of their masses, and inversely as the square of their distance
from each other." We know that we ourselves and the things about us are
pulled toward the earth by a force (weight) which is called, in the
Latin that Newton wrote, _gravitas_, and the word marks well the true
significance of the law of gravitation. Newton did not discover a new
force in the heavens, but he extended an old and familiar one from a
limited terrestrial sphere of action to an unlimited and celestial one,
and furnished a precise statement of the way in which the force
operates. Whether a body be hot or cold, wet or dry, solid, liquid, or
gaseous, is of no account in determining the force which it exerts,
since this depends solely upon mass and distance.

The student should perhaps be warned against straining too far the
language which it is customary to employ in this connection. The law of
gravitation is certainly a far-reaching one, and it may operate in every
remotest corner of the universe precisely as stated above, but
additional information about those corners would be welcome to
supplement our rather scanty stock of knowledge concerning what happens
there. We may not controvert the words of a popular preacher who says,
"When I lift my hand I move the stars in Ursa Major," but we should not
wish to stand sponsor for them, even though they are justified by a
rigorous interpretation of the Newtonian law.

The word _mass_, in the statement of the law of gravitation, means the
quantity of matter contained in the body, and if we represent by the
letters _m´_ and _m´´_ the respective quantities of matter contained in
the two bodies whose distance from each other is _r_, we shall have, in
accordance with the law of gravitation, the following mathematical
expression for the force, _F_, which acts between them:

    F = k {m´m´´/r^{2}}.

This equation, which is the general mathematical expression for the law
of gravitation, may be made to yield some curious results. Thus, if we
select two bullets, each having a mass of 1 gram, and place them so that
their centers are 1 centimeter apart, the above expression for the force
exerted between them becomes

    F = k {(1 × 1)/1^{2}} = k,

from which it appears that the coefficient _k_ is the force exerted
between these bodies. This is called the gravitation constant, and it
evidently furnishes a measure of the specific intensity with which one
particle of matter attracts another. Elaborate experiments which have
been made to determine the amount of this force show that it is
surprisingly small, for in the case of the two bullets whose mass of 1
gram each is supposed to be concentrated into an indefinitely small
space, gravity would have to operate between them continuously for more
than forty minutes in order to pull them together, although they were
separated by only 1 centimeter to start with, and nothing save their own
inertia opposed their movements. It is only when one or both of the
masses _m´_, _m´´_ are very great that the force of gravity becomes
large, and the weight of bodies at the surface of the earth is
considerable because of the great quantity of matter which goes to make
up the earth. Many of the heavenly bodies are much more massive than the
earth, as the mathematical astronomers have found by applying the law of
gravitation to determine numerically their masses, or, in more popular
language, to "weigh" them.

The student should observe that the two terms mass and weight are not
synonymous; mass is defined above as the quantity of matter contained in
a body, while weight is the force with which the earth attracts that
body, and in accordance with the law of gravitation its weight depends
upon its distance from the center of the earth, while its mass is quite
independent of its position with respect to the earth.

By the third law of motion the earth is pulled toward a falling body
just as strongly as the body is pulled toward the earth--i. e., by a
force equal to the weight of the body. How much does the earth rise
toward the body?

38. THE MOTION OF A PLANET.--In Fig. 20 _S_ represents the sun and _P_ a
planet or other celestial body, which for the moment is moving along the
straight line _P 1_. In accordance with the first law of motion it would
continue to move along this line with uniform velocity if no external
force acted upon it; but such a force, the sun's attraction, is acting,
and by virtue of this attraction the body is pulled aside from the line
_P 1_.

Knowing the velocity and direction of the body's motion and the force
with which the sun attracts it, the mathematician is able to apply
Newton's laws of motion so as to determine the path of the body, and a
few of the possible orbits are shown in the figure where the short cross
stroke marks the point of each orbit which is nearest to the sun. This
point is called the _perihelion_.

Without any formal application of mathematics we may readily see that
the swifter the motion of the body at _P_ the shorter will be the time
during which it is subjected to the sun's attraction at close range, and
therefore the force exerted by the sun, and the resulting change of
motion, will be small, as in the orbits _P 1_ and _P 2_.

On the other hand, _P 5_ and _P 6_ represent orbits in which the
velocity at _P_ was comparatively small, and the resulting change of
motion greater than would be possible for a more swiftly moving body.

What would be the orbit if the velocity at _P_ were reduced to nothing
at all?

What would be the effect if the body starting at _P_ moved directly away
from _1_?

[Illustration: FIG. 20.--Different kinds of orbits.]

The student should not fail to observe that the sun's attraction tends
to pull the body at _P_ forward along its path, and therefore increases
its velocity, and that this influence continues until the planet reaches
perihelion, at which point it attains its greatest velocity, and the
force of the sun's attraction is wholly expended in changing the
direction of its motion. After the planet has passed perihelion the
sun begins to pull backward and to retard the motion in just the same
measure that before perihelion passage it increased it, so that the
two halves of the orbit on opposite sides of a line drawn from the
perihelion through the sun are exactly alike. We may here note the
explanation of Kepler's second law: when the planet is near the sun it
moves faster, and the radius vector changes its direction more rapidly
than when the planet is remote from the sun on account of the greater
force with which it is attracted, and the exact relation between the
rates at which the radius vector turns in different parts of the orbit,
as given by the second law, depends upon the changes in this force.

When the velocity is not too great, the sun's backward pull, after a
planet has passed perihelion, finally overcomes it and turns the planet
toward the sun again, in such a way that it comes back to the point _P_,
moving in the same direction and with the same speed as before--i. e.,
it has gone around the sun in an orbit like _P 6_ or _P 4_, an ellipse,
along which it will continue to move ever after. But we must not fail to
note that this return into the same orbit is a consequence of the last
line in the statement of the law of gravitation (p. 54), and that, if
the magnitude of this force were inversely as the cube of the distance
or any other proportion than the square, the orbit would be something
very different. If the velocity is too great for the sun's attraction to
overcome, the orbit will be a hyperbola, like _P 2_, along which the
body will move away never to return, while a velocity just at the limit
of what the sun can control gives an orbit like _P 3_, a parabola, along
which the body moves with _parabolic velocity_, which is ever
diminishing as the body gets farther from the sun, but is always just
sufficient to keep it from returning. If the earth's velocity could be
increased 41 per cent, from 19 up to 27 miles per second, it would have
parabolic velocity, and would quit the sun's company.

The summation of the whole matter is that the orbit in which a body
moves around the sun, or past the sun, depends upon its velocity and if
this velocity and the direction of the motion at any one point in the
orbit are known the whole orbit is determined by them, and the position
of the planet in its orbit for past as well as future times can be
determined through the application of Newton's laws; and the same is
true for any other heavenly body--moon, comet, meteor, etc. It is in
this way that astronomers are able to predict, years in advance, in what
particular part of the sky a given planet will appear at a given time.

It is sometimes a source of wonder that the planets move in ellipses
instead of circles, but it is easily seen from Fig. 20 that the planet,
_P_, could not by any possibility move in a circle, since the direction
of its motion at _P_ is not at right angles with the line joining it to
the sun as it must be in a circular orbit, and even if it were
perpendicular to the radius vector the planet must needs have exactly
the right velocity given to it at this point, since either more or less
speed would change the circle into an ellipse. In order to produce
circular motion there must be a balancing of conditions as nice as is
required to make a pin stand upon its point, and the really surprising
thing is that the orbits of the planets should be so nearly circular as
they are. If the orbit of the earth were drawn accurately to scale, the
untrained eye would not detect the slightest deviation from a true
circle, and even the orbit of Mercury (Fig. 17), which is much more
eccentric than that of the earth, might almost pass for a circle.

[Illustration: FIG. 21. An impossible orbit.]

The orbit _P 2_, which lies between the parabola and the straight line,
is called in geometry a hyperbola, and Newton succeeded in proving from
the law of gravitation that a body might move under the sun's attraction
in a hyperbola as well as in a parabola or ellipse; but it must move in
some one of these curves; no other orbit is possible.[1] Thus it would
not be possible for a body moving under the law of gravitation to
describe about the sun any such orbit as is shown in Fig. 21. If the
body passes a second time through any point of its orbit, such as _P_ in
the figure, then it must retrace, time after time, the whole path that
it first traversed in getting from _P_ around to _P_ again--i. e., the
orbit must be an ellipse.

  [1] The circle and straight line are considered to be special cases
      of these curves, which, taken collectively, are called the conic

Newton also proved that Kepler's three laws are mere corollaries from
the law of gravitation, and that to be strictly correct the third law
must be slightly altered so as to take into account the masses of the
planets. These are, however, so small in comparison with that of the
sun, that the correction is of comparatively little moment.

39. PERTURBATIONS.--In what precedes we have considered the motion of a
planet under the influence of no other force than the sun's attraction,
while in fact, as the law of gravitation asserts, every other body in
the universe is in some measure attracting it and changing its motion.
The resulting disturbances in the motion of the attracted body are
called _perturbations_, but for the most part these are insignificant,
because the bodies by whose disturbing attractions they are caused are
either very small or very remote, and it is only when our moving planet,
_P_, comes under the influence of some great disturbing power like
Jupiter or one of the other planets that the perturbations caused by
their influence need to be taken into account.

The problem of the motion of three bodies--sun, Jupiter, planet--which
must then be dealt with is vastly more complicated than that which we
have considered, and the ablest mathematicians and astronomers have not
been able to furnish a complete solution for it, although they have
worked upon the problem for two centuries, and have developed an immense
amount of detailed information concerning it.


In general each planet works ceaselessly upon the orbit of every other,
changing its size and shape and position, backward and forward in
accordance with the law of gravitation, and it is a question of serious
moment how far this process may extend. If the diameter of the earth's
orbit were very much increased or diminished by the perturbing action of
the other planets, the amount of heat received from the sun would be
correspondingly changed, and the earth, perhaps, be rendered unfit
for the support of life. The tipping of the plane of the earth's orbit
into a new position might also produce serious consequences; but the
great French mathematician of a century ago, Laplace, succeeded in
proving from the law of gravitation that although both of these changes
are actually in progress they can not, at least for millions of years,
go far enough to prove of serious consequence, and the same is true for
all the other planets, unless here and there an asteroid may prove an
exception to the rule.

The precession (Chapter V) is a striking illustration of a perturbation
of slightly different character from the above, and another is found in
connection with the plane of the moon's orbit. It will be remembered
that the moon in its motion among the stars never goes far from the
ecliptic, but in a complete circuit of the heavens crosses it twice,
once in going from south to north and once in the opposite direction.
The points at which it crosses the ecliptic are called the _nodes_, and
under the perturbing influence of the sun these nodes move westward
along the ecliptic about twenty degrees per year, an extraordinarily
rapid perturbation, and one of great consequence in the theory of

[Illustration: FIG. 22.--A planet subject to great perturbations by

40. WEIGHING THE PLANETS.--Although these perturbations can not be
considered dangerous, they are interesting since they furnish a method
for weighing the planets which produce them. From the law of gravitation
we learn that the ability of a planet to produce perturbations depends
directly upon its mass, since the force _F_ which it exerts contains
this mass, _m´_, as a factor. So, too, the divisor _r^{2}_ in the
expression for the force shows that the distance between the disturbing
and disturbed bodies is a matter of great consequence, for the smaller
the distance the greater the force. When, therefore, the mass of a
planet such as Jupiter is to be determined from the perturbations it
produces, it is customary to select some such opportunity as is
presented in Fig. 22, where one of the small planets, called asteroids,
is represented as moving in a very eccentric orbit, which at one point
approaches close to the orbit of Jupiter, and at another place comes
near to the orbit of the earth. For the most part Jupiter will not exert
any very great disturbing influence upon a planet moving in such an
orbit as this, since it is only at rare intervals that the asteroid and
Jupiter approach so close to each other, as is shown in the figure. The
time during which the asteroid is little affected by the attraction of
Jupiter is used to study the motion given to it by the sun's
attraction--that is, to determine carefully the undisturbed orbit in
which it moves; but there comes a time at which the asteroid passes
close to Jupiter, as shown in the figure, and the orbital motion which
the sun imparts to it will then be greatly disturbed, and when the
planet next comes round to the part of its orbit near the earth the
effect of these disturbances upon its apparent position in the sky will
be exaggerated by its close proximity to the earth. If now the
astronomer observes the actual position of the asteroid in the sky, its
right ascension and declination, and compares these with the position
assigned to the planet by the law of gravitation when the attraction of
Jupiter is ignored, the differences between the observed right
ascensions and declinations and those computed upon the theory of
undisturbed motion will measure the influence that Jupiter has had upon
the asteroid, and the amount by which Jupiter has shifted it, compared
with the amount by which the sun has moved it--that is, with the motion
in its orbit--furnishes the mass of Jupiter expressed as a fractional
part of the mass of the sun.

There has been determined in this manner the mass of every planet in the
solar system which is large enough to produce any appreciable
perturbation, and all these masses prove to be exceedingly small
fractions of the mass of the sun, as may be seen from the following
table, in which is given opposite the name of each planet the number by
which the mass of the sun must be divided in order to get the mass of
the planet:

    Mercury      7,000,000 (?)
    Venus          408,000
    Earth          329,000
    Mars         3,093,500
    Jupiter          1,047.4
    Saturn           3,502
    Uranus          22,800
    Neptune         19,700

It is to be especially noted that the mass given for each planet
includes the mass of all the satellites which attend it, since their
influence was felt in the perturbations from which the mass was derived.
Thus the mass assigned to the earth is the combined mass of earth and

41. DISCOVERY OF NEPTUNE.--The most famous example of perturbations is
found in connection with the discovery, in the year 1846, of Neptune,
the outermost planet of the solar system. For many years the motion of
Uranus, his next neighbor, had proved a puzzle to astronomers. In
accordance with Kepler's first law this planet should move in an ellipse
having the sun at one of its foci, but no ellipse could be found which
exactly fitted its observed path among the stars, although, to be sure,
the misfit was not very pronounced. Astronomers surmised that the small
deviations of Uranus from the best path which theory combined with
observation could assign, were due to perturbations in its motion
caused by an unknown planet more remote from the sun--a thing easy to
conjecture but hard to prove, and harder still to find the unknown
disturber. But almost simultaneously two young men, Adams in England and
Le Verrier in France, attacked the problem quite independently of each
other, and carried it to a successful solution, showing that if the
irregularities in the motion of Uranus were indeed caused by an unknown
planet, then that planet must, in September, 1846, be in the direction
of the constellation Aquarius; and there it was found on September 23d
by the astronomers of the Berlin Observatory whom Le Verrier had invited
to search for it, and found within a degree of the exact point which the
law of gravitation in his hands had assigned to it.

This working backward from the perturbations experienced by Uranus to
the cause which produced them is justly regarded as one of the greatest
scientific achievements of the human intellect, and it is worthy of note
that we are approaching the time at which it may be repeated, for
Neptune now behaves much as did Uranus three quarters of a century ago,
and the most plausible explanation which can be offered for these
anomalies in its path is that the bounds of the solar system must be
again enlarged to include another disturbing planet.

42. THE SHAPE OF A PLANET.--There is an effect of gravitation not yet
touched upon, which is of considerable interest and wide application in
astronomy--viz., its influence in determining the shape of the heavenly
bodies. The earth is a globe because every part of it is drawn toward
the center by the attraction of the other parts, and if this attraction
on its surface were everywhere of equal force the material of the earth
would be crushed by it into a truly spherical form, no matter what may
have been the shape in which it was originally made. But such is not the
real condition of the earth, for its diurnal rotation develops in every
particle of its body a force which is sometimes called _centrifugal_,
but which is really nothing more than the inertia of its particles,
which tend at every moment to keep unchanged the direction of their
motion and which thus resist the attraction that pulls them into a
circular path marked out by the earth's rotation, just as a stone tied
at the end of a string and swung swiftly in a circle pulls upon the
string and opposes the constraint which keeps it moving in a circle. A
few experiments with such a stone will show that the faster it goes the
harder does it pull upon the string, and the same is true of each
particle of the earth, the swiftly moving ones near the equator having a
greater centrifugal force than the slow ones near the poles. At the
equator the centrifugal force is directly opposed to the force of
gravity, and in effect diminishes it, so that, comparatively, there is
an excess of gravity at the poles which compresses the earth along its
axis and causes it to bulge out at the equator until a balance is thus
restored. As we have learned from the study of geography, in the case of
the earth, this compression amounts to about 27 miles, but in the larger
planets, Jupiter and Saturn, it is much greater, amounting to several
thousand miles.

But rotation is not the only influence that tends to pull a planet out
of shape. The attraction which the earth exerts upon the moon is
stronger on the near side and weaker on the far side of our satellite
than at its center, and this difference of attraction tends to warp the
moon, as is illustrated in Fig. 23 where _1_, _2_, and _3_ represent
pieces of iron of equal mass placed in line on a table near a horseshoe
magnet, _H_. Each piece of iron is attracted by the magnet and is held
back by a weight to which it is fastened by means of a cord running over
a pulley, _P_, at the edge of the table. These weights are all to be
supposed equally heavy and each of them pulls upon its piece of iron
with a force just sufficient to balance the attraction of the magnet for
the middle piece, No. _2_. It is clear that under this arrangement No.
_2_ will move neither to the right nor to the left, since the forces
exerted upon it by the magnet and the weight just balance each other.
Upon No. _1_, however, the magnet pulls harder than upon No. _2_,
because it is nearer and its pull therefore more than balances the force
exerted by the weight, so that No. _1_ will be pulled away from No. _2_
and will stretch the elastic cords, which are represented by the lines
joining _1_ and _2_, until their tension, together with the force
exerted by the weight, just balances the attraction of the magnet. For
No. _3_, the force exerted by the magnet is less than that of the
weight, and it will also be pulled away from No. _2_ until its elastic
cords are stretched to the proper tension. The net result is that the
three blocks which, without the magnet's influence, would be held close
together by the elastic cords, are pulled apart by this outside force as
far as the resistance of the cords will permit.

[Illustration: FIG. 23.--Tide-raising forces.]

An entirely analogous set of forces produces a similar effect upon the
shape of the moon. The elastic cords of Fig. 23 stand for the attraction
of gravitation by which all the parts of the moon are bound together.
The magnet represents the earth pulling with unequal force upon
different parts of the moon. The weights are the inertia of the moon in
its orbital motion which, as we have seen in a previous section, upon
the whole just balances the earth's attraction and keeps the moon from
falling into it. The effect of these forces is to stretch out the
moon along a line pointing toward the earth, just as the blocks were
stretched out along the line of the magnet, and to make this diameter
of the moon slightly but permanently longer than the others.

[Illustration: FIG. 24.--The tides.]

THE TIDES.--Similarly the moon and the sun attract opposite sides of the
earth with different forces and feebly tend to pull it out of shape. But
here a new element comes into play: the earth turns so rapidly upon its
axis that its solid parts have no time in which to yield sensibly to the
strains, which shift rapidly from one diameter to another as different
parts of the earth are turned toward the moon, and it is chiefly the
waters of the sea which respond to the distorting effect of the sun's
and moon's attraction. These are heaped up on opposite sides of the
earth so as to produce a slight elongation of its diameter, and Fig. 24
shows how by the earth's rotation this swelling of the waters is swept
out from under the moon and is pulled back by the moon until it finally
takes up some such position as that shown in the figure where the effect
of the earth's rotation in carrying it one way is just balanced by the
moon's attraction urging it back on line with the moon. This heaping up
of the waters is called a _tide_. If _I_ in the figure represents a
little island in the sea the waters which surround it will of course
accompany it in its diurnal rotation about the earth's axis, but
whenever the island comes back to the position _I_, the waters will
swell up as a part of the tidal wave and will encroach upon the land in
what is called high tide or flood tide. So too when they reach _I´´_,
half a day later, they will again rise in flood tide, and midway between
these points, at _I´_, the waters must subside, giving low or ebb tide.

The height of the tide raised by the moon in the open sea is only a very
few feet, and the tide raised by the sun is even less, but along the
coast of a continent, in bays and angles of the shore, it often happens
that a broad but low tidal wave is forced into a narrow corner, and then
the rise of the water may be many feet, especially when the solar tide
and the lunar tide come in together, as they do twice in every month, at
new and full moon. Why do they come together at these times instead of
some other?

Small as are these tidal effects, it is worth noting that they may in
certain cases be very much greater--e. g., if the moon were as massive
as is the sun its tidal effect would be some millions of times greater
than it now is and would suffice to grind the earth into fragments.
Although the earth escapes this fate, some other bodies are not so
fortunate, and we shall see in later chapters some evidence of their

43. THE SCOPE OF THE LAW OF GRAVITATION.--In all the domain of physical
science there is no other law so famous as the Newtonian law of
gravitation; none other that has been so dwelt upon, studied, and
elaborated by astronomers and mathematicians, and perhaps none that can
be considered so indisputably proved. Over and over again mathematical
analysis, based upon this law, has pointed out conclusions which, though
hitherto unsuspected, have afterward been found true, as when Newton
himself derived as a corollary from this law that the earth ought to be
flattened at the poles--a thing not known at that time, and not proved
by actual measurement until long afterward. It is, in fact, this
capacity for predicting the unknown and for explaining in minutest
detail the complicated phenomena of the heavens and the earth that
constitutes the real proof of the law of gravitation, and it is
therefore worth while to note that at the present time there are a very
few points at which the law fails to furnish a satisfactory account of
things observed. Chief among these is the case of the planet Mercury,
the long diameter of whose orbit is slowly turning around in a way for
which the law of gravitation as yet furnishes no explanation. Whether
this is because the law itself is inaccurate or incomplete, or whether
it only marks a case in which astronomers have not yet properly applied
the law and traced out its consequences, we do not know; but whether it
be the one or the other, this and other similar cases show that even
here, in its most perfect chapter, astronomy still remains an incomplete



44. THE SIZE OF THE EARTH.--The student is presumed to have learned, in
his study of geography, that the earth is a globe about 8,000 miles in
diameter and, without dwelling upon the "proofs" which are commonly
given for these statements, we proceed to consider the principles upon
which the measurement of the earth's size and shape are based.

[Illustration: FIG. 25.--Measuring the size of the earth.]

In Fig. 25 the circle represents a meridian section of the earth; _P P´_
is the axis about which it rotates, and the dotted lines represent a
beam of light coming from a star in the plane of the meridian, and so
distant that the dotted lines are all practically parallel to each
other. The several radii drawn through the points _1_, _2_, _3_,
represent the direction of the vertical at these points, and the angles
which these radii produced, make with the rays of starlight are each
equal to the angular distance of the star from the zenith of the place
at the moment the star crosses the meridian. We have already seen, in
Chapter II, how these angles may be measured, and it is apparent from
the figure that the difference between any two of these angles--e. g.,
the angles at _1_ and _2_--is equal to the angle at the center, _O_,
between the points _1_ and _2_. By measuring these angular distances of
the star from the zenith, the astronomer finds the angles at the center
of the earth between the stations _1_, _2_, _3_, etc., at which his
observations are made. If the meridian were a perfect circle the change
of zenith distance of the star, as one traveled along a meridian from
the equator to the pole, would be perfectly uniform--the same number of
degrees for each hundred miles traveled--and observations made in many
parts of the earth show that this is very nearly true, but that, on the
whole, as we approach the pole it is necessary to travel a little
greater distance than is required for a given change in the angle at the
equator. The earth is, in fact, flattened at the poles to the amount of
about 27 miles in the length of its diameter, and by this amount, as
well as by smaller variations due to mountains and valleys, the shape of
the earth differs from a perfect sphere. These astronomical measurements
of the curvature of the earth's surface furnish by far the most
satisfactory proof that it is very approximately a sphere, and furnish
as its equatorial diameter 7,926 miles.

Neglecting the _compression_, as it is called, i. e., the 27 miles by
which the equatorial diameter exceeds the polar, the size of the earth
may easily be found by measuring the distance _1_--_2_ along the
surface and by combining with this the angle _1 O 2_ obtained through
measuring the meridian altitudes of any star as seen from _1_ and _2_.
Draw on paper an angle equal to the measured difference of altitude and
find how far you must go from its vertex in order to have the distance
between the sides, measured along an arc of a circle, equal to the
measured distance between _1_ and _2_. This distance from the vertex
will be the earth's radius.

EXERCISE 19.--Measure the diameter of the earth by the method given
above. In order that this may be done satisfactorily, the two stations
at which observations are made must be separated by a considerable
distance--i. e., 200 miles. They need not be on the same meridian, but
if they are on different meridians in place of the actual distance
between them, there must be used the projection of that distance upon
the meridian--i. e., the north and south part of the distance.

By co-operation between schools in the Northern and Southern States,
using a good map to obtain the required distances, the diameter of the
earth may be measured with the plumb-line apparatus described in Chapter
II and determined within a small percentage of its true value.

45. THE MASS OF THE EARTH.--We have seen in Chapter IV the possibility
of determining the masses of the planets as fractional parts of the
sun's mass, but nothing was there shown, or could be shown, about
measuring these masses after the common fashion in kilogrammes or tons.
To do this we must first get the mass of the earth in tons or
kilogrammes, and while the principles involved in this determination are
simple enough, their actual application is delicate and difficult.

[Illustration: FIG. 26.--Illustrating the principles involved in
weighing the earth.]

In Fig. 26 we suppose a long plumb line to be suspended above the
surface of the earth and to be attracted toward the center of the earth,
_C_, by a force whose intensity is (Chapter IV)

    F = k mE/R^{2},

where _E_ denotes the mass of the earth, which is to be determined by
experiment, and _R_ is the radius of the earth, 3,963 miles. If there is
no disturbing influence present, the plumb line will point directly
downward, but if a massive ball of lead or other heavy substance is
placed at one side, _1_, it will attract the plumb line with a force
equal to

    f = k mB/r^{2},

where _r_ is the distance of its center from the plumb bob and _B_ is
its mass which we may suppose, for illustration, to be a ton. In
consequence of this attraction the plumb line will be pulled a little to
one side, as shown by the dotted line, and if we represent by _l_ the
length of the plumb line and by _d_ the distance between the original
and the disturbed positions of the plumb bob we may write the proportion

    F : f :: l : d;

and introducing the values of _F_ and _f_ given above, and solving for
_E_ the proportion thus transformed, we find

    E = B × l/d × (R/r)^{2}.

In this equation the mass of the ball, _B_, the length of the plumb
line, _l_, the distance between the center of the ball and the center of
the plumb bob, _r_, and the radius of the earth, _R_, can all be
measured directly, and _d_, the amount by which the plumb bob is pulled
to one side by the ball, is readily found by shifting the ball over to
the other side, at _2_, and measuring with a microscope how far the
plumb bob moves. This distance will, of course, be equal to _2 d_.

By methods involving these principles, but applied in a manner more
complicated as well as more precise, the mass of the earth is found to
be, in tons, 6,642 × 10^{18}--i. e., 6,642 followed by 18 ciphers, or in
kilogrammes 60,258 × 10^{20}. The earth's atmosphere makes up about a
millionth part of this mass.

If the length of the plumb line were 100 feet, the weight of the ball a
ton, and the distance between the two positions of the ball, _1_ and
_2_, six feet, how many inches, _d_, would the plumb bob be pulled out
of place?

Find from the mass of the earth and the data of § 40 the mass of the sun
in tons. Find also the mass of Mars. The computation can be very greatly
abridged by the use of logarithms.

46. PRECESSION.--That the earth is isolated in space and has no support
upon which to rest, is sufficiently shown by the fact that the stars are
visible upon every side of it, and no support can be seen stretching out
toward them. We must then consider the earth to be a globe traveling
freely about the sun in a circuit which it completes once every year,
and rotating once in every twenty-four hours about an axis which remains
at all seasons directed very nearly toward the star Polaris. The student
should be able to show from his own observations of the sun that, with
reference to the stars, the direction of the sun from the earth changes
about a degree a day. Does this prove that the earth revolves about the

But it is only in appearance that the pole maintains its fixed position
among the stars. If photographs are taken year after year, after the
manner of Exercise 7, it will be found that slowly the pole is moving
(nearly) toward Polaris, and making this star describe a smaller and
smaller circle in its diurnal path, while stars on the other side of the
pole (in right ascension 12h.) become more distant from it and describe
larger circles in their diurnal motion; but the process takes place so
slowly that the space of a lifetime is required for the motion of the
pole to equal the angular diameter of the full moon.

Spin a top and note how its rapid whirl about its axis corresponds to
the earth's diurnal rotation. When the axis about which the top spins is
truly vertical the top "sleeps"; but if the axis is tipped ever so
little away from the vertical it begins to wobble, so that if we imagine
the axis prolonged out to the sky and provided with a pencil point as a
marker, this would trace a circle around the zenith, along which the
pole of the top would move, and a little observation will show that the
more the top is tipped from the vertical the larger does this circle
become and the more rapidly does the wobbling take place. Were it not
for the spinning of the top about its axis, it would promptly fall over
when tipped from the vertical position, but the spin combines with the
force which pulls the top over and produces the wobbling motion. Spin
the top in opposite directions, with the hands of a watch and contrary
to the hands of a watch, and note the effect which is produced upon the

The earth presents many points of resemblance to the top. Its diurnal
rotation is the spin about the axis. This axis is tipped 23.5° away from
the perpendicular to its orbit (obliquity of the ecliptic) just as the
axis of the top is tipped away from the vertical line. In consequence of
its rapid spin, the body of the earth bulges out at the equator (27
miles), and the sun and moon, by virtue of their attraction (see Chapter
IV), lay hold of this protuberance and pull it down toward the plane of
the earth's orbit, so that if it were not for the spin this force would
straighten the axis up and set it perpendicular to the orbit plane. But
here, as in the case of the top, the spin and the tipping force combine
to produce a wobble which is called precession, and whose effect we
recognize in the shifting position of the pole among the stars. The
motion of precession is very much slower than the wobbling of the top,
since the tipping force for the earth is relatively very small, and a
period of nearly 26,000 years is required for a complete circuit of the
pole about its center of motion. Friction ultimately stops both the spin
and the wobble of the top, but this influence seems wholly absent in the
case of the earth, and both rotation and precession go on unchanged from
century to century, save for certain minor forces which for a time
change the direction or rate of the precessional motion, first in one
way and then in another, without in the long run producing any results
of consequence.

The center of motion, about which the pole travels in a small circle
having an angular radius of 23.5°, is at that point of the heavens
toward which a perpendicular to the plane of the earth's orbit points,
and may be found on the star map in right ascension 18h. 0m. and
declination 66.5°.

EXERCISE 20.--Find this point on the map, and draw as well as you can
the path of the pole about it. The motion of the pole along its path is
toward the constellation Cepheus. Mark the position of the pole along
this path at intervals of 1,000 years, and refer to these positions in
dealing with some of the following questions:

Does the wobbling of the top occur in the same direction as the motion
of precession? Do the tipping forces applied to the earth and top act in
the same direction? What will be the polar star 12,000 years hence? The
Great Pyramid of Egypt is thought to have been used as an observatory
when Alpha Draconis was the bright star nearest the pole. How long ago
was that?

The motion of the pole of course carries the equator and the equinoxes
with it, and thus slowly changes the right ascensions and declinations
of all the stars. On this account it is frequently called the precession
of the equinoxes, and this motion of the equinox, slow though it is, is
a matter of some consequence in connection with chronology and the
length of the year.

Will the precession ever bring back the right ascensions and
declinations to be again what they now are?

In what direction is the pole moving with respect to the Big Dipper?
Will its motion ever bring it exactly to Polaris? How far away from
Polaris will the precession carry the pole? What other bright stars will
be brought near the pole by the precession?

47. THE WARMING OF THE EARTH.--Winter and summer alike the day is on the
average warmer than the night, and it is easy to see that this surplus
of heat comes from the sun by day and is lost by night through radiation
into the void which surrounds the earth; just as the heat contained in a
mass of molten iron is radiated away and the iron cooled when it is
taken out from the furnace and placed amid colder surroundings. The
earth's loss of heat by radiation goes on ceaselessly day and night,
and were it not for the influx of solar heat this radiation would
steadily diminish the temperature toward what is called the "absolute
zero"--i. e., a state in which all heat has been taken away and beyond
which there can be no greater degree of cold. This must not be
confounded with the zero temperatures shown by our thermometers,
since it lies nearly 500° below the zero of the Fahrenheit scale (-273°
Centigrade), a temperature which by comparison makes the coldest winter
weather seem warm, although the ordinary thermometer may register
many degrees below its zero. The heat radiated by the sun into the
surrounding space on every side of it is another example of the same
cooling process, a hot body giving up its heat to the colder space about
it, and it is the minute fraction of this heat poured out by the sun,
and in small part intercepted by the earth, which warms the latter and
produces what we call weather, climate, the seasons, etc.

Observe the fluctuations, the ebb and flow, which are inherent in this
process. From sunset to sunrise there is nothing to compensate the
steady outflow of heat, and air and ground grow steadily colder, but
with the sunrise there comes an influx of solar heat, feeble at first
because it strikes the earth's surface very obliquely, but becoming more
and more efficient as the sun rises higher in the sky. But as the air
and the ground grow warm during the morning hours they part more and
more readily and rapidly with their store of heat, just as a steam pipe
or a cup of coffee radiates heat more rapidly when very hot. The warmest
hour of the day is reached when these opposing tendencies of income and
expenditure of heat are just balanced; and barring such disturbing
factors as wind and clouds, the gain in temperature usually extends to
the time--an hour or two beyond noon--at which the diminishing altitude
of the sun renders his rays less efficient, when radiation gains the
upper hand and the temperature becomes for a short time stationary, and
then commences to fall steadily until the next sunrise.

We have here an example of what is called a periodic change--i. e., one
which, within a definite and uniform period (24 hours), oscillates from
a minimum up to a maximum temperature and then back again to a minimum,
repeating substantially the same variation day after day. But it must be
understood that minor causes not taken into account above, such as
winds, water, etc., produce other fluctuations from day to day which
sometimes obscure or even obliterate the diurnal variation of
temperature caused by the sun.

Expose the back of your hand to the sun, holding the hand in such a
position that the sunlight strikes perpendicularly upon it; then turn
the hand so that the light falls quite obliquely upon it and note how
much more vigorous is the warming effect of the sun in the first
position than in the second. It is chiefly this difference of angle that
makes the sun's warmth more effective when he is high up in the sky than
when he is near the horizon, and more effective in summer than in

We have seen in Chapter III that the sun's motion among the stars takes
place along a path which carries it alternately north and south of the
equator to a distance of 23.5°, and the stars show by their earlier
risings and later settings, as we pass from the equator toward the north
pole of the heavens, that as the sun moves northward from the equator,
each day in the northern hemisphere will become a little longer, each
night a little shorter, and every day the sun will rise higher toward
the zenith until this process culminates toward the end of June, when
the sun begins to move southward, bringing shorter days and smaller
altitudes until the Christmas season, when again it is reversed and the
sun moves northward. We have here another periodic variation, which runs
its complete course in a period of a year, and it is easy to see that
this variation must have a marked effect on the warming of the earth,
the long days and great altitudes of summer producing the greater warmth
of that season, while the shorter days and lower altitudes of December,
by diminishing the daily supply of solar heat, bring on the winter's
cold. The succession of the seasons, winter following summer and summer
winter, is caused by the varying altitude of the sun, and this in turn
is due to the obliquity of the ecliptic, or, what is the same thing, the
amount by which the axis of the earth is tipped from being perpendicular
to the plane of its orbit, and the seasons are simply a periodic change
in the warming of the earth, quite comparable with the diurnal change
but of longer period.

It is evident that the period within which the succession of winter and
summer is completed, the year, as we commonly call it, must equal the
time required by the sun to go from the vernal equinox around to the
vernal equinox again, since this furnishes a complete cycle of the sun's
motions north and south from the equator. On account of the westward
motion of the equinox (precession) this is not quite the same as the
time required for a complete revolution of the earth in its orbit, but
is a little shorter (20m. 23s.), since the equinox moves back to meet
the sun.

48. RELATION OF THE SUN TO CLIMATE.--It is clear that both the northern
and southern hemispheres of the earth must have substantially the same
kind of seasons, since the motion of the sun north and south affects
both alike; but when the sun is north of the equator and warming our
hemisphere most effectively, his light falls more obliquely upon the
other hemisphere, the days there are short and winter reigns at the
time we are enjoying summer, while six months later the conditions are

In those parts of the earth near the equator--the torrid zone--there is
no such marked change from cold to warm as we experience, because, as
the sun never gets more than 23.5° away from the celestial equator, on
every day of the year he mounts high in the tropic skies, always coming
within 23.5° of the zenith, and usually closer than this, so that there
is no such periodic change in the heat supply as is experienced in
higher latitudes, and within the tropics the temperature is therefore
both higher and more uniform than in our latitude.

In the frigid zones, on the contrary, the sun never rises high in the
sky; at the poles his greatest altitude is only 23.5°, and during the
winter season he does not rise at all, so that the temperature is here
low the whole year round, and during the winter season, when for weeks
or months at a time the supply of solar light is entirely cut off, the
temperature falls to a degree unknown in more favored climes.

If the obliquity of the ecliptic were made 10° greater, what would be
the effect upon the seasons in the temperate zones? What if it were made
10° less?

Does the precession of the equinoxes have any effect upon the seasons or
upon the climate of different parts of the earth?

If the axis of the earth pointed toward Arcturus instead of Polaris,
would the seasons be any different from what they are now?

49. THE ATMOSPHERE.--Although we live upon its surface, we are not
outside the earth, but at the bottom of a sea of air which forms the
earth's outermost layer and extends above our heads to a height of many
miles. The study of most of the phenomena of the atmosphere belongs to
that branch of physics called meteorology, but there are a few matters
which fairly come within our consideration of the earth as a planet. We
can not see the stars save as we look through this atmosphere, and the
light which comes through it is bent and oftentimes distorted so as to
present serious obstacles to any accurate telescopic study of the
heavenly bodies. Frequently this disturbance is visible to the naked
eye, and the stars are said to twinkle--i. e., to quiver and change
color many times per second, solely in consequence of a disturbed
condition of the air and not from anything which goes on in the star.
This effect is more marked low down in the sky than near the zenith, and
it is worth noting that the planets show very little of it because the
light they send to the earth comes from a disk of sensible area, while a
star, being much smaller and farther from the earth, has its disk
reduced practically to a mere point whose light is more easily affected
by local disturbances in the atmosphere than is the broader beam which
comes from the planets' disk.

50. REFRACTION.--At all times, whether the stars twinkle or not, their
light is bent in its passage through the atmosphere, so that the stars
appear to stand higher up in the sky than their true positions. This
effect, which the astronomer calls refraction, must be allowed for in
observations of the more precise class, although save at low altitudes
its amount is a very small fraction of a degree, but near the horizon it
is much exaggerated in amount and becomes easily visible to the naked
eye by distorting the disks of the sun and moon from circles into ovals
with their long diameters horizontal. The refraction lifts both upper
and lower edge of the sun, but lifts the lower edge more than the upper,
thus shortening the vertical diameter. See Fig. 27, which shows not only
this effect, but also the reflection of the sun from the curved surface
of the sea, still further flattening the image. If the surface of the
water were flat, the reflected image would have the same shape as the
sun's disk, and its altered appearance is sometimes cited as a proof
that the earth's surface is curved.

The total amount of the refraction at the horizon is a little more than
half a degree, and since the diameters of the sun and moon subtend an
angle of about half a degree, we have the remarkable result that in
reality the whole disk of either sun or moon is below the horizon at the
instant that the lower edge appears to touch the horizon and sunset or
moonset begins. The same effect exists at sunrise, and as a consequence
the duration of sunshine or of moonshine is on the average about six
minutes longer each day than it would be if there were no atmosphere and
no refraction. A partial offset to this benefit is found in the fact
that the atmosphere absorbs the light of the heavenly bodies, so that
stars appear much less bright when near the horizon than when they are
higher up in the sky, and by reason of this absorption the setting sun
can be looked at with the naked eye without the discomfort which its
dazzling luster causes at noon.

[Illustration: FIG. 27.--Flattening of the sun's disk by refraction and
by reflection from the surface of the sea.]

51. THE TWILIGHT.--Another effect of the atmosphere, even more marked
than the preceding, is the twilight. As at sunrise the mountain top
catches the rays of the coming sun before they reach the lowland, and at
sunset it keeps them after they have faded from the regions below, so
the particles of dust and vapor, which always float in the atmosphere,
catch the sunlight and reflect it to the surface of the earth while the
sun is still below the horizon, giving at the beginning and end of day
that vague and diffuse light which we call twilight.

[Illustration: FIG. 28.--Twilight phenomena.]

Fig. 28 shows a part of the earth surrounded by such a dust-laden
atmosphere, which is illuminated on the left by the rays of the sun, but
which, on the right of the figure, lies in the shadow cast by the earth.
To an observer placed at _1_ the sun is just setting, and all the
atmosphere above him is illumined with its rays, which furnish a bright
twilight. When, by the earth's rotation, this observer has been carried
to _2_, all the region to the east of his zenith lies in the shadow,
while to the west there is a part of the atmosphere from which there
still comes a twilight, but now comparatively faint, because the lower
part of the atmosphere about our observer lies in the shadow, and it is
mainly its upper regions from which the light comes, and here the dust
and moisture are much less abundant than in the lower strata. Still
later, when the observer has been carried by the earth's rotation to the
point _3_, every vestige of twilight will have vanished from his sky,
because all of the illuminated part of the atmosphere is now below his
horizon, which is represented by the line _3 L_. In the figure the sun
is represented to be 78° below this horizon line at the end of twilight,
but this is a gross exaggeration, made for the sake of clearness in the
drawing--in fact, twilight is usually said to end when the sun is 18°
below the horizon.

Let the student redraw Fig. 28 on a large scale, so that the points _1_
and _3_ shall be only 18° apart, as seen from the earth's center. He
will find that the point _L_ is brought down much closer to the surface
of the earth, and measuring the length of the line _2 L_, he should find
for the "height of the atmosphere" about one-eightieth part of the
radius of the earth--i. e., a little less than 50 miles. This, however,
is not the true height of the atmosphere. The air extends far beyond
this, but the particles of dust and vapor which are capable of sending
sunlight down to the earth seem all to lie below this limit.

The student should not fail to watch the eastern sky after sunset, and
see the shadow of the earth rise up and fill it while the twilight arch
retreats steadily toward the west.

[Illustration: FIG. 29.--The cause of long and short twilights.]

_Duration of twilight._--Since twilight ends when the sun is 18° below
the horizon, any circumstance which makes the sun go down rapidly will
shorten the duration of twilight, and anything which retards the
downward motion of the sun will correspondingly prolong it. Chief among
influences of this kind is the angle which the sun's course makes with
the horizon. If it goes straight down, as at _a_, Fig. 29, a much
shorter time will suffice to carry it to a depression of 18° than is
needed in the case shown at _b_ in the same figure, where the motion is
very oblique to the horizon. If we consider different latitudes and
different seasons of the year, we shall find every possible variety of
circumstance from _a_ to _b_, and corresponding to these, the duration
of twilight varies from an all-night duration in the summers of Scotland
and more northern lands to an hour or less in the mountains of Peru. For
the sake of graphical effect, the shortness of tropical twilight is
somewhat exaggerated by Coleridge in the lines,

    "The sun's rim dips; the stars rush out:
     At one stride comes the dark."
                            _The Ancient Mariner._

In the United States the longest twilights come at the end of June, and
last for a little more than two hours, while the shortest ones are in
March and September, amounting to a little more than an hour and a half;
but at all times the last half hour of twilight is hardly to be
distinguished from night, so small is the quantity of reflecting matter
in the upper regions of the atmosphere. For practical convenience it is
customary to assume in the courts of law that twilight ends an hour
after sunset.

How long does twilight last at the north pole?

_The Aurora._--One other phenomenon of the atmosphere may be mentioned,
only to point out that it is not of an astronomical character. The
Aurora, or northern lights, is as purely an affair of the earth as is a
thunderstorm, and its explanation belongs to the subject of terrestrial



52. SOLAR TIME.--To measure any quantity we need a unit in terms of
which it must be expressed. Angles are measured in degrees, and the
degree is the unit for angular measurement. For most scientific purposes
the centimeter is adopted as the unit with which to measure distances,
and similarly a day is the fundamental unit for the measurement of time.
Hours, minutes, and seconds are aliquot parts of this unit convenient
for use in dealing with shorter periods than a day, and the week, month,
and year which we use in our calendars are multiples of the day.

Strictly speaking, a day is not the time required by the earth to make
one revolution upon its axis, but it is best defined as the amount of
time required for a particular part of the sky to make the complete
circuit from the meridian of a particular place through west and east
back to the meridian again. The day begins at the moment when this
specified part of the sky is on the meridian, and "the time" at any
moment is the hour angle of this particular part of the sky--i. e., the
number of hours, minutes, etc., that have elapsed since it was on the

The student has already become familiar with the kind of day which is
based upon the motion of the vernal equinox, and which furnishes
sidereal time, and he has seen that sidereal time, while very convenient
in dealing with the motions of the stars, is decidedly inconvenient for
the ordinary affairs of life since in the reckoning of the hours it
takes no account of daylight and darkness. One can not tell off-hand
whether 10 hours, sidereal time, falls in the day or in the night. We
must in some way obtain a day and a system of time reckoning based upon
the apparent diurnal motion of the sun, and we may, if we choose, take
the sun itself as the point in the heavens whose transit over the
meridian shall mark the beginning and the end of the day. In this system
"the time" is the number of hours, minutes, etc., which have elapsed
since the sun was on the meridian, and this is the kind of time which is
shown by a sun dial, and which was in general use, years ago, before
clocks and watches became common. Since the sun moves among the stars
about a degree per day, it is easily seen that the rotating earth will
have to turn farther in order to carry any particular meridian from the
sun around to the sun again, than to carry it from a star around to the
same star, or from the vernal equinox around to the vernal equinox
again; just as the minute hand of a clock turns farther in going from
the hour hand round to the hour hand again than it turns in going from
XII to XII. These solar days and hours and minutes are therefore a
little longer than the corresponding sidereal ones, and this furnishes
the explanation why the stars come to the meridian a little earlier, by
solar time, every night than on the night before, and why sidereal time
gains steadily upon solar time, this gain amounting to approximately
3m. 56.5s. per day, or exactly one day per year, since the sun makes the
complete circuit of the constellations once in a year.

With the general introduction of clocks and watches into use about a
century ago this kind of solar time went out of common use, since no
well-regulated clock could keep the time correctly. The earth in its
orbital motion around the sun goes faster in some parts of its orbit
than in others, and in consequence the sun appears to move more rapidly
among the stars in winter than in summer; moreover, on account of the
convergence of hour circles as we go away from the equator, the same
amount of motion along the ecliptic produces more effect in winter and
summer when the sun is north or south, than it does in the spring and
autumn when the sun is near the equator, and as a combined result of
these causes and other minor ones true solar time, as it is called, is
itself not uniform, but falls behind the uniform lapse of sidereal time
at a variable rate, sometimes quicker, sometimes slower. A true solar
day, from noon to noon, is 51 seconds shorter in September than in

[Illustration: FIG. 30.--The equation of time.]

53. MEAN SOLAR TIME.--To remedy these inconveniences there has been
invented and brought into common use what is called _mean solar time_,
which is perfectly uniform in its lapse and which, by comparison with
sidereal time, loses exactly one day per year. "The time" in this system
never differs much from true solar time, and the difference between the
two for any particular day may be found in any good almanac, or may be
read from the curve in Fig. 30, in which the part of the curve above the
line marked _0m_ shows how many minutes mean solar time is faster than
true solar time. The correct name for this difference between the two
kinds of solar time is the _equation of time_, but in the almanacs it is
frequently marked "sun fast" or "sun slow." In sidereal time and true
solar time the distinction between A. M. hours (_ante meridiem_ =
before the sun reaches the meridian) and P. M. hours (_post meridiem_ =
after the sun has passed the meridian) is not observed, "the time" being
counted from 0 hours to 24 hours, commencing when the sun or vernal
equinox is on the meridian. Occasionally the attempt is made to
introduce into common use this mode of reckoning the hours, beginning
the day (date) at midnight and counting the hours consecutively up to
24, when the next date is reached and a new start made. Such a system
would simplify railway time tables and similar publications; but the
American public is slow to adopt it, although the system has come into
practical use in Canada and Spain.

When the mean solar time is known. Let _W_ represent the time shown by
an ordinary watch, and represent by _S_ the corresponding sidereal time
and by _D_ the number of days that have elapsed from March 23d to the
date in question. Then

    S = W + 69/70 × D × 4.

The last term is expressed in minutes, and should be reduced to hours
and minutes. Thus at 4 P. M. on July 4th--

                _D_ = 103 days.
    69/70 × _D_ × 4 = 406m.
                    = 6h. 46m.
                _W_ = 4h. 0m.
                _S_ = 10h. 46m.

The daily gain of sidereal upon mean solar time is 69/70 of 4 minutes,
and March 23d is the date on which sidereal and mean solar time are
together, taking the average of one year with another, but it varies a
little from year to year on account of the extra day introduced in leap

RULE II. When the stars in the northern sky can be seen. Find β
Cassiopeiæ, and imagine a line drawn from it to Polaris, and another
line from Polaris to the zenith. The sidereal time is equal to the angle
between these lines, provided that that angle must be measured from the
zenith toward the west. Turn the angle from degrees into hours by
dividing by 15.

55. THE EARTH'S ROTATION.--We are familiar with the fact that a watch
may run faster at one time than at another, and it is worth while to
inquire if the same is not true of our chief timepiece--the earth. It is
assumed in the sections upon the measurement of time that the earth
turns about its axis with absolute uniformity, so that mean solar time
never gains or loses even the smallest fraction of a second. Whether
this be absolutely true or not, no one has ever succeeded in finding
convincing proof of a variation large enough to be measured, although it
has recently been shown that the axis about which it rotates is not
perfectly fixed within the body of the earth. The solid body of the
earth wriggles about this axis like a fish upon a hook, so that the
position of the north pole upon the earth's surface changes within a
year to the extent of 40 or 50 feet (15 meters) without ever getting
more than this distance away from its average position. This is probably
caused by the periodical shifting of masses of air and water from one
part of the earth to another as the seasons change, and it seems
probable that these changes will produce some small effect upon the
rotation of the earth. But in spite of these, for any such moderate
interval of time as a year or a century, so far as present knowledge
goes, we may regard the earth's rotation as uniform and undisturbed. For
longer intervals--e. g., 1,000,000 or 10,000,000 years--the question is
a very different one, and we shall have to meet it again in another

56. LONGITUDE AND TIME.--In what precedes there has been constant
reference to the meridian. The day begins when the sun is on the
meridian. Solar time is the angular distance of the sun past the
meridian. Sidereal time was determined by observing transits of stars
over a meridian line actually laid out upon the ground, etc. But every
place upon the earth has its own meridian from which "the time" may be
reckoned, and in Fig. 31, where the rays of sunlight are represented as
falling upon a part of the earth's equator through which the meridians
of New York, Chicago, and San Francisco pass, it is evident that these
rays make different angles with the meridians, and that the sun is
farther from the meridian of New York than from that of San Francisco by
an amount just equal to the angle at _O_ between these meridians. This
angle is called by geographers the difference of longitude between the
two places, and the student should note that the word longitude is here
used in a different sense from that on page 36. From Fig. 31 we obtain

_Theorem._--The difference between "the times" at any two meridians is
equal to their difference of longitude, and the time at the eastern
meridian is greater than at the western meridian. Astronomers usually
express differences of longitude in hours instead of degrees. 1h. = 15°.

The name given to any kind of time should distinguish all the elements
which enter into it--e. g., New York sidereal time means the hour angle
of the vernal equinox measured from the meridian of New York, Chicago
true solar time is the hour angle of the sun reckoned from the meridian
of Chicago, etc.

[Illustration: FIG. 31.--Longitude and time]

[Illustration: FIG. 32.--Standard time.]

57. STANDARD TIME.--The requirements of railroad traffic have led to the
use throughout the United States and Canada of four "standard times,"
each of which is a mean solar time some integral number of hours slower
than the time of the meridian passing through the Royal Observatory at
Greenwich, England.

    Eastern time is 5 hours slower than that of Greenwich.
    Central  "      6  "      "     "    "         "
    Mountain "      7  "      "     "    "         "
    Pacific  "      8  "      "     "    "         "

In Fig. 32 the broken lines indicate roughly the parts of the United
States and Canada in which these several kinds of time are used, and
illustrate how irregular are the boundaries of these parts.

Standard time is sent daily into all of the more important telegraph
offices of the United States, and serves to regulate watches and clocks,
to the almost complete exclusion of local time.

58. TO DETERMINE THE LONGITUDE.--With an ordinary watch observe the time
of the sun's transit over your local meridian, and correct the observed
time for the equation of time by means of the curve in Fig. 30. The
difference between the corrected time and 12 o'clock will be the
correction of your watch referred to local mean solar time. Compare your
watch with the time signals in the nearest telegraph office and find its
correction referred to standard time. The difference between the two
corrections is the difference between your longitude and that of the
standard meridian.

N. B.--Don't tamper with the watch by trying to "set it right." No harm
will be done if it is wrong, provided you take due account of the
correction as indicated above.

If the correction of the watch changed between your observation and the
comparison in the telegraph office, what effect would it have upon the
longitude determination? How can you avoid this effect?

59. CHRONOLOGY.--The Century Dictionary defines chronology as "the
science of time"--that is, "the method of measuring or computing time
by regular divisions or periods according to the revolutions of the sun
or moon."

We have already seen that for the measurement of short intervals of time
the day and its subdivisions--hours, minutes, seconds--furnish a very
complete and convenient system. But for longer periods, extending to
hundreds and thousands of days, a larger unit of time is required, and
for the most part these longer units have in all ages and among all
peoples been based upon astronomical considerations. But to this there
is one marked exception. The week is a simple multiple of the day, as
the dime is a multiple of the cent, and while it may have had its origin
in the changing phases of the moon this is at best doubtful, since it
does not follow these with any considerable accuracy. If the still
longer units of time--the month and the year--had equally been made to
consist of an integral number of days much confusion and
misunderstanding might have been avoided, and the annals of ancient
times would have presented fewer pitfalls to the historian than is now
the case. The month is plainly connected with the motion of the moon
among the stars. The year is, of course, based upon the motion of the
sun through the heavens and the change of seasons which is thus
produced; although, as commonly employed, it is not quite the same as
the time required by the earth to make one complete revolution in its
orbit. This time of one revolution is called a sidereal year, while, as
we have already seen in Chapter V, the year which measures the course of
the seasons is shorter than this on account of the precession of the
equinoxes. It is called a tropical year with reference to the circuit
which the sun makes from one tropic to the other and back again.

We can readily understand why primitive peoples should adopt as units of
time these natural periods, but in so doing they incurred much the same
kind of difficulty that we should experience in trying to use both
English and American money in the ordinary transactions of life. How
many dollars make a pound sterling? How shall we make change with
English shillings and American dimes, etc.? How much is one unit worth
in terms of the other?

One of the Greek poets[2] has left us a quaint account of the confusion
which existed in his time with regard to the place of months and moons
in the calendar:

    "The moon by us to you her greeting sends,
    But bids us say that she's an ill-used moon
    And takes it much amiss that you will still
    Shuffle her days and turn them topsy-turvy,
    So that when gods, who know their feast days well,
    By your false count are sent home supperless,
    They scold and storm at her for your neglect."

  [2] Aristophanes, The Clouds, Whewell's translation.

60. DAY, MONTH, AND YEAR.--If the day, the month, and the year are to be
used concurrently, it is necessary to determine how many days are
contained in the month and year, and when this has been done by the
astronomer the numbers are found to be very awkward and inconvenient for
daily use; and much of the history of chronology consists in an account
of the various devices by which ingenious men have sought to use
integral numbers to replace the cumbrous decimal fractions which follow.

According to Professor Harkness, for the epoch 1900 A. D.--

    One tropical year = 365.242197 mean solar days.
     "     "      "   = 365d. 5h. 48m. 45.8s.
    One lunation      = 29.530588 mean solar days.
     "     "          = 29d. 12h. 44m. 2.8s.

The word _lunation_ means the average interval from one new moon to the
next one--i. e., the time required by the moon to go from conjunction
with the sun round to conjunction again.

A very ancient device was to call a year equal to 365 days, and to have
months alternately of 29 and 30 days in length, but this was
unsatisfactory in more than one way. At the end of four years this
artificial calendar would be about one day ahead of the true one, at the
end of forty years ten days in error, and within a single lifetime the
seasons would have appreciably changed their position in the year, April
weather being due in March, according to the calendar. So, too, the year
under this arrangement did not consist of any integral number of months,
12 months of the average length of 29.5 days being 354 days, and 13
months 383.5 days, thus making any particular month change its position
from the beginning to the middle and the end of the year within a
comparatively short time. Some peoples gave up the astronomical year as
an independent unit and adopted a conventional year of 12 lunar months,
354 days, which is now in use in certain Mohammedan countries, where it
is known as the wandering year, with reference to the changing positions
of the seasons in such a year. Others held to the astronomical year and
adopted a system of conventional months, such that twelve of them would
just make up a year, as is done to this day in our own calendar, whose
months of arbitrary length we are compelled to remember by some such
jingle as the following:

    "Thirty days hath September,
    April, June, and November;
    All the rest have thirty-one
    Save February,
    Which alone hath twenty-eight,
    Till leap year gives it twenty-nine."

61. THE CALENDAR.--The foundations of our calendar may fairly be
ascribed to Julius Cæsar, who, under the advice of the Egyptian
astronomer Sosigines, adopted the old Egyptian device of a leap year,
whereby every fourth year was to consist of 366 days, while ordinary
years were only 365 days long. He also placed the beginning of the year
at the first of January, instead of in March, where it had formerly
been, and gave his own name, Julius, to the month which we now call
July. August was afterward named in honor of his successor, Augustus.
The names of the earlier months of the year are drawn from Roman
mythology; those of the later months, September, October, etc., meaning
seventh month, eighth month, represent the places of these months in the
year, before Cæsar's reformation, and also their places in some of the
subsequent calendars, for the widest diversity of practice existed
during mediæval times with regard to the day on which the new year
should begin, Christmas, Easter, March 25th, and others having been
employed at different times and places.

The system of leap years introduced by Cæsar makes the average length of
a year 365.25 days, which differs by about eleven minutes from the true
length of the tropical year, a difference so small that for ordinary
purposes no better approximation to the true length of the year need be
desired. But _any_ deviation from the true length, however small, must
in the course of time shift the seasons, the vernal and autumnal
equinox, to another part of the year, and the ecclesiastical authorities
of mediæval Europe found here ground for objection to Cæsar's calendar,
since the great Church festival of Easter has its date determined with
reference to the vernal equinox, and with the lapse of centuries Easter
became more and more displaced in the calendar, until Pope Gregory XIII,
late in the sixteenth century, decreed another reformation, whereby ten
days were dropped from the calendar, the day after March 11th being
called March 21st, to bring back the vernal equinox to the date on which
it fell in A. D. 325, the time of the Council of Nicæa, which Gregory
adopted as the fundamental epoch of his calendar.

The calendar having thus been brought back into agreement with that of
old time, Gregory purposed to keep it in such agreement for the future
by modifying Cæsar's leap-year rule so that it should run: Every year
whose number is divisible by 4 shall be a leap year except those years
whose numbers are divisible by 100 but not divisible by 400. These
latter years--e. g., 1900--are counted as common years. The calendar
thus altered is called Gregorian to distinguish it from the older,
Julian calendar, and it found speedy acceptance in those civilized
countries whose Church adhered to Rome; but the Protestant powers were
slow to adopt it, and it was introduced into England and her American
colonies by act of Parliament in the year 1752, nearly two centuries
after Gregory's time. In Russia the Julian calendar has remained in
common use to our own day, but in commercial affairs it is there
customary to write the date according to both calendars--e. g., July
4/16, and at the present time strenuous exertions are making in that
country for the adoption of the Gregorian calendar to the complete
exclusion of the Julian one.

The Julian and Gregorian calendars are frequently represented by the
abbreviations O. S. and N. S., old style, new style, and as the older
historical dates are usually expressed in O. S., it is sometimes
convenient to transform a date from the one calendar to the other. This
is readily done by the formula

    G = J + (N - 2) - N/4,

where _G_ and _J_ are the respective dates, _N_ is the number of the
century, and the remainder is to be neglected in the division by 4. For
September 3, 1752, O. S., we have

        J = Sept. 3
    N - 2 = + 15
    - N/4 = -  4
        G = Sept. 14

and September 14 is the date fixed by act of Parliament to correspond to
September 3, 1752, O. S. Columbus discovered America on October 12,
1492, O. S. What is the corresponding date in the Gregorian calendar?

62. THE DAY OF THE WEEK.--A problem similar to the above but more
complicated consists in finding the day of the week on which any given
date of the Gregorian calendar falls--e. g., October 21, 1492.

The formula for this case is

    7q + r = Y + D + (Y - 1)/4 - (Y - 1)/100 + (Y - 1)/400

where _Y_ denotes the given year, _D_ the number of the day (date) in
that year, and _q_ and _r_ are respectively the quotient and the
remainder obtained by dividing the second member of the equation by 7.
If _r_ = 1 the date falls on Sunday, etc., and if _r_ = 0 the day is
Saturday. For the example suggested above we have

      Jan.  31
      Feb.  29
      Mch.  31
      April 30
      May   31
      June  30
      July  31
      Aug.  31
      Sept. 30
      Oct.  21
       D = 295

                  Y =  1492
                + D = + 295
    + (Y - 1) ÷   4 = + 372
    - (Y - 1) ÷ 100 = -  14
    + (Y - 1) ÷ 400 = +   3
                    7) 2148

                  _q_ =   306
                  _r_ =     6 = Friday.

Find from some history the day of the week on which Columbus first saw
America, and compare this with the above.

On what day of the week did last Christmas fall? On what day of the week
were you born? In the formula for the day of the week why does _q_ have
the coefficient 7? What principles in the calendar give rise to the
divisors 4, 100, 400?

For much curious and interesting information about methods of reckoning
the lapse of time the student may consult the articles Calendar and
Chronology in any good encyclopædia.




63. THE NATURE OF ECLIPSES.--Every planet has a shadow which travels
with the planet along its orbit, always pointing directly away from the
sun, and cutting off from a certain region of space the sunlight which
otherwise would fill it. For the most part these shadows are invisible,
but occasionally one of them falls upon a planet or some other body
which shines by reflected sunlight, and, cutting off its supply of
light, produces the striking phenomenon which we call an eclipse. The
satellites of Jupiter, Saturn, and Mars are eclipsed whenever they
plunge into the shadows cast by their respective planets, and Jupiter
himself is partially eclipsed when one of his own satellites passes
between him and the sun, and casts upon his broad surface a shadow too
small to cover more than a fraction of it.

But the eclipses of most interest to us are those of the sun and moon,
called respectively solar and lunar eclipses. In Fig. 33 the full moon,
_M´_, is shown immersed in the shadow cast by the earth, and therefore
eclipsed, and in the same figure the new moon, _M_, is shown as casting
its shadow upon the earth and producing an eclipse of the sun. From a
mere inspection of the figure we may learn that an eclipse of the sun
can occur only at new moon--i. e., when the moon is on line between the
earth and sun--and an eclipse of the moon can occur only at full moon.
Why? Also, the eclipsed moon, _M´_, will present substantially the same
appearance from every part of the earth where it is at all visible--the
same from North America as from South America--but the eclipsed sun
will present very different aspects from different parts of the earth.
Thus, at _L_, within the moon's shadow, the sunlight will be entirely
cut off, producing what is called a total eclipse. At points of the
earth's surface near _J_ and _K_ there will be no interference whatever
with the sunlight, and no eclipse, since the moon is quite off the line
joining these regions to any part of the sun. At places between _J_ and
_L_ or _K_ and _L_ the moon will cut off a part of the sun's light, but
not all of it, and will produce what is called a partial eclipse, which,
as seen from the northern parts of the earth, will be an eclipse of the
lower (southern) part of the sun, and as seen from the southern
hemisphere will be an eclipse of the northern part of the sun.

[Illustration: FIG. 33.--Different kinds of eclipse.]

The moon revolves around the earth in a plane, which, in the figure, we
suppose to be perpendicular to the surface of the paper, and to pass
through the sun along the line _M´ M_ produced. But it frequently
happens that this plane is turned to one side of the sun, along some
such line as _P Q_, and in this case the full moon would cut through the
edge of the earth's shadow without being at any time wholly immersed in
it, giving a partial eclipse of the moon, as is shown in the figure.

In what parts of the earth would this eclipse be visible? What kinds of
solar eclipse would be produced by the new moon at _Q_? In what parts of
the earth would they be visible?

64. THE SHADOW CONE.--The shape and position of the earth's shadow are
indicated in Fig. 33 by the lines drawn tangent to the circles which
represent the sun and earth, since it is only between these lines that
the earth interferes with the free radiation of sunlight, and since both
sun and earth are spheres, and the earth is much the smaller of the two,
it is evident that the earth's shadow must be, in geometrical language,
a cone whose base is at the earth, and whose vertex lies far to the
right of the figure--in other words, the earth's shadow, although very
long, tapers off finally to a point and ends. So, too, the shadow of the
moon is a cone, having its base at the moon and its vertex turned away
from the sun, and, as shown in the figure, just about long enough to
reach the earth.

It is easily shown, by the theorem of similar triangles in connection
with the known size of the earth and sun, that the distance from the
center of the earth to the vertex of its shadow is always equal to the
distance of the earth from the sun divided by 108, and, similarly, that
the length of the moon's shadow is equal to the distance of the moon
from the sun divided by 400, the moon's shadow being the smaller and
shorter of the two, because the moon is smaller than the earth. The
radius of the moon's orbit is just about 1/400th part of the radius of
the earth's orbit--i. e., the distance of the moon from the earth is
1/400th part of the distance of the earth from the sun, and it is this
"chance" agreement between the length of the moon's shadow and the
distance of the moon from the earth which makes the tip of the moon's
shadow fall very near the earth at the time of solar eclipses. Indeed,
the elliptical shape of the moon's orbit produces considerable
variations in the distance of the moon from the earth, and in
consequence of these variations the vertex of the shadow sometimes falls
short of reaching the earth, and sometimes even projects considerably
beyond its farther side. When the moon's distance is too great for the
shadow to bridge the space between earth and moon there can be no total
eclipse of the sun, for there is no shadow which can fall upon the
earth, even though the moon does come directly between earth and sun.
But there is then produced a peculiar kind of partial eclipse called
_annular_, or ring-shaped, because the moon, although eclipsing the
central parts of the sun, is not large enough to cover the whole of it,
but leaves the sun's edge visible as a ring of light, which completely
surrounds the moon. Although, strictly speaking, this is only a partial
eclipse, it is customary to put total and annular eclipses together in
one class, which is called central eclipses, since in these eclipses the
line of centers of sun and moon strikes the earth, while in ordinary
partial eclipses it passes to one side of the earth without striking it.
In this latter case we have to consider another cone called the
_penumbra_--i. e., partial shadow--which is shown in Fig. 33 by the
broken lines tangent to the sun and moon, and crossing at the point _V_,
which is the vertex of this cone. This penumbral cone includes within
its surface all that region of space within which the moon cuts off any
of the sunlight, and of course it includes the shadow cone which
produces total eclipses. Wherever the penumbra falls there will be a
solar eclipse of some kind, and the nearer the place is to the axis of
the penumbra, the more nearly total will be the eclipse. Since the moon
stands about midway between the earth and the vertex of the penumbra,
the diameter of the penumbra where it strikes the earth will be about
twice as great as the diameter of the moon, and the student should be
able to show from this that the region of the earth's surface within
which a partial solar eclipse is visible extends in a straight line
about 2,100 miles on either side of the region where the eclipse is
total. Measured along the curved surface of the earth, this distance is
frequently much greater.

Is it true that if at any time the axis of the shadow cone comes within
2,100 miles of the earth's surface a partial eclipse will be visible in
those parts of the earth nearest the axis of the shadow?

difference between lunar and solar eclipses which has been already
suggested, may be learned from Fig. 33. The full moon, _M´_, will be
seen eclipsed from every part of the earth where it is visible at all at
the time of the eclipse--that is, from the whole night side of the
earth; while the eclipsed sun will be seen eclipsed only from those
parts of the day side of the earth upon which the moon's shadow or
penumbra falls. Since the point of the shadow at best but little more
than reaches to the earth, the amount of space upon the earth which it
can cover at any one moment is very small, seldom more than 100 to 200
miles in length, and it is only within the space thus actually covered
by the shadow that the sun is at any given moment totally eclipsed, but
within this region the sun disappears, absolutely, behind the solid body
of the moon, leaving to view only such outlying parts and appendages as
are too large for the moon to cover. At a lunar eclipse, on the other
hand, the earth coming between sun and moon cuts off the light from the
latter, but, curiously enough, does not cut it off so completely that
the moon disappears altogether from sight even in mid-eclipse. The
explanation of this continued visibility is furnished by the broken
lines extending, in Fig. 33, from the earth through the moon. These
represent sunlight, which, entering the earth's atmosphere near the edge
of the earth (edge as seen from sun and moon), passes through it and
emerges in a changed direction, refracted, into the shadow cone and
feebly illumines the moon's surface with a ruddy light like that often
shown in our red sunsets. Eclipse and sunset alike show that when the
sun's light shines through dense layers of air it is the red rays which
come through most freely, and the attentive observer may often see at a
clear sunset something which corresponds exactly to the bending of the
sunlight into the shadow cone; just before the sun reaches the horizon
its disk is distorted from a circle into an oval whose horizontal
diameter is longer than the vertical one (see § 50).

QUERY.--At a total lunar eclipse what would be the effect upon the
appearance of the moon if the atmosphere around the edge of the earth
were heavily laden with clouds?

66. THE TRACK OF THE SHADOW.--We may regard the moon's shadow cone as a
huge pencil attached to the moon, moving with it along its orbit in the
direction of the arrowhead (Fig. 34), and as it moves drawing a black
line across the face of the earth at the time of total eclipse. This
black line is the path of the shadow and marks out those regions within
which the eclipse will be total at some stage of its progress. If the
point of the shadow just reaches the earth its trace will have no
sensible width, while, if the moon is nearer, the point of the cone will
be broken off, and, like a blunt pencil, it will draw a broad streak
across the earth, and this under the most favorable circumstances may
have a breadth of a little more than 160 miles and a length of 10,000 or
12,000 miles. The student should be able to show from the known distance
of the moon (240,000 miles) and the known interval between consecutive
new moons (29.5 days) that on the average the moon's shadow sweeps past
the earth at the rate of 2,100 miles per hour, and that in a general way
this motion is from west to east, since that is the direction of the
moon's motion in its orbit. The actual velocity with which the moon's
shadow moves past a given station may, however, be considerably greater
or less than this, since on the one hand when the shadow falls very
obliquely, as when the eclipse occurs near sunrise or sunset, the
shifting of the shadow will be very much greater than the actual motion
of the moon which produces it, and on the other hand the earth in
revolving upon its axis carries the spectator and the ground upon which
he stands along the same direction in which the shadow is moving. At the
equator, with the sun and moon overhead, this motion of the earth
subtracts about 1,000 miles per hour from the velocity with which the
shadow passes by. It is chiefly on this account, the diminished velocity
with which the shadow passes by, that total solar eclipses last longer
in the tropics than in higher latitudes, but even under the most
favorable circumstances the duration of totality does not reach eight
minutes at any one place, although it may take the shadow several hours
to sweep the entire length of its path across the earth.

According to Whitmell the greatest possible duration of a total solar
eclipse is 7m. 40s., and it can attain this limit only when the eclipse
occurs near the beginning of July and is visible at a place 5° north of
the equator.

The duration of a lunar eclipse depends mainly upon the position of the
moon with respect to the earth's shadow. If it strikes the shadow
centrally, as at _M´_, Fig. 33, a total eclipse may last for about two
hours, with an additional hour at the beginning and end, during which
the moon is entering and leaving the earth's shadow. If the moon meets
the shadow at one side of the axis, as at _P_, the total phase of the
eclipse may fail altogether, and between these extremes the duration of
totality may be anything from two hours downward.

[Illustration: FIG. 34.--Relation of the lunar nodes to eclipses.]

sometimes encounters the earth's shadow centrally and more frequently at
full moon passes by without touching it at all, we resort to Fig. 34,
which represents a part of the orbit of the earth about the sun, with
dates showing the time in each year at which the earth passes the part
of its orbit thus marked. The orbit of the moon about the earth, _M M´_,
is also shown, with the new moon, _M_, casting its shadow toward the
earth and the full moon, _M´_, apparently immersed in the earth's
shadow. But here appearances are deceptive, and the student who has
made the observations set forth in Chapter III has learned for himself
a fact of which careful account must now be taken. The apparent paths of
the moon and sun among the stars are great circles which lie near each
other, but are not exactly the same; and since these great circles are
only the intersections of the sky with the planes of the earth's orbit
and the moon's orbit, we see that these planes are slightly inclined to
each other and must therefore intersect along some line passing through
the center of the earth. This line, _N´ N´´_, is shown in the figure,
and if we suppose the surface of the paper to represent the plane of the
earth's orbit, we shall have to suppose the moon's orbit to be tipped
around this line, so that the left side of the orbit lies above and the
right side below the surface of the paper. But since the earth's shadow
lies in the plane of its orbit--i. e., in the surface of the paper--the
full moon of March, _M´_, must have passed below the shadow, and the new
moon, _M_, must have cast its shadow above the earth, so that neither a
lunar nor a solar eclipse could occur in that month. But toward the end
of May the earth and moon have reached a position where the line
_N´ N´´_ points almost directly toward the sun, in line with the shadow
cones which hide it. Note that the line _N´ N´´_ remains very nearly
parallel to its original position, while the earth is moving along its
orbit. The full moon will now be very near this line and therefore very
close to the plane of the earth's orbit, if not actually in it, and must
pass through the shadow of the earth and be eclipsed. So also the new
moon will cast its shadow in the plane of the ecliptic, and this shadow,
falling upon the earth, produced the total solar eclipse of May 28,

_N´ N´´_ is called the line of nodes of the moon's orbit (§ 39), and the
two positions of the earth in its orbit, diametrically opposite each
other, at which _N´ N´´_ points exactly toward the sun, we shall call
the _nodes_ of the lunar orbit. Strictly speaking, the nodes are those
points of the sky against which the moon's center is projected at the
moment when in its orbital motion it cuts through the plane of the
earth's orbit. Bearing in mind these definitions, we may condense much
of what precedes into the proposition: Eclipses of either sun or moon
can occur only when the earth is at or near one of the nodes of the
moon's orbit. Corresponding to these positions of the earth there are in
each year two seasons, about six months apart, at which times, and at
these only, eclipses can occur. Thus in the year 1900 the earth passed
these two points on June 2d and November 24th respectively, and the
following list of eclipses which occurred in that year shows that all of
them were within a few days of one or the other of these dates:

        _Eclipses of the Year 1900_

    Total solar eclipse          May 28th.
    Partial lunar eclipse        June 12th.
    Annular (solar) eclipse      November 21st.

68. ECLIPSE LIMITS.--If the earth is exactly at the node at the time of
new moon, the moon's shadow will fall centrally upon it and will produce
an eclipse visible within the torrid zone, since this is that part of
the earth's surface nearest the plane of its orbit. If the earth is near
but not at the node, the new moon will stand a little north or south of
the plane of the earth's orbit, and its shadow will strike the earth
farther north or south than before, producing an eclipse in the
temperate or frigid zones; or the shadow may even pass entirely above or
below the earth, producing no eclipse whatever, or at most a partial
eclipse visible near the north or south pole. Just how many days' motion
the earth may be away from the node and still permit an eclipse is shown
in the following brief table of eclipse limits, as they are called:

                  _Solar Eclipse Limits_

  If at any new moon the earth is

  Less than 10 days away from a node, a central eclipse is certain.
  Between 10 and 16 days   "  "  "    some kind of eclipse is certain.
  Between 16 and 19 days   "  "  "    a partial eclipse is possible.
  More than 19 days        "  "  "    no eclipse is possible.

                _Lunar Eclipse Limits_

  If at any full moon the earth is

  Less than 4 days away from a node, a total eclipse is certain.
  Between 4 and 10 days   "  "  "    some kind of eclipse is certain.
  Between 10 and 14 days  "  "  "    a partial eclipse is possible.
  More than 14 days       "  "  "    no eclipse is possible.

From this table of eclipse limits we may draw some interesting
conclusions about the frequency with which eclipses occur.

69. NUMBER OF ECLIPSES IN A YEAR.--Whenever the earth passes a node of
the moon's orbit a new moon must occur at some time during the 2 × 16
days that the earth remains inside the limits where some kind of eclipse
is certain, and there must therefore be an eclipse of the sun every time
the earth passes a node of the moon's orbit. But, since there are two
nodes past which the earth moves at least once in each year, there must
be at least two solar eclipses every year. Can there be more than two?
On the average, will central or partial eclipses be the more numerous?

A similar line of reasoning will not hold true for eclipses of the moon,
since it is quite possible that no full moon should occur during the 20
days required by the earth to move past the node from the western to the
eastern limit. This omission of a full moon while the earth is within
the eclipse limits sometimes happens at both nodes in the same year, and
then we have a year with no eclipse of the moon. The student may note in
the list of eclipses for 1900 that the partial lunar eclipse of June
12th occurred 10 days after the earth passed the node, and was therefore
within the doubtful zone where eclipses may occur and may fail, and
corresponding to this position the eclipse was a very small one, only a
thousandth part of the moon's diameter dipping into the shadow of the
earth. By so much the year 1900 escaped being an illustration of a year
in which no lunar eclipse occurred.

A partial eclipse of the moon will usually occur about a fortnight
before or after a total eclipse of the sun, since the full moon will
then be within the eclipse limit at the opposite node. A partial eclipse
of the sun will always occur about a fortnight before or after a total
eclipse of the moon.

[Illustration: FIG. 35.--The eclipse of May 28, 1900.]

70. ECLIPSE MAPS.--It is the custom of astronomers to prepare, in
advance of the more important eclipses, maps showing the trace of the
moon's shadow across the earth, and indicating the times of beginning
and ending of the eclipses, as is shown in Fig. 35. While the actual
construction of such a map requires much technical knowledge, the
principles involved are simple enough: the straight line passed through
the center of sun and moon is the axis of the shadow cone, and the map
contains little more than a graphical representation of when and where
this cone meets the surface of the earth. Thus in the map, the "Path of
Total Eclipse" is the trace of the shadow cone across the face of the
earth, and the width of this path shows that the earth encountered the
shadow considerably inside the vertex of the cone. The general direction
of the path is from west to east, and the slight sinuousities which it
presents are for the most part due to unavoidable distortion of the
map caused by the attempt to represent the curved surface of the earth
upon the flat surface of the paper. On either side of the Path of Total
Eclipse is the region within which the eclipse was only partial, and the
broken lines marked Begins at 3h., Ends at 3h., show the intersection of
the penumbral cone with the surface of the earth at 3 P. M., Greenwich
time. These two lines inclose every part of the earth's surface from
which at that time any eclipse whatever could be seen, and at this
moment the partial eclipse was just beginning at every point on the
eastern edge of the penumbra and just ending at every point on the
western edge, while at the center of the penumbra, on the Path of Total
Eclipse, lay the shadow of the moon, an oval patch whose greatest
diameter was but little more than 60 miles in length, and within which
lay every part of the earth where the eclipse was total at that moment.

The position of the penumbra at other hours is also shown on the map,
although with more distortion, because it then meets the surface of the
earth more obliquely, and from these lines it is easy to obtain the time
of beginning and end of the eclipse at any desired place, and to
estimate by the distance of the place from the Path of Total Eclipse how
much of the sun's face was obscured.

Let the student make these "predictions" for Washington, Chicago,
London, and Algiers.

The points in the map marked First Contact, Last Contact, show the
places at which the penumbral cone first touched the earth and finally
left it. According to computations made as a basis for the construction
of the map the Greenwich time of First Contact was 0h. 12.5m. and of
Last Contact 5h. 35.6m., and the difference between these two times
gives the total duration of the eclipse upon the earth--i. e., 5 hours
23.1 minutes.

[Illustration: FIG. 36.--Central eclipses for the first two decades of
the twentieth century. OPPOLZER.]

71. FUTURE ECLIPSES.--An eclipse map of a different kind is shown in
Fig. 36, which represents the shadow paths of all the central eclipses
of the sun, visible during the period 1900-1918 A. D., in those parts of
the earth north of the south temperate zone. Each continuous black line
shows the path of the shadow in a total eclipse, from its beginning, at
sunrise, at the western end of the line to its end, sunset, at the
eastern end, the little circle near the middle of the line showing the
place at which the eclipse was total at noon. The broken lines represent
similar data for the annular eclipses. This map is one of a series
prepared by the Austrian astronomer, Oppolzer, showing the path of every
such eclipse from the year 1200 B. C. to 2160 A. D., a period of more
than three thousand years.

If we examine the dates of the eclipses shown in this map we shall find
that they are not limited to the particular seasons, May and November,
in which those of the year 1900 occurred, but are scattered through all
the months of the year, from January to December. This shows at once
that the line of nodes, _N´ N´´_, of Fig. 34, does not remain in a fixed
position, but turns round in the plane of the earth's orbit so that in
different years the earth reaches the node in different months. The
precession has already furnished us an illustration of a similar change,
the slow rotation of the earth's axis, producing a corresponding
shifting of the line in which the planes of the equator and ecliptic
intersect; and in much the same way, through the disturbing influence of
the sun's attraction, the line _N´ N´´_ is made to revolve westward,
opposite to the arrowheads in Fig. 34, at the rate of nearly 20° per
year, so that the earth comes to each node about 19 days earlier in each
year than in the year preceding, and the eclipse season in each year
comes on the average about 19 days earlier than in the year before,
although there is a good deal of irregularity in the amount of change in
particular years.

72. RECURRENCE OF ECLIPSES.--Before the beginning of the Christian era
astronomers had found out a rough-and-ready method of predicting
eclipses, which is still of interest and value. The substance of the
method is that if we start with any eclipse whatever--e. g., the eclipse
of May 28, 1900--and reckon forward or backward from that date a period
of 18 years and 10 or 11 days, we shall find another eclipse quite
similar in its general characteristics to the one with which we started.
Thus, from the map of eclipses (Fig. 36), we find that a total solar
eclipse will occur on June 8, 1918, 18 years and 11 days after the one
illustrated in Fig. 35. This period of 18 years and 11 days is called
_saros_, an ancient word which means cycle or repetition, and since
every eclipse is repeated after the lapse of a saros, we may find the
dates of all the eclipses of 1918 by adding 11 days to the dates given
in the table of eclipses for 1900 (§ 67), and it is to be especially
noted that each eclipse of 1918 will be like its predecessor of 1900 in
character--lunar, solar, partial, total, etc. The eclipses of any year
may be predicted by a similar reference to those which occurred eighteen
years earlier. Consult a file of old almanacs.

The exact length of a saros is 223 lunar months, each of which is a
little more than 29.5 days long, and if we multiply the exact value of
this last number (see § 60) by 223, we shall find for the product
6,585.32 days, which is equal to 18 years 11.32 days when there are four
leap years included in the 18, or 18 years 10.32 days when the number of
leap years is five; and in applying the saros to the prediction of
eclipses, due heed must be paid to the number of intervening leap years.
To explain why eclipses are repeated at the end of the saros, we note
that the occurrence of an eclipse depends solely upon the relative
positions of the earth, moon, and node of the moon's orbit, and the
eclipse will be repeated as often as these three come back to the
position which first produced it. This happens at the end of every
saros, since the saros is, approximately, the least common multiple of
the length of the year, the length of the lunar month, and the length of
time required by the line of nodes to make a complete revolution around
the ecliptic. If the saros were exactly a multiple of these three
periods, every eclipse would be repeated over and over again for
thousands of years; but such is not the case, the saros is not an exact
multiple of a year, nor is it an exact multiple of the time required for
a revolution of the line of nodes, and in consequence the restitution
which comes at the end of the saros is not a perfect one. The earth at
the 223d new moon is in fact about half a day's motion farther west,
relative to the node, than it was at the beginning, and the resulting
eclipse, while very similar, is not precisely the same as before. After
another 18 years, at the second repetition, the earth is a day farther
from the node than at first, and the eclipse differs still more in
character, etc. This is shown in Fig. 37, which represents the apparent
positions of the disks of the sun and moon as seen from the center of
the earth at the end of each sixth saros, 108 years, where the upper row
of figures represents the number of repetitions of the eclipse from the
beginning, marked _0_, to the end, _72_. The solar eclipse limits, 10,
16, 19 days, are also shown, and all those eclipses which fall between
the 10-day limits will be central as seen from some part of the earth,
those between 16 and 19 partial wherever seen, while between 10 and 16
they may be either total or partial. Compare the figure with the
following description given by Professor Newcomb: "A series of such
eclipses commences with a very small eclipse near one pole of the earth.
Gradually increasing for about eleven recurrences, it will become
central near the same pole. Forty or more central eclipses will then
recur, the central line moving slowly toward the other pole. The series
will then become partial, and finally cease. The entire duration of the
series will be more than a thousand years. A new series commences, on
the average, at intervals of thirty years."

[Illustration: FIG. 37.--Graphical illustration of the saros.]

A similar figure may be constructed to represent the recurrence of lunar
eclipses; but here, in consequence of the smaller eclipse limits, we
shall find that a series is of shorter duration, a little over eight
centuries as compared with twelve centuries, which is the average
duration of a series of solar eclipses.

One further matter connected with the saros deserves attention. During
the period of 6,585.32 days the earth has 6,585 times turned toward the
sun the same face upon which the moon's shadow fell at the beginning of
the saros, but at the end of the saros the odd 0.32 of a day gives the
earth time to make about a third of a revolution more before the eclipse
is repeated, and in consequence the eclipse is seen in a different
region of the earth, on the average about 116° farther west in
longitude. Compare in Fig. 36 the regions in which the eclipses of 1900
and 1918 are visible.

Is this change in the region where the repeated eclipse is visible, true
of lunar eclipses as well as solar?

73. USE OF ECLIPSES.--At all times and among all peoples eclipses, and
particularly total eclipses of the sun, have been reckoned among the
most impressive phenomena of Nature. In early times and among
uncultivated people they were usually regarded with apprehension, often
amounting to a terror and frenzy, which civilized travelers have not
scrupled to use for their own purposes with the aid of the eclipse
predictions contained in their almanacs, threatening at the proper time
to destroy the sun or moon, and pointing to the advancing eclipse as
proof that their threats were not vain. In our own day and our own land
these feelings of awe have not quite disappeared, but for the most part
eclipses are now awaited with an interest and pleasure which, contrasted
with the former feelings of mankind, furnish one of the most striking
illustrations of the effect of scientific knowledge in transforming
human fear and misery into a sense of security and enjoyment.

But to the astronomer an eclipse is more than a beautiful illustration
of the working of natural laws; it is in varying degree an opportunity
of adding to his store of knowledge respecting the heavenly bodies. The
region immediately surrounding the sun is at most times closed to
research by the blinding glare of the sun's own light, so that a planet
as large as the moon might exist here unseen were it not for the
occasional opportunity presented by a total eclipse which shuts off the
excessive light and permits not only a search for unknown planets but
for anything and everything which may exist around the sun. More than
one astronomer has reported the discovery of such planets, and at least
one of these has found a name and a description in some of the books,
but at the present time most astronomers are very skeptical about the
existence of any such object of considerable size, although there is
some reason to believe that an enormous number of little bodies, ranging
in size from grains of sand upward, do move in this region, as yet
unseen and offering to the future problems for investigation.

But in other directions the study of this region at the times of total
eclipse has yielded far larger returns, and in the chapter on the sun we
shall have to consider the marvelous appearances presented by the solar
prominences and by the corona, an appendage of the sun which reaches out
from his surface for millions of miles but is never seen save at an
eclipse. Photographs of the corona are taken by astronomers at every
opportunity, and reproductions of some of these may be found in Chapter

Annular eclipses and lunar eclipses are of comparatively little
consequence, but any recorded eclipse may become of value in connection
with chronology. We date our letters in a particular year of the
twentieth century, and commonly suppose that the years are reckoned from
the birth of Christ; but this is an error, for the eclipses which were
observed of old and by the chroniclers have been associated with events
of his life, when examined by the astronomers are found quite
inconsistent with astronomic theory. They are, however, reconciled with
it if we assume that our system of dates has its origin four years after
the birth of Christ, or, in other words, that Christ was born in the
year 4 B. C. A mistake was doubtless made at the time the Christian era
was introduced into chronology. At many other points the chance record
of an eclipse in the early annals of civilization furnishes a similar
means of controlling and correcting the dates assigned by the historian
to events long past.



74. TWO FAMILIAR INSTRUMENTS.--In previous chapters we have seen that a
clock and a divided circle (protractor) are needed for the observations
which an astronomer makes, and it is worth while to note here that the
geography of the sky and the science of celestial motions depend
fundamentally upon these two instruments. The protractor is a simple
instrument, a humble member of the family of divided circles, but untold
labor and ingenuity have been expended on this family to make possible
the construction of a circle so accurately divided that with it angles
may be measured to the tenth of a second instead of to the tenth of a
degree--i. e., 3,600 times as accurate as the protractor furnishes.

The building of a good clock is equally important and has cost a like
amount of labor and pains, so that it is a far cry from Galileo and his
discovery that a pendulum "keeps time" to the modern clock with its
accurate construction and elaborate provision against disturbing
influences of every kind. Every such timepiece, whether it be of the
nutmeg variety which sells for a dollar, or whether it be the standard
clock of a great national observatory, is made up of the same essential
parts that fall naturally into four classes, which we may compare with
the departments of a well-ordered factory: I. A timekeeping department,
the pendulum or balance spring, whose oscillations must all be of equal
duration. II. A power department, the weights or mainspring, which,
when wound, store up the power applied from outside and give it out
piecemeal as required to keep the first department running. III. A
publication department, the dial and hands, which give out the time
furnished by Department I. IV. A transportation department, the wheels,
which connect the other three and serve as a means of transmitting power
and time from one to the other. The case of either clock or watch is
merely the roof which shelters it and forms no department of its
industry. Of these departments the first is by far the most important,
and its good or bad performance makes or mars the credit of the clock.
Beware of meddling with the balance wheel of your watch.

75. RADIANT ENERGY.--But we have now to consider other instruments which
in practice supplement or displace the simple apparatus hitherto
employed. Among the most important of these modern instruments are the
telescope, the spectroscope, and the photographic camera; and since all
these instruments deal with the light which comes from the stars to the
earth, we must for their proper understanding take account of the nature
of that light, or, more strictly speaking, we must take account of the
radiant energy emitted by the sun and stars, which energy, coming from
the sun, is translated by our nerves into the two different sensations
of light and heat. The radiant energy which comes from the stars is not
fundamentally different from that of the sun, but the amount of energy
furnished by any star is so small that it is unable to produce through
our nerves any sensible perception of heat, and for the same reason the
vast majority of stars are invisible to the unaided eye; they do not
furnish a sufficient amount of energy to affect the optic nerves. A hot
brick taken into the hand reveals its presence by the two different
sensations of heat and pressure (weight); but as there is only one brick
to produce the two sensations, so there is only one energy to produce
through its action upon different nerves the two sensations of light
and heat, and this energy is called _radiant_ because it appears to
stream forth radially from everything which has the capacity of emitting
it. For the detailed study of radiant energy the student is referred to
that branch of science called physics; but some of its elementary
principles may be learned through the following simple experiment, which
the student should not fail to perform for himself:

Drop a bullet or other similar object into a bucket of water and observe
the circular waves which spread from the place where it enters the
water. These waves are a form of radiant energy, but differing from
light or heat in that they are visibly confined to a single plane, the
surface of the water, instead of filling the entire surrounding space.
By varying the size of the bucket, the depth of the water, the weight of
the bullet, etc., different kinds of waves, big and little, may be
produced; but every such set of waves may be described and defined in
all its principal characteristics by means of three numbers--viz., the
vertical height of the waves from hollow to crest; the distance of one
wave from the next; and the velocity with which the waves travel across
the water. The last of these quantities is called the velocity of
propagation; the second is called the wave length; one half of the first
is called the amplitude; and all these terms find important applications
in the theory of light and heat.

The energy of the falling bullet, the disturbance which it produced on
entering the water, was carried by the waves from the center to the edge
of the bucket but not beyond, for the wave can go only so far as the
water extends. The transfer of energy in this way requires a perfectly
continuous medium through which the waves may travel, and the whole
visible universe is supposed to be filled with something called _ether_,
which serves everywhere as a medium for the transmission of radiant
energy just as the water in the experiment served as a medium for
transmitting in waves the energy furnished to it by the falling bullet.
The student may think of this energy as being transmitted in spherical
waves through the ether, every glowing body, such as a star, a candle
flame, an arc lamp, a hot coal, etc., being the origin and center of
such systems of waves, and determining by its own physical and chemical
properties the wave length and amplitude of the wave systems given off.

The intensity of any light depends upon the amplitude of the
corresponding vibration, and its color depends upon the wave length. By
ingenious devices which need not be here described it has been found
possible to measure the wave length corresponding to different
colors--e. g., all of the colors of the rainbow, and some of these wave
lengths expressed in tenth meters are as follows: A tenth meter is the
length obtained by dividing a meter into 10^{10} equal parts. 10^{10} =

                 Color.               Wave length.

    Extreme limit of visible violet      3,900
    Middle of the violet                 4,060
      "      "    blue                   4,730
      "      "    green                  5,270
      "      "    yellow                 5,810
      "      "    orange                 5,970
      "      "    red                    7,000
    Extreme limit of visible red         7,600


The phrase "extreme limit of visible violet" or red used above must be
understood to mean that in general the eye is not able to detect radiant
energy having a wave length less than 3,900 or greater than 7,600 tenth
meters. Radiant energy, however, exists in waves of both greater and
shorter length than the above, and may be readily detected by apparatus
not subject to the limitations of the human eye--e. g., a common
thermometer will show a rise of temperature when its bulb is exposed to
radiant energy of wave length much greater than 7,600 tenth meters,
and a photographic plate will be strongly affected by energy of
shorter wave length than 3,900 tenth meters.

76. REFLECTION AND CONDENSATION OF WAVES.--When the waves produced by
dropping a bullet into a bucket of water meet the sides of the bucket,
they appear to rebound and are reflected back toward the center, and if
the bullet is dropped very near the center of the bucket the reflected
waves will meet simultaneously at this point and produce there by their
combined action a wave higher than that which was reflected at the walls
of the bucket. There has been a condensation of energy produced by the
reflection, and this increased energy is shown by the greater amplitude
of the wave. The student should not fail to notice that each portion of
the wave has traveled out and back over the radius of the bucket, and
that they meet simultaneously at the center because of this equality of
the paths over which they travel, and the resulting equality of time
required to go out and back. If the bullet were dropped at one side of
the center, would the reflected waves produce _at any point_ a
condensation of energy?

If the bucket were of elliptical instead of circular cross section and
the bullet were dropped at one focus of the ellipse there would be
produced a condensation of reflected energy at the other focus, since
the sum of the paths traversed by each portion of the wave before and
after reflection is equal to the sum of the paths traversed by every
other portion, and all parts of the wave reach the second focus at the
same time. Upon what geometrical principle does this depend?

The condensation of wave energy in the circular and elliptical buckets
are special cases under the general principle that such a condensation
will be produced at any point which is so placed that different parts of
the wave front reach it simultaneously, whether by reflection or by some
other means, as shown below.

The student will note that for the sake of greater precision we here
say _wave front_ instead of wave. If in any wave we imagine a line drawn
along the crest, so as to touch every drop which at that moment is
exactly at the crest, we shall have what is called a wave front, and
similarly a line drawn through the trough between two waves, or through
any set of drops similarly placed on a wave, constitutes a wave front.

77. MIRRORS AND LENSES.--That form of radiant energy which we recognize
as light and heat may be reflected and condensed precisely as are the
waves of water in the exercise considered above, but owing to the
extreme shortness of the wave length in this case the reflecting surface
should be very smooth and highly polished. A piece of glass hollowed out
in the center by grinding, and with a light film of silver chemically
deposited upon the hollow surface and carefully polished, is often used
by astronomers for this purpose, and is called a concave mirror.

The radiant energy coming from a star or other distant object and
falling upon the silvered face of such a mirror is reflected and
condensed at a point a little in front of the mirror, and there forms an
image of the star, which may be seen with the unaided eye, if it is held
in the right place, or may be examined through a magnifying glass.
Similarly, an image of the sun, a planet, or a distant terrestrial
object is formed by the mirror, which condenses at its appropriate place
the radiant energy proceeding from each and every point in the surface
of the object, and this, in common phrase, produces an image of the

Another device more frequently used by astronomers for the production of
images (condensation of energy) is a lens which in its simplest form is
a round piece of glass, thick in the center and thin at the edge, with a
cross section, such as is shown at _A B_ in Fig. 38. If we suppose _E G
D_ to represent a small part of a wave front coming from a very distant
source of radiant energy, such as a star, this wave front will be
practically a plane surface represented by the straight line _E D_, but
in passing through the lens this surface will become warped, since light
travels slower in glass than in air, and the central part of the beam,
_G_, in its onward motion will be retarded by the thick center of the
lens, more than _E_ or _D_ will be retarded by the comparatively thin
outer edges of _A B_. On the right of the lens the wave front therefore
will be transformed into a curved surface whose exact character depends
upon the shape of the lens and the kind of glass of which it is made. By
properly choosing these the new wave front may be made a part of a
sphere having its center at the point _F_ and the whole energy of the
wave front, _E G D_, will then be condensed at _F_, because this point
is equally distant from all parts of the warped wave front, and
therefore is in a position to receive them simultaneously. The distance
of _F_ from _A B_ is called the focal length of the lens, and _F_ itself
is called the focus. The significance of this last word (Latin, _focus_
= fireplace) will become painfully apparent to the student if he will
hold a common reading glass between his hand and the sun in such a way
that the focus falls upon his hand.

[Illustration: FIG. 38.--Illustrating the theory of lenses.]

All the energy transmitted by the lens in the direction _G F_ is
concentrated upon a very small area at _F_, and an image of the
object--e. g., a star, from which the light came--is formed here. Other
stars situated near the one in question will also send beams of light
along slightly different directions to the lens, and these will be
concentrated, each in its appropriate place, in the _focal plane_,
_F H_, passed through the focus, _F_, perpendicular to the line, _F G_,
and we shall find in this plane a picture of all the stars or other
objects within the range of the lens.

[Illustration: FIG. 39.--Essential parts of a reflecting telescope.]

78. TELESCOPES.--The simplest kind of telescope consists of a concave
mirror to produce images, and a magnifying glass, called an _eyepiece_,
through which to examine them; but for convenience' sake, so that the
observer may not stand in his own light, a small mirror is frequently
added to this combination, as at _H_ in Fig. 39, where the lines
represent the directions along which the energy is propagated. By
reflection from this mirror the focal plane and the images are shifted
to _F_, where they may be examined from one side through the magnifying
glass _E_.

[Illustration: FIG. 40.--A simple form of refracting telescope.]

Such a combination of parts is called a _reflecting_ telescope, while
one in which the images are produced by a lens or combination of lenses
is called a _refracting_ telescope, the adjective having reference to
the bending, refraction, produced by the glass upon the direction in
which the energy is propagated. The customary arrangement of parts in
such a telescope is shown in Fig. 40, where the part marked _O_ is
called the objective and _V E_ (the magnifying glass) is the eyepiece,
or ocular, as it is sometimes called.

Most objects with which we have to deal in using a telescope send to it
not light of one color only, but a mixture of light of many colors,
many different wave lengths, some of which are refracted more than
others by the glass of which the lens is composed, and in consequence of
these different amounts of refraction a single lens does not furnish a
single image of a star, but gives a confused jumble of red and yellow
and blue images much inferior in sharpness of outline (definition) to
the images made by a good concave mirror. To remedy this defect it is
customary to make the objective of two or more pieces of glass of
different densities and ground to different shapes as is shown at _O_ in
Fig. 40. The two pieces of glass thus mounted in one frame constitute a
compound lens having its own focal plane, shown at _F_ in the figure,
and similarly the lenses composing the eyepiece have a focal plane
between the eyepiece and the objective which must also fall at _F_, and
in the use of a telescope the eyepiece must be pushed out or in until
its focal plane coincides with that of the objective. This process,
which is called focusing, is what is accomplished in the ordinary opera
glass by turning a screw placed between the two tubes, and it must be
carefully done with every telescope in order to obtain distinct vision.

79. MAGNIFYING POWER.--The amount by which a given telescope magnifies
depends upon the focal length of the objective (or mirror) and the focal
length of the eyepiece, and is equal to the ratio of these two
quantities. Thus in Fig. 40 the distance of the objective from the focal
plane _F_ is about 16 times as great as the distance of the eyepiece
from the same plane, and the magnifying power of this telescope is
therefore 16 diameters. A magnifying power of 16 diameters means that
the diameter of any object seen in the telescope looks 16 times as large
as it appears without the telescope, and is nearly equivalent to saying
that the object appears only one sixteenth as far off. Sometimes the
magnifying power is assumed to be the number of times that the _area_ of
an object seems increased; and since areas are proportional to the
squares of lines, the magnifying power of 16 diameters might be called
a power of 256. Every large telescope is provided with several eyepieces
of different focal lengths, ranging from a quarter of an inch to two and
a half inches, which are used to furnish different magnifying powers as
may be required for the different kinds of work undertaken with the
instrument. Higher powers can be used with large telescopes than with
small ones, but it is seldom advantageous to use with any telescope an
eyepiece giving a higher power than 60 diameters for each inch of
diameter of the objective.

The part played by the eyepiece in determining magnifying power will be
readily understood from the following experiment:

Make a pin hole in a piece of cardboard. Bring a printed page so close
to one eye that you can no longer see the letters distinctly, and then
place the pin hole between the eye and the page. The letters which were
before blurred may now be seen plainly through the pin hole, even when
the page is brought nearer to the eye than before. As it is brought
nearer, notice how the letters seem to become larger, solely because
they are nearer. A pin hole is the simplest kind of a magnifier, and the
eyepiece in a telescope plays the same part as does the pin hole in the
experiment; it enables the eye to be brought nearer to the image, and
the shorter the focal length of the eyepiece the nearer is the eye
brought to the image and the higher is the magnifying power.

80. THE EQUATORIAL MOUNTING.--Telescopes are of all sizes, from the
modest opera glass which may be carried in the pocket and which requires
no other support than the hand, to the giant which must have a special
roof to shelter it and elaborate machinery to support and direct it
toward the sky. But for even the largest telescopes this machinery
consists of the following parts, which are illustrated, with exception
of the last one, in the small equatorial telescope shown in Fig. 41. It
is not customary to place a driving clock on so small a telescope as

(_a_) A supporting pier or tripod.

(_b_) An axis placed parallel to the axis of the earth.

(_c_) Another axis at right angles to _b_ and capable of revolving upon
_b_ as an axle.

(_d_) The telescope tube attached to _c_ and capable of revolving about

(_e_) Graduated circles attached to _c_ and _b_ to measure the amount by
which the telescope is turned on these axes.

(_f_) A driving clock so connected with _b_ as to make _c_ (and _d_)
revolve about _b_ with an angular velocity equal and opposite to that
with which the earth turns upon its axis.

[Illustration: FIG. 41.--A simple equatorial mounting.]

[Illustration: FIG. 42.--Equatorial mounting of the great telescope of
the Yerkes Observatory.]

Such a support is called an equatorial mounting, and the student should
note from the figure that the circles, _e_, measure the hour angle and
declination of any star toward which the telescope is directed, and
conversely if the telescope be so set that these circles indicate the
hour angle and declination of any given star, the telescope will then
point toward that star. In this way it is easy to find with the
telescope any moderately bright star, even in broad daylight, although
it is then absolutely invisible to the naked eye. The rotation of the
earth about its axis will speedily carry the telescope away from the
star, but if the driving clock be started, its effect is to turn the
telescope toward the west just as fast as the earth's rotation carries
it toward the east, and by these compensating motions to keep it
directed toward the star. In Fig. 42, which represents the largest and
one of the most perfect refracting telescopes ever built, let the
student pick out and identify the several parts of the mounting above
described. A part of the driving clock may be seen within the head of
the pier. In Fig. 43 trace out the corresponding parts in the mounting
of a reflecting telescope.

[Illustration: FIG. 43.--The reflecting telescope of the Paris

A telescope is often only a subordinate part of some instrument or
apparatus, and then its style of mounting is determined by the
requirements of the special case; but when the telescope is the chief
thing, and the remainder of the apparatus is subordinate to it, the
equatorial mounting is almost always adopted, although sometimes the
arrangement of the parts is very different in appearance from any of
those shown above. Beware of the popular error that an object held close
in front of a telescope can be seen by an observer at the eyepiece. The
numerous stories of astronomers who saw spiders crawling over the
objective of their telescope, and imagined they were beholding strange
objects in the sky, are all fictitious, since nothing on or near the
objective could possibly be seen through the telescope.

81. PHOTOGRAPHY.--A photographic camera consists of a lens and a device
for holding at its focus a specially prepared plate or film. This plate
carries a chemical deposit which is very sensitive to the action of
light, and which may be made to preserve the imprint of any picture
which the lens forms upon it. If such a sensitive plate is placed at the
focus of a reflecting telescope, the combination becomes a camera
available for astronomical photography, and at the present time the
tendency is strong in nearly every branch of astronomical research to
substitute the sensitive plate in place of the observer at a telescope.
A refracting telescope may also be used for astronomical photography,
and is very much used, but some complications occur here on account of
the resolution of the light into its constituent colors in passing
through the objective. Fig. 44 shows such a telescope, or rather two
telescopes, one photographic, the other visual, supported side by side
upon the same equatorial mounting.

[Illustration: FIG. 44.--Photographic telescope of the Paris

One of the great advantages of photography is found in connection with
what is called--

82. PERSONAL EQUATION.--It is a remarkable fact, first investigated by
the German astronomer Bessel, three quarters of a century ago, that
where extreme accuracy is required the human senses can not be
implicitly relied upon. The most skillful observers will not agree
exactly in their measurement of an angle or in estimating the exact
instant at which a star crossed the meridian; the most skillful artists
can not draw identical pictures of the same object, etc.

These minor deceptions of the senses are included in the term _personal
equation_, which is a famous phrase in astronomy, denoting that the
observations of any given person require to be corrected by means of
some equation involving his personality.

General health, digestion, nerves, fatigue, all influence the personal
equation, and it was in reference to such matters that one of the most
eminent of living astronomers has given this description of his habits
of observing:

"In order to avoid every physiological disturbance, I have adopted the
rule to abstain for one or two hours before commencing observations from
every laborious occupation; never to go to the telescope with stomach
loaded with food; to abstain from everything which could affect the
nervous system, from narcotics and alcohol, and especially from the
abuse of coffee, which I have found to be exceedingly prejudicial to the
accuracy of observation."[3] A regimen suggestive of preparation for an
athletic contest rather than for the more quiet labors of an astronomer.

  [3] Schiaparelli, Osservazioni sulle Stelle Doppie.

83. VISUAL AND PHOTOGRAPHIC WORK.--The photographic plate has no stomach
and no nerves, and is thus free from many of the sources of error which
inhere in visual observations, and in special classes of work it
possesses other marked advantages, such as rapidity when many stars are
to be dealt with simultaneously, permanence of record, and owing to the
cumulative effect of long exposure of the plate it is possible to
photograph with a given telescope stars far too faint to be seen through
it. On the other hand, the eye has the advantage in some respects, such
as studying the minute details of a fairly bright object--e. g., the
surface of a planet, or the sun's corona and, for the present at least,
neither method of observing can exclude the other. For a remarkable case
of discordance between the results of photographic and visual
observations compare the pictures of the great nebula in the
constellation Andromeda, which are given in Chapter XIV. A partial
explanation of these discordances and other similar ones is that the eye
is most strongly affected by greenish-yellow light, while the
photographic plate responds most strongly to violet light; the
photograph, therefore, represents things which the eye has little
capacity for seeing, and _vice versa_.

84. THE SPECTROSCOPE.--In some respects the spectroscope is the exact
counterpart of the telescope. The latter condenses radiant energy and
the former disperses it. As a measuring instrument the telescope is
mainly concerned with the direction from which light comes, and the
different colors of which that light is composed affect it only as an
obstacle to be overcome in its construction. On the other hand, with the
spectroscope the direction from which the radiant energy comes is of
minor consequence, and the all-important consideration is the intrinsic
character of that radiation. What colors are present in the light and in
what proportions? What can these colors be made to tell about the nature
and condition of the body from which they come, be it sun, or star, or
some terrestrial source of light, such as an arc lamp, a candle flame,
or a furnace in blast? These are some of the characteristic questions of
the spectrum analysis, and, as the name implies, they are solved by
analyzing the radiant energy into its component parts, setting down the
blue light in one place, the yellow in another, the red in still
another, etc., and interpreting this array of colors by means of
principles which we shall have to consider. Something of this process of
color analysis may be seen in the brilliant hues shown by a soap bubble,
or reflected from a piece of mother-of-pearl, and still more strikingly
exhibited in the rainbow, produced by raindrops which break up the
sunlight into its component colors and arrange them each in its
appropriate place. Any of these natural methods of decomposing light
might be employed in the construction of a spectroscope, but in
spectroscopes which are used for analyzing the light from feeble
sources, such as a star, or a candle flame, a glass prism of triangular
cross section is usually employed to resolve the light into its
component colors, which it does by refracting it as shown at the edges
of the lens in Fig. 38.

[Illustration: FIG. 45.--Resolution of light into its component colors.]

The course of a beam of light in passing through such a prism is shown
in Fig. 45. Note that the bending of the light from its original course
into a new one, which is here shown as produced by the prism, is quite
similar to the bending shown at the edges of a lens and comes from the
same cause, the slower velocity of light in glass than in air. It takes
the light-waves as long to move over the path _A B_ in glass as over the
longer path _1_, _2_, _3_, _4_, of which only the middle section lies in
the glass.

Not only does the prism bend the beam of light transmitted by it, but it
bends in different degree light of different colors, as is shown in the
figure, where the beam at the left of the prism is supposed to be made
up of a mixture of blue and red light, while at the right of the prism
the greater deviation imparted to the blue quite separates the colors,
so that they fall at different places on the screen, _S S_. The compound
light has been analyzed into its constituents, and in the same way every
other color would be put down at its appropriate place on the screen,
and a beam of white light falling upon the prism would be resolved by it
into a sequence of colors, falling upon the screen in the order red,
orange, yellow, green, blue, indigo, violet. The initial letters of
these names make the word _Roygbiv_, and by means of it their order is
easily remembered.

[Illustration: FIG. 46.--Principal parts of a spectroscope.]

If the light which is to be examined comes from a star the analysis made
by the prism is complete, and when viewed through a telescope the image
of the star is seen to be drawn out into a band of light, which is
called a _spectrum_, and is red at one end and violet or blue at the
other, with all the colors of the rainbow intervening in proper order
between these extremes. Such a prism placed in front of the objective of
a telescope is called an objective prism, and has been used for stellar
work with marked success at the Harvard College Observatory. But if the
light to be analyzed comes from an object having an appreciable extent
of surface, such as the sun or a planet, the objective prism can not be
successfully employed, since each point of the surface will produce its
own spectrum, and these will appear in the _view telescope_ superposed
and confused one with another in a very objectionable manner. To avoid
this difficulty there is placed between the prism and the source of
light an opaque screen, _S_, with a very narrow slit cut in it, through
which all the light to be analyzed must pass and must also go through a
lens, _A_, placed between the slit and the prism, as shown in Fig. 46.
The slit and lens, together with the tube in which they are usually
supported, are called a _collimator_. By this device a very limited
amount of light is permitted to pass from the object through the slit
and lens to the prism and is there resolved into a spectrum, which is in
effect a series of images of the slit in light of different colors,
placed side by side so close as to make practically a continuous ribbon
of light whose width is the length of each individual picture of the
slit. The length of the ribbon (dispersion) depends mainly upon the
shape of the prism and the kind of glass of which it is made, and it may
be very greatly increased and the efficiency of the spectroscope
enhanced by putting two, three, or more prisms in place of the single
one above described. When the amount of light is very great, as in the
case of the sun or an electric arc lamp, it is advantageous to alter
slightly the arrangement of the spectroscope and to substitute in place
of the prism a grating--i. e., a metallic mirror with a great number of
fine parallel lines ruled upon its surface at equal intervals, one from
another. It is by virtue of such a system of fine parallel grooves that
mother-of-pearl displays its beautiful color effects, and a brilliant
spectrum of great purity and high dispersion is furnished by a grating
ruled with from 10,000 to 20,000 lines to the inch. Fig. 47 represents,
rather crudely, a part of the spectrum of an arc light furnished by such
a grating, or rather it shows three different spectra arranged side by
side, and looking something like a rude ladder. The sides of the ladder
are the spectra furnished by the incandescent carbons of the lamp, and
the cross pieces are the spectrum of the electric arc filling the space
between the carbons. Fig. 48 shows a continuation of the same spectra
into a region where the radiant energy is invisible to the eye, but is
capable of being photographed.

[Illustration: FIG. 47.--Green and blue part of the spectrum of an
electric arc light.]

It is only when a lens is placed between the lamp and the slit of the
spectroscope that the three spectra are shown distinct from each other
as in the figure. The purpose of the lens is to make a picture of the
lamp upon the slit, so that all the radiant energy from any one point of
the arc may be brought to one part of the slit, and thus appear in the
resulting spectrum separated from the energy which comes from every
other part of the arc. Such an instrument is called an _analyzing
spectroscope_ while one without the lens is called an _integrating
spectroscope_, since it furnishes to each point of the slit a sample of
the radiant energy coming from every part of the source of light, and
thus produces only an average spectrum of that source without
distinction of its parts. When a spectroscope is attached to a
telescope, as is often done (see Fig. 49), the eyepiece is removed to
make way for it, and the telescope objective takes the part of the
analyzing lens. A camera is frequently combined with such an apparatus
to photograph the spectra it furnishes, and the whole instrument is then
called a _spectrograph_.

[Illustration: FIG. 48.--Violet and ultraviolet parts of spectrum of
an arc lamp.]

[Illustration: FIG. 49.--A spectroscope attached to the Yerkes

85. SPECTRUM ANALYSIS.--Having seen the mechanism of the spectroscope by
which the light incident upon it is resolved into its constituent parts
and drawn out into a series of colors arranged in the order of their
wave lengths, we have now to consider the interpretation which is to be
placed upon the various kinds of spectra which may be seen, and here we
rely upon the experience of physicists and chemists, from whom we learn
as follows:

The radiant energy which is analyzed by the spectroscope has its source
in the atoms and molecules which make up the luminous body from which
the energy is radiated, and these atoms and molecules are able to
impress upon the ether their own peculiarities in the shape of waves of
different length and amplitude. We have seen that by varying the
conditions of the experiment different kinds of waves may be produced in
a bucket of water; and as a study of these waves might furnish an index
to the conditions which produced them, so the study of the waves
peculiar to the light which comes from any source may be made to give
information about the molecules which make up that source. Thus the
molecules of iron produce a system of waves peculiar to themselves and
which can be duplicated by nothing else, and every other substance gives
off its own peculiar type of energy, presenting a limited and definite
number of wave lengths dependent upon the nature and condition of its
molecules. If these molecules are free to behave in their own
characteristic fashion without disturbance or crowding, they emit light
of these wave lengths only, and we find in the spectrum a series of
bright lines, pictures of the slit produced by light of these particular
wave lengths, while between these bright lines lie dark spaces showing
the absence from the radiant energy of light of intermediate wave
lengths. Such a spectrum is shown in the central portion of Fig. 47,
which, as we have already seen, is produced by the space between the
carbons of the arc lamp. On the other hand, if the molecules are closely
packed together under pressure they so interfere with each other as to
give off a jumble of energy of all wave lengths, and this is translated
by the spectroscope into a continuous ribbon of light with no dark
spaces intervening, as in the upper and lower parts of Figs. 47 and 48,
produced by the incandescent solid carbons of the lamp. These two types
are known as the continuous and discontinuous spectrum, and we may lay
down the following principle regarding them:

A discontinuous spectrum, or bright-line spectrum as it is familiarly
called, indicates that the molecules of the source of light are not
crowded together, and therefore the light must come from an incandescent
gas. A continuous spectrum shows only that the molecules are crowded
together, or are so numerous that the body to which they belong is not
transparent and gives no further information. The body may be solid,
liquid, or gaseous, but in the latter case the gas must be under
considerable pressure or of great extent.

A second principle is: The lines which appear in a spectrum are
characteristic of the source from which the light came--e. g., the
double line in the yellow part of the spectrum at the extreme left in
Fig. 47 is produced by sodium vapor in and around the electric arc and
is never produced by anything but sodium. When by laboratory experiments
we have learned the particular set of lines corresponding to iron, we
may treat the presence of these lines in another spectrum as proof that
iron is present in the source from which the light came, whether that
source be a white-hot poker in the next room or a star immeasurably
distant. The evidence that iron is present lies in the nature of the
light, and there is no reason to suppose that nature to be altered on
the way from star to earth. It may, however, be altered by something
happening to the source from which it comes--e. g., changing temperature
or pressure may affect, and does affect, the spectrum which such a
substance as iron emits, and we must be prepared to find the same
substance presenting different spectra under different conditions, only
these conditions must be greatly altered in order to produce radical
changes in the spectrum.

[Illustration: FIG. 50.--The chief lines in the spectrum of

86. WAVE LENGTHS.--To identify a line as belonging to and produced by
iron or any other substance, its position in the spectrum--i. e., its
wave length--must be very accurately determined, and for the
identification of a substance by means of its spectrum it is often
necessary to determine accurately the wave lengths of many lines. A
complicated spectrum may consist of hundreds or thousands of lines, due
to the presence of many different substances in the source of light, and
unless great care is taken in assigning the exact position of these
lines in the spectrum, confusion and wrong identifications are sure to
result. For the measurement of the required wave length a tenth meter
(§ 75) is the unit employed, and a scale of wave lengths expressed in
this unit is presented in Fig. 50. The accuracy with which some of these
wave lengths are determined is truly astounding; a ten-billionth of an
inch! These numerical wave lengths save all necessity for referring to
the color of any part of the spectrum, and pictures of spectra for
scientific use are not usually printed in colors.

87. ABSORPTION SPECTRA.--There is another kind of spectrum, of greater
importance than either of those above considered, which is well
illustrated by the spectrum of sunlight (Fig. 50). This is a nearly
continuous spectrum crossed by numerous _dark_ lines due to absorption
of radiant energy in a comparatively cool gas through which it passes on
its way to the spectroscope. Fraunhofer, who made the first careful
study of spectra, designated some of the more conspicuous of these lines
by letters of the alphabet which are shown in the plate, and which are
still in common use as names for the lines, not only in the spectrum of
sunlight but wherever they occur in other spectra. Thus the double line
marked _D_, wave length 5893, falls at precisely the same place in the
spectrum as does the double (sodium) line which we have already seen in
the yellow part of the arc-light spectrum, which line is also called _D_
and bears a very intimate relation to the dark _D_ line of the solar

The student who has access to colored crayons should color one edge of
Fig. 50 in accordance with the lettering there given and, so far as
possible, he should make the transition from one color to the next a
gradual one, as it is in the rainbow.

Fig. 50 is far from being a complete representation of the spectrum of
sunlight. Not only does this spectrum extend both to the right and to
the left into regions invisible to the human eye, but within the limits
of the figure, instead of the seventy-five lines there shown, there are
literally thousands upon thousands of lines, of which only the most
conspicuous can be shown in such a cut as this.

The dark lines which appear in the spectrum of sunlight can, under
proper conditions, be made to appear in the spectrum of an arc light,
and Fig. 51 shows a magnified representation of a small part of such a
spectrum adjacent to the _D_ (sodium) lines. Down the middle of each of
these lines runs a black streak whose position (wave length) is
precisely that of the _D_ lines in the spectrum of sunlight, and whose
presence is explained as follows:

The very hot sodium vapor at the center of the arc gives off its
characteristic light, which, shining through the outer and cooler layers
of sodium vapor, is partially absorbed by these, resulting in a fine
dark line corresponding exactly in position and wave length to the
bright lines, and seen against these as a background, since the higher
temperature at the center of the arc tends to broaden the bright lines
and make them diffuse. Similarly the dark lines in the spectrum of the
sun (Fig. 50) point to the existence of a surrounding envelope of
relatively cool gases, which absorb from the sunlight precisely those
kinds of radiant energy which they would themselves emit if
incandescent. The resulting dark lines in the spectrum are to be
interpreted by the same set of principles which we have above applied to
the bright lines of a discontinuous spectrum, and they may be used to
determine the chemical composition of the sun, just as the bright lines
serve to determine the chemical elements present in the electric arc.
With reference to the mode of their formation, bright-line and dark-line
spectra are sometimes called respectively _emission_ and _absorption_

[Illustration: FIG. 51.--The lines reversed.]

88. TYPES OF SPECTRUM.--The sun presents by far the most complex
spectrum known, and Fig. 50 shows only a small number of the more
conspicuous lines which appear in it. Spectra of stars, _per contra_,
appear relatively simple, since their feeble light is insufficient to
bring out faint details. In Chapters XIII and XIV there are shown types
of the different kinds of spectra given by starlight, and these are to
be interpreted by the principles above established. Thus the spectrum of
the bright star β Aurigæ shows a continuous spectrum crossed by a few
heavy absorption lines which are known from laboratory experiments to be
produced only by hydrogen. There must therefore be an atmosphere of
relatively cool hydrogen surrounding this star. The spectrum of Pollux
is quite similar to that of the sun and is to be interpreted as showing
a physical condition similar to that of the sun, while the spectrum of α
Herculis is quite different from either of the others. In subsequent
chapters we shall have occasion to consider more fully these different
types of spectrum.

89. THE DOPPLER PRINCIPLE.--This important principle of the spectrum
analysis is most readily appreciated through the following experiment:

Listen to the whistle of a locomotive rapidly approaching, and observe
how the pitch changes and the note becomes more grave as the locomotive
passes by and commences to recede. During the approach of the whistle
each successive sound wave has a shorter distance to travel in coming to
the ear of the listener than had its predecessor, and in consequence the
waves appear to come in quicker succession, producing a higher note with
a correspondingly shorter wave length than would be heard if the same
whistle were blown with the locomotive at rest. On the other hand, the
wave length is increased and the pitch of the note lowered by the
receding motion of the whistle. A similar effect is produced upon the
wave length of light by a rapid change of distance between the source
from which it comes and the instrument which receives it, so that a
diminishing distance diminishes very slightly the wave length of every
line in the spectrum produced by the light, and an increasing distance
increases these wave lengths, and this holds true whether the change of
distance is produced by motion of the source of light or by motion of
the instrument which receives it.

This change of wave length is sometimes described by saying that when a
body is rapidly approaching, the lines of its spectrum are all displaced
toward the violet end of the spectrum, and are correspondingly displaced
toward the red end by a receding motion. The amount of this shifting,
when it can be measured, measures the velocity of the body along the
line of sight, but the observations are exceedingly delicate, and it is
only in recent years that it has been found possible to make them with
precision. For this purpose there is made to pass through the
spectroscope light from an artificial source which contains one or more
chemical elements known to be present in the star which is to be
observed, and the corresponding lines in the spectrum of this light and
in the spectrum of the star are examined to determine whether they
exactly match in position, or show, as they sometimes do, a slight
displacement, as if one spectrum had been slipped past the other. The
difficulty of the observations lies in the extremely small amount of
this slipping, which rarely if ever in the case of a moving star amounts
to one sixth part of the interval between the close parallel lines
marked _D_ in Fig. 50. The spectral lines furnished by the headlight of
a locomotive running at the rate of a hundred miles per hour would be
displaced by this motion less than one six-thousandth part of the space
between the _D_ lines, an amount absolutely imperceptible in the most
powerful spectroscope yet constructed. But many of the celestial bodies
have velocities so much greater than a hundred miles per hour that these
may be detected and measured by means of the Doppler principle.

90. OTHER INSTRUMENTS.--Other instruments of importance to the
astronomer, but of which only casual mention can here be made, are the
meridian-circle; the transit, one form of which is shown in Fig. 52, and
the zenith telescope, which furnish refined methods for making
observations similar in kind to those which the student has already
learned to make with plumb line and protractor; the sextant, which is
pre-eminently the sailor's instrument for finding the latitude and
longitude at sea, by measuring the altitudes of sun and stars above the
sea horizon; the heliometer, which serves for the very accurate
measurement of small angles, such as the angular distance between two
stars not more than one or two degrees apart; and the photometer, which
is used for measuring the amount of light received from the celestial

[Illustration: FIG. 52.--A combined transit instrument and zenith



made the observations of the moon which are indicated in Chapter III has
in hand data from which much may be learned about the earth's satellite.
Perhaps the most striking feature brought out by them is the motion of
the moon among the stars, always from west toward east, accompanied by
that endless series of changes in shape and brightness--new moon, first
quarter, full moon, etc.--whose successive stages we represent by the
words, the phase of the moon. From his own observation the student
should be able to verify, at least approximately, the following
statements, although the degree of numerical precision contained in some
of them can be reached only by more elaborate apparatus and longer study
than he has given to the subject:

A. The phase of the moon depends upon the distance apart of sun and moon
in the sky, new moon coming when they are together, and full moon when
they are as far apart as possible.

[Illustration: THE MOON, ONE DAY AFTER FIRST QUARTER. From a photograph
made at the Paris Observatory.]

B. The moon is essentially a round, dark body, giving off no light of
its own, but shining solely by reflected sunlight. The proof of this is
that whenever we see a part of the moon which is turned away from the
sun it looks dark--e. g., at new moon, sun and moon are in nearly the
same direction from us and we see little or nothing of the moon, since
the side upon which the sun shines is turned away from us. At full moon
the earth is in line between sun and moon, and we see, round and
bright, the face upon which the sun shines. At other phases, such as the
quarters, the moon turns toward the earth a part of its night hemisphere
and a part of its day hemisphere, but in general only that part which
belongs to the day side of the moon is visible and the peculiar curved
line which forms the boundary--the "ragged edge," or _terminator_, as it
is called, is the dividing line between day and night upon the moon.

A partial exception to what precedes is found for a few days after new
moon when the moon and sun are not very far apart in the sky, for then
the whole round disk of the moon may often be seen, a small part of it
brightly illuminated by the sun and the larger part feebly illuminated
by sunlight which fell first upon the earth and was by it reflected back
to the moon, giving the pleasing effect which is sometimes called the
old moon in the new moon's arms. The new moon--i. e., the part illumined
by the sun--usually appears to belong to a sphere of larger radius than
the old moon, but this is purely a trick played by the eyes of the
observer, and the effect disappears altogether in a telescope. Is there
any similar effect in the few days before new moon?

C. The moon makes the circuit of the sky from a given star around to the
same star again in a little more than 27 days (27.32166), but the
interval between successive new moons--i. e., from the sun around to the
sun again--is more than 29 days (29.53059). This last interval, which is
called a lunar month or _synodical_ month, indicates what we have
learned before--that the sun has changed its place among the stars
during the month, so that it takes the moon an extra two days to
overtake him after having made the circuit of the sky, just as it takes
the minute hand of a clock an extra 5 minutes to catch up with the hour
hand after having made a complete circuit of the dial.

D. Wherever the moon may be in the sky, it turns always the same face
toward the earth, as is shown by the fact that the dark markings which
appear on its surface stand always upon (nearly) the same part of its
disk. It does not always turn the same face toward the sun, for the
boundary line between the illumined and unillumined parts of the moon
shifts from one side to the other as the phase changes, dividing at each
moment day from night upon the moon and illustrating by its slow
progress that upon the moon the day and the month are of equal length
(29.5 terrestrial days), instead of being time units of different
lengths as with us.

[Illustration: FIG. 53.--Motion of moon and earth relative to the sun.]

92. THE MOON'S MOTION.--The student should compare the results of his
own observations, as well as the preceding section, with Fig. 53, in
which the lines with dates printed on them are all supposed to radiate
from the sun and to represent the direction from the sun of earth and
moon upon the given dates which are arbitrarily assumed for the sake of
illustration, any other set would do equally well. The black dots, small
and large, represent the moon revolving about the earth, but having the
circular path shown in Fig. 34 (ellipse) transformed by the earth's
forward motion into the peculiar sinuous line here shown. With respect
to both earth and sun, the moon's orbit deviates but little from a
circle, since the sinuous curve of Fig. 53 follows very closely the
earth's orbit around the sun and is almost identical with it. For
clearness of representation the distance between earth and moon in the
figure has been made ten times too great, and to get a proper idea of
the moon's orbit with reference to the sun, we must suppose the moon
moved up toward the earth until its distance from the line of the
earth's orbit is only a tenth part of what it is in the figure. When
this is done, the moon's path becomes almost indistinguishable from that
of the earth, as may be seen in the figure, where the attempt has been
made to show both lines, and it is to be especially noted that this
real orbit of the moon is everywhere concave toward the sun.

The phase presented by the moon at different parts of its path is
indicated by the row of circles at the right, and the student should
show why a new moon is associated with June 30th and a full moon with
July 15th, etc. What was the date of first quarter? Third quarter?

We may find in Fig. 53 another effect of the same kind as that noted
above in C. Between noon, June 30th, and noon, July 3d, the earth makes
upon its axis three complete revolutions with respect to the sun, but
the meridian which points toward the moon at noon on June 30th will not
point toward it at noon on July 3d, since the moon has moved into a new
position and is now 37° away from the meridian. Verify this statement by
measuring, in Fig. 53, with the protractor, the moon's angular distance
from the meridian at noon on July 3d. When will the meridian overtake
the moon?

93. HARVEST MOON.--The interval between two successive transits of the
meridian past the moon is called a lunar day, and the student should
show from the figure that on the average a lunar day is 51 minutes
longer than a solar day--i. e., upon the average each day the moon comes
to the meridian 51 minutes of solar time later than on the day before.
It is also true that on the average the moon rises and sets 51 minutes
later each day than on the day before. But there is a good deal of
irregularity in the retardation of the time of moonrise and moonset,
since the time of rising depends largely upon the particular point of
the horizon at which the moon appears, and between two days this point
may change so much on account of the moon's orbital motion as to make
the retardation considerably greater or less than its average value. In
northern latitudes this effect is particularly marked in the month of
September, when the eastern horizon is nearly parallel with the moon's
apparent path in the sky, and near the time of full moon in that month
the moon rises on several successive nights at nearly the same hour, and
in less degree the same is true for October. This highly convenient
arrangement of moonlight has caused the full moons of these two months
to be christened respectively the Harvest Moon and the Hunter's Moon.

94. SIZE AND MASS OF THE MOON.--It has been shown in Chapter I how the
distance of the moon from the earth may be measured and its diameter
determined by means of angles, and without enlarging upon the details of
these observations, we note as their result that the moon is a globe
2,163 miles in diameter, and distant from the earth on the average about
240,000 miles. But, as we have seen in Chapter VII, this distance
changes to the extent of a few thousand miles, sometimes less, sometimes
greater, mainly on account of the elliptic shape of the moon's orbit
about the earth, but also in part from the disturbing influence of other
bodies, such as the sun, which pull the moon to and fro, backward and
forward, to quite an appreciable extent.

From the known diameter of the moon it is a matter of elementary
geometry to derive in miles the area of its surface and its volume or
solid contents. Leaving this as an exercise for the student, we adopt
the earth as the standard of comparison and find that the diameter of
the moon is rather more than a quarter, 4/15, that of the earth, the
area of its surface is a trifle more than 1/14 that of the earth, and
its volume a little more than 1/49 of the earth's. So much is pure
geometry, but we may combine with it some mechanical principles which
enable us to go a step farther and to "weigh" the moon--i. e., determine
its mass and the average density of the material of which it is made.

We have seen that the moon moves around the sun in a path differing but
little from the smooth curve shown in Fig. 53, with arrows indicating
the direction of motion, and it would follow absolutely such a smooth
path were it not for the attraction of the earth, and in less degree of
some of the other planets, which swing it about first to one side then
to the other. But action and reaction are equal; the moon pulls as
strongly upon the earth as does the earth upon the moon, and if earth
and moon were of equal mass, the deviation of the earth from the smooth
curve in the figure would be just as large as that of the moon. It is
shown in the figure that the moon does displace the earth from this
curve, and we have only to measure the amount of this displacement of
the earth and compare it with the displacement suffered by the moon to
find how much the mass of the one exceeds that of the other. It may be
seen from the figure that at first quarter, about July 7th, the earth is
thrust ahead in the direction of its orbital motion, while at the third
quarter, July 22d, it is pulled back by the action of the moon, and at
all times it is more or less displaced by this action, so that, in order
to be strictly correct, we must amend our former statement about the
moon moving around the earth and make it read, Both earth and moon
revolve around a point on line between their centers. This point is
called their _center of gravity_, and the earth and the moon both move
in ellipses having this center of gravity at their common focus. Compare
this with Kepler's First Law. These ellipses are similarly shaped, but
of very different size, corresponding to Newton's third law of motion
(Chapter IV), so that the action of the earth in causing the small moon
to move around a large orbit is just equal to the reaction of the moon
in causing the larger earth to move in the smaller orbit. This is
equivalent to saying that the dimensions of the two orbits are inversely
proportional to the masses of the earth and the moon.

By observing throughout the month the direction from the earth to the
sun or to a near planet, such as Mars or Venus, astronomers have
determined that the diameter of the ellipse in which the earth moves is
about 5,850 miles, so that the distance of the earth from the center of
gravity is 2,925 miles, and the distance of the moon from it is
240,000-2,925 = 237,075. We may now write in the form of a proportion--

    Mass of earth : Mass of moon :: 237,075 : 2,925,

and find from it that the mass of the earth is 81 times as great as the
mass of the moon--i. e., leaving kind and quality out of account, there
is enough material in the earth to make 81 moons. We may note in this
connection that the diameter of the earth, 7,926 miles, is greater than
the diameter of the monthly orbit in which the moon causes it to move,
and therefore the center of gravity of earth and moon always lies inside
the body of the earth, about 1,000 miles below the surface.

95. DENSITY OF THE MOON.--It is believed that in a general way the moon
is made of much the same kind of material which goes to make up the
earth--metals, minerals, rocks, etc.--and a part of the evidence upon
which this belief is based lies in the density of the moon. By density
of a substance we mean the amount of it which is contained in a given
volume--i. e., the weight of a bushel or a cubic centimeter of the
stuff. The density of chalk is twice as great as the density of water,
because a cubic centimeter of chalk weighs twice as much as an equal
volume of water, and similarly in other cases the density is found by
dividing the mass or weight of the body by the mass or weight of an
equal volume of water.

We know the mass of the earth (§ 45), and knowing the mass of a cubic
foot of water, it is easy, although a trifle tedious, to compute what
would be the mass of a volume of water equal in size to the earth. The
quotient obtained by dividing one of these masses by the other (mass of
earth ÷ mass of water) is the average density of the material composing
the earth, and we find numerically that this is 5.6--i. e., it would
take 5.6 water earths to attract as strongly as does the real one. From
direct experiment we know that the average density of the principal
rocks which make up the crust of the earth is only about half of this,
showing that the deep-lying central parts of the earth are denser than
the surface parts, as we should expect them to be, because they have to
bear the weight of all that lies above them and are compressed by it.

Turning now to the moon, we find in the same way as for the earth that
its average density is 3.4 as great as that of water.

96. FORCE OF GRAVITY UPON THE MOON.--This number, 3.4, compared with the
5.6 which we found for the earth, shows that on the whole the moon is
made of lighter stuff than is the body of the earth, and this again is
much what we should expect to find, for weight, the force which tends to
compress the substance of the moon, is less there than here. The weight
of a cubic yard of rock at the surface of either earth or moon is the
force with which the earth or moon attracts it, and this by the law of
gravitation is for the earth--

    W = k × (m m´) / (3963)^{2};

and for the moon--

    w = k × {m (m´/81)} / (1081)^{2};

from which we find by division--

    w = (W / 81) (3963 / 1081)^{2} = (W / 6) (approximately).

The cubic yard of rock, which upon the earth weighs two tons, would, if
transported to the moon, weigh only one third of a ton, and would have
only one sixth as much influence in compressing the rocks below it as it
had upon the earth. Note that this rock when transported to the moon
would be still attracted by the earth and would have weight toward the
earth, but it is not this of which we are speaking; by its weight in
the moon we mean the force with which the moon attracts it. Making due
allowance for the difference in compression produced by weight, we may
say that in general, so far as density goes, the moon is very like a
piece of the earth of equal mass set off by itself alone.

97. ALBEDO.--In another respect the lunar stuff is like that of which
the earth is made: it reflects the sunlight in much the same way and to
the same amount. The contrast of light and dark areas on the moon's
surface shows, as we shall see in another section, the presence of
different substances upon the moon which reflect the sunlight in
different degrees. This capacity for reflecting a greater or less
percentage of the incident sunlight is called _albedo_ (Latin,
whiteness), and the brilliancy of the full moon might lead one to
suppose that its albedo is very great, like that of snow or those masses
of summer cloud which we call thunderheads. But this is only an effect
of contrast with the dark background of the sky. The same moon by day
looks pale, and its albedo is, in fact, not very different from that of
our common rocks--weather-beaten sandstone according to Sir John
Herschel--so that it would be possible to build an artificial moon of
rock or brick which would shine in the sunlight much as does the real

The effect produced by the differences of albedo upon the moon's face is
commonly called the "man in the moon," but, like the images presented by
glowing coals, the face in the moon is anything which we choose to make
it. Among the Chinese it is said to be a monkey pounding rice; in India,
a rabbit; in Persia, the earth reflected as in a mirror, etc.

98. LIBRATIONS.--We have already learned that the moon turns always the
same face toward the earth, and we have now to modify this statement and
to find that here, as in so many other cases, the thing we learn first
is only approximately true and needs to be limited or added to or
modified in some way. In general, Nature is too complex to be completely
understood at first sight or to be perfectly represented by a simple
statement. In Fig. 55 we have two photographs of the moon, taken nearly
three years apart, the right-hand one a little after first quarter and
the left-hand one a little before third quarter. They therefore
represent different parts of the moon's surface, but along the ragged
edge the same region is shown on both photographs, and features common
to both pictures may readily be found--e. g., the three rings which form
a right-angled triangle about one third of the way down from the top of
the cut, and the curved mountain chain just below these. If the moon
turned exactly the same face toward us in the two pictures, the distance
of any one of these markings from any part of the moon's edge must be
the same in both pictures; but careful measurement will show that this
is not the case, and that in the left-hand picture the upper edge of the
moon is tipped toward us and the lower edge away from us, as if the
whole moon had been rotated slightly about a horizontal line and must be
turned back a little (about 7°) in order to match perfectly the other
part of the picture.

This turning is called a _libration_, and it should be borne in mind
that the moon librates not only in the direction above measured, north
and south, but also at right angles to this, east and west, so that we
are able to see a little farther around every part of the moon's edge
than would be possible if it turned toward us at all times exactly the
same face. But in spite of the librations there remains on the farther
side of the moon an area of 6,000,000 square miles which is forever
hidden from us, and of whose character we have no direct knowledge,
although there is no reason to suppose it very different from that which
is visible, despite the fact that some of the books contain quaint
speculations to the contrary. The continent of South America is just
about equal in extent to this unknown region, while North America is a
fair equivalent for all the rest of the moon's surface, both those
central parts which are constantly visible, and the zone around the edge
whose parts sometimes come into sight and are sometimes hidden.

An interesting consequence of the peculiar rotation of the moon is that
from our side of it the earth is always visible. Sun, stars, and planets
rise and set there as well as here, but to an observer on the moon the
earth swings always overhead, shifting its position a few degrees one
way or the other on account of the libration but running through its
succession of phases, new earth, first quarter, etc., without ever going
below the horizon, provided the observer is anywhere near the center of
the moon's disk.

[Illustration: FIG. 54.--Illustrating the moon's rotation.]

99. CAUSE OF LIBRATIONS.--That the moon should librate is by no means so
remarkable a fact as that it should at all times turn very nearly the
same face toward the earth. This latter fact can have but one meaning:
the moon revolves about an axis as does the earth, but the time required
for this revolution is just equal to the time required to make a
revolution in its orbit. Place two coins upon a table with their heads
turned toward the north, as in Fig. 54, and move the smaller one around
the larger in such a way that its face shall always look away from the
larger one. In making one revolution in its orbit the head on this small
coin will be successively directed toward every point of the compass,
and when it returns to its initial position the small coin will have
made just one revolution about an axis perpendicular to the plane of its
orbit. In no other way can it be made to face always away from the
figure at the center of its orbit while moving around it.

We are now in a position to understand the moon's librations, for, if
the small coin at any time moves faster or slower in its orbit than it
turns about its axis, a new side will be turned toward the center, and
the same may happen if the central coin itself shifts into a new
position. This is what happens to the moon, for its orbital motion, like
that of Mercury (Fig. 17), is alternately fast and slow, and in addition
to this there are present other minor influences, such as the fact that
its rotation axis is not exactly perpendicular to the plane of its
orbit; in addition to this the observer upon the earth is daily carried
by its rotation from one point of view to another, etc., so that it is
only in a general way that the rotation upon the axis and motion in the
orbit keep pace with each other. In a general way a cable keeps a ship
anchored in the same place, although wind and waves may cause it to
"librate" about the anchor.

How the moon came to have this exact equality between its times of
revolution and rotation constitutes a chapter of its history upon which
we shall not now enter; but the equality having once been established,
the mechanism by which it is preserved is simple enough.

The attraction of the earth for the moon has very slightly pulled the
latter out of shape (§ 42), so that the particular diameter, which
points toward the earth, is a little longer than any other, and thus
serves as a handle which the earth lays hold of and pulls down into its
lowest possible position--i. e., the position in which it points toward
the center of the earth. Just how long this handle is, remains unknown,
but it may be shown from the law of gravitation that less than a hundred
yards of elongation would suffice for the work it has to do.

100. THE MOON AS A WORLD.--Thus far we have considered the moon as a
satellite of the earth, dependent upon the earth, and interesting
chiefly because of its relation to it. But the moon is something more
than this; it is a world in itself, very different from the earth,
although not wholly unlike it. The most characteristic feature of the
earth's surface is its division into land and water, and nothing of this
kind can be found upon the moon. It is true that the first generation of
astronomers who studied the moon with telescopes fancied that the large
dark patches shown in Fig. 55 were bodies of water, and named them
oceans, seas, lakes, and ponds, and to the present day we keep those
names, although it is long since recognized that these parts of the
moon's surface are as dry as any other. Their dark appearance indicates
a different kind of material from that composing the lighter parts of
the moon, material with a different albedo, just as upon the earth we
have light-colored and dark-colored rocks, marble and slate, which seen
from the moon must present similar contrasts of brightness. Although
these dark patches are almost the only features distinguishable with the
unaided eye, it is far otherwise in the telescope or the photograph,
especially along the ragged edge where great numbers of rings can be
seen, which are apparently depressions in the moon and are called
craters. These we find in great number all over the moon, but, as the
figure shows, they are seen to the best advantage near the
_terminator_--i. e., the dividing line between day and night, since the
long shadows cast here by the rising or setting sun bring out the
details of the surface better than elsewhere. Carefully examine Fig. 55
with reference to these features.

[Illustration: FIG. 55.--The moon at first and last quarter. Lick
Observatory photographs.]

Another feature which exists upon both earth and moon, although far less
common there than here, is illustrated in the chain of mountains visible
near the terminator, a little above the center of the moon in both parts
of Fig. 55. This particular range of mountains, which is called the
Lunar Apennines, is by far the most prominent one upon the moon,
although others, the Alps and Caucasus, exist. But for the most part the
lunar mountains stand alone, each by itself, instead of being grouped
into ranges, as on the earth. Note in the figure that some of the lunar
mountains stretch out into the night side of the moon, their peaks
projecting up into the sunlight, and thus becoming visible, while the
lowlands are buried in the shadow.

A subordinate feature of the moon's surface is the system of _rays_
which seem to radiate like spokes from some of the larger craters,
extending over hill and valley sometimes for hundreds of miles. A
suggestion of these rays may be seen in Fig. 55, extending from the
great crater Copernicus a little southwest of the end of the Apennines,
but their most perfect development is to be seen at the time of full
moon around the crater Tycho, which lies near the south pole of the
moon. Look for them with an opera glass.

Another and even less conspicuous feature is furnished by the rills,
which, under favorable conditions of illumination, appear like long
cracks on the moon's surface, perhaps analogous to the cañons of our
Western country.

101. THE MAP OF THE MOON.--Fig. 55 furnishes a fairly good map of a
limited portion of the moon near the terminator, but at the edges little
or no detail can be seen. This is always true; the whole of the moon can
not be seen to advantage at any one time, and to remedy this we need to
construct from many photographs or drawings a map which shall represent
the several parts of the moon as they appear at their best. Fig. 56
shows such a map photographed from a relief model of the moon, and
representing the principal features of the lunar surface in a way they
can never be seen simultaneously. Perhaps its most striking feature is
the shape of the craters, which are shown round in the central parts of
the map and oval at the edges, with their long diameters parallel to the
moon's edge. This is, of course, an effect of the curvature of the
moon's surface, for we look very obliquely at the edge portions, and
thus see their formations much foreshortened in the direction of the
moon's radius.

[Illustration: FIG. 56.--Relief map of the moon's surface.--After

The north and south poles of the moon are at the top and bottom of the
map respectively, and a mere inspection of the regions around them will
show how much more rugged is the southern hemisphere of the moon than
the northern. It furnishes, too, some indication of how numerous are the
lunar craters, and how in crowded regions they overlap one another.

The student should pick out upon the map those features which he has
learned to know in the photograph (Fig. 55)--the Apennines, Copernicus,
and the continuation of the Apennines, extending into the dark part of
the moon.

[Illustration: FIG. 57.--Mare Imbrium. Photographed by G. W. RITCHEY.]

102. SIZE OF THE LUNAR FEATURES.--We may measure distances here in the
same way as upon a terrestrial map, remembering that near the edges the
scale of the map is very much distorted parallel to the moon's diameter,
and measurements must not be taken in this direction, but may be taken
parallel to the edge. Measuring with a millimeter scale, we find on the
map for the diameter of the crater Copernicus, 2.1 millimeters. To turn
this into the diameter of the real Copernicus in miles, we measure upon
the same map the diameter of the moon, 79.7 millimeters, and then have
the proportion--

    Diameter of Copernicus in miles : 2,163 :: 2.1 : 79.7,

which when solved gives 57 miles. The real diameter of Copernicus is a
trifle over 56 miles. At the eastern edge of the moon, opposite the
Apennines, is a large oval spot called the Mare Crisium (Latin, _ma-re_
= sea). Measure its length. The large crater to the northwest of the
Apennines is called Archimedes. Measure its diameter both in the map and
in the photograph (Fig. 55), and see how the two results agree. The true
diameter of this crater, east and west, is very approximately 50 miles.
The great smooth surface to the west of Archimedes is the Mare Imbrium.
Is it larger or smaller than Lake Superior? Fig. 57 is from a photograph
of the Mare Imbrium, and the amount of detail here shown at the bottom
of the sea is a sufficient indication that, in this case at least, the
water has been drawn off, if indeed any was ever present.

[Illustration: FIG. 58.--Mare Crisium. Lick Observatory photographs.]

Fig. 58 is a representation of the Mare Crisium at a time when night was
beginning to encroach upon its eastern border, and it serves well to
show the rugged character of the ring-shaped wall which incloses this

With these pictures of the smoother parts of the moon's surface we may
compare Fig. 59, which shows a region near the north pole of the moon,
and Fig. 60, giving an early morning view of Archimedes and the
Apennines. Note how long and sharp are the shadows.

[Illustration: FIG. 59.--Illustrating the rugged character of the moon's
surface.--NASMYTH and CARPENTER.]

103. THE MOON'S ATMOSPHERE.--Upon the earth the sun casts no shadows so
sharp and black as those of Fig. 60, because his rays are here scattered
and reflected in all directions by the dust and vapors of the
atmosphere (§ 51), so that the place from which direct sunlight is cut
off is at least partially illumined by this reflected light. The shadows
of Fig. 60 show that upon the moon it must be otherwise, and suggest
that if the moon has any atmosphere whatever, its density must be
utterly insignificant in comparison with that of the earth. In its
motion around the earth the moon frequently eclipses stars (_occults_ is
the technical word), and if the moon had an atmosphere such as is shown
in Fig. 61, the light from the star _A_ must shine through this
atmosphere just before the moon's advancing body cuts it off, and it
must be refracted by the atmosphere so that the star would appear in a
slightly different direction (nearer to _B_) than before. The earth's
atmosphere refracts the starlight under such circumstances by more than
a degree, but no one has been able to find in the case of the moon any
effect of this kind amounting to even a fraction of a second of arc.
While this hardly justifies the statement sometimes made that the moon
has no atmosphere, we shall be entirely safe in saying that if it has
one at all its density is less than a thousandth part of that of the
earth's atmosphere. Quite in keeping with this absence of an atmosphere
is the fact that clouds never float over the surface of the moon. Its
features always stand out hard and clear, without any of that haze and
softness of outline which our atmosphere introduces into all terrestrial

[Illustration: FIG. 60.--Archimedes and Apennines. NASMYTH and

104. HEIGHT OF THE LUNAR MOUNTAINS.--Attention has already been called
to the detached mountain peaks, which in Fig. 55 prolong the range of
Apennines into the lunar night. These are the beginnings of the Caucasus
mountains, and from the photograph we may measure as follows the height
to which they rise above the surrounding level of the moon: Fig. 62
represents a part of the lunar surface along the boundary line between
night and day, the horizontal line at the top of the figure representing
a level ray of sunlight which just touches the moon at _T_ and barely
illuminates the top of the mountain, _M_, whose height, _h_, is to be
determined. If we let _R_ stand for the radius of the moon and _s_ for
the distance, _T M_, we shall have in the right-angled triangle _M T C_,

    R^{2} + s^{2} = (R + h)^{2},

and we need only to measure _s_--that is, the distance from the
terminator to the detached mountain peak--to make this equation
determine _h_, since _R_ is already known, being half the diameter of
the moon--1,081 miles. Practically it is more convenient to use instead
of this equation another form, which the student who is expert in
algebra may show to be very nearly equivalent to it:

        _h_ (miles) = s^{2} / 2163,
    or  _h_ (feet) = 2.44 s^{2}.

The distance _s_ must be expressed in miles in all of these equations.
In Fig. 55 the distance from the terminator to the first detached peak
of the Caucasus mountains is 1.7 millimeters = 52 miles, from which we
find the height of the mountain to be 1.25 miles, or 6,600 feet.

[Illustration: FIG. 61.--Occultations and the moon's atmosphere.]

[Illustration: FIG. 62.--Determining the height of a lunar mountain.]

Two things, however, need to be borne in mind in this connection. On the
earth we measure the heights of mountains _above sea level_, while on
the moon there is no sea, and our 6,600 feet is simply the height of the
mountain top above the level of that particular point in the terminator,
from which we measure its distance. So too it is evident from the
appearance of things, that the sunlight, instead of just touching the
top of the particular mountain whose height we have measured, really
extends some little distance down from its summit, and the 6,600 feet is
therefore the elevation of the lowest point on the mountains to which
the sunlight reaches. The peak itself may be several hundred feet
higher, and our photograph must be taken at the exact moment when this
peak appears in the lunar morning or disappears in the evening if we are
to measure the altitude of the mountain's summit. Measure the height of
the most northern visible mountain of the Caucasus range. This is one of
the outlying spurs of the great mountain Calippus, whose principal peak,
19,000 feet high, is shown in Fig. 55 as the brightest part of the
Caucasus range.

The highest peak of the lunar Apennines, Huyghens, has an altitude of
18,000 feet, and the Leibnitz and Doerfel Mountains, near the south pole
of the moon, reach an altitude 50 per cent greater than this, and are
probably the highest peaks on the moon. This falls very little short of
the highest mountain on the earth, although the moon is much smaller
than the earth, and these mountains are considerably higher than
anything on the western continent of the earth.

The vagueness of outline of the terminator makes it difficult to measure
from it with precision, and somewhat more accurate determinations of the
heights of lunar mountains can be obtained by measuring the length of
the shadows which they cast, and the depths of craters may also be
measured by means of the shadows which fall into them.

105. CRATERS.--Fig. 63 shows a typical lunar crater, and conveys a good
idea of the ruggedness of the lunar landscape. Compare the appearance of
this crater with the following generalizations, which are based upon the
accurate measurement of many such:

A. A crater is a real depression in the surface of the moon, surrounded
usually by an elevated ring which rises above the general level of the
region outside, while the bottom of the crater is about an equal
distance below that level.

B. Craters are shallow, their diameters ranging from five times to more
than fifty times their depth. Archimedes, whose diameter we found to be
50 miles, has an average depth of about 4,000 feet below the crest of
its surrounding wall, and is relatively a shallow crater.

[Illustration: FIG. 63.--A typical lunar crater.--NASMYTH and

C. Craters frequently have one or more hills rising within them which,
however, rarely, if ever, reach up to the level of the surrounding wall.

D. Whatever may have been the mode of their formation, the craters can
not have been produced by scooping out material from the center and
piling it up to make the wall, for in three cases out of four the volume
of the excavation is greater than the volume of material contained in
the wall.

106. MOON AND EARTH.--We have gone far enough now to appreciate both the
likeness and the unlikeness of the moon and earth. They may fairly
enough be likened to offspring of the same parent who have followed very
different careers, and in the fullness of time find themselves in very
different circumstances. The most serious point of difference in these
circumstances is the atmosphere, which gives to the earth a wealth of
phenomena altogether lacking in the moon. Clouds, wind, rain, snow,
dew, frost, and hail are all dependent upon the atmosphere and can not
be found where it is not. There can be nothing upon the moon at all like
that great group of changes which we call weather, and the unruffled
aspect of the moon's face contrasts sharply with the succession of cloud
and sunshine which the earth would present if seen from the moon.

The atmosphere is the chief agent in the propagation of sound, and
without it the moon must be wrapped in silence more absolute than can be
found upon the surface of the earth. So, too, the absence of an
atmosphere shows that there can be no water or other liquid upon the
moon, for if so it would immediately evaporate and produce a gaseous
envelope which we have seen does not exist. With air and water absent
there can be of course no vegetation or life of any kind upon the moon,
and we are compelled to regard it as an arid desert, utterly waste.

107. TEMPERATURE OF THE MOON.--A characteristic feature of terrestrial
deserts, which is possessed in exaggerated degree by the moon, is the
great extremes of temperature to which they and it are subject. Owing to
its slow rotation about its axis, a point on the moon receives the solar
radiation uninterruptedly for more than a fortnight, and that too
unmitigated by any cloud or vaporous covering. Then for a like period it
is turned away from the sun and allowed to cool off, radiating into
interplanetary space without hindrance its accumulated store of heat. It
is easy to see that the range of temperature between day and night must
be much greater under these circumstances than it is with us where
shorter days and clouded skies render day and night more nearly alike,
to say nothing of the ocean whose waters serve as a great balance wheel
for equalizing temperatures. Just how hot or how cold the moon becomes
is hard to determine, and very different estimates are to be found in
the books. Perhaps the most reliable of these are furnished by the
recent researches of Professor Very, whose experiments lead him to
conclude that "its rocky surface at midday, in latitudes where the sun
is high, is probably hotter than boiling water and only the most
terrible of earth's deserts, where the burning sands blister the skin,
and men, beasts, and birds drop dead, can approach a noontide on the
cloudless surface of our satellite. Only the extreme polar latitudes of
the moon can have an endurable temperature by day, to say nothing of the
night, when we should have to become troglodytes to preserve ourselves
from such intense cold."

While the night temperature of the moon, even very soon after sunset,
sinks to something like 200° below zero on the centigrade scale, or 320°
below zero on the Fahrenheit scale, the lowest known temperature upon
the earth, according to General Greely, is 90° Fahr. below zero,
recorded in Siberia in January, 1885.

Winter and summer are not markedly different upon the moon, since its
rotation axis is nearly perpendicular to the plane of the earth's orbit
about the sun, and the sun never goes far north or south of the moon's
equator. The month is the one cycle within which all seasonal changes in
its physical condition appear to run their complete course.

108. CHANGES IN THE MOON.--It is evidently idle to look for any such
changes in the condition of the moon's surface as with us mark the
progress of the seasons or the spread of civilization over the
wilderness. But minor changes there may be, and it would seem that the
violent oscillations of temperature from day to night ought to have some
effect in breaking down and crumbling the sharp peaks and crags which
are there so common and so pronounced. For a century past astronomers
have searched carefully for changes of this kind--the filling up of some
crater or the fall of a mountain peak; but while some things of this
kind have been reported from time to time, the evidence in their behalf
has not been altogether conclusive. At the present time it is an open
question whether changes of this sort large enough to be seen from the
earth are in progress. A crater much less than a mile wide can be seen
in the telescope, but it is not easy to tell whether so minute an object
has changed in size or shape during a year or a decade, and even if
changes are seen they may be apparent rather than real. Fig. 64 contains
two views of the crater Archimedes, taken under a morning and an
afternoon sun respectively, and shows a very pronounced difference
between the two which proceeds solely from a difference of illumination.
In the presence of such large fictitious changes astronomers are slow to
accept smaller ones as real.

[Illustration: FIG. 64.--Archimedes in the lunar morning and

It is this absence of change that is responsible for the rugged and
sharp-cut features of the moon which continue substantially as they were
made, while upon the earth rain and frost are continually wearing down
the mountains and spreading their substance upon the lowland in an
unending process of smoothing off the roughnesses of its surface. Upon
the moon this process is almost if not wholly wanting, and the moon
abides to-day much more like its primitive condition than is the earth.

109. THE MOON'S INFLUENCE UPON THE EARTH.--There is a widespread popular
belief that in many ways the moon exercises a considerable influence
upon terrestrial affairs: that it affects the weather for good or ill,
that crops must be planted and harvested, pigs must be killed, and
timber cut at the right time of the moon, etc. Our common word lunatic
means moonstruck--i. e., one upon whom the moon has shone while
sleeping. There is not the slightest scientific basis for any of these
beliefs, and astronomers everywhere class them with tales of witchcraft,
magic, and popular delusion. For the most part the moon's influence upon
the earth is limited to the light which it sends and the effect of its
gravitation, chiefly exhibited in the ocean tides. We receive from the
moon a very small amount of second-hand solar heat and there is also a
trifling magnetic influence, but neither of these last effects comes
within the range of ordinary observation, and we shall not go far wrong
in saying that, save the moonlight and the tides, every supposed lunar
influence upon the earth is either fictitious or too small to be readily



110. DEPENDENCE OF THE EARTH UPON THE SUN.--There is no better
introduction to the study of the sun than Byron's Ode to Darkness,
beginning with the lines--

    "I dreamed a dream
    That was not all a dream.
    The bright sun was extinguished,"

and proceeding to depict in vivid words the consequences of this
extinction. The most matter-of-fact language of science agrees with the
words of the poet in declaring the earth's dependence upon the sun for
all those varied forms of energy which make it a fit abode for living
beings. The winds blow and the rivers run; the crops grow, are gathered
and consumed, by virtue of the solar energy. Factory, locomotive, beast,
bird, and the human body furnish types of machines run by energy derived
from the sun; and the student will find it an instructive exercise to
search for kinds of terrestrial energy which are not derived either
directly or indirectly from the sun. There are a few such, but they are
neither numerous nor important.

111. THE SUN'S DISTANCE FROM THE EARTH.--To the astronomer the sun
presents problems of the highest consequence and apparently of very
diverse character, but all tending toward the same goal: the framing of
a mechanical explanation of the sun considered as a machine; what it is,
and how it does its work. In the forefront of these problems stand those
numerical determinations of distance, size, mass, density, etc., which
we have already encountered in connection with the moon, but which must
here be dealt with in a different manner, because the immensely greater
distance of the sun makes impossible the resort to any such simple
method as the triangle used for determining the moon's distance. It
would be like determining the distance of a steeple a mile away by
observing its direction first from one eye, then from the other; too
short a base for the triangle. In one respect, however, we stand upon a
better footing than in the case of the moon, for the mass of the earth
has already been found (Chapter IV) as a fractional part of the sun's
mass, and we have only to invert the fraction in order to find that the
sun's mass is 329,000 times that of the earth and moon combined, or
333,000 times that of the earth alone.

If we could rely implicitly upon this number we might make it determine
for us the distance of the sun through the law of gravitation as
follows: It was suggested in § 38 that Newton proved Kepler's three laws
to be imperfect corollaries from the law of gravitation, requiring a
little amendment to make them strictly correct, and below we give in the
form of an equation Kepler's statement of the Third Law together with
Newton's amendment of it. In these equations--

_T_ = Periodic time of any planet;

_a_ = One half the major axis of its orbit;

_m_ = Its mass;

_M_ = The mass of the sun;

_k_ = The gravitation constant corresponding to the particular set of
units in which _T_, _a_, _m_, and _M_ are expressed.

  (Kepler) a^{3}/T^{2} = h;
  (Newton) a^{3}/T^{2} = k (M + m).

Kepler's idea was: For every planet which moves around the sun, _a^{3}_
divided by _T^{2}_ always gives the same quotient, _h_; and he did not
concern himself with the significance of this quotient further than to
note that if the particular _a_ and _T_ which belong to any
planet--e. g., the earth--be taken as the units of length and time, then
the quotient will be 1. Newton, on the other hand, attached a meaning to
the quotient, and showed that it is equal to the product obtained by
multiplying the sum of the two masses, planet and sun, by a number which
is always the same when we are dealing with the action of gravitation,
whether it be between the sun and planet, or between moon and earth, or
between the earth and a roast of beef in the butcher's scales, provided
only that we use always the same units with which to measure times,
distances, and masses.

Numerically, Newton's correction to Kepler's Third Law does not amount
to much in the motion of the planets. Jupiter, which shows the greatest
effect, makes the circuit of his orbit in 4,333 days instead of 4,335,
which it would require if Kepler's law were strictly true. But in
another respect the change is of the utmost importance, since it enables
us to extend Kepler's law, which relates solely to the sun and its
planets, to other attracting bodies, such as the earth, moon, and stars.
Thus for the moon's motion around the earth we write--

    (240,000^{3})/(27.32^{2}) = k (1 + 1/81),

from which we may find that, with the units here employed, the earth's
mass as the unit of mass, the mean solar day as the unit of time, and
the mile as the unit of distance--

    k = 1830 × 10^{10}.

If we introduce this value of _k_ into the corresponding equation, which
represents the motion of the earth around the sun, we shall have--

    a^{3}/(365.25)^{2} = 1830 × 10^{10} (333,000 + 1),

where the large number in the parenthesis represents the number of times
the mass of the sun is greater than the mass of the earth. We shall find
by solving this equation that _a_, the mean distance of the sun from the
earth, is very approximately 93,000,000 miles.

best appreciated by a reference to Fig. 17. It appears here that the
earth makes its nearest approach to the orbit of Mars in the month of
August, and if in any August Mars happens to be in opposition, its
distance from the earth will be very much less than the distance of the
sun from the earth, and may be measured by methods not unlike those
which served for the moon. If now the orbits of Mars and the earth were
circles having their centers at the sun this distance between them,
which we may represent by _D_, would be the difference of the radii of
these orbits--

    D = a´´ - a´,

where the accents ´´, ´ represent Mars and the earth respectively.
Kepler's Third Law furnishes the relation--

    (a´´)^{3}/(T´´)^{2} = (a´)^{3}/(T´)^{2};

and since the periodic times of the earth and Mars, _T´_, _T´´_, are
known to a high degree of accuracy, these two equations are sufficient
to determine the two unknown quantities, _a´_, _a´´_--i. e., the
distance of the sun from Mars as well as from the earth. The first of
these equations is, of course, not strictly true, on account of the
elliptical shape of the orbits, but this can be allowed for easily

In practice it is found better to apply this method of determining the
sun's distance through observations of an asteroid rather than
observations of Mars, and great interest has been aroused among
astronomers by the discovery, in 1898, of an asteroid, or planet, Eros,
which at times comes much closer to the earth than does Mars or any
other heavenly body except the moon, and which will at future
oppositions furnish a more accurate determination of the sun's distance
than any hitherto available. Observations for this purpose are being
made at the present time (October, 1900).

Many other methods of measuring the sun's distance have been devised by
astronomers, some of them extremely ingenious and interesting, but every
one of them has its weak point--e. g., the determination of the mass of
the earth in the first method given above and the measurement of _D_ in
the second method, so that even the best results at present are
uncertain to the extent of 200,000 miles or more, and astronomers,
instead of relying upon any one method, must use all of them, and take
an average of their results. According to Professor Harkness, this
average value is 92,796,950 miles, and it seems certain that a line of
this length drawn from the earth toward the sun would end somewhere
within the body of the sun, but whether on the nearer or the farther
side of the center, or exactly at it, no man knows.

114. PARALLAX AND DISTANCE.--It is quite customary among astronomers to
speak of the sun's parallax, instead of its distance from the earth,
meaning by parallax its difference of direction as seen from the center
and surface of the earth--i. e., the angle subtended at the sun by a
radius of the earth placed at right angles to the line of sight. The
greater the sun's distance the smaller will this angle be, and it
therefore makes a substitute for the distance which has the advantage of
being represented by a small number, 8".8, instead of a large one.

The books abound with illustrations intended to help the reader
comprehend how great is a distance of 93,000,000 miles, but a single one
of these must suffice here. To ride 100 miles a day 365 days in the year
would be counted a good bicycling record, but the rider who started at
the beginning of the Christian era and rode at that rate toward the sun
from the year 1 A. D. down to the present moment would not yet have
reached his destination, although his journey would be about three
quarters done. He would have crossed the orbit of Venus about the time
of Charlemagne, and that of Mercury soon after the discovery of America.

115. SIZE AND DENSITY OF THE SUN.--Knowing the distance of the sun, it
is easy to find from the angle subtended by its diameter (32 minutes of
arc) that the length of that diameter is 865,000 miles. We recall in
this connection that the diameter of the moon's _orbit_ is only 480,000
miles, but little more than half the diameter of the sun, thus affording
abundant room inside the sun, and to spare, for the moon to perform the
monthly revolution about its orbit, as shown in Fig. 65.

[Illustration: FIG. 65.--The sun's size.--YOUNG.]

In the same manner in which the density of the moon was found from its
mass and diameter, the student may find from the mass and diameter of
the sun given above that its mean density is 1.4 times that of water.
This is about the same as the density of gravel or soft coal, and is
just about one quarter of the average density of the earth.

We recall that the small density of the moon was accounted for by the
diminished weight of objects upon it, but this explanation can not hold
in the case of the sun, for not only is the density less but the force
of gravity (weight) is there 28 times as great as upon the earth. The
athlete who here weighs 175 pounds, if transported to the surface of the
sun would weigh more than an elephant does here, and would find his
bones break under his own weight if his muscles were strong enough to
hold him upright. The tremendous pressure exerted by gravity at the
surface of the sun must be surpassed below the surface, and as it does
not pack the material together and make it dense, we are driven to one
of two conclusions: Either the stuff of which the sun is made is
altogether unlike that of the earth, not so readily compressed by
pressure, or there is some opposing influence at work which more than
balances the effect of gravity and makes the solar stuff much lighter
than the terrestrial.

116. MATERIAL OF WHICH THE SUN IS MADE.--As to the first of these
alternatives, the spectroscope comes to our aid and shows in the sun's
spectrum (Fig. 50) the characteristic line marked _D_, which we know
always indicates the presence of sodium and identifies at least one
terrestrial substance as present in the sun in considerable quantity.
The lines marked _C_ and _F_ are produced by hydrogen, which is one of
the constituents of water, _E_ shows calcium to be present in the sun,
_b_ magnesium, etc. In this way it has been shown that about one half of
our terrestrial elements, mainly the metallic ones, are present as gases
on or near the sun's surface, but it must not be inferred that elements
not found in this way are absent from the sun. They may be there,
probably are there, but the spectroscopic proof of their presence is
more difficult to obtain. Professor Rowland, who has been prominent in
the study of the solar spectrum, says: "Were the whole earth heated to
the temperature of the sun, its spectrum would probably resemble that of
the sun very closely."

Some of the common terrestrial elements found in the sun are:

      Oxygen (?)

Whatever differences of chemical structure may exist between the sun and
the earth, it seems that we must regard these bodies as more like than
unlike to each other in substance, and we are brought back to the second
of our alternatives: there must be some influence opposing the force of
gravity and making the substance of the sun light instead of heavy, and
we need not seek far to find it in--

117. THE HEAT OF THE SUN.--That the sun is hot is too evident to require
proof, and it is a familiar fact that heat expands most substances and
makes them less dense. The sun's heat falling upon the earth expands it
and diminishes its density in some small degree, and we have only to
imagine this process of expansion continued until the earth's diameter
becomes 58 per cent larger than it now is, to find the earth's density
reduced to a level with that of the sun. Just how much the temperature
of the earth must be raised to produce this amount of expansion we do
not know, neither do we know accurately the temperature of the sun, but
there can be no doubt that heat is the cause of the sun's low density
and that the corresponding temperature is very high.

Before we inquire more closely into the sun's temperature, it will be
well to draw a sharp distinction between the two terms heat and
temperature, which are often used as if they meant the same thing. Heat
is a form of energy which may be found in varying degree in every
substance, whether warm or cold--a block of ice contains a considerable
amount of heat--while temperature corresponds to our sensations of warm
and cold, and measures the extent to which heat is concentrated in the
body. It is the amount of heat per molecule of the body. A barrel of
warm water contains more heat than the flame of a match, but its
temperature is not so high. Bearing in mind this distinction, we seek to
determine not the amount of heat contained in the sun but the sun's
temperature, and this involves the same difficulty as does the question,
What is the temperature of a locomotive? It is one thing in the fire box
and another thing in the driving wheels, and still another at the
headlight; and so with the sun, its temperature is certainly different
in different parts--one thing at the center and another at the surface.
Even those parts which we see are covered by a veil of gases which
produce by absorption the dark lines of the solar spectrum, and
seriously interfere both with the emission of energy from the sun and
with our attempts at measuring the temperature of those parts of the
surface from which that energy streams.

In view of these and other difficulties we need not be surprised that
the wildest discordance has been found in estimates of the solar
temperature made by different investigators, who have assigned to it
values ranging from 1,400° C. to more than 5,000,000° C. Quite recently,
however, improved methods and a better understanding of the problem have
brought about a better agreement of results, and it now seems probable
that the temperature of the visible surface of the sun lies somewhere
between 5,000° and 10,000° C., say 15,000° of the Fahrenheit scale.

118. DETERMINING THE SUN'S TEMPERATURE.--One ingenious method which has
been used for determining this temperature is based upon the principle
stated above, that every object, whether warm or cold, contains heat and
gives it off in the form of radiant energy. The radiation from a body
whose temperature is lower than 500° C. is made up exclusively of energy
whose wave length is greater than 7,600 tenth meters, and is therefore
invisible to the eye, although a thermometer or even the human hand can
often detect it as radiant heat. A brick wall in the summer sunshine
gives off energy which can be felt as heat but can not be seen. When
such a body is further heated it continues to send off the same kinds
(wave lengths) of energy as before, but new and shorter waves are added
to its radiation, and when it begins to emit energy of wave length 7,500
or 7,600 tenth meters, it also begins to shine with a dull-red light,
which presently becomes brighter and less ruddy and changes to white as
the temperature rises, and waves of still shorter length are thereby
added to the radiation. We say, in common speech, the body becomes first
red hot and then white hot, and we thus recognize in a general way that
the kind or color of the radiation which a body gives off is an index to
its temperature. The greater the proportion of energy of short wave
lengths the higher is the temperature of the radiating body. In sunlight
the maximum of brilliancy to the eye lies at or near the wave length,
5,600 tenth meters, but the greatest intensity of radiation of all kinds
(light included) is estimated to fall somewhere between green and blue
in the spectrum at or near the wave length 5,000 tenth meters, and if we
can apply to this wave length Paschen's law--temperature reckoned in
degrees centigrade from the absolute zero is always equal to the
quotient obtained by dividing the number 27,000,000 by the wave length
corresponding to maximum radiation--we shall find at once for the
absolute temperature of the sun's surface 5,400° C.

Paschen's law has been shown to hold true, at least approximately, for
lower temperatures and longer wave lengths than are here involved, but
as it is not yet certain that it is strictly true and holds for all
temperatures, too great reliance must not be attached to the numerical
result furnished by it.

[Illustration: FIG. 66.--The sun, August 11, 1894. Photographed at the
Goodsell Observatory.]

[Illustration: FIG. 67.--The sun, August 14, 1894. Photographed at the
Goodsell Observatory.]

119. THE SUN'S SURFACE.--A marked contrast exists between the faces of
sun and moon in respect of the amount of detail to be seen upon them,
the sun showing nothing whatever to correspond with the mountains,
craters, and seas of the moon. The unaided eye in general finds in the
sun only a blank bright circle as smooth and unmarked as the surface of
still water, and even the telescope at first sight seems to show but
little more. There may usually be found upon the sun's face a certain
number of black patches called _sun spots_, such as are shown in Figs.
66 to 69, and occasionally these are large enough to be seen through a
smoked glass without the aid of a telescope. When seen near the edge of
the sun they are quite frequently accompanied, as in Fig. 69, by vague
patches called _faculæ_ (Latin, _facula_ = a little torch), which look a
little brighter than the surrounding parts of the sun. So, too, a good
photograph of the sun usually shows that the central parts of the disk
are rather brighter than the edge, as indeed we should expect them to
be, since the absorption lines in the sun's spectrum have already taught
us that the visible surface of the sun is enveloped by invisible vapors
which in some measure absorb the emitted light and render it feebler at
the edge where it passes through a greater thickness of this envelope
than at the center. See Fig. 70, where it is shown that the energy
coming from the edge of the sun to the earth has to traverse a much
longer path inside the vapors than does that coming from the center.

[Illustration: FIG. 68.--The sun, August 18, 1894. Photographed at the
Goodsell Observatory.]

Examine the sun spots in the four photographs, Figs. 66 to 69, and note
that the two spots which appear at the extreme left of the first
photograph, very much distorted and foreshortened by the curvature of
the sun's surface, are seen in a different part of the second picture,
and are not only more conspicuous but show better their true shape.


120. THE SUN'S ROTATION.--The changed position of these spots shows that
the sun rotates about an axis at right angles to the direction of the
spot's motion, and the position of this axis is shown in the figure by a
faint line ruled obliquely across the face of the sun nearly north and
south in each of the four photographs. This rotation in the space of
three days has carried the spots from the edge halfway to the center of
the disk, and the student should note the progress of the spots in the
two later photographs, that of August 21st showing them just ready to
disappear around the farther edge of the sun.

[Illustration: FIG. 69.--The sun, August 21, 1894. Photographed at the
Goodsell Observatory.]

Plot accurately in one of these figures the positions of the spots as
shown in the other three, and observe whether the path of the spots
across the sun's face is a straight line. Is there any reason why it
should not be straight?

These four pictures may be made to illustrate many things about the sun.
Thus the sun's axis is not parallel to that of the earth, for the
letters _N S_ mark the direction of a north and south line across the
face of the sun, and this line, of course, is parallel to the earth's
axis, while it is evidently not parallel to the sun's axis. The group of
spots took more than ten days to move across the sun's face, and as at
least an equal time must be required to move around the opposite side of
the sun, it is evident that the period of the sun's rotation is
something more than 20 days. It is, in fact, rather more than 25 days,
for this same group of spots reappeared again on the left-hand edge of
the sun on September 5th.

[Illustration: FIG. 70.--Absorption at the sun's edge.]

121. SUN SPOTS.--Another significant fact comes out plainly from the
photographs. The spots are not permanent features of the sun's face,
since they changed their size and shape very appreciably in the few days
covered by the pictures. Compare particularly the photographs of August
14th and August 18th, where the spots are least distorted by the
curvature of the sun's surface. By September 16th this group of spots
had disappeared absolutely from the sun's face, although when at its
largest the group extended more than 80,000 miles in length, and several
of the individual spots were large enough to contain the earth if it had
been dropped upon them. From Fig. 67 determine in miles the length of
the group on August 14th. Fig. 71 shows an enlarged view of these spots
as they appeared on August 17th, and in this we find some details not so
well shown in the preceding pictures. The larger spots consist of a
black part called the _nucleus_ or _umbra_ (Latin, shadow), which is
surrounded by an irregular border called the _penumbra_ (partial
shadow), which is intermediate in brightness between the nucleus and
the surrounding parts of the sun. It should not be inferred from the
picture that the nucleus is really black or even dark. It shines, in
fact, with a brilliancy greater than that of an electric lamp, but the
background furnished by the sun's surface is so much brighter that by
contrast with it the nucleus and penumbra appear relatively dark.

[Illustration: FIG. 71.--Sun spots, August 17, 1894. Goodsell

[Illustration: FIG. 72.--Sun spot of March 5, 1873.--From LANGLEY, The
New Astronomy. By permission of the publishers.]

The bright shining surface of the sun, the background for the spots, is
called the _photosphere_ (Greek, light sphere), and, as Fig. 71 shows,
it assumes under a suitable magnifying power a mottled aspect quite
different from the featureless expanse shown in the earlier pictures.
The photosphere is, in fact, a layer of little clouds with darker
spaces between them, and the fine detail of these clouds, their
complicated structure, and the way in which, when projected against the
background of a sun spot, they produce its penumbra, are all brought out
in Fig. 72. Note that the little patch in one corner of this picture
represents North and South America drawn to the same scale as the sun

[Illustration: FIG. 73.--Spectroheliograph, showing distribution of
faculæ upon the sun.--HALE.]

[Illustration: FIG. 74.--Eclipse of July 20, 1878.--TROUVELOT.]

122. FACULÆ.--We have seen in Fig. 69 a few of the bright spots called
faculæ. At the telescope or in the ordinary photograph these can be seen
only at the edge of the sun, because elsewhere the background furnished
by the photosphere is so bright that they are lost in it. It is
possible, however, by an ingenious application of the spectroscope to
break up the sunlight into a spectrum in such a way as to diminish the
brightness of this background, much more than the brightness of the
faculæ is diminished, and in this way to obtain a photograph of the
sun's surface which shall show them wherever they occur, and such a
photograph, showing faintly the spectral lines, is reproduced in Fig.
73. The faculæ are the bright patches which stretch inconspicuously
across the face of the sun, in two rather irregular belts with a
comparatively empty lane between them. This lane lies along the sun's
equator, and it is upon either side of it between latitudes 5° and 40°
that faculæ seem to be produced. It is significant of their connection
with sun spots that the spots occur in these particular zones and are
rarely found outside them.

[Illustration: FIG. 75.--Eclipse of April 16, 1893.--SCHAEBERLE.]

123. INVISIBLE PARTS OF THE SUN. THE CORONA.--Thus far we have been
dealing with parts of the sun that may be seen and photographed under
all ordinary conditions. But outside of and surrounding these parts is
an envelope, or rather several envelopes, of much greater extent than
the visible sun. These envelopes are for the most part invisible save at
those times when the brighter central portions of the sun are hidden in
a total eclipse.

[Illustration: FIG. 76.--Eclipse of January 21, 1898.--CAMPBELL.]

Fig. 74 is from a drawing, and Figs. 75 and 76 are from eclipse
photographs showing this region, in which the most conspicuous object
is the halo of soft light called the _corona_, that completely surrounds
the sun but is seen to be of differing shapes and differing extent at
the several eclipses here shown, although a large part of these apparent
differences is due to technical difficulties in photographing, and
reproducing an object with outlines so vague as those of the corona. The
outline of the corona is so indefinite and its outer portions so faint
that it is impossible to assign to it precise dimensions, but at its
greatest extent it reaches out for several millions of miles and fills a
space more than twenty times as large as the visible part of the sun.
Despite its huge bulk, it is of most unsubstantial character, an airy
nothing through which comets have been known to force their way around
the sun from one side to the other, literally for millions of miles,
without having their course influenced or their velocity checked to any
appreciable extent. This would hardly be possible if the density even at
the bottom of the corona were greater than that of the best vacuum which
we are able to produce in laboratory experiments. It seems odd that a
vacuum should give off so bright a light as the coronal pictures show,
and the exact character of that light and the nature of the corona are
still subjects of dispute among astronomers, although it is generally
agreed that, in part at least, its light is ordinary sunlight faintly
reflected from the widely scattered molecules composing the substance of
the corona. It is also probable that in part the light has its origin in
the corona itself. A curious and at present unconfirmed result announced
by one of the observers of the eclipse of May 28, 1900, is that _the
corona is not hot_, its effective temperature being lower than that of
the instrument used for the observation.

[Illustration: FIG. 77.--Solar prominence of March 25, 1895.--HALE.]

124. THE CHROMOSPHERE.--Between the corona and the photosphere there is
a thin separating layer called the _chromosphere_ (Greek, color sphere),
because when seen at an eclipse it shines with a brilliant red light
quite unlike anything else upon the sun save the _prominences_ which are
themselves only parts of the chromosphere temporarily thrown above its
surface, as in a fountain a jet of water is thrown up from the basin and
remains for a few moments suspended in mid-air. Not infrequently in such
a fountain foreign matter is swept up by the rush of the water--dirt,
twigs, small fish, etc.--and in like manner the prominences often carry
along with them parts of the underlying layers of the sun, photosphere,
faculæ, etc., which reveal their presence in the prominence by adding
their characteristic lines to the spectrum, like that of the
chromosphere, which the prominence presents when they are absent. None
of the eclipse photographs (Figs. 74 to 76) show the chromosphere,
because the color effect is lacking in them, but a great curving
prominence may be seen near the bottom of Fig. 75, and smaller ones at
other parts of the sun's edge.

[Illustration: FIG. 78.--A solar prominence.--HALE.]

125. PROMINENCES.--Fig. 77 shows upon a larger scale one of these
prominences rising to a height of 160,000 miles above the photosphere;
and another photograph, taken 18 minutes later, but not reproduced here,
showed the same prominence grown in this brief interval to a stature of
280,000 miles. These pictures were not taken during an eclipse, but in
full sunlight, using the same spectroscopic apparatus which was employed
in connection with the faculæ to diminish the brightness of the
background without much enfeebling the brilliancy of the prominence
itself. The dark base from which the prominence seems to spring is not
the sun's edge, but a part of the apparatus used to cut off the direct

Fig. 78 contains a series of photographs of another prominence taken
within an interval of 1 hour 47 minutes and showing changes in size and
shape which are much more nearly typical of the ordinary prominence than
was the very unusual change in the case of Fig. 77.

[Illustration: FIG. 79.--Contrasted forms of solar

The preceding pictures are from photographs, and with them the student
may compare Fig. 79, which is constructed from drawings made at the
spectroscope by the German astronomer Zoellner. The changes here shown
are most marked in the prominence at the left, which is shaped like a
broken tree trunk, and which appears to be vibrating from one side to
the other like a reed shaken in the wind. Such a prominence is
frequently called an _eruptive_ one, a name suggested by its appearance
of having been blown out from the sun by something like an explosion,
while the prominence at the right in this series of drawings, which
appears much less agitated, is called by contrast with the other a
_quiescent_ prominence. These quiescent prominences are, as a rule, much
longer-lived than the eruptive ones. One more picture of prominences
(Fig. 80) is introduced to show the continuous stretch of chromosphere
out of which they spring.

[Illustration: FIG. 80.--Prominences and chromosphere.--HALE.]

Prominences are seen only at the edge of the sun, because it is there
alone that the necessary background can be obtained, but they must occur
at the center of the sun and elsewhere quite as well as at the edge, and
it is probable that quiescent prominences are distributed over all parts
of the sun's surface, but eruptive prominences show a strong tendency
toward the regions of sun spots and faculæ as if all three were
intimately related phenomena.

126. THE SUN AS A MACHINE.--Thus far we have considered the anatomy of
the sun, dissecting it into its several parts, and our next step should
be a consideration of its physiology, the relation of the parts to each
other, and their function in carrying on the work of the solar organism,
but this step, unfortunately, must be a lame one. The science of
astronomy to-day possesses no comprehensive and well-established theory
of this kind, but looks to the future for the solution of this the
greatest pending problem of solar physics. Progress has been made
toward its solution, and among the steps of this progress that we shall
have to consider, the first and most important is the conception of the
sun as a kind of heat engine.

In a steam engine coal is burned under the boiler, and its chemical
energy, transformed into heat, is taken up by the water and delivered,
through steam as a medium, to the engine, which again transforms and
gives it out as mechanical work in the turning of shafts, the driving of
machinery, etc. Now, the function of the sun is exactly opposite to that
of the engine and boiler: it gives out, instead of receiving, radiant
energy; but, like the engine, it must be fed from some source; it can
not be run upon nothing at all any more than the engine can run day
after day without fresh supplies of fuel under its boiler. We know that
for some thousands of years the sun has been furnishing light and heat
to the earth in practically unvarying amount, and not to the earth
alone, but it has been pouring forth these forms of energy in every
direction, without apparent regard to either use or economy. Of all the
radiant energy given off by the sun, only two parts out of every
thousand million fall upon any planet of the solar system, and of this
small fraction the earth takes about one tenth for the maintenance of
its varied forms of life and action. Astronomers and physicists have
sought on every hand for an explanation of the means by which this
tremendous output of energy is maintained century after century without
sensible diminution, and have come with almost one mind to the
conclusion that the gravitative forces which reside in the sun's own
mass furnish the only adequate explanation for it, although they may be
in some small measure re-enforced by minor influences, such as the fall
of meteoric dust and stones into the sun.

Every boy who has inflated a bicycle tire with a hand pump knows that
the pump grows warm during the operation, on account of the compression
of the air within the cylinder. A part of the muscular force (energy)
expended in working the pump reappears in the heat which warms both air
and pump, and a similar process is forever going on in the sun, only in
place of muscular force we must there substitute the tremendous
attraction of gravitation, 28 times as great as upon the earth. "The
matter in the interior of the sun must be as a shuttlecock between the
stupendous pressure and the enormously high temperature," the one
tending to compress and the other to expand it, but with this important
difference between them: the temperature steadily tends to fall as the
heat energy is wasted away, while the gravitative force suffers no
corresponding diminution, and in the long run must gain the upper hand,
causing the sun to shrink and become more dense. It is this progressive
shrinking and compression of its molecules into a smaller space which
supplies the energy contained in the sun's output of light and heat.
According to Lord Kelvin, each centimeter of shrinkage in the sun's
diameter furnishes the energy required to keep up its radiation for
something more than an hour, and, on account of the sun's great
distance, the shrinkage might go on at this rate for many centuries
without producing any measurable effect in the sun's appearance.

127. GASEOUS CONSTITUTION OF THE SUN.--But Helmholtz's dynamical theory
of the maintenance of the sun's heat, which we are here considering,
includes one essential feature that is not sufficiently stated above. In
order that the explanation may hold true, it is necessary that the sun
should be in the main a gaseous body, composed from center to
circumference of gases instead of solid or liquid parts. Pumping air
warms the bicycle pump in a way that pumping water or oil will not.

The high temperature of the sun itself furnishes sufficient reason for
supposing the solar material to be in the gaseous state, but the gas
composing those parts of the sun below the photosphere must be very
different in some of its characteristics from the air or other gases
with which we are familiar at the earth, since its average density is
1,000 times as great as that of air, and its consistence and mechanical
behavior must be more like that of honey or tar than that of any gas
with which we are familiar. It is worth noting, however, that if a hole
were dug into the crust of the earth to a depth of 15 or 20 miles the
air at the bottom of the hole would be compressed by that above it to a
density comparable with that of the solar gases.

128. THE SUN'S CIRCULATION.--It is plain that under the conditions which
exist in the sun the outer portions, which can radiate their heat freely
into space, must be cooler than the inner central parts, and this
difference of temperature must set up currents of hot matter drifting
upward and outward from within the sun and counter currents of cooler
matter settling down to take its place. So, too, there must be some
level at which the free radiation into outer space chills the hot matter
sufficiently to condense its less refractory gases into clouds made up
of liquid drops, just as on a cloudy day there is a level in our own
atmosphere at which the vapor of water condenses into liquid drops which
form the thin shell of clouds that hovers above the earth's surface,
while above and below is the gaseous atmosphere. In the case of the sun
this cloud layer is always present and is that part which we have
learned to call the photosphere. Above the photosphere lies the
chromosphere, composed of gases less easily liquefied, hydrogen is the
chief one, while between photosphere and chromosphere is a thin layer of
metallic vapors, perhaps indistinguishable from the top crust of the
photosphere itself, which by absorbing the light given off from the
liquid photosphere produces the greater part of the Fraunhofer lines in
the solar spectrum.

From time to time the hot matter struggling up from below breaks through
the photosphere and, carrying with it a certain amount of the metallic
vapors, is launched into the upper and cooler regions of the sun,
where, parting with its heat, it falls back again upon the photosphere
and is absorbed into it. It is altogether probable that the corona is
chiefly composed of fine particles ejected from the sun with velocities
sufficient to carry them to a height of millions of miles, or even
sufficient to carry them off never to return. The matter of the corona
must certainly be in a state of the most lively agitation, its particles
being alternately hurled up from the photosphere and falling back again
like fireworks, the particles which make up the corona of to-day being
quite a different set from those of yesterday or last week. It seems
beyond question that the prominences and faculæ too are produced in some
way by this up-and-down circulation of the sun's matter, and that any
mechanical explanation of the sun must be worked out along these lines;
but the problem is an exceedingly difficult one, and must include and
explain many other features of the sun's activity of which only a few
can be considered here.

129. THE SUN-SPOT PERIOD.--Sun spots come and go, and at best any
particular spot is but short-lived, rarely lasting more than a month or
two, and more often its duration is a matter of only a few days. They
are not equally numerous at all times, but, like swarms of locusts, they
seem to come and abound for a season and then almost to disappear, as if
the forces which produced them were of a periodic character alternately
active and quiet. The effect of this periodic activity since 1870 is
shown in Fig. 81, where the horizontal line is a scale of times, and the
distance of the curve above this line for any year shows the relative
number of spots which appeared upon the sun in that year. This indicates
very plainly that 1870, 1883, and 1893 were years of great sun-spot
activity, while 1879 and 1889 were years in which few spots appeared.
The older records, covering a period of two centuries, show the same
fluctuations in the frequency of sun spots and from these records
curves (which may be found in Young's, The Sun) have been plotted,
showing a succession of waves extending back for many years.

[Illustration: FIG. 81.--The curve of sun-spot frequency.]

The sun-spot period is the interval of time from the crest or hollow of
one wave to the corresponding part of the next one, and on the average
this appears to be a little more than eleven years, but is subject to
considerable variation. In accordance with this period there is drawn in
broken lines at the right of Fig. 81 a predicted continuation of the
sun-spot curve for the first decade of the twentieth century. The
irregularity shown by the three preceding waves is such that we must not
expect the actual course of future sun spots to correspond very closely
to the prediction here made; but in a general way 1901 and 1911 will
probably be years of few sun spots, while they will be numerous in 1905,
but whether more or less numerous than at preceding epochs of greatest
frequency can not be foretold with any approach to certainty so long as
we remain in our present ignorance of the causes which make the sun-spot

Determine from Fig. 81 as accurately as possible the length of the
sun-spot period. It is hard to tell the exact position of a crest or
hollow of the curve. Would it do to draw a horizontal line midway
between top and bottom of the curve and determine the length of the
period from its intersections with the curve--e. g., in 1874 and 1885?

[Illustration: FIG. 82.--Illustrating change of the sun-spot zones.]

130. THE SUN-SPOT ZONES.--It has been already noted that sun spots are
found only in certain zones of latitude upon the sun, and that faculæ
and eruptive prominences abound in these zones more than elsewhere,
although not strictly confined to them. We have now to note a
peculiarity of these zones which ought to furnish a clew to the sun's
mechanism, although up to the present time it has not been successfully
traced out. Just before a sun-spot minimum the few spots which appear
are for the most part clustered near the sun's equator. As these spots
die out two new groups appear, one north the other south of the sun's
equator and about 25° or 30° distant from it, and as the period advances
toward a maximum these groups shift their positions more and more toward
the equator, thus approaching each other but leaving between them a
vacant lane, which becomes steadily narrower until at the close of the
period, when the next minimum is at hand, it reaches its narrowest
dimensions, but does not altogether close up even then. In Fig. 82 these
relations are shown for the period falling between 1879 and 1890, by
means of the horizontal lines; for each year one line in the northern
and one in the southern hemisphere of the sun, their lengths being
proportional to the number of spots which appeared in the corresponding
hemisphere during the year, and their positions on the sun's disk
showing the average latitude of the spots in question. It is very
apparent from the figure that during this decade the sun's southern
hemisphere was much more active than the northern one in the production
of spots, and this appears to be generally the case, although the
difference is not usually as great as in this particular decade.

131. INFLUENCE OF THE SUN-SPOT PERIOD.--Sun spots are certainly less hot
than the surrounding parts of the sun's surface, and, in view of the
intimate dependence of the earth upon the solar radiation, it would be
in no way surprising if their presence or absence from the sun's face
should make itself felt in some degree upon the earth, raising and
lowering its temperature and quite possibly affecting it in other ways.
Ingenious men have suggested many such kinds of influence, which,
according to their investigations, appear to run in cycles of eleven
years. Abundant and scanty harvests, cyclones, tornadoes, epidemics,
rainfall, etc., are among these alleged effects, and it is possible that
there may be a real connection between any or all of them and the
sun-spot period, but for the most part astronomers are inclined to hold
that there is only one case in which the evidence is strong enough to
really establish a connection of this kind. The magnetic condition of
the earth and its disturbances, which are called magnetic storms, do
certainly follow in a very marked manner the course of sun-spot
activity, and perhaps there should be added to this the statement that
auroras (northern lights) stand in close relation to these magnetic
disturbances and are most frequent at the times of sun-spot maxima.

Upon the sun, however, the influence of the spot period is not limited
to things in and near the photosphere, but extends to the outermost
limits of the corona. Determine from Fig. 81 the particular part of the
sun-spot period corresponding to the date of each picture of the corona
and note how the pictures which were taken near times of sun-spot minima
present a general agreement in the shape and extent of the corona, while
the pictures taken at a time of maximum activity of the sun spots show a
very differently shaped and much smaller corona.

132. THE LAW OF THE SUN'S ROTATION.--We have seen in a previous part of
the chapter how the time required by the sun to make a complete rotation
upon its axis may be determined from photographs showing the progress of
a spot or group of spots across its disk, and we have now to add that
when this is done systematically by means of many spots situated in
different solar latitudes it leads to a very peculiar and extraordinary
result. Each particular parallel of latitude has its own period of
rotation different from that of its neighbors on either side, so that
there can be no such thing as a fixed geography of the sun's surface.
Every part of it is constantly taking up a new position with respect to
every other part, much as if the Gulf of Mexico should be south of the
United States this year, southeast of it next year, and at the end of a
decade should have shifted around to the opposite side of the earth from
us. A meridian of longitude drawn down the Mississippi Valley remains
always a straight line, or, rather, great circle, upon the surface of
the earth, while Fig. 83 shows what would become of such a meridian
drawn through the equatorial parts of the sun's disk. In the first
diagram it appears as a straight line running down the middle of the
sun's disk. Twenty-five days later, when the same face of the sun comes
back into view again, after making a complete revolution about the axis,
the equatorial parts will have moved so much faster and farther than
those in higher latitudes that the meridian will be warped as in the
second diagram, and still more warped after another and another
revolution, as shown in the figure.

[Illustration: FIG. 83.--Effect of the sun's peculiar rotation in
warping a meridian, originally straight.]

At least such is the case if the spots truly represent the way in which
the sun turns round. There is, however, a possibility that the spots
themselves drift with varying speeds across the face of the sun, and
that the differences which we find in their rates of motion belong to
them rather than to the photosphere. Just what happens in the regions
near the poles is hard to say, for the sun spots only extend about
halfway from the equator to the poles, and the spectroscope, which may
be made to furnish a certain amount of information bearing upon the
case, is not as yet altogether conclusive, nor are the faculæ which have
also been observed for this purpose.

The simple theory that the solar phenomena are caused by an interchange
of hotter and cooler matter between the photosphere and the lower strata
of the sun furnishes in its present shape little or no explanation of
such features as the sun-spot period, the variations in the corona, the
peculiar character of the sun's rotation, etc., and we have still
unsolved in the mechanical theory of the sun one of the noblest problems
of astronomy, and one upon which both observers and theoretical
astronomers are assiduously working at the present time. A close watch
is kept upon sun spots and prominences, the corona is observed at every
total eclipse, and numerous are the ingenious methods which are being
suggested and tried for observing it without an eclipse in ordinary
daylight. Attempts, more or less plausible, have been made and are now
pending to explain photosphere, spots and the reversing layer by means
of the refraction of light within the sun's outer envelope of gases, and
it seems altogether probable, in view of these combined activities, that
a considerable addition to our store of knowledge concerning the sun may
be expected in the not distant future.



133. PLANETS.--Circling about the sun, under the influence of his
attraction, is a family of planets each member of which is, like the
moon, a dark body shining by reflected sunlight, and therefore
presenting phases; although only two of them, Mercury and Venus, run
through the complete series--new, first quarter, full, last
quarter--which the moon presents. The way in which their orbits are
grouped about the sun has been considered in Chapter III, and Figs. 16
and 17 of that chapter may be completed so as to represent all of the
planets by drawing in Fig. 16 two circles with radii of 7.9 and 12.4
centimeters respectively, to represent the orbits of the planets Uranus
and Neptune, which are more remote from the sun than Saturn, and by
introducing a little inside the orbit of Jupiter about 500 ellipses of
different sizes, shapes, and positions to represent a group of minor
planets or asteroids as they are often called. It is convenient to
regard these asteroids as composing by themselves a class of very small
planets, while the remaining 8 larger planets fall naturally into two
other classes, a group of medium-sized ones--Mercury, Venus, Earth, and
Mars--called inner planets by reason of their nearness to the sun; and
the outer planets--Jupiter, Saturn, Uranus, Neptune--each of which is
much larger and more massive than any planet of the inner group. Compare
in Figs. 84 and 85 their relative sizes. The earth, _E_, is introduced
into Fig. 85 as a connecting link between the two figures.

Some of these planets, like the earth, are attended by one or more
moons, technically called satellites, which also shine by reflected
sunlight and which move about their respective planets in accordance
with the law of gravitation, much as the moon moves around the earth.

[Illustration: FIG. 84.--The inner planets and the moon.]

[Illustration: FIG. 85.--The outer planets.]

134. DISTANCES OF THE PLANETS FROM THE SUN.--It is a comparatively
simple matter to observe these planets year after year as they move
among the stars, and to find from these observations how long each one
of them requires to make its circuit around the sun--that is, its
periodic time, _T_, which figures in Kepler's Third Law, and when these
periodic times have been ascertained, to use them in connection with
that law to determine the mean distance of each planet from the sun.
Thus, Jupiter requires 4,333 days to move completely around its orbit;
and comparing this with the periodic time and mean distance of the earth
we find--

    a^{3} / (4333^{2}) = (93,000,000^{3}) / (365.25^{2}),

which when solved gives as the mean distance of Jupiter from the sun,
483,730,000 miles, or 5.20 times as distant as the earth. If we make a
similar computation for each planet, we shall find that their distances
from the sun show a remarkable agreement with an artificial series of
numbers called Bode's law. We write down the numbers contained in the
first line of figures below, each of which, after the second, is
obtained by doubling the preceding one, add 4 to each number and point
off one place of decimals; the resulting number is (approximately) the
distance of the corresponding planet from the sun.

 Mercury.  Venus.  Earth.  Mars.    Jupiter.  Saturn.  Uranus.  Neptune.
    0        3       6      12   24   48        96       192      384
    4        4       4       4    4    4         4         4        4
   0.4      0.7     1.0    1.6  2.8  5.2      10.0      19.6     38.8
   0.4      0.7     1.0    1.5  2.8  5.2       9.5      19.2     30.1

The last line of figures shows the real distance of the planet as
determined from Kepler's law, the earth's mean distance from the sun
being taken as the unit for this purpose. With exception of Neptune, the
agreement between Bode's law and the true distances is very striking,
but most remarkable is the presence in the series of a number, 2.8, with
no planet corresponding to it. This led astronomers at the time Bode
published the law, something more than a century ago, to give new heed
to a suggestion made long before by Kepler, that there might be an
unknown planet moving between the orbits of Mars and Jupiter, and a
number of them agreed to search for such a planet, each in a part of the
sky assigned him for that purpose. But they were anticipated by Piazzi,
an Italian, who found the new planet, by accident, on the first day of
the nineteenth century, moving at a distance from the sun represented by
the number 2.77.

This planet was the first of the asteroids, and in the century that has
elapsed hundreds of them have been discovered, while at the present time
no year passes by without several more being added to the number. While
some of these are nearer to the sun than is the first one discovered,
and others are farther from it, their average distance is fairly
represented by the number 2.8.

Why Bode's law should hold true, or even so nearly true as it does, is
an unexplained riddle, and many astronomers are inclined to call it no
law at all, but only a chance coincidence--an illustration of the
"inherent capacity of figures to be juggled with"; but if so, it is
passing strange that it should represent the distance of the asteroids
and of Uranus, which was also an undiscovered planet at the time the law
was published.

135. THE PLANETS COMPARED WITH EACH OTHER.--When we pass from general
considerations to a study of the individual peculiarities of the
planets, we find great differences in the extent of knowledge concerning
them, and the reason for this is not far to seek. Neptune and Uranus, at
the outskirts of the solar system, are so remote from us and so feebly
illumined by the sun that any detailed study of them can go but little
beyond determining the numbers which represent their size, mass,
density, the character of their orbits, etc. The asteroids are so small
that in the telescope they look like mere points of light, absolutely
indistinguishable in appearance from the fainter stars. Mercury,
although closer at hand and presenting a disk of considerable size,
always stands so near the sun that its observation is difficult on this
account. Something of the same kind is true for Venus, although in much
less degree; while Mars, Jupiter, and Saturn are comparatively easy
objects for telescopic study, and our knowledge of them, while far from
complete, is considerably greater than for the other planets.

Figs. 84 and 85 show the relative sizes of the planets composing the
inner and outer groups respectively, and furnish the numerical data
concerning their diameters, masses, densities, etc., which are of most
importance in judging of their physical condition. Each planet, save
Saturn, is represented by two circles, of which the outer is drawn
proportional to the size of the planet, and the inner shows the amount
of material that must be subtracted from the interior in order that the
remaining shell shall just float in water. Note the great difference in
thickness of shell between the two groups. Saturn, having a mean density
less than that of water, must have something loaded upon it, instead of
removed, in order that it should float just submerged.


136. APPEARANCE.--Commencing our consideration of the individual planets
with Jupiter, which is by far the largest of them, exceeding both in
bulk and mass all the others combined, we have in Fig. 86 four
representations of Jupiter and his family of satellites as they may be
seen in a very small telescope--e. g., an opera glass--save that the
little dots which here represent the satellites are numbered _1_, _2_,
_3_, _4_, in order to preserve their identity in the successive

The chief interest of these pictures lies in the satellites, but,
reserving them for future consideration, we note that the planet itself
resembles in shape the full moon, although in respect of brightness it
sends to us less than 1/6000 part as much light as the moon. From a
consideration of the motion of Jupiter and the earth in Fig. 16, show
that Jupiter can not present any such phases as does the moon, but that
its disk must be at all times nearly full. As seen from Saturn, what
kind of phases would Jupiter present?

137. THE BELTS.--Even upon the small scale of Fig. 86 we detect the most
characteristic feature of Jupiter's appearance in the telescope, the two
bands extending across his face parallel to the line of the satellites,
and in Fig. 87 these same dark bands may be recognized amid the
abundance of detail which is here brought out by a large telescope.
Photography does not succeed as a means of reproducing this detail, and
for it we have to rely upon the skill of the artist astronomer. The
lettering shows the Pacific Standard time at which the sketches were
made, and also the longitude of the meridian of Jupiter passing down the
center of the planet's disk.

[Illustration: FIG. 86.--Jupiter and his satellites.]

[Illustration: FIG. 87.--Drawings of Jupiter made at the 36-inch
telescope of the Lick Observatory.--KEELER.]

The dark bands are called technically the belts of Jupiter; and a
comparison of these belts in the second and third pictures of the group,
in which nearly the same face of the planet is turned toward us, will
show that they are subject to considerable changes of form and position
even within the space of a few days. So, too, by a comparison of such
markings as the round white spots in the upper parts of the disks, and
the indentations in the edges of the belts, we may recognize that the
planet is in the act of turning round, and must therefore have an axis
about which it turns, and poles, an equator, etc. The belts are in fact
parallel to the planet's equator; and generalizing from what appears in
the pictures, we may say that there is always a strongly marked belt on
each side of the equator with a lighter colored streak between them,
and that farther from the equator are other belts variable in number,
less conspicuous, and less permanent than the two first seen. Compare
the position of the principal belts with the position of the zones of
sun-spot activity in the sun. A feature of the planet's surface, which
can not be here reproduced, is the rich color effect to be found upon
it. The principal belts are a brick-red or salmon color, the intervening
spaces in general white but richly mottled, and streaked with purples,
browns, and greens.

The drawings show the planet as it appeared in the telescope, inverted,
and they must be turned upside down if we wish the points of the compass
to appear as upon a terrestrial map. Bearing this in mind, note in the
last picture the great oval spot in the southern hemisphere of Jupiter.
This is a famous marking, known from its color as the _great red spot_,
which appeared first in 1878 and has persisted to the present day
(1900), sometimes the most conspicuous marking on the planet, at others
reduced to a mere ghost of itself, almost invisible save for the
indentation which it makes in the southern edge of the belt near it.

138. ROTATION AND FLATTENING AT THE POLES.--One further significant fact
with respect to Jupiter may be obtained from a careful measurement of
the drawings; the planet is flattened at the poles, so that its polar
diameter is about one sixteenth part shorter than the equatorial
diameter. The flattening of the earth amounts to only one
three-hundredth part, and the marked difference between these two
numbers finds its explanation in the greater swiftness of Jupiter's
rotation about its axis, since in both cases it is this rotation which
makes the flattening.

It is not easy to determine the precise dimensions of the planet, since
this involves a knowledge both of its distance from us and of the angle
subtended by its diameter, but the most recent determinations of this
kind assign as the equatorial diameter 90,200 miles, and for the polar
diameter 84,400 miles. Determine from either of these numbers the size
of the great red spot.

The earth turns on its axis once in 24 hours but no such definite time
can be assigned to Jupiter, which, like the sun, seems to have different
rotation periods in different latitudes--9h. 50m. in the equatorial belt
and 9h. 56m. in the dark belts and higher latitudes. There is some
indication that the larger part of the visible surface rotates in 9h.
55.6m., while a broad stream along the equator flows eastward some 270
miles per hour, and thus comes back to the center of the planet, as seen
from the earth, five or six minutes earlier than the parts which do not
share in this motion. Judged by terrestrial standards, 270 miles per
hour is a great velocity, but Jupiter is constructed on a colossal
scale, and, too, we have to compare this movement, not to a current
flowing in the ocean, but to a wind blowing in the upper regions of the
earth's atmosphere. The visible surface of Jupiter is only the top of a
cloud formation, and contains nothing solid or permanent, if indeed
there is anything solid even at the core of the planet. The great red
spot during the first dozen years of its existence, instead of remaining
fixed relative to the surrounding formations, drifted two thirds of the
way around the planet, and having come to a standstill about 1891, it is
now slowly retracing its path.

139. PHYSICAL CONDITION.--For a better understanding of the physical
condition of Jupiter, we have now to consider some independent lines of
evidence which agree in pointing to the conclusion that Jupiter,
although classed with the earth as a planet, is in its essential
character much more like the sun.

_Appearance._--The formations which we see in Fig. 87 look like clouds.
They gather and disappear, and the only element of permanence about them
is their tendency to group themselves along zones of latitude. If we
measure the light reflected from the planet we find that its albedo is
very high, like that of snow or our own cumulus clouds, and it is of
course greater from the light parts of the disk than from the darker
bands. The spectroscope shows that the sunlight reflected from these
darker belts is like that reflected from the lighter parts, save that a
larger portion of the blue and violet rays has been absorbed out of it,
thus producing the ruddy tint of the belts, as sunset colors are
produced on the earth, and showing that here the light has penetrated
farther into the planet's atmosphere before being thrown back by
reflection from lower-lying cloud surfaces. The dark bands are therefore
to be regarded as rifts in the clouds, reaching down to some
considerable distance and indicating an atmosphere of great depth. The
great red spot, 28,000 miles long, and obviously thrusting back the
white clouds on every side of it, year after year, can hardly be a mere
patch on the face of the planet, but indicates some considerable depth
of atmosphere.

_Density._--So, too, the small mean density of the planet, only 1.3
times that of water and actually less than the density of the sun,
suggests that the larger part of the planet's bulk may be made of gases
and clouds, with very little solid matter even at the center; but here
we get into a difficulty from which there seems but one escape. The
force of gravity at the visible surface of Jupiter may be found from its
mass and dimensions to be 2.6 times as great as at the surface of the
earth, and the pressure exerted upon its atmosphere by this force ought
to compress the lower strata into something more dense than we find in
the planet. Some idea of this compression may be obtained from Fig. 88,
where the line marked _E_ shows approximately how the density of the air
increases as we move from its upper strata down toward the surface of
the earth through a distance of 16 miles, the density at any level being
proportional to the distance of the curved line from the straight one
near it. The line marked _J_ in the same figure shows how the density
would increase if the force of gravity were as great here as it is in
Jupiter, and indicates a much greater rate of increase. Starting from
the upper surface of the cloud in Jupiter's atmosphere, if we descend,
not 16 miles, but 1,600 or 16,000, what must the density of the
atmosphere become and how is this to be reconciled with what we know to
be the very small mean density of the planet?

We are here in a dilemma between density on the one hand and the effects
of gravity on the other, and the only escape from it lies in the
assumption that the interior of Jupiter is tremendously hot, and that
this heat expands the substance of the planet in spite of the pressure
to which it is subject, making a large planet with a low density,
possibly gaseous at the very center, but in its outer part surrounded by
a shell of clouds condensed from the gases by radiating their heat into
the cold of outer space.

[Illustration: FIG. 88.--Increase of density in the atmospheres of
Jupiter and the earth.]

This is essentially the same physical condition that we found for the
sun, and we may add, as further points of resemblance between it and
Jupiter, that there seems to be a circulation of matter from the hot
interior of the planet to its cooler surface that is more pronounced in
the southern hemisphere than in the northern, and that has its periods
of maximum and minimum activity, which, curiously enough, seem to
coincide with periods of maximum and minimum sun-spot development. Of
this, however, we can not be entirely sure, since it is only in recent
years that it has been studied with sufficient care, and further
observations are required to show whether the agreement is something
more than an accidental and short-lived coincidence.

_Temperature._--The temperature of Jupiter must, of course, be much
lower than that of the sun, since the surface which we see is not
luminous like the sun's; but below the clouds it is not improbable that
Jupiter may be incandescent, white hot, and it is surmised with some
show of probability that a little of its light escapes through the
clouds from time to time, and helps to produce the striking brilliancy
with which this planet shines.

140. THE SATELLITES OF JUPITER.--The satellites bear much the same
relation to Jupiter that the moon bears to the earth, revolving about
the planet in accordance with the law of gravitation, and conforming to
Kepler's three laws, as do the planets in their courses about the sun.
Observe in Fig. 86 the position of satellite No. _1_ on the four dates,
and note how it oscillates back and forth from left to right of Jupiter,
apparently making a complete revolution in about two days, while No. _4_
moves steadily from left to right during the entire period, and has
evidently made only a fraction of a revolution in the time covered by
the pictures. This quicker motion, of course, means that No. _1_ is
nearer to Jupiter than No. _4_, and the numbers given to the satellites
show the order of their distances from the planet. The peculiar way in
which the satellites are grouped, always standing nearly in a straight
line, shows that their orbits must lie nearly in the same plane, and
that this plane, which is also the plane of the planets' equator, is
turned edgewise toward the earth.

These satellites enjoy the distinction of being the first objects ever
discovered with the telescope, having been found by Galileo almost
immediately after its invention, A. D. 1610. It is quite possible that
before this time they may have been seen with the naked eye, for in more
recent years reports are current that they have been seen under
favorable circumstances by sharp-eyed persons, and very little
telescopic aid is required to show them. Look for them with an opera or
field glass. They bear the names Io, Europa, Ganymede, Callisto, which,
however, are rarely used, and, following the custom of astronomers, we
shall designate them by the Roman numerals I, II, III, IV.

[Illustration: FIG. 89.--Orbits of Jupiter's satellites.]

For nearly three centuries (1610 to 1892) astronomers spoke of the four
satellites of Jupiter; but in September, 1892, a fifth one was added to
the number by Professor Barnard, who, observing with the largest
telescope then extant, found very close to Jupiter a tiny object only
1/600 part as bright as the other satellites, but, like them, revolving
around Jupiter, a permanent member of his system. This is called the
fifth satellite, and Fig. 89 shows the orbits of these satellites around
Jupiter, which is here represented on the same scale as the orbits
themselves. The broken line just inside the orbit of I represents the
size of the moon's orbit. The cut shows also the periodic times of the
satellites expressed in days, and furnishes in this respect a striking
illustration of the great mass of Jupiter. Satellite I is a little
farther from Jupiter than is the moon from the earth, but under the
influence of a greater attraction it makes the circuit of its orbit in
1.77 days, instead of taking 29.53 days, as does the moon. Determine
from the figure by the method employed in § 111 how much more massive is
Jupiter than the earth.

Small as these satellites seem in Fig. 86, they are really bodies of
considerable size, as appears from Fig. 90, where their dimensions are
compared with those of the earth and moon, save that the fifth satellite
is not included. This one is so small as to escape all attempts at
measuring its diameter, but, judging from the amount of light it
reflects, the period printed with the legend of the figure represents a
gross exaggeration of this satellite's size.

[Illustration: FIG. 90.--Jupiter's satellites compared with the earth
and moon.]

Like the moon, each of these satellites may fairly be considered a world
in itself, and as such a fitting object of detailed study, but,
unfortunately, their great distance from us makes it impossible, even
with the most powerful telescope, to see more upon their surfaces than
occasional vague markings, which hardly suffice to show the rotations of
the satellites upon their axes.

One striking feature, however, comes out from a study of their influence
in disturbing each other's motion about Jupiter. Their masses and the
resulting densities of the satellites are smaller than we should have
expected to find, the density being less than that of the moon, and
averaging only a little greater than the density of Jupiter itself. At
the surface of the third satellite the force of gravity is but little
less than on the moon, although the moon's density is nearly twice as
great as that of III, and there can be no question here of accounting
for the low density through expansion by great heat, as in the case of
the sun and Jupiter. It has been surmised that these satellites are not
solid bodies, like the earth and moon, but only shoals of rock and
stone, loosely piled together and kept from packing into a solid mass by
the action of Jupiter in raising tides within them. But the explanation
can hardly be regarded as an accepted article of astronomical belief,
although it is supported by some observations which tend to show that
the apparent shapes of the satellites change under the influence of the
tidal forces impressed upon them.

141. ECLIPSES OF THE SATELLITES.--It may be seen from Fig. 89 that in
their motion around the planet Jupiter's satellites must from time to
time pass through his shadow and be eclipsed, and that the shadows of
the satellites will occasionally fall upon the planet, producing to an
observer upon Jupiter an eclipse of the sun, but to an observer on the
earth presenting only the appearance of a round black spot moving slowly
across the face of the planet. Occasionally also a satellite will pass
exactly between the earth and Jupiter, and may be seen projected against
the planet as a background. All of these phenomena are duly predicted
and observed by astronomers, but the eclipses are the only ones we need
consider here. The importance of these eclipses was early recognized,
and astronomers endeavored to construct a theory of their recurrence
which would permit accurate predictions of them to be made. But in this
they met with no great success, for while it was easy enough to foretell
on what night an eclipse of a given satellite would occur, and even to
assign the hour of the night, it was not possible to make the predicted
minute agree with the actual time of eclipse until after Roemer, a
Danish astronomer of the seventeenth century, found where lay the
trouble. His discovery was, that whenever the earth was on the side of
its orbit toward Jupiter the eclipses really occurred before the
predicted time, and when the earth was on the far side of its orbit they
came a few minutes later than the predicted time. He correctly inferred
that this was to be explained, not by any influence which the earth
exerted upon Jupiter and his satellites, but through the fact that the
light by which we see the satellite and its eclipse requires an
appreciable time to cross the intervening space, and a longer time when
the earth is far from Jupiter than when it is near.

For half a century Roemer's views found little credence, but we know now
that he was right, and that on the average the eclipses come 8m. 18s.
early when the earth is nearest to Jupiter, and 8m. 18s. late when it is
on the opposite side of its orbit. This is equivalent to saying that
light takes 8m. 18s. to cover the distance from the sun to the earth, so
that at any moment we see the sun not as it then is, but as it was 8
minutes earlier. It has been found possible in recent years to measure
by direct experiment the velocity with which light travels--186,337
miles per second--and multiplying this number by the 498s. (= 8m. 18s.)
we obtain a new determination of the sun's distance from the earth. The
product of the two numbers is 92,795,826, in very fair agreement with
the 93,000,000 miles found in Chapter X; but, as noted there, this
method, like every other, has its weak side, and the result may be a
good many thousands of miles in error.

It is worthy of note in this connection that both methods of obtaining
the sun's distance which were given in Chapter X involve Kepler's Third
Law, while the result obtained from Jupiter's satellites is entirely
independent of this law, and the agreement of the several results is
therefore good evidence both for the truth of Kepler's laws and for the
soundness of Roemer's explanation of the eclipses. This mode of proof,
by comparing the numerical results furnished by two or more different
principles, and showing that they agree or disagree, is of wide
application and great importance in physical science.


142. THE RING OF SATURN.--In respect of size and mass Saturn stands next
to Jupiter, and although far inferior to him in these respects, it
contains more material than all the remaining planets combined. But the
unique feature of Saturn which distinguishes it from every other known
body in the heavens is its ring, which was long a puzzle to the
astronomers who first studied the planet with a telescope (one of them
called Saturn a planet with ears), but, was after nearly half a century
correctly understood and described by Huyghens, whose Latin text we
translate into--"It is surrounded by a ring, thin, flat, nowhere
touching it, and making quite an angle with the ecliptic."

[Illustration: FIG. 91.--Aspects of Saturn's rings.]

Compare with this description Fig. 91, which shows some of the
appearances presented by the ring at different positions of Saturn in
its orbit. It was their varying aspects that led Huyghens to insert the
last words of his description, for, if the plane of the ring coincided
with the plane of the earth's orbit, then at all times the ring must be
turned edgewise toward the earth, as shown in the middle picture of the
group. Fig. 92 shows the sun and the orbit of the earth placed near the
center of Saturn's orbit, across whose circumference are ruled some
oblique lines representing the plane of the ring, the right end always
tilted up, no matter where the planet is in its orbit. It is evident
that an observer upon the earth will see the _N_ side of the ring when
the planet is at _N_ and the _S_ side when it is at _S_, as is shown in
the first and third pictures of Fig. 91, while midway between these
positions the edge of the ring will be presented to the earth.

[Illustration: FIG. 92.--Aspects of the ring in their relation to
Saturn's orbital motion.]

The last occasion of this kind was in October, 1891, and with the large
telescope of the Washburn Observatory the writer at that time saw
Saturn without a trace of a ring surrounding it. The ring is so thin
that it disappears altogether when turned edgewise. The names of the
zodiacal constellations are inserted in Fig. 92 in their proper
direction from the sun, and from these we learn that the ring will
disappear, or be exceedingly narrow, whenever Saturn is in the
constellation Pisces or near the boundary line between Leo and Virgo. It
will be broad and show its northern side when Saturn is in Scorpius or
Sagittarius, and its southern face when the planet is in Gemini. What
will be its appearance in 1907 at the date marked in the figure?

143. NATURE OF THE RING.--It is apparent from Figs. 91 and 93 that
Saturn's ring is really made up of two or more rings lying one inside of
the other and completely separated by a dark space which, though narrow,
is as clean and sharp as if cut with a knife. Also, the inner edge of
the ring fades off into an obscure border called the _dusky ring_ or
_crape ring_. This requires a pretty good telescope to show it, as may
be inferred from the fact that it escaped notice for more than two
centuries during which the planet was assiduously studied with
telescopes, and was discovered at the Harvard College Observatory as
recently as 1850.

Although the rings appear oval in all of the pictures, this is mainly an
effect of perspective, and they are in fact nearly circular with the
planet at their center. The extreme diameter of the ring is 172,000
miles, and from this number, by methods already explained (Chapter IX),
the student should obtain the width of the rings, their distance from
the ball of the planet, and the diameter of the ball. As to thickness,
it is evident, from the disappearance of the ring when its edge is
turned toward the earth, that it is very thin in comparison with its
diameter, probably not more than 100 miles thick, although no exact
measurement of this can be made.

[Illustration: FIG. 93.--Saturn.]

From theoretical reasons based upon the law of gravitation astronomers
have held that the rings of Saturn could not possibly be solid or
liquid bodies. The strains impressed upon them by the planet's
attraction would tear into fragments steel rings made after their size
and shape. Quite recently Professor Keeler has shown, by applying the
spectroscope (Doppler's principle) to determine the velocity of the
ring's rotation about Saturn, that the inner parts of the ring move, as
Kepler's Third Law requires, more rapidly than do the outer parts, thus
furnishing a direct proof that they are not solid, and leaving no doubt
that they are made up of separate fragments, each moving about the
planet in its own orbit, like an independent satellite, but standing so
close to its neighbors that the whole space reflects the sunlight as
completely as if it were solid. With this understanding of the rings it
is easy to see why they are so thin. Like Jupiter, Saturn is greatly
flattened at the poles, and this flattening, or rather the protuberant
mass about the equator, lays hold of every satellite near the planet and
exerts upon it a direct force tending to thrust it down into the plane
of the planet's equator and hold it there. The ring lies in the plane of
Saturn's equator because each particle is constrained to move there.

The division of the ring into two parts, an outer and an inner ring, is
usually explained as follows: Saturn is surrounded by a numerous brood
of satellites, which by their attractions produce perturbations in the
material composing the rings, and the dividing line between the outer
and inner rings falls at the place where by the law of gravitation the
perturbations would have their greatest effect. The dividing line
between the rings is therefore a narrow lane, 2,400 miles wide, from
which the fragments have been swept clean away by the perturbing action
of the satellites. Less conspicuous divisions are seen from time to time
in other parts of the ring, where the perturbations, though less, are
still appreciable. But it is open to some question whether this
explanation is sufficient.

The curious darkness of the inner or crape ring is easily explained.
The particles composing it are not packed together so closely as in the
outer ring, and therefore reflect less sunlight. Indeed, so sparsely
strewn are the particles in this ring that it is in great measure
transparent to the sunlight, as is shown by a recorded observation of
one of the satellites which was distinctly although faintly seen while
moving through the shadow of the dark ring, but disappeared in total
eclipse when it entered the shadow cast by the bright ring.

144. THE BALL OF SATURN.--The ball of the planet is in most respects a
smaller copy of Jupiter. With an equatorial diameter of 76,000 miles, a
polar diameter of 69,000 miles, and a mass 95 times that of the earth,
its density is found to be the least of any planet in the solar system,
only 0.70 of the density of water, and about one half as great as is the
density of Jupiter. The force of gravity at its surface is only a little
greater (1.18) than on the earth; and this, in connection with the low
density, leads, as in the case of Jupiter, to the conclusion that the
planet must be mainly composed of gases and vapors, very hot within, but
inclosed by a shell of clouds which cuts off their glow from our eyes.

Like Jupiter in another respect, the planet turns very swiftly upon its
axis, making a revolution in 10 hours 14 minutes, but up to the present
it remains unknown whether different parts of the surface have different
rotation times.

145. THE SATELLITES.--Saturn is attended by a family of nine satellites,
a larger number than belongs to any other planet, but with one exception
they are exceedingly small and difficult to observe save with a very
large telescope. Indeed, the latest one is said to have been discovered
in 1898 by means of the image which it impressed upon a photographic
plate, and it has never been _seen_.

Titan, the largest of them, is distant 771,000 miles from the planet and
bears much the same relation to Saturn that Satellite III bears to
Jupiter, the similarity in distance, size and mass being rather
striking, although, of course, the smaller mass of Saturn as compared
with Jupiter makes the periodic time of Titan--15 days 23 hours--much
greater than that of III. Can you apply Kepler's Third Law to the motion
of Titan so as to determine from the data given above, the time required
for a particle at the outer or inner edge of the ring to revolve once
around Saturn?

Japetus, the second satellite in point of size, whose distance from
Saturn is about ten times as great as the moon's distance from the
earth, presents the remarkable peculiarity of being always brighter in
one part of its orbit than in another, three or four times as bright
when west of Saturn as when east of it. This probably indicates that,
like our own moon, the satellite turns always the same face toward its
planet, and further, that one side of the satellite reflects the
sunlight much better than the other side--i. e., has a higher albedo.
With these two assumptions it is easily seen that the satellite will
always turn toward the earth one face when west, and the other face when
east of Saturn, and thus give the observed difference of brightness.


146. CHIEF CHARACTERISTICS.--The two remaining large planets are
interesting chiefly as modern additions to the known members of the
sun's family. The circumstances leading to the discovery of Neptune have
been touched upon in Chapter IV, and for Uranus we need only note that
it was found by accident in the year 1781 by William Herschel, who for
some time after the discovery considered it to be only a comet. It was
the first planet ever discovered, all of its predecessors having been
known from prehistoric times.

[Illustration: WILLIAM HERSCHEL (1738-1822).]

Uranus has four satellites, all of them very faint, which present only
one feature of special importance. Instead of moving in orbits which are
approximately parallel to the plane of the ecliptic, as do the
satellites of the inner planets, their orbit planes are tipped up nearly
perpendicular to the planes of the orbits of both Uranus and the earth.
The one satellite which Neptune possesses has the same peculiarity in
even greater degree, for its motion around the planet takes place in the
direction opposite to that in which all the planets move around the sun,
much as if the orbit of the satellite had been tipped over through an
angle of 150°. Turn a watch face down and note how the hands go round in
the direction opposite to that in which they moved before the face was
turned through 180°.

Both Uranus and Neptune are too distant to allow much detail to be seen
upon their surfaces, but the presence of broad absorption bands in their
spectra shows that they must possess dense atmospheres quite different
in constitution from the atmosphere of the earth. In respect of density
and the force of gravity at their surfaces, they are not very unlike
Saturn, although their density is greater and gravity less than his,
leading to the supposition that they are for the most part gaseous
bodies, but cooler and probably more nearly solid than either Jupiter or

Under favorable circumstances Uranus may be seen with the naked eye by
one who knows just where to look for it. Neptune is never visible save
in a telescope.

147. THE INNER PLANETS.--In sharp contrast with the giant planets which
we have been considering stands the group of four inner planets, or five
if we count the moon as an independent body, which resemble each other
in being all small, dense, and solid bodies, which by comparison with
the great distances separating the outer planets may fairly be described
as huddled together close to the sun. Their relative sizes are shown in
Fig. 84, together with the numerical data concerning size, mass,
density, etc., which we have already found important for the
understanding of a planet's physical condition.


[Illustration: FIG. 94.--The phases of Venus.--ANTONIADI.]

148. APPEARANCE.--Omitting the earth, Venus is by far the most
conspicuous member of this group, and when at its brightest is, with
exception of the sun and moon, the most brilliant object in the sky, and
may be seen with the naked eye in broad daylight if the observer knows
just where to look for it. But its brilliancy is subject to considerable
variations on account of its changing distance from the earth, and the
apparent size of its disk varies for the same reason, as may be seen
from Fig. 94. These drawings bring out well the phases of the planet,
and the student should determine from Fig. 17 what are the relative
positions in their orbits of the earth and Venus at which the planet
would present each of these phases. As a guide to this, observe that the
dark part of Venus's earthward side is always proportional in area to
the angle at Venus between the earth and sun. In the first picture of
Fig. 94 about two thirds of the surface corresponding to the full
hemisphere of the planet is dark, and the angle at Venus between earth
and sun is therefore two thirds of 180°--i. e., 120°. In Fig. 17 find a
place on the orbit of Venus from which if lines be drawn to the sun and
earth, as there shown, the angle between them will be 120°. Make a
similar construction for the fourth picture in Fig. 94. Which of these
two positions is farther from the earth? How do the distances compare
with the apparent size of Venus in the two pictures? What is the phase
of Venus to-day?

The irregularities in the shading of the illuminated parts of the disk
are too conspicuous in Fig. 94, on account of difficulties of
reproduction; these shadings are at the best hard to see in the
telescope, and distinct permanent markings upon the planet are wholly
lacking. This absence of markings makes almost impossible a
determination of the planet's time of rotation about its axis, and
astronomers are divided in this respect into two parties, one of which
maintains that Venus, like the earth, turns upon its axis in some period
not very different from 24 hours, while the other contends that, like
the moon, it turns always the same face toward the center of its orbit,
making a rotation upon its axis in the same period in which it makes a
revolution about the sun. The reason why no permanent markings are to be
seen on this planet is easily found. Like Jupiter and Saturn, its
atmosphere is at all times heavily cloud-laden, so that we seldom, if
ever, see down to the level of its solid parts. There is, however, no
reason here to suppose the interior parts hot and gaseous. It is much
more probable that Venus, like the earth, possesses a solid crust whose
temperature we should expect to be considerably higher than that of the
earth, because Venus is nearer the sun. But the cloud layer in its
atmosphere must modify the temperature in some degree, and we have
practically no knowledge of the real temperature conditions at the
surface of the planet.

It is the clouds of Venus which in great measure are responsible for its
marked brilliancy, since they are an excellent medium for reflecting the
sunlight, and give to its surface an albedo greater than that of any
other planet, although Saturn is nearly equal to it.

Of course, the presence of such cloud formations indicates that Venus is
surrounded by a dense atmosphere, and we have independent evidence of
this in the shape of its disk when the planet is very nearly between the
earth and sun. The illuminated part, from tip to tip of the horns, then
stretches more than halfway around the planet's circumference, and shows
that a certain amount of light must have been refracted through its
atmosphere, thus making the horns of the crescent appear unduly
prolonged. This atmosphere is shown by the spectroscope to be not unlike
that of the earth, although, possibly, more dense.


149. CHIEF CHARACTERISTICS.--Mercury, on account of its nearness to the
sun, is at all times a difficult object to observe, and Copernicus, who
spent most of his life in Poland, is said, despite all his efforts, to
have gone to his grave without ever seeing it. In our more southern
latitude it can usually be seen for about a fortnight at the time of
each elongation--i. e., when at its greatest angular distance from the
sun--and the student should find from Fig. 16 the time at which the next
elongation occurs and look for the planet, shining like a star of the
first magnitude, low down in the sky just after sunset or before
sunrise, according as the elongation is to the east or west of the sun.
When seen in the morning sky the planet grows brighter day after day
until it disappears in the sun's rays, while in the evening sky its
brilliancy as steadily diminishes until the planet is lost. It should
therefore be looked for in the evening as soon as possible after it
emerges from the sun's rays.

Mercury, as the smallest of the planets, is best compared with the
moon, which it does not greatly surpass in size and which it strongly
resembles in other respects. Careful comparisons of the amount of light
reflected by the planet in different parts of its orbit show not only
that its albedo agrees very closely with that of the moon, but also that
its light changes with the varying phase of the planet in almost exactly
the same way as the amount of moonlight changes. We may therefore infer
that its surface is like that of the moon, a rough and solid one, with
few or no clouds hanging over it, and most probably covered with very
little or no atmosphere. Like Venus, its rotation period is uncertain,
with the balance of probability favoring the view that it rotates upon
its axis once in 88 days, and therefore always turns the same face
toward the sun.

If such is the case, its climate must be very peculiar: one side roasted
in a perpetual day, where the direct heating power of the sun's rays,
when the planet is at perihelion, is ten times as great as on the moon,
and which six weeks later, when the planet is at its farthest from the
sun, has fallen off to less than half of this. On the opposite side of
the planet there must reign perpetual night and perpetual cold,
mitigated by some slight access of warmth from the day side, and perhaps
feebly imitating the rapid change of season which takes place on the day
side of the planet. This view, however, takes no account of a possible
deviation of the planet's axis from being perpendicular to the plane of
its orbit, or of the librations which must be produced by the great
eccentricity of the orbit, either of which would complicate without
entirely destroying the ideal conditions outlined above.


150. APPEARANCE.--The one remaining member of the inner group, Mars, has
in recent years received more attention than any other planet, and the
newspapers and magazines have announced marvelous things concerning it:
that it is inhabited by a race of beings superior in intelligence to
men; that the work of their hands may be seen upon the face of the
planet; that we should endeavor to communicate with them, if indeed they
are not already sending messages to us, etc.--all of which is certainly
important, if true, but it rests upon a very slender foundation of
evidence, a part of which we shall have to consider.

Beginning with facts of which there is no doubt, this ruddy-colored
planet, which usually shines about as brightly as a star of the first
magnitude, sometimes displays more than tenfold this brilliancy,
surpassing every other planet save Venus and presenting at these times
especially favorable opportunities for the study of its surface. The
explanation of this increase of brilliancy is, of course, that the
planet approaches unusually near to the earth, and we have already seen
from a consideration of Fig. 17 that this can only happen in the months
of August and September. The last favorable epoch of this kind was in
1894. From Fig. 17 the student should determine when the next one will

[Illustration: FIG. 95.--Mars.--SCHAEBERLE.]

Fig. 95 presents nine drawings of the planet made at one of the epochs
of close approach to the earth, and shows that its face bears certain
faint markings which, though inconspicuous, are fixed and permanent
features of the planet. The dark triangular projection in the lower
half of the second drawing was seen and sketched by Huyghens, 1659
A. D. In Fig. 96 some of these markings are shown much more plainly, but
Fig. 95 gives a better idea of their usual appearance in the telescope.

[Illustration: FIG. 96.--Four views of Mars differing 90° in

151. ROTATION.--It may be seen readily enough, from a comparison of the
first two sketches of Fig. 95, that the planet rotates about an axis,
and from a more extensive study it is found to be very like the earth in
this respect, turning once in 24h. 37m. around an axis tipped from being
perpendicular to the plane of its orbit about a degree and a half more
than is the earth's axis. Since it is this inclination of the axis which
is the cause of changing seasons upon the earth, there must be similar
changes, winter and summer, as well as day and night, upon Mars, only
each season is longer there than here in the same proportion that its
year is longer than ours--i. e., nearly two to one. It is summer in the
northern hemisphere of Mars whenever the sun, as seen from Mars, stands
in that constellation which is nearest the point of the sky toward which
the planet's axis points. But this axis points toward the constellation
Cygnus, and Alpha Cygni is the bright star nearest the north pole of
Mars. As Pisces is the zodiacal constellation nearest to Cygnus, it must
be summer in the northern hemisphere of Mars when the sun is in Pisces,
or, turning the proposition about, it must be summer in the _southern_
hemisphere of Mars when the planet, as seen from the sun, lies in the
direction of Pisces.

152. THE POLAR CAPS.--One effect of the changing seasons upon Mars is
shown in Fig. 97, where we have a series of drawings of the region about
its south pole made in 1894, on dates between May 21st and December
10th. Show from Fig. 17 that during this time it was summer in the
region here shown. Mars crossed the prime radius in 1894 on September
5th. The striking thing in these pictures is the white spot surrounding
the pole, which shrinks in size from the beginning to near the end of
the series, and then disappears altogether. The spot came back again a
year later, and like a similar spot at the north pole of the planet it
waxes in the winter and wanes during the summer of Mars in endless

[Illustration: FIG. 97.--The south polar cap of Mars in 1894.--BARNARD.]

Sir W. Herschel, who studied these appearances a century ago, compared
them with the snow fields which every winter spread out from the region
around the terrestrial pole, and in the summer melt and shrink, although
with us they do not entirely disappear. This explanation of the polar
caps of Mars has been generally accepted among astronomers, and from it
we may draw one interesting conclusion: the temperature upon Mars
between summer and winter oscillates above and below the freezing point
of water, as it does in the temperate zones of the earth. But this
conclusion plunges us into a serious difficulty. The temperature of the
earth is made by the sun, and at the distance of Mars from the sun the
heating effect of the latter is reduced to less than half what it is at
the earth, so that, if Mars is to be kept at the same temperature as the
earth, there must be some peculiar means for storing the solar heat and
using it more economically than is done here. Possibly there is some
such mechanism, although no one has yet found it, and some astronomers
are very confident that it does not exist, and assert that the
comparison of the polar caps with snow fields is misleading, and that
the temperature upon Mars must be at least 100°, and perhaps 200° or
more, below zero.

153. ATMOSPHERE AND CLIMATE.--In this connection one feature of Mars is
of importance. The markings upon its surface are always visible when
turned toward the earth, thus showing that the atmosphere contains no
such amount of cloud as does our own, but on the whole is decidedly
clear and sunny, and presumably much less dense than ours. We have seen
in comparing the earth and the moon how important is the service which
the earth's atmosphere renders in storing the sun's heat and checking
those great vicissitudes of temperature to which the moon is subject;
and with this in mind we must regard the smaller density and cloudless
character of the atmosphere of Mars as unfavorable to the maintenance
there of a temperature like that of the earth. Indeed, this
cloudlessness must mean one of two things: either the temperature is so
low that vapors can not exist in any considerable quantity, or the
surface of Mars is so dry that there is little water or other liquid to
be evaporated. The latter alternative is adopted by those astronomers
who look upon the polar caps as true snow fields, which serve as the
chief reservoir of the planet's water supply, and who find in Fig. 98
evidence that as the snow melts and the water flows away over the flat,
dry surface of the planet, vegetation springs up, as shown by the dark
markings on the disk, and gradually dies out with the advancing season.
Note that in the first of these pictures the season upon Mars
corresponds to the end of May with us, and in the last picture to the
beginning of August, a period during which in much of our western
country the luxuriant vegetation of spring is burned out by the
scorching sun. From this point of view the permanent dark spots are the
low-lying parts of the planet's surface, in which at all times there is
a sufficient accumulation of water to support vegetable life.

[Illustration: FIG. 98.--The same face of Mars at three different

154. THE CANALS.--In Fig. 98 the lower part of the disk of Mars shows
certain faint dark lines which are generally called canals, and in Plate
III there is given a map of Mars showing many of these canals running in
narrow, dusky streaks across the face of the planet according to a
pattern almost as geometrical as that of a spider's web. This must not
be taken for a picture of the planet's appearance in a telescope. No man
ever saw Mars look like this, but the map is useful as a plain
representation of things dimly seen. Some of the regions of this map are
marked Mare (sea), in accordance with the older view which regarded the
darker parts of the planet--and of the moon--as bodies of water, but
this is now known to be an error in both cases. The curved surface of a
planet can not be accurately reproduced upon the flat surface of paper,
but is always more or less distorted by the various methods of
"projecting" it which are in use. Compare the map of Mars in Plate III
with Fig. 99, in which the projection represents very well the
equatorial parts of the planet, but enormously exaggerates the region
around the poles.

It is a remarkable feature of the canals that they all begin and end in
one of these dark parts of the planet's surface; they show no loose ends
lying on the bright parts of the planet. Another even more remarkable
feature is that while the larger canals are permanent features of the
planet's surface, they at times appear "doubled"--i. e., in place of one
canal two parallel ones side by side, lasting for a time and then giving
place again to a single canal.

It is exceedingly difficult to frame any reasonable explanation of these
canals and the varied appearances which they present. The source of the
wild speculations about Mars, to which reference is made above, is to be
found in the suggestion frequently made, half in jest and half in
earnest, that the canals are artificial water courses constructed upon a
scale vastly exceeding any public works upon the earth, and testifying
to the presence in Mars of an advanced civilization. The distinguished
Italian astronomer, Schiaparelli, who has studied these formations
longer than any one else, seems inclined to regard them as water courses
lined on either side by vegetation, which flourishes as far back from
the central channel as water can be supplied from it--a plausible enough
explanation if the fundamental difficulty about temperature can be

[Illustration: FIG. 99.--A chart of Mars, 1898-'99.--CERULLI.]


155. SATELLITES.--In 1877, one of the times of near approach, Professor
Hall, of Washington, discovered two tiny satellites revolving about Mars
in orbits so small that the nearer one, Phobos, presents the remarkable
anomaly of completing the circuit of its orbit in less time than the
planet takes for a rotation about its axis. This satellite, in fact,
makes three revolutions in its orbit while the planet turns once upon
its axis, and it therefore rises in the west and sets in the east, as
seen from Mars, going from one horizon to the other in a little less
than 6 hours. The other satellite, Deimos, takes a few hours more than a
day to make the circuit of its orbit, but the difference is so small
that it remains continuously above the horizon of any given place upon
Mars for more than 60 hours at a time, and during this period runs twice
through its complete set of phases--new, first quarter, full, etc. In
ordinary telescopes these satellites can be seen only under especially
favorable circumstances, and are far too small to permit of any direct
measurement of their size. The amount of light which they reflect has
been compared with that of Mars and found to be as much inferior to it
as is Polaris to two full moons, and, judging from this comparison,
their diameters can not much exceed a half dozen miles, unless their
albedo is far less than that of Mars, which does not seem probable.


156. MINOR PLANETS.--These may be dismissed with few words. There are
about 500 of them known, all discovered since the beginning of the
nineteenth century, and new ones are still found every year. No one
pretends to remember the names which have been assigned them, and they
are commonly represented by a number inclosed in a circle, showing the
order in which they were discovered--e. g., [circle 1] = Ceres,
[circle 433] = Eros, etc. For the most part they are little more than
chips, world fragments, adrift in space, and naturally it was the larger
and brighter of them that were first discovered. The size of the first
four of them--Ceres, Pallas, Juno, and Vesta--compared with the size of
the moon, according to Professor Barnard, is shown in Fig. 100. The
great majority of them must be much smaller than the smallest of these,
perhaps not more than a score of miles in diameter.

A few of the asteroids present problems of special interest, such as
Eros, on account of its close approach to the earth; Polyhymnia, whose
very eccentric orbit makes it a valuable means for determining the mass
of Jupiter, etc.; but these are special cases and the average asteroid
now receives scant attention, although half a century ago, when only a
few of them were known, they were regarded with much interest, and the
discovery of a new one was an event of some consequence.

It was then a favorite speculation that they were in fact fragments of
an ill-fated planet which once filled the gap between the orbits of Mars
and Jupiter, but which, by some mischance, had been blown into pieces.
This is now known to be well-nigh impossible, for every fragment which
after the explosion moved in an elliptical orbit, as all the asteroids
do move, would be brought back once in every revolution to the place of
the explosion, and all the asteroid orbits must therefore intersect at
this place. But there is no such common point of intersection.

[Illustration: FIG. 100.--The size of the first four

157. LIFE ON THE PLANETS.--There is a belief firmly grounded in the
popular mind, and not without its advocates among professional
astronomers, that the planets are inhabited by living and intelligent
beings, and it seems proper at the close of this chapter to inquire
briefly how far the facts and principles here developed are consistent
with this belief, and what support, if any, they lend to it.

At the outset we must observe that the word life is an elastic term,
hard to define in any satisfactory way, and yet standing for something
which we know here upon the earth. It is this idea, our familiar though
crude knowledge of life, which lies at the root of the matter. Life, if
it exists in another planet, must be in its essential character like
life upon the earth, and must at least possess those features which are
common to all forms of terrestrial life. It is an abuse of language to
say that life in Mars may be utterly unlike life in the earth; if it is
absolutely unlike, it is not life, whatever else it may be. Now, every
form of life found upon the earth has for its physical basis a certain
chemical compound, called protoplasm, which can exist and perpetuate
itself only within a narrow range of temperature, roughly speaking,
between 0° and 100° centigrade, although these limits can be
considerably overstepped for short periods of time. Moreover, this
protoplasm can be active only in the presence of water, or water vapor,
and we may therefore establish as the necessary conditions for the
continued existence and reproduction of life in any place that its
temperature must not be permanently above 100° or below 0°, C., and
water must be present in that place in some form.

With these conditions before us it is plain that life can not exist in
the sun on account of its high temperature. It is conceivable that
active and intelligent beings, salamanders, might exist there, but they
could not properly be said to live. In Jupiter and Saturn the same
condition of high temperature prevails, and probably also in Uranus and
Neptune, so that it seems highly improbable that any of these planets
should be the home of life.

Of the inner planets, Mercury and the moon seem destitute of any
considerable atmospheres, and are therefore lacking in the supply of
water necessary for life, and the same is almost certainly true of all
the asteroids. There remain Venus, Mars, and the satellites of the outer
planets, which latter, however, we must drop from consideration as being
too little known. On Venus there is an atmosphere probably containing
vapor of water, and it is well within the range of possibility that
liquid water should exist upon the surface of this planet and that its
temperature should fall within the prescribed limits. It would, however,
be straining our actual knowledge to affirm that such is the case, or to
insist that if such were the case, life would necessarily exist upon the

On Mars we encounter the fundamental difficulty of temperature already
noted in § 152. If in some unknown way the temperature is maintained
sufficiently high for the polar caps to be real snow, thawing and
forming again with the progress of the seasons, the necessary conditions
of life would seem to be fulfilled here and life if once introduced upon
the planet might abide and flourish. But of positive proof that such is
the case we have none.

On the whole, our survey lends little encouragement to the belief in
planetary life, for aside from the earth, of all the hundreds of bodies
in the solar system, not one is found in which the necessary conditions
of life are certainly fulfilled, and only two exist in which there is a
reasonable probability that these conditions may be satisfied.



158. VISITORS IN THE SOLAR SYSTEM.--All of the objects--sun, moon,
planets, stars--which we have thus far had to consider, are permanent
citizens of the sky, and we have no reason to suppose that their present
appearance differs appreciably from what it was 1,000 years or 10,000
years ago. But there is another class of objects--comets, meteors--which
appear unexpectedly, are visible for a time, and then vanish and are
seen no more. On account of this temporary character the astronomers of
ancient and mediæval times for the most part refused to regard them as
celestial bodies but classed them along with clouds, fogs,
Jack-o'-lanterns, and fireflies, as exhalations from the swamps or the
volcano; admitting them to be indeed important as harbingers of evil to
mankind, but having no especial significance for the astronomer.

The comet of 1618 A. D. inspired the lines--

    "Eight things there be a Comet brings,
      When it on high doth horrid range:
    Wind, Famine, Plague, and Death to Kings,
      War, Earthquakes, Floods, and Direful Change,"

which, according to White (History of the Doctrine of Comets), were to
be taught in all seriousness to peasants and school children.

It was by slow degrees, and only after direct measurements of parallax
had shown some of them to be more distant than the moon, that the tide
of old opinion was turned and comets were transferred from the sublunary
to the celestial sphere, and in more recent times meteors also have
been recognized as coming to us from outside the earth. A meteor, or
shooting star as it is often called, is one of the commonest of
phenomena, and one can hardly watch the sky for an hour on any clear and
moonless night without seeing several of those quick flashes of light
which look as if some star had suddenly left its place, dashed swiftly
across a portion of the sky and then vanished. It is this misleading
appearance that probably is responsible for the name shooting star.

[Illustration: FIG. 101.--Donati's comet.--BOND.]

159. COMETS.--Comets are less common and much longer-lived than meteors,
lasting usually for several weeks, and may be visible night after night
for many months, but never for many years, at a time. During the last
decade there is no year in which less than three comets have appeared,
and 1898 is distinguished by the discovery of ten of these bodies, the
largest number ever found in one year. On the average, we may expect a
new comet to be found about once in every ten weeks, but for the most
part they are small affairs, visible only in the telescope, and a fine
large one, like Donati's comet of 1858 (Fig. 101), or the Great Comet of
September, 1882, which was visible in broad daylight close beside the
sun, is a rare spectacle, and as striking and impressive as it is rare.

[Illustration: FIG. 102.--Some famous comets.]

Note in Fig. 102 the great variety of aspect presented by some of the
more famous comets, which are here represented upon a very small scale.

Fig. 103 is from a photograph of one of the faint comets of the year
1893, which appears here as a rather feeble streak of light amid the
stars which are scattered over the background of the picture. An
apparently detached portion of this comet is shown at the extreme left
of the picture, looking almost like another independent comet. The
clean, straight line running diagonally across the picture is the flash
of a bright meteor that chanced to pass within the range of the camera
while the comet was being photographed.

A more striking representation of a moderately bright telescopic comet
is contained in Figs. 104 and 105, which present two different views of
the same comet, showing a considerable change in its appearance. A
striking feature of Fig. 105 is the star images, which are here drawn
out into short lines all parallel with each other. During the exposure
of 2h. 20m. required to imprint this picture upon the photographic
plate, the comet was continually changing its position among the stars
on account of its orbital motion, and the plate was therefore moved
from time to time, so as to follow the comet and make its image always
fall at the same place. Hence the plate was continually shifted relative
to the stars whose images, drawn out into lines, show the direction in
which the plate was moved--i. e., the direction in which the comet was
moving across the sky. The same effect is shown in the other
photographs, but less conspicuously than here on account of their
shorter exposure times.

These pictures all show that one end of the comet is brighter and
apparently more dense than the other, and it is customary to call this
bright part the _head_ of the comet, while the brushlike appendage that
streams away from it is called the comet's _tail_.

[Illustration: FIG. 103.--Brooks's comet, November 13, 1893. BARNARD.]

160. THE PARTS OF A COMET.--It is not every comet that has a tail,
though all the large ones do, and in Fig. 103 the detached piece of
cometary matter at the left of the picture represents very well the
appearance of a tailless comet, a rather large but not very bright star
of a fuzzy or hairy appearance. The word comet means long-haired or
hairy star. Something of this vagueness of outline is found in all
comets, whose exact boundaries are hard to define, instead of being
sharp and clean-cut like those of a planet or satellite. Often,
however, there is found in the head of a comet a much more solid
appearing part, like the round white ball at the center of Fig. 106,
which is called the nucleus of the comet, and appears to be in some sort
the center from which its activities radiate. As shown in Figs. 106 and
107, the nucleus is sometimes surrounded by what are called envelopes,
which have the appearance of successive wrappings or halos placed about
it, and odd, spurlike projections, called jets, are sometimes found in
connection with the envelopes or in place of them. These figures also
show what is quite a common characteristic of large comets, a dark
streak running down the axis of the tail, showing that the tail is
hollow, a mere shell surrounding empty space.

[Illustration: FIG. 104.--Swift's comet, April 17, 1892.--BARNARD.]

The amount of detail shown in Figs. 106 and 107 is, however, quite
exceptional, and the ordinary comet is much more like Fig. 103 or 104.
Even a great comet when it first appears is not unlike the detached
fragment in Fig. 103, a faint and roundish patch of foggy light which
grows through successive stages to its maximum estate, developing a
tail, nucleus, envelopes, etc., only to lose them again as it shrinks
and finally disappears.

[Illustration: FIG. 105.--Swift's comet, April 24, 1892.--BARNARD.]

161. THE ORBITS OF COMETS.--It will be remembered that Newton found, as
a theoretical consequence of the law of gravitation, that a body moving
under the influence of the sun's attraction might have as its orbit any
one of the conic sections, ellipse, parabola, or hyperbola, and among
the 400 and more comet orbits which have been determined every one of
these orbit forms appears, but curiously enough there is not a hyperbola
among them which, if drawn upon paper, could be distinguished by the
unaided eye from a parabola, and the ellipses are all so long and
narrow, not one of them being so nearly round as is the most eccentric
planet orbit, that astronomers are accustomed to look upon the parabola
as being the normal type of comet orbit, and to regard a comet whose
motion differs much from a parabola as being abnormal and calling for
some special explanation.

The fact that comet orbits are parabolas, or differ but little from
them, explains at once the temporary character and speedy disappearance
of these bodies. They are visitors to the solar system and visible for
only a short time, because the parabola in which they travel is not a
closed curve, and the comet, having passed once along that portion of it
near the earth and the sun, moves off along a path which ever thereafter
takes it farther and farther away, beyond the limit of visibility. The
development of the comet during the time it is visible, the growth and
disappearance of tail, nucleus, etc., depend upon its changing distance
from the sun, the highest development and most complex structure being
presented when it is nearest to the sun.

[Illustration: FIG. 106.--Head of Coggia's comet, July 13,

Fig. 108 shows the path of the Great Comet of 1882 during the period in
which it was seen, from September 3, 1882, to May 26, 1883. These
dates--IX, 3, and V, 26--are marked in the figure opposite the parts of
the orbit in which the comet stood at those times. Similarly, the
positions of the earth in its orbit at the beginning of September,
October, November, etc., are marked by the Roman numerals IX, X, XI,
etc. The line _S V_ shows the direction from the sun to the vernal
equinox, and _S_ Ω is the line along which the plane of the
comet's orbit intersects the plane of the earth's orbit--i. e., it is
the line of nodes of the comet orbit. Since the comet approached the sun
from the south side of the ecliptic, all of its orbit, save the little
segment which falls to the left of _S_ Ω, lies below (south) of
the plane of the earth's orbit, and the part which would be hidden if
this plane were opaque is represented by a broken line.

[Illustration: FIG. 107.--Head of Donati's comet, September 30, October
2, 1858.--BOND.]

162. ELEMENTS OF A COMET'S ORBIT.--There is a theorem of geometry to the
effect that through any three points not in the same straight line one
circle, and only one, can be drawn. Corresponding to this there is a
theorem of celestial mechanics, that through any three positions of a
comet one conic section, and only one, can be passed along which the
comet can move in accordance with the law of gravitation. This conic
section is, of course, its orbit, and at the discovery of a comet
astronomers always hasten to observe its position in the sky on
different nights in order to obtain the three positions (right
ascensions and declinations) necessary for determining the particular
orbit in which it moves. The circle, to which reference was made above,
is completely ascertained and defined when we know its radius and the
position of its center. A parabola is not so simply defined, and five
numbers, called the _elements_ of its orbit, are required to fix
accurately a comet's path around the sun. Two of these relate to the
position of the line of nodes and the angle which the orbit plane makes
with the plane of the ecliptic; a third fixes the direction of the axis
of the orbit in its plane, and the remaining two, which are of more
interest to us, are the date at which the comet makes its nearest
approach to the sun (_perihelion passage_) and its distance from the sun
at that date (_perihelion distance_). The date, September 17th, placed
near the center of Fig. 108, is the former of these elements, while the
latter, which is too small to be accurately measured here, may be found
from Fig. 109 to be 0.82 of the sun's diameter, or, in terms of the
earth's distance from the sun, 0.008.

[Illustration: FIG. 108.--Orbits of the earth and the Great Comet of

Fig. 109 shows on a large scale the shape of that part of the orbit near
the sun and gives the successive positions of the comet, at intervals of
2/10 of a day, on September 16th and 17th, showing that in less than 10
hours--17.0 to 17.4--the comet swung around the sun through an angle of
more than 240°. When at its perihelion it was moving with a velocity of
300 miles per second! This very unusual velocity was due to the comet's
extraordinarily close approach to the sun. The earth's velocity in its
orbit is only 19 miles per second, and the velocity of any comet at any
distance from the sun, provided its orbit is a parabola, may be found
by dividing this number by the square root of half the comet's
distance--e. g., 300 miles per second equals 19 ÷ √ 0.004.

[Illustration: FIG. 109.--Motion of the Great Comet of 1883 in passing
around the sun.]

Most of the visible comets have their perihelion distances included
between 1/3 and 4/3 of the earth's distance from the sun, but
occasionally one is found, like the second comet of 1885, whose nearest
approach to the sun lies far outside the earth's orbit, in this case
halfway out to the orbit of Jupiter; but such a comet must be a very
large one in order to be seen at all from the earth. There is, however,
some reason for believing that the number of comets which move around
the sun without ever coming inside the orbit of Jupiter, or even that of
Saturn, is much larger than the number of those which come close enough
to be discovered from the earth. In any case we are reminded of Kepler's
saying, that comets in the sky are as plentiful as fishes in the sea,
which seems to be very little exaggerated when we consider that,
according to Kleiber, out of all the comets which enter the solar system
probably not more than 2 or 3 per cent are ever discovered.

[Illustration: FIG. 110.--The Great Comet of 1843.]

163. DIMENSIONS OF COMETS.--The comet whose orbit is shown in Figs. 108
and 109 is the finest and largest that has appeared in recent years. Its
tail, which at its maximum extent would have more than bridged the space
between sun and earth (100,000,000 miles), is made very much too short
in Fig. 109, but when at its best was probably not inferior to that of
the Great Comet of 1843, shown in Fig. 110. As we shall see later,
there is a peculiar and special relationship between these two comets.

The head of the comet of 1882 was not especially large--about twice the
diameter of the ball of Saturn--but its nucleus, according to an
estimate made by Dr. Elkin when it was very near perihelion, was as
large as the moon. The head of the comet shown in Fig. 107 was too large
to be put in the space between the earth and the moon, and the Great
Comet of 1811 had a head considerably larger than the sun itself. From
these colossal sizes down to the smallest shred just visible in the
telescope, comets of all dimensions may be found, but the smaller the
comet the less the chance of its being discovered, and a comet as small
as the earth would probably go unobserved unless it approached very
close to us.

164. THE MASS OF A COMET.--There is no known case in which the mass of a
comet has ever been measured, yet nothing about them is more sure than
that they are bodies with mass which is attracted by the sun and the
planets, and which in its turn attracts both sun and planets and
produces perturbations in their motion. These perturbations are,
however, too small to be measured, although the corresponding
perturbations in the comet's motion are sometimes enormous, and since
these mutual perturbations are proportional to the masses of comet and
planet, we are forced to say that, by comparison with even such small
bodies as the moon or Mercury, the mass of a comet is utterly
insignificant, certainly not as great as a ten-thousandth part of the
mass of the earth. In the case of the Great Comet of 1882, if we leave
its hundred million miles of tail out of account and suppose the entire
mass condensed into its head, we find by a little computation that the
average density of the head under these circumstances must have been
less than 1/1500 of the density of air. In ordinary laboratory practice
this would be called a pretty good vacuum. A striking observation made
on September 17, 1882, goes to confirm the very small density of this
comet. It is shown in Fig. 109 that early on that day the comet crossed
the line joining earth and sun, and therefore passed in transit over the
sun's disk. Two observers at the Cape of Good Hope saw the comet
approach the sun, and followed it with their telescopes until the
nucleus actually reached the edge of the sun and disappeared, behind it
as they supposed, for no trace of the comet, not even its nucleus, could
be seen against the sun, although it was carefully looked for. Now, the
figure shows that the comet passed between the earth and sun, and its
densest parts were therefore too attenuated to cut off any perceptible
fraction of the sun's rays. In other cases stars have been seen through
the head of a comet, shining apparently with undimmed luster, although
in some cases they seem to have been slightly refracted out of their
true positions.

165. METEORS.--Before proceeding further with the study of comets it is
well to turn aside and consider their humbler relatives, the shooting
stars. On some clear evening, when the moon is absent from the sky,
watch the heavens for an hour and count the meteors visible during that
time. Note their paths, the part of the sky where they appear and where
they disappear, their brightness, and whether they all move with equal
swiftness. Out of such simple observations with the unaided eye there
has grown a large and important branch of astronomical science, some
parts of which we shall briefly summarize here.

A particular meteor is a local phenomenon seen over only a small part of
the earth's surface, although occasionally a very big and bright one may
travel and be visible over a considerable territory. Such a one in
December, 1876, swept over the United States from Kansas to
Pennsylvania, and was seen from eleven different States. But the
ordinary shooting star is much less conspicuous, and, as we know from
simultaneous observations made at neighboring places, it makes its
appearance at a height of some 75 miles above the earth's surface,
occupies something like a second in moving over its path, and then
disappears at a height of about 50 miles or more, although occasionally
a big one comes down to the very surface of the earth with force
sufficient to bury itself in the ground, from which it may be dug up,
handled, weighed, and turned over to the chemist to be analyzed. The
pieces thus found show that the big meteors, at least, are masses of
stone or mineral; iron is quite commonly found in them, as are a
considerable number of other terrestrial substances combined in rather
peculiar ways. But no chemical element not found on the earth has ever
been discovered in a meteor.

166. NATURE OF METEORS.--The swiftness with which the meteors sweep down
shows that they must come from outside the earth, for even half their
velocity, if given to them by some terrestrial volcano or other
explosive agent, would send them completely away from the earth never to
return. We must therefore look upon them as so many projectiles,
bullets, fired against the earth from some outside source and arrested
in their motion by the earth's atmosphere, which serves as a cushion to
protect the ground from the bombardment which would otherwise prove in
the highest degree dangerous to both property and life. The speed of the
meteor is checked by the resistance which the atmosphere offers to its
motion, and the energy represented by that speed is transformed into
heat, which in less than a second raises the meteor and the surrounding
air to incandescence, melts the meteor either wholly or in part, and
usually destroys its identity, leaving only an impalpable dust, which
cools off as it settles slowly through the lower atmosphere to the
ground. The heating effect of the air's resistance is proportional to
the square of the meteor's velocity, and even at such a moderate speed
as 1 mile per second the effect upon the meteor is the same as if it
stood still in a bath of red-hot air. Now, the actual velocity of
meteors through the air is often 30 or 40 times as great as this, and
the corresponding effect of the air in raising its temperature is more
than 1,000 times that of red heat. Small wonder that the meteor is
brought to lively incandescence and consumed even in a fraction of a

167. THE NUMBER OF METEORS.--A single observer may expect to see in the
evening hours about one meteor every 10 minutes on the average,
although, of course, in this respect much irregularity may occur. Later
in the night they become more frequent, and after 2 A. M. there are
about three times as many to be seen as in the evening hours. But no one
person can keep a watch upon the whole sky, high and low, in front and
behind, and experience shows that by increasing the number of observers
and assigning to each a particular part of the sky, the total number of
meteors counted may be increased about five-fold. So, too, the observers
at any one place can keep an effective watch upon only those meteors
which come into the earth's atmosphere within some moderate distance of
their station, say 50 or 100 miles, and to watch every part of that
atmosphere would require a large number of stations, estimated at
something more than 10,000, scattered systematically over the whole face
of the earth. If we piece together the several numbers above considered,
taking 14 as a fair average of the hourly number of meteors to be seen
by a single observer at all hours of the night, we shall find for
the total number of meteors encountered by the earth in 24 hours,
14 × 5 × 10,000 × 24 = 16,800,000. Without laying too much stress upon
this particular number, we may fairly say that the meteors picked up by
the earth every day are to be reckoned by millions, and since they come
at all seasons of the year, we shall have to admit that the region
through which the earth moves, instead of being empty space, is really a
dust cloud, each individual particle of dust being a prospective meteor.

On the average these individual particles are very small and very far
apart; a cloud of silver dimes each about 250 miles from its nearest
neighbor is perhaps a fair representation of their average mass and
distance from each other, but, of course, great variations are to be
expected both in the size and in the frequency of the particles. There
must be great numbers of them that are too small to make shooting stars
visible to the naked eye, and such are occasionally seen darting by
chance across the field of view of a telescope.

168. THE ZODIACAL LIGHT is an effect probably due to the reflection of
sunlight from the myriads of these tiny meteors which occupy the space
inside the earth's orbit. It is a faint and diffuse stream of light,
something like the Milky Way, which may be seen in the early evening or
morning stretching up from the sunrise or sunset point of the horizon
along the ecliptic and following its course for many degrees, possibly
around the entire circumference of the sky. It may be seen at any season
of the year, although it shows to the best advantage in spring evenings
and autumn mornings. Look for it.

169. GREAT METEORS.--But there are other meteors, veritable fireballs in
appearance, far more conspicuous and imposing than the ordinary shooting
star. Such a one exploded over the city of Madrid, Spain, on the morning
of February 10, 1896, giving in broad sunlight "a brilliant flash which
was followed ninety seconds later by a succession of terrific noises
like the discharge of a battery of artillery." Fig. 111 shows a large
meteor which was seen in California in the early evening of July 27,
1894, and which left behind it a luminous trail or cloud visible for
more than half an hour.

Not infrequently large meteors are found traveling together, two or
three or more in company, making their appearance simultaneously as did
the California meteor of October 22, 1896, which is described as triple,
the trio following one another like a train of cars, and Arago cites an
instance, from the year 1830, where within a short space of time some
forty brilliant meteors crossed the sky, all moving in the same
direction with a whistling noise and displaying in their flight all the
colors of the rainbow.

The mass of great meteors such as these must be measured in hundreds if
not thousands of pounds, and stories are current, although not very well
authenticated, of even larger ones, many tons in weight, having been
found partially buried in the ground. Of meteors which have been
actually seen to fall from the sky, the largest single fragment
recovered weighs about 500 pounds, but it is only a fragment of the
original meteor, which must have been much more massive before it was
broken up by collision with the atmosphere.

[Illustration: FIG. 111.--The California meteor of July 27, 1894.]

170. THE VELOCITY OF METEORS.--Every meteor, big or little, is subject
to the law of gravitation, and before it encounters the earth must be
moving in some kind of orbit having the sun at its focus, the particular
species of orbit--ellipse, parabola, hyperbola--depending upon the
velocity and direction of its motion. Now, the direction in which a
meteor is moving can be determined without serious difficulty from
observations of its apparent path across the sky made by two or more
observers, but the velocity can not be so readily found, since the
meteors go too fast for any ordinary process of timing. But by
photographing one of them two or three times on the same plate, with an
interval of only a tenth of a second between exposures, Dr. Elkin has
succeeded in showing, in a few cases, that their velocities varied from
20 to 25 miles per second, and must have been considerably greater than
this before the meteors encountered the earth's atmosphere. This is a
greater velocity than that of the earth in its orbit, 19 miles per
second, as might have been anticipated, since the mere fact that meteors
can be seen at all in the evening hours shows that some of them at least
must travel considerably faster than the earth, for, counting in the
direction of the earth's motion, the region of sunset and evening is
always on the rear side of the earth, and meteors in order to strike
this region must overtake it by their swifter motion. We have here, in
fact, the reason why meteors are especially abundant in the morning
hours; at this time the observer is on the front side of the earth which
catches swift and slow meteors alike, while the rear is pelted only by
the swifter ones which follow it.

A comparison of the relative number of morning and evening meteors makes
it probable that the average meteor moves, relative to the sun, with a
velocity of about 26 miles per second, which is very approximately the
average velocity of comets when they are at the earth's distance from
the sun. Astronomers, therefore, consider meteors as well as comets to
have the parabola and the elongated ellipse as their characteristic

171. METEOR SHOWERS--THE RADIANT.--There is evident among meteors a
distinct tendency for individuals, to the number of hundreds or even
hundreds of millions, to travel together in flocks or swarms, all going
the same way in orbits almost exactly alike. This gregarious tendency is
made manifest not only by the fact that from time to time there are
unusually abundant meteoric displays, but also by a striking peculiarity
of their behavior at such times. The meteors all seem to come from a
particular part of the heavens, as if here were a hole in the sky
through which they were introduced, and from which they flow away in
every direction, even those which do not visibly start from this place
having paths among the stars which, if prolonged backward, would pass
through it. The cause of this appearance may be understood from Fig.
112, which represents a group of meteors moving together along parallel
paths toward an observer at _D_. Traveling unseen above the earth until
they encounter the upper strata of its atmosphere, they here become
incandescent and speed on in parallel paths, _1_, _2_, _3_, _4_, _5_,
_6_, which, as seen by the observer, are projected back against the sky
into luminous streaks that, as is shown by the arrowheads, _b_, _c_,
_d_, all seem to radiate from the point _a_--i. e., from the point in
the sky whose direction from the observer is parallel to the paths of
the meteors.

[Illustration: FIG. 112.--Explanation of the radiant of a meteoric

Such a display is called a meteor shower, and the point _a_ is called
its radiant. Note how those meteors which appear near the radiant all
have short paths, while those remote from it in the sky have longer
ones. Query: As the night wears on and the stars shift toward the west,
will the radiant share in their motion or will it be left behind? Would
the luminous part of the path of any of these meteors pass across the
radiant from one side to the other? Is such a crossing of the radiant
possible under any circumstances? Fig. 113 shows how the meteor paths
are grouped around the radiant of a strongly marked shower. Select from
it the meteors which do not belong to this shower.

[Illustration: FIG. 113.--The radiant of a meteoric shower, showing
also the paths of three meteors which do not belong to this

Many hundreds of these radiants have been observed in the sky, each of
which represents an orbit along which a group of meteors moves, and the
relation of one of these orbits to that of the earth is shown in Fig.
114. The orbit of the meteors is an ellipse extending out beyond the
orbit of Uranus, but so eccentric that a part of it comes inside the
orbit of the earth, and the figure shows only that part of it which lies
nearest the sun. The Roman numerals which are placed along the earth's
orbit show the position of the earth at the beginning of the tenth
month, eleventh month, etc. The meteors flow along their orbit in a long
procession, whose direction of motion is indicated by the arrow heads,
and the earth, coming in the opposite direction, plunges into this
stream and receives the meteor shower when it reaches the intersection
of the two orbits. The long arrow at the left of the figure represents
the direction of motion of another meteor shower which encounters the
earth at this point.

[Illustration: FIG. 114.--The orbits of the earth and the November

Can you determine from the figure answers to the following questions? On
what day of the year will the earth meet each of these showers? Will the
radiant points of the showers lie above or below the plane of the
earth's orbit? Will these meteors strike the front or the rear of the
earth? Can they be seen in the evening hours?

From many of the radiants year after year, upon the same day or week in
each year, there comes a swarm of shooting stars, showing that there
must be a continuous procession of meteors moving along this orbit, so
that some are always ready to strike the earth whenever it reaches the
intersection of its orbit with theirs. Such is the explanation of the
shower which appears each year in the first half of August, and whose
meteors are sometimes called Perseids, because their radiant lies in the
constellation Perseus, and a similar explanation holds for all the star
showers which are repeated year after year.

172. THE LEONIDS.--There is, however, a kind of star shower, of which
the Leonids (radiant in Leo) is the most conspicuous type, in which the
shower, although repeated from year to year, is much more striking in
some years than in others. Thus, to quote from the historian: "In 1833
the shower was well observed along the whole eastern coast of North
America from the Gulf of Mexico to Halifax. The meteors were most
numerous at about 5 A. M. on November 13th, and the rising sun could not
blot out all traces of the phenomena, for large meteors were seen now
and then in full daylight. Within the scope that the eye could contain,
more than twenty could be seen at a time shooting in every direction.
Not a cloud obscured the broad expanse, and millions of meteors sped
their way across in every point of the compass. Their coruscations were
bright, gleaming, and incessant, and they fell thick as the flakes in
the early snows of December." But, so far as is known, none of them
reached the ground. An illiterate man on the following day remarked:
"The stars continued to fall until none were left. I am anxious to see
how the heavens will appear this evening, for I believe we shall see no
more stars."

An eyewitness in the Southern States thus describes the effect of this
shower upon the plantation negroes: "Upward of a hundred lay prostrate
upon the ground, some speechless and some with the bitterest cries, but
with their hands upraised, imploring God to save the world and them. The
scene was truly awful, for never did rain fall much thicker than the
meteors fell toward the earth--east, west, north, and south it was the
same." In the preceding year a similar but feebler shower from the same
radiant created much alarm in France, and through the old historic
records its repetitions may be traced back at intervals of 33 or 34
years, although with many interruptions, to October 12, 902, O. S., when
"an immense number of falling stars were seen to spread themselves over
the face of the sky like rain."

Such a star shower differs from the one repeated every year chiefly in
the fact that its meteors, instead of being drawn out into a long
procession, are mainly clustered in a single flock which may be long
enough to require two or three or four years to pass a given point of
its orbit, but which is far from extending entirely around it, so that
meteors from this source are abundant only in those years in which the
flock is at or near the intersection of its orbit with that of the
earth. The fact that the Leonid shower is repeated at intervals of 33 or
34 years (it appeared in 1799, 1832-'33, 1866-'67) shows that this is
the "periodic time" in its orbit, which latter must of course be an
ellipse, and presumably a long and narrow one. It is this orbit which is
shown in Fig. 114, and the student should note in this figure that if
the meteor stream at the point where it cuts through the plane of the
earth's orbit were either nearer to or farther from the sun than is the
earth there could be no shower; the earth and the meteors would pass by
without a collision. Now, the meteors in their motion are subject to
perturbations, particularly by the large planets Jupiter, Saturn, and
Uranus, which slightly change the meteor orbit, and it seems certain
that the changes thus produced will sometimes thrust the swarm inside or
outside the orbit of the earth, and thus cause a failure of the shower
at times when it is expected. The meteors were due at the crossing of
the orbits in November, 1899 and 1900, and, although a few were then
seen, the shower was far from being a brilliant one, and its failure was
doubtless caused by the outer planets, which switched the meteors aside
from the path in which they had been moving for a century. Whether they
will be again switched back so as to produce future showers is at the
present time uncertain.

173. CAPTURE OF THE LEONIDS.--But a far more striking effect of
perturbations is to be found in Fig. 115, which shows the relation of
the Leonid orbit to those of the principal planets, and illustrates a
curious chapter in the history of the meteor swarm that has been worked
out by mathematical analysis, and is probably a pretty good account of
what actually befell them. Early in the second century of the Christian
era this flock of meteors came down toward the sun from outer space,
moving along a parabolic orbit which would have carried it just inside
the orbit of Jupiter, and then have sent it off to return no more. But
such was not to be its fate. As it approached the orbit of Uranus, in
the year 126 A. D., that planet chanced to be very near at hand and
perturbed the motion of the meteors to such an extent that the character
of their orbit was completely changed into the ellipse shown in the
figure, and in this new orbit they have moved from that time to this,
permanent instead of transient members of the solar system. The
perturbations, however, did not end with the year in which the meteors
were captured and annexed to the solar system, but ever since that time
Jupiter, Saturn, and Uranus have been pulling together upon the orbit,
and have gradually turned it around into its present position as shown
in the figure, and it is chiefly this shifting of the orbit's position
in the thousand years that have elapsed since 902 A. D. that makes the
meteor shower now come in November instead of in October as it did

[Illustration: FIG. 115.--Supposed capture of the November meteors by

174. BREAKING UP A METEOR SWARM.--How closely packed together these
meteors were at the time of their annexation to the solar system is
unknown, but it is certain that ever since that time the sun has been
exerting upon them a tidal influence tending to break up the swarm and
distribute its particles around the orbit, as the Perseids are
distributed, and, given sufficient time, it will accomplish this, but up
to the present the work is only partly done. A certain number of the
meteors have gained so much over the slower moving ones as to have made
an extra circuit of the orbit and overtaken the rear of the procession,
so that there is a thin stream of them extending entirely around the
orbit and furnishing in every November a Leonid shower; but by far the
larger part of the meteors still cling together, although drawn out into
a stream or ribbon, which, though very thin, is so long that it takes
some three years to pass through the perihelion of its orbit. It is only
when the earth plunges through this ribbon, as it should in 1899, 1900,
1901, that brilliant Leonid showers can be expected.

175. RELATION OF COMETS AND METEORS.--It appears from the foregoing that
meteors and comets move in similar orbits, and we have now to push the
analogy a little further and note that in some instances at least they
move in identically the same orbit, or at least in orbits so like that
an appreciable difference between them is hardly to be found. Thus a
comet which was discovered and observed early in the year 1866, moves in
the same orbit with the Leonid meteors, passing its perihelion about ten
months ahead of the main body of the meteors. If it were set back in its
orbit by ten months' motion, _it would be a part of the meteor swarm_.
Similarly, the Perseid meteors have a comet moving in their orbit
actually immersed in the stream of meteor particles, and several other
of the more conspicuous star showers have comets attending them.

Perhaps the most remarkable case of this character is that of a shower
which comes in the latter part of November from the constellation
Andromeda, and which from its association with the comet called Biela
(after the name of its discoverer) is frequently referred to as the
Bielid shower. This comet, an inconspicuous one moving in an unusually
small elliptical orbit, had been observed at various times from 1772
down to 1846 without presenting anything remarkable in its appearance;
but about the beginning of the latter year, with very little warning, it
broke in two, and for three months the pieces were watched by
astronomers moving off, side by side, something more than half as far
apart as are the earth and moon. It disappeared, made the circuit of its
orbit, and six years later came back, with the fragments nearly ten
times as far apart as before, and after a short stay near the earth once
more disappeared in the distance, never to be seen again, although the
fragments should have returned to perihelion at least half a dozen times
since then. In one respect the orbit of the comet was remarkable: it
passed through the place in which the earth stands on November 27th of
each year, so that if the comet were at that particular part of its
orbit on any November 27th, a collision between it and the earth would
be inevitable. So far as is known, no such collision with the comet has
ever occurred, but the Bielid meteors which are strung along its orbit
do encounter the earth on that date, in greater or less abundance in
different years, and are watched with much interest by the astronomers
who look upon them as the final appearance of the _débris_ of a worn-out

176. PERIODIC COMETS.--The Biela comet is a specimen of the type which
astronomers call periodic comets--i. e., those which move in small
ellipses and have correspondingly short periodic times, so that they
return frequently and regularly to perihelion. The comets which
accompany the other meteor swarms--Leonids, Perseids, etc.--also belong
to this class as do some 30 or 40 others which have periodic times less
than a century. As has been already indicated, these deviations from the
normal parabolic orbit call for some special explanation, and the
substance of that explanation is contained in the account of the Leonid
meteors and their capture by Uranus. Any comet may be thus captured by
the attraction of a planet near which it passes. It is only necessary
that the perturbing action of the planet should result in a diminution
of the comet's velocity, for we have already learned that it is this
velocity which determines the character of the orbit, and anything less
than the velocity appropriate to a parabola must produce an
ellipse--i. e., a closed orbit around which the body will revolve time
after time in endless succession. We note in Fig. 115 that when the
Leonid swarm encountered Uranus it passed _in front of_ the planet and
had its velocity diminished and its orbit changed into an ellipse
thereby. It might have passed behind Uranus, it would have passed behind
had it come a little later, and the effect would then have been just the
opposite. Its velocity would have been increased, its orbit changed to a
hyperbola, and it would have left the solar system more rapidly than it
came into it, thrust out instead of held in by the disturbing planet. Of
such cases we can expect no record to remain, but the captured comet is
its own witness to what has happened, and bears imprinted upon its orbit
the brand of the planet which slowed down its motion. Thus in Fig. 115
the changed orbit of the meteors has its _aphelion_ (part remotest from
the sun) quite close to the orbit of Uranus, and one of its nodes, ℧,
the point in which it cuts through the plane of the ecliptic from north
to south side, is also very near to the same orbit. It is these two
marks, aphelion and node, which by their position identify Uranus as the
planet instrumental in capturing the meteor swarm, and the date of the
capture is found by working back with their respective periodic times to
an epoch at which planet and comet were simultaneously near this node.

Jupiter, by reason of his great mass, is an especially efficient
capturer of comets, and Fig. 116 shows his group of captives, his
family of comets as they are sometimes called. The several orbits are
marked with the names commonly given to the comets. Frequently this is
the name of their discoverer, but often a different system is
followed--e. g., the name 1886, IV, means the fourth comet to pass
through perihelion in the year 1886. The other great planets--Saturn,
Uranus, Neptune--have also their families of captured comets, and
according to Schulhof, who does not entirely agree with the common
opinion about captured comets, the earth has caught no less than nine of
these bodies.

[Illustration: FIG. 116.--Jupiter's family of comets.]

177. COMET GROUPS.--But there is another kind of comet family, or comet
group as it is called, which deserves some notice, and which is best
exemplified by the Great Comet of 1882 and its relatives. No less than
four other comets are known to be traveling in substantially the same
orbit with this one, the group consisting of comets 1668, I; 1843, I;
1880, I; 1882, II; 1887, I. The orbit itself is not quite a parabola,
but a very elongated ellipse, whose major axis and corresponding
periodic time can not be very accurately determined from the available
data, but it certainly extends far beyond the orbit of Neptune, and
requires not less than 500 years for the comet to complete a revolution
in it. It was for a time supposed that some one of the recent comets of
this group of five might be a return of the comet of 1668 brought back
ahead of time by unknown perturbations. There is still a possibility of
this, but it is quite out of the question to suppose that the last four
members of the group are anything other than separate and distinct
comets moving in practically the same orbit. This common orbit suggests
a common origin for the comets, but leaves us to conjecture how they
became separated.

The observed orbits of these five comets present some slight
discordances among themselves, but if we suppose each comet to move in
the average of the observed paths it is a simple matter to fix their
several positions at the present time. They have all receded from the
sun nearly on line toward the bright star Sirius, and were all of them,
at the beginning of the year 1900, standing nearly motionless inside of
a space not bigger than the sun and distant from the sun about 150 radii
of the earth's orbit. The great rapidity with which they swept through
that part of their orbit near the sun (see § 162) is being compensated
by the present extreme slowness of their motions, so that the comets of
1668 and 1882, whose passages through the solar system were separated by
an interval of more than two centuries, now stand together near the
aphelion of their orbits, separated by a distance only 50 per cent
greater than the diameter of the moon's orbit, and they will continue
substantially in this position for some two or three centuries to come.

The slowness with which these bodies move when far from the sun is
strikingly illustrated by an equation of celestial mechanics which for
parabolic orbits takes the place of Kepler's Third Law--viz.:

    r^3 / T^2 = 178,

where _T_ is the time, in years, required for the comet to move from its
perihelion to any remote part of the orbit, whose distance from the sun
is represented, in radii of the earth's orbit, by _r_. If the comet of
1668 had moved in a parabola instead of the ellipse supposed above, how
many years would have been required to reach its present distance from
the sun?

178. RELATION OF COMETS TO THE SOLAR SYSTEM.--The orbits of these comets
illustrate a tendency which is becoming ever more strongly marked.
Because comet orbits are nearly parabolas, it used to be assumed that
they were exactly parabolic, and this carried with it the conclusion
that comets have their origin outside the solar system. It may be so,
and this view is in some degree supported by the fact that these nearly
parabolic orbits of both comets and meteors are tipped at all possible
angles to the plane of the ecliptic instead of lying near it as do the
orbits of the planets; and by the further fact that, unlike the planets,
the comets show no marked tendency to move around their orbits in the
direction in which the sun rotates upon his axis. There is, in fact, the
utmost confusion among them in this respect, some going one way and some
another. The law of the solar system (gravitation) is impressed upon
their movements, but its order is not.

But as observations grow more numerous and more precise, and comet
orbits are determined with increasing accuracy, there is a steady gain
in the number of elliptic orbits at the expense of the parabolic ones,
and if comets are of extraneous origin we must admit that a very
considerable percentage of them have their velocities slowed down
within the solar system, perhaps not so much by the attraction of the
planets as by the resistance offered to their motion by meteor particles
and swarms along their paths. A striking instance of what may befall a
comet in this way is shown in Fig. 117, where the tail of a comet
appears sadly distorted and broken by what is presumed to have been a
collision with a meteor swarm. A more famous case of impeded motion is
offered by the comet which bears the name of Encke. This has a periodic
time less than that of any other known comet, and at intervals of forty
months comes back to perihelion, each time moving in a little smaller
orbit than before, unquestionably on account of some resistance which it
has suffered.

[Illustration: FIG. 117.--Brooks's comet, October 21, 1893.--BARNARD.]

179. THE DEVELOPMENT OF A COMET.--We saw in § 174 that the sun's action
upon a meteor swarm tends to break it up into a long stream, and the
same tendency to break up is true of comets whose attenuated substance
presents scant resistance to this force. According to the mathematical
analysis of Roche, if the comet stood still the sun's tidal force would
tend first to draw it out on line with the sun, just as the earth's
tidal force pulled the moon out of shape (§ 42), and then it would cause
the lighter part of the comet's substance to flow away from both ends of
this long diameter. This destructive action of the sun is not limited to
comets and meteor streams, for it tends to tear the earth and moon to
pieces as well; but the densities and the resulting mutual attractions
of their parts are far too great to permit this to be accomplished.

As a curiosity of mathematical analysis we may note that a spherical
cloud of meteors, or dust particles weighing a gramme each, and placed
at the earth's distance from the sun, will be broken up and dissipated
by the sun's tidal action if the average distance between the particles
exceeds two yards. Now, the earth is far more dense than such a cloud,
whose extreme tenuity, however, suggests what we have already learned of
the small density of comets, and prepares us in their case for an
outflow of particles at both ends of the diameter directed toward the
sun. Something of this kind actually occurs, for the tail of a comet
streams out on the side opposite to the sun, and in general points away
from the sun, as is shown in Fig. 109, and the envelopes and jets rise
up toward the sun; but an inspection of Fig. 106 will show that the tail
and the envelope are too unlike to be produced by one and the same set
of forces.

It was long ago suggested that the sun possibly exerts upon a comet's
substance a repelling force in addition to the attracting force which we
call gravity. We think naturally in this connection of the repelling
force which a charge of electricity exerts upon a similar charge placed
on a neighboring body, and we note that if both sun and comet carried a
considerable store of electricity upon their surfaces this would furnish
just such a repelling force as seems indicated by the phenomena of
comets' tails; for the force of gravity would operate between the
substance of sun and comet, and on the whole would be the controlling
force, while the electric charges would produce a repulsion, relatively
feeble for the big particles and strong for the little ones, since an
electric charge lies wholly on the surface, while gravity permeates the
whole mass of a body, and the ratio of volume (gravity) to surface
(electric charge) increases rapidly with increasing size. The repelling
force would thrust back toward the comet those particles which flowed
out toward the sun, while it would urge forward those which flowed away
from it, thus producing the difference in appearance between tail and
envelopes, the latter being regarded from this standpoint as stunted
tails strongly curved backward. In recent years the Russian astronomer
Bredichin has made a careful study of the shape and positions of comets'
tails and finds that they fit with mathematical precision to the
theories of electric repulsion.

180. COMET TAILS.--According to Bredichin, a comet's tail is formed by
something like the following process: In the head of the comet itself a
certain part of its matter is broken up into fine bits, single molecules
perhaps, which, as they no longer cling together, may be described as in
the condition of vapor. By the repellent action of both sun and comet
these molecules are cast out from the head of the comet and stream away
in the direction opposite to the sun with different velocities, the
heavy ones slowly and the light ones faster, much as particles of smoke
stream away from a smokestack, making for the comet a tail which like a
trail of smoke is composed of constantly changing particles. The result
of this process is shown in Fig. 118, where the positions of the comet
in its orbit on successive days are marked by the Roman numerals, and
the broken lines represent the paths of molecules _m^{I}_, _m^{II}_,
_m^{III}_, etc., expelled from it on their several dates and traveling
thereafter in orbits determined by the combined effect of the sun's
attraction, the sun's repulsion, and the comet's repulsion. The comet's
attraction (gravity) is too small to be taken into account. The line
drawn upward from _VI_ represents the positions of these molecules on
the sixth day, and shows that all of them are arranged in a tail
pointing nearly away from the sun. A similar construction for the other
dates gives the corresponding positions of the tail, always pointing
away from the sun.

[Illustration: FIG. 118.--Formation of a comet's tail.]

Only the lightest kind of molecules--e. g., hydrogen--could drift away
from the comet so rapidly as is here shown. The heavier ones, such as
carbon and iron, would be repelled as strongly by the electric forces,
but they would be more strongly pulled back by the gravitative forces,
thus producing a much slower separation between them and the head of the
comet. Construct a figure such as the above, in which the molecules
shall recede from the comet only one eighth as fast as in Fig. 118, and
note what a different position it gives to the comet's tail. Instead of
pointing directly away from the sun, it will be bent strongly to one
side, as is the large plume-shaped tail of the Donati comet shown in
Fig. 101. But observe that this comet has also a nearly straight tail,
like the theoretical one of Fig. 118. We have here two distinct types of
comet tails, and according to Bredichin there is still another but
unusual type, even more strongly bent to one side of the line joining
comet and sun, and appearing quite short and stubby. The existence of
these three types, and their peculiarities of shape and position, are
all satisfactorily accounted for by the supposition that they are made
of different materials. The relative molecular weights of hydrogen, some
of the hydrocarbons, and iron, are such that tails composed of these
molecules would behave just as do the actual tails observed and
classified into these three types. The spectroscope shows that these
materials--hydrogen, hydrocarbons, and iron--are present in comets, and
leaves little room for doubt of the essential soundness of Bredichin's

181. DISINTEGRATION OF COMETS.--We must regard the tail as waste matter
cast off from the comet's head, and although the amount of this matter
is very small, it must in some measure diminish the comet's mass. This
process is, of course, most active at the time of perihelion passage,
and if the comet returns to perihelion time after time, as the periodic
ones which move in elliptic orbits must do, this waste of material may
become a serious matter, leading ultimately to the comet's destruction.
It is significant in this connection that the periodic comets are all
small and inconspicuous, not one of them showing a tail of any
considerable dimensions, and it appears probable that they are far
advanced along the road which, in the case of Biela's comet, led to its
disintegration. Their fragments are in part strewn through the solar
system, making some small fraction of its cloud of cosmic dust, and in
part they have been carried away from the sun and scattered throughout
the universe along hyperbolic orbits impressed upon them at the time
they left the comet.

But it is not through the tail only that the disintegrating process is
worked out. While Biela's comet is perhaps the most striking instance in
which the head has broken up, it is by no means the only one. The Great
Comet of 1882 cast off a considerable number of fragments which moved
away as independent though small comets and other more recent comets
have been seen to do the same. An even more striking phenomenon was the
gradual breaking up of the nucleus of the same comet, 1882, II, into a
half dozen nuclei arranged in line like beads upon a string, and
pointing along the axis of the tail. See Fig. 119, which shows the
series of changes observed in the head of this comet.

182. COMETS AND THE SPECTROSCOPE.--The spectrum presented by comets was
long a puzzle, and still retains something of that character, although
much progress has been made toward an understanding of it. In general it
consists of two quite distinct parts--first, a faint background of
continuous spectrum due to ordinary sunlight reflected from the comet;
and, second, superposed upon this, three bright bands like the carbon
band shown at the middle of Fig. 48, only not so sharply defined. These
bands make a discontinuous spectrum quite similar to that given off by
compounds of hydrogen and carbon, and of course indicate that a part of
the comet's light originates in the body itself, which must therefore be
incandescent, or at least must contain some incandescent portions.

[Illustration: FIG. 119.--The head of the Great Comet of

By heating hydrocarbons in our laboratories until they become
incandescent, something like the comet spectrum may be artificially
produced, but the best approximation to it is obtained by passing a
disruptive electrical discharge through a tube in which fragments of
meteors have been placed. A flash of lightning is a disruptive
electrical discharge upon a grand scale. Now, meteors and electric
phenomena have been independently brought to our notice in connection
with comets, and with this suggestion it is easy to frame a general idea
of the physical condition of these objects--for example, a cloud of
meteors of different sizes so loosely clustered that the average density
of the swarm is very low indeed; the several particles in motion
relative to each other, as well as to the sun, and disturbed in that
motion by the sun's tidal action. Each particle carries its own electric
charge, which may be of higher or lower tension than that of its
neighbor, and is ready to leap across the intervening gap whenever two
particles approach each other. To these conditions add the inductive
effect of the sun's electric charge, which tends to produce a particular
and artificial distribution of electricity among the comet's particles,
and we may expect to find an endless succession of sparks, tiny
lightning flashes, springing from one particle to another, most frequent
and most vivid when the comet is near the sun, but never strong enough
to be separately visible. Their number is, however, great enough to make
the comet in part self-luminous with three kinds of light--i. e., the
three bright bands of its spectrum, whose wave lengths show in the comet
the same elements and compounds of the elements--carbon, hydrogen, and
oxygen--which chemical analysis finds in the fallen meteor. It is not to
be supposed that these are the only chemical elements in the comet, as
they certainly are not the only ones in the meteor. They are the easy
ones to detect under ordinary circumstances, but in special cases, like
that of the Great Comet of 1882, whose near approach to the sun rendered
its whole substance incandescent, the spectrum glows with additional
bright lines of sodium, iron, etc.

183. COLLISIONS.--A question sometimes asked, What would be the effect
of a collision between the earth and a comet? finds its answer in the
results reached in the preceding sections. There would be a star
shower, more or less brilliant according to the number and size of the
pieces which made up the comet's head. If these were like the remains of
the Biela comet, the shower might even be a very tame one; but a
collision with a great comet would certainly produce a brilliant
meteoric display if its head came in contact with the earth. If the
comet were built of small pieces whose individual weights did not exceed
a few ounces or pounds, the earth's atmosphere would prove a perfect
shield against their attacks, reducing the pieces to harmless dust
before they could reach the ground, and leaving the earth uninjured by
the encounter, although the comet might suffer sadly from it. But big
stones in the comet, meteors too massive to be consumed in their flight
through the air, might work a very different effect, and by their
bombardment play sad havoc with parts of the earth's surface, although
any such result as the wrecking of the earth, or the destruction of all
life upon it, does not seem probable. The 40 meteors of § 169 may stand
for a collision with a small comet. Consult the Bible (Joshua x, 11) for
an example of what might happen with a larger one.



184. THE CONSTELLATIONS.--In the earlier chapters the student has
learned to distinguish between wandering stars (planets) and those fixed
luminaries which remain year after year in the same constellation,
shining for the most part with unvarying brilliancy, and presenting the
most perfect known image of immutability. Homer and Job and prehistoric
man saw Orion and the Pleiades much as we see them to-day, although the
precession, by changing their relation to the pole of the heavens, has
altered their risings and settings, and it may be that their luster has
changed in some degree as they grew old with the passing centuries.

[Illustration: FIG. 120.--Illustrating the division of the sky into

The division of the sky into constellations dates back to the most
primitive times, long before the Christian era, and the crooked and
irregular boundaries of these constellations, shown by the dotted lines
in Fig. 120, such as no modern astronomer would devise, are an
inheritance from antiquity, confounded and made worse in its descent to
our day. The boundaries assigned to constellations near the south pole
are much more smooth and regular, since this part of the sky, invisible
to the peoples from whom we inherit, was not studied and mapped until
more modern times. The old traditions associated with each constellation
a figure, often drawn from classical mythology, which was supposed to be
suggested by the grouping of the stars: thus Ursa Major is a great bear,
stalking across the sky, with the handle of the Dipper for his tail; Leo
is a lion; Cassiopeia, a lady in a chair; Andromeda, a maiden chained
to a rock, etc.; but for the most part the resemblances are far-fetched
and quite too fanciful to be followed by the ordinary eye.

185. THE NUMBER OF STARS.--"As numerous as the stars of heaven" is a
familiar figure of speech for expressing the idea of countless number,
but as applied to the visible stars of the sky the words convey quite a
wrong impression, for, under ordinary circumstances, in a clear sky
every star to be seen may be counted in the course of a few hours, since
they do not exceed 3,000 or 4,000, the exact number depending upon
atmospheric conditions and the keenness of the individual eye. Test your
own vision by counting the stars of the Pleiades. Six are easily seen,
and you may possibly find as many as ten or twelve; but however many are
seen, there will be a vague impression of more just beyond the limit of
visibility, and doubtless this impression is partly responsible for the
popular exaggeration of the number of the stars. In fact, much more than
half of what we call starlight comes from stars which are separately too
small to be seen, but whose number is so great as to more than make up
for their individual faintness.

The Milky Way is just such a cloud of faint stars, and the student who
can obtain access to a small telescope, or even an opera glass, should
not fail to turn it toward the Milky Way and see for himself how that
vague stream of light breaks up into shining points, each an independent
star. These faint stars, which are found in every part of the sky as
well as in the Milky Way, are usually called _telescopic_, in
recognition of the fact that they can be seen only in the telescope,
while the other brighter ones are known as _lucid stars_.

186. MAGNITUDES.--The telescopic stars show among themselves an even
greater range of brightness than do the lucid ones, and the system of
magnitudes (§ 9) has accordingly been extended to include them, the
faintest star visible in the greatest telescope of the present time
being of the sixteenth or seventeenth magnitude, while, as we have
already learned, stars on the dividing line between the telescopic and
the lucid ones are of the sixth magnitude. To compare the amount of
light received from the stars with that from the planets, and
particularly from the sun and moon, it has been found necessary to
prolong the scale of magnitudes backward into the negative numbers, and
we speak of the sun as having a stellar magnitude represented by the
number -26.5. The full moon's stellar magnitude is -12, and the planets
range from -3 (Venus) to +8 (Neptune). Even a very few of the stars are
so bright that negative magnitudes must be used to represent their true
relation to the fainter ones. Sirius, for example, the brightest of the
fixed stars, is of the -1 magnitude, and such stars as Arcturus and Vega
are of the 0 magnitude.

The relation of these magnitudes to each other has been so chosen that a
star of any one magnitude is very approximately 2.5 times as bright as
one of the next fainter magnitude, and this ratio furnishes a convenient
method of comparing the amount of light received from different stars.
Thus the brightness of Venus is 2.5 × 2.5 times that of Sirius. The full
moon is 2.5^{9} times as bright as Venus, etc.; only it should be
observed that the number 2.5 is not exactly the value of the _light
ratio_ between two consecutive magnitudes. Strictly this ratio is the
100^{1/5} = 2.5119+, so that to be entirely accurate we must say that
a difference of five magnitudes gives a hundredfold difference of
brightness. In mathematical symbols, if _B_ represents the ratio of
brightness (quantity of light) of two stars whose magnitudes are _m_ and
_n_, then

    B = (100)^{(m-n)/5}

How much brighter is an ordinary first-magnitude star, such as Aldebaran
or Spica, than a star just visible to the naked eye? How many of the
faintest stars visible in a great telescope would be required to make
one star just visible to the unaided eye? How many full moons must be
put in the sky in order to give an illumination as bright as daylight?
How large a part of the visible hemisphere would they occupy?

187. CLASSIFICATION BY MAGNITUDES.--The brightness of all the lucid
stars has been carefully measured with an instrument (photometer)
designed for that special purpose, and the following table shows,
according to the Harvard Photometry, the number of stars in the whole
sky, from pole to pole, which are brighter than the several magnitudes
named in the table:

    The number of stars brighter than magnitude 1.0 is    11
        "       "          "      "       "     2.0 "     39
        "       "          "      "       "     3.0 "    142
        "       "          "      "       "     4.0 "    463
        "       "          "      "       "     5.0 "  1,483
        "       "          "      "       "     6.0 "  4,326

It must not be inferred from this table that there are in the whole sky
only 4,326 stars visible to the naked eye. The actual number is probably
50 or 60 per cent greater than this, and the normal human eye sees stars
as faint as the magnitude 6.4 or 6.5, the discordance between this
number and the previous statement, that the sixth magnitude is the limit
of the naked-eye vision, having been introduced in the attempt to make
precise and accurate a classification into magnitudes which was at first
only rough and approximate. This same striving after accuracy leads to
the introduction of fractional numbers to represent gradations of
brightness intermediate between whole magnitudes. Thus of the 2,843
stars included between the fifth and sixth magnitudes a certain
proportion are said to be of the 5.1 magnitude, 5.2 magnitude, and so on
to the 5.9 magnitude, even hundredths of a magnitude being sometimes

We have found the number of stars included between the fifth and sixth
magnitudes by subtracting from the last number of the preceding table
the number immediately preceding it, and similarly we may find the
number included between each other pair of consecutive magnitudes, as

    Magnitude       0    1    2     3     4       5       6
    Number of stars   11   28   103   321   1,020   2,843
    4 × 3^{m}         12   36   108   324     972   2,916

In the last line each number after the first is found by multiplying the
preceding one by 3, and the approximate agreement of each such number
with that printed above it shows that on the whole, as far as the table
goes, the fainter stars are approximately three times as numerous as
those a magnitude brighter.

The magnitudes of the telescopic stars have not yet been measured
completely, and their exact number is unknown; but if we apply our
principle of a threefold increase for each successive magnitude, we
shall find for the fainter stars--those of the tenth and twelfth
magnitudes--prodigious numbers which run up into the millions, and even
these are probably too small, since down to the ninth or tenth magnitude
it is certain that the number of the telescopic stars increases from
magnitude to magnitude in more than a threefold ratio. This is balanced
in some degree by the less rapid increase which is known to exist in
magnitudes still fainter; and applying our formula without regard to
these variations in the rate of increase, we obtain as a rude
approximation to the total number of stars down to the fifteenth
magnitude, 86,000,000. The Herschels, father and son, actually counted
the number of stars visible in nearly 8,000 sample regions of the sky,
and, inferring the character of the whole sky from these samples, we
find it to contain 58,500,000 stars; but the magnitude of the faintest
star visible in their telescope, and included in their count, is rather

How many first-magnitude stars would be needed to give as much light as
do the 2,843 stars of magnitude 5.0 to 6.0? How many tenth-magnitude
stars are required to give the same amount of light?

To the modern man it seems natural to ascribe the different brilliancies
of the stars to their different distances from us; but such was not the
case 2,000 years ago, when each fixed star was commonly thought to be
fastened to a "crystal sphere," which carried them with it, all at the
same distance from us, as it turned about the earth. In breaking away
from this erroneous idea and learning to think of the sky itself as only
an atmospheric illusion through which we look to stars at very different
distances beyond, it was easy to fall into the opposite error and to
think of the stars as being much alike one with another, and, like
pebbles on the beach, scattered throughout space with some rough degree
of uniformity, so that in every direction there should be found in equal
measure stars near at hand and stars far off, each shining with a luster
proportioned to its remoteness.

188. DISTANCES OF THE STARS.--Now, in order to separate the true from
the false in this last mode of thinking about the stars, we need some
knowledge of their real distances from the earth, and in seeking it we
encounter what is perhaps the most delicate and difficult problem in the
whole range of observational astronomy. As shown in Fig. 121, the
principles involved in determining these distances are not fundamentally
different from those employed in determining the moon's distance from
the earth. Thus, the ellipse at the left of the figure represents the
earth's orbit and the position of the earth at different times of the
year. The direction of the star _A_ at these several times is shown by
lines drawn through _A_ and prolonged to the background apparently
furnished by the sky. A similar construction is made for the star _B_,
and it is readily seen that owing to the changing position of the
observer as he moves around the earth's orbit, both _A_ and _B_ will
appear to move upon the background in orbits shaped like that of the
earth as seen from the star, but having their size dependent upon the
star's distance, the apparent orbit of _A_ being larger than that of
_B_, because _A_ is nearer the earth. By measuring the angular distance
between _A_ and _B_ at opposite seasons of the year (e. g., the angles
_A--Jan.--B_, and _A--July--B_) the astronomer determines from the
change in this angle how much larger is the one path than the other, and
thus concludes how much nearer is _A_ than _B_. Strictly, the difference
between the January and July angles is equal to the difference between
the angles subtended at _A_ and _B_ by the diameter of the earth's
orbit, and if _B_ were so far away that the angle _Jan.--B--July_ were
nothing at all we should get immediately from the observations the angle
_Jan.--A--July_, which would suffice to determine the stars' distance.
Supposing the diameter of the earth's orbit and the angle at _A_ to be
known, can you make a graphical construction that will determine the
distance of _A_ from the earth?

[Illustration: FIG. 121.--Determining a star's parallax.]

The angle subtended at _A_ by the radius of the earth's orbit--i. e.,
1/2 (_Jan.--A--July_)--is called the star's parallax, and this is
commonly used by astronomers as a measure of the star's distance instead
of expressing it in linear units such as miles or radii of the earth's
orbit. The distance of a star is equal to the radius of the earth's
orbit divided by the parallax, in seconds of arc, and multiplied by the
number 206265.

A weak point of this method of measuring stellar distances is that it
always gives what is called a relative parallax--i. e., the difference
between the parallaxes of _A_ and _B_; and while it is customary to
select for _B_ a star or stars supposed to be much farther off than _A_,
it may happen, and sometimes does happen, that these comparison stars as
they are called are as near or nearer than _A_, and give a negative
parallax--i. e., the difference between the angles at _A_ and _B_ proves
to be negative, as it must whenever the star _B_ is nearer than _A_.

The first really successful determinations of stellar parallax were made
by Struve and Bessel a little prior to 1840, and since that time the
distances of perhaps 100 stars have been measured with some degree of
reliability, although the parallaxes themselves are so small--never as
great as 1''--that it is extremely difficult to avoid falling into
error, since even for the nearest star the problem of its distance is
equivalent to finding the distance of an object more than 5 miles away
by looking at it first with one eye and then with the other. Too short a
base line.

189. THE SUN AND HIS NEIGHBORS.--The distances of the sun's nearer
neighbors among the stars are shown in Fig. 122, where the two circles
having the sun at their center represent distances from it equal
respectively to 1,000,000 and 2,000,000 times the distance between earth
and sun. In the figure the direction of each star from the sun
corresponds to its right ascension, as shown by the Roman numerals about
the outer circle; the true direction of the star from the sun can not,
of course, be shown upon the flat surface of the paper, but it may be
found by elevating or depressing the star from the surface of the paper
through an angle, as seen from the sun, equal to its declination, as
shown in the fifth column of the following table,

                       _The Sun's Nearest Neighbors_

  No.| STAR.            |Magnitude.| R. A. |Dec. | Parallax.|Distance.
  1  | α Centauri       |   0.7    | 14.5h.| -60°|   0.75"  |  0.27
     |                  |          |       |     |          |
  2  | Ll. 21,185       |   6.8    | 11.0  | +37 |   0.45   |  0.46
     |                  |          |       |     |          |
  3  | 61 Cygni         |   5.0    | 21.0  | +38 |   0.40   |  0.51
     |                  |          |       |     |          |
  4  | η Herculis       |   3.6    | 16.7  | +39 |   0.40   |  0.51
     |                  |          |       |     |          |
  5  | Sirius           |  -1.4    |  6.7  | -17 |   0.37   |  0.56
     |                  |          |       |     |          |
  6  | Σ 2,398          |   8.2    | 18.7  | +59 |   0.35   |  0.58
     |                  |          |       |     |          |
  7  | Procyon          |   0.5    |  7.6  | + 5 |   0.34   |  0.60
     |                  |          |       |     |          |
  8  | γ Draconis       |   4.8    | 17.5  | +55 |   0.30   |  0.68
     |                  |          |       |     |          |
  9  | Gr. 34           |   7.9    |  0.2  | +43 |   0.29   |  0.71
     |                  |          |       |     |          |
  10 | Lac. 9,352       |   7.5    | 23.0  | -36 |   0.28   |  0.74
     |                  |          |       |     |          |
  11 | σ Draconis       |   4.8    | 19.5  | +69 |   0.25   |  0.82
     |                  |          |       |     |          |
  12 | A. O. 17,415-6   |   9.0    | 17.6  | +68 |   0.25   |  0.82
     |                  |          |       |     |          |
  13 | η Cassiopeiæ     |   3.4    |  0.7  | +57 |   0.25   |  0.82
     |                  |          |       |     |          |
  14 | Altair           |   1.0    | 19.8  | + 9 |   0.21   |  0.97
     |                  |          |       |     |          |
  15 | ε Indi           |   5.2    | 21.9  | -57 |   0.20   |  1.03
     |                  |          |       |     |          |
  16 | Gr. 1,618        |   6.7    | 10.1  | +50 |   0.20   |  1.03
     |                  |          |       |     |          |
  17 | 10 Ursæ Majoris  |   4.2    |  8.9  | +42 |   0.20   |  1.03
     |                  |          |       |     |          |
  18 | Castor           |   1.5    |  7.5  | +32 |   0.20   |  1.03
     |                  |          |       |     |          |
  19 | Ll. 21,258       |   8.5    | 11.0  | +44 |   0.20   |  1.03
     |                  |          |       |     |          |
  20 | ο^{2} Eridani    |   4.5    |  4.2  | - 8 |   0.19   |  1.08
     |                  |          |       |     |          |
  21 | A. O. 11,677     |   9.0    | 11.2  | +66 |   0.19   |  1.08
     |                  |          |       |     |          |
  22 | Ll. 18,115       |   8.0    |  9.1  | +53 |   0.18   |  1.14
     |                  |          |       |     |          |
  23 | B. D. 36°, 3,883 |   7.1    | 20.0  | +36 |   0.18   |  1.14
     |                  |          |       |     |          |
  24 | Gr. 1,618        |   6.5    | 10.1  | +50 |   0.17   |  1.21
     |                  |          |       |     |          |
  25 | β Cassiopeiæ     |   2.3    |  0.1  | +59 |   0.16   |  1.28
     |                  |          |       |     |          |
  26 | 70 Ophiuchi      |   4.4    | 18.0  | + 2 |   0.16   |  1.28
     |                  |          |       |     |          |
  27 | Σ 1,516          |   6.5    | 11.2  | +74 |   0.15   |  1.38
     |                  |          |       |     |          |
  28 | Gr. 1,830        |   6.6    | 11.8  | +39 |   0.15   |  1.38
     |                  |          |       |     |          |
  29 | μ Cassiopeiæ     |   5.4    |  1.0  | +54 |   0.14   |  1.47
     |                  |          |       |     |          |
  30 | ε Eridani        |   4.4    |  3.5  | -10 |   0.14   |  1.47
     |                  |          |       |     |          |
  31 | ι Ursæ Majoris   |   3.2    |  8.9  | +48 |   0.13   |  1.58
     |                  |          |       |     |          |
  32 | β Hydri          |   2.9    |  0.3  | -78 |   0.13   |  1.58
     |                  |          |       |     |          |
  33 | Fomalhaut        |   1.0    | 22.9  | -30 |   0.13   |  1.58
     |                  |          |       |     |          |
  34 | Br. 3,077        |   6.0    | 23.1  | +57 |   0.13   |  1.58
     |                  |          |       |     |          |
  35 | ε Cygni          |   2.5    | 20.8  | +33 |   0.12   |  1.71
     |                  |          |       |     |          |
  36 | β Comæ           |   4.5    | 13.1  | +28 |   0.11   |  1.87
     |                  |          |       |     |          |
  37 | ψ^{5} Aurigæ     |   8.8    |  6.6  | +44 |   0.11   |  1.87
     |                  |          |       |     |          |
  38 | π Herculis       |   3.3    | 17.2  | +37 |   0.11   |  1.87
     |                  |          |       |     |          |
  39 | Aldebaran        |   1.1    |  4.5  | +16 |   0.10   |  2.06
     |                  |          |       |     |          |
  40 | Capella          |   0.1    |  5.1  | +46 |   0.10   |  2.06
     |                  |          |       |     |          |
  41 | B. D. 35°, 4,003 |   9.2    | 20.1  | +35 |   0.10   |  2.06
     |                  |          |       |     |          |
  42 | Gr. 1,646        |   6.3    | 10.3  | +49 |   0.10   |  2.06
     |                  |          |       |     |          |
  43 | γ Cygni          |   2.3    | 20.3  | +40 |   0.10   |  2.06
     |                  |          |       |     |          |
  44 | Regulus          |   1.2    | 10.0  | +12 |   0.10   |  2.06
     |                  |          |       |     |          |
  45 | Vega             |   0.2    | 18.6  | +39 |   0.10   |  2.06

in which the numbers in the first column are those placed adjacent to
the stars in the diagram to identify them.

[Illustration: FIG. 122.--Stellar neighbors of the sun.]

190. LIGHT YEARS.--The radius of the inner circle in Fig. 122, 1,000,000
times the earth's distance from the sun, is a convenient unit in which
to express the stellar distances, and in the preceding table the
distances of the stars from the sun are expressed in terms of this
unit. To express them in miles the numbers in the table must be
multiplied by 93,000,000,000,000. The nearest star, α Centauri, is
25,000,000,000,000 miles away. But there is another unit in more common
use--i. e., the distance traveled over by light in the period of one
year. We have already found (§ 141) that it requires light 8m. 18s. to
come from the sun to the earth, and it is a simple matter to find from
this datum that in a year light moves over a space equal to 63,368 radii
of the earth's orbit. This distance is called a _light year_, and the
distance of the same star, α Centauri, expressed in terms of this
unit, is 4.26 years--i. e., it takes light that long to come from the
star to the earth.

In Fig. 122 the stellar magnitudes of the stars are indicated by the
size of the dots--the bigger the dot the brighter the star--and a mere
inspection of the figure will serve to show that within a radius of 30
light years from the sun bright stars and faint ones are mixed up
together, and that, so far as distance is concerned, the sun is only a
member of this swarm of stars, whose distances apart, each from its
nearest neighbor, are of the same order of magnitude as those which
separate the sun from the three or four stars nearest it.

Fig. 122 is not to be supposed complete. Doubtless other stars will be
found whose distance from the sun is less than 2,000,000 radii of the
earth's orbit, but it is not probable that they will ever suffice to
more than double or perhaps treble the number here shown. The vast
majority of the stars lie far beyond the limits of the figure.

191. PROPER MOTIONS.--It is evident that these stars are too far apart
for their mutual attractions to have much influence one upon another,
and that we have here a case in which, according to § 34, each star is
free to keep unchanged its state of rest or motion with unvarying
velocity along a straight line. Their very name, _fixed stars_, implies
that they are at rest, and so astronomers long believed. Hipparchus (125
B. C.) and Ptolemy (130 A. D.) observed and recorded many allineations
among the stars, in order to give to future generations a means of
settling this very question of a possible motion of the stars and a
resulting change in their relative positions upon the sky. For example,
they found at the beginning of the Christian era that the four stars,
Capella, ε Persei, α and β Arietis, stood in a straight line--i. e.,
upon a great circle of the sky. Verify this by direct reference to the
sky, and see how nearly these stars have kept the same position for
nearly twenty centuries. Three of them may be identified from the star
maps, and the fourth, ε Persei, is a third-magnitude star between
Capella and the other two.

Other allineations given by Ptolemy are: Spica, Arcturus and β Bootis;
Spica, δ Corvi and γ Corvi; α Libræ, Arcturus and ζ Ursæ Majoris.
Arcturus does not now fit very well to these alignments, and nearly two
centuries ago it, together with Aldebaran and Sirius, was on other
grounds suspected to have changed its place in the sky since the days of
Ptolemy. This discovery, long since fully confirmed, gave a great
impetus to observing with all possible accuracy the right ascensions and
declinations of the stars, with a view to finding other cases of what
was called _proper motion_--i. e., a motion peculiar to the individual
star as contrasted with the change of right ascension and declination
produced for all stars by the precession.

Since the middle of the eighteenth century there have been made many
thousands of observations of this kind, whose results have gone into
star charts and star catalogues, and which are now being supplemented by
a photographic survey of the sky that is intended to record permanently
upon photographic plates the position and magnitude of every star in the
heavens down to the fourteenth magnitude, with a view to ultimately
determining all their proper motions.

The complete achievement of this result is, of course, a thing of the
remote future, but sufficient progress in determining these motions has
been made during the past century and a half to show that nearly every
lucid star possesses some proper motion, although in most cases it is
very small, there being less than 100 known stars in which it amounts
to so much as 1" per annum--i. e., a rate of motion across the sky which
would require nearly the whole Christian era to alter a star's direction
from us by so much as the moon's angular diameter. The most rapid known
proper motion is that of a telescopic star midway between the equator
and the south pole, which changes its position at the rate of nearly 9"
per annum, and the next greatest is that of another telescopic star, in
the northern sky, No. 28 of Fig. 122. It is not until we reach the tenth
place in a list of large proper motions that we find a bright lucid
star, No. 1 of Fig. 122. It is a significant fact that for the most part
the stars with large proper motions are precisely the ones shown in Fig.
122, which is designed to show stars near the earth. This connection
between nearness and rapidity of proper motions is indeed what we should
expect to find, since a given amount of real motion of the star along
its orbit will produce a larger angular displacement, proper motion, the
nearer the star is to the earth, and this fact has guided astronomers in
selecting the stars to be observed for parallax, the proper motion being
determined first and the parallax afterward.

192. THE PATHS OF THE STARS.--We have already seen reason for thinking
that the orbit along which a star moves is practically a straight line,
and from a study of proper motions, particularly their directions across
the sky, it appears that these orbits point in all possible ways--north,
south, east, and west--so that some of them are doubtless directed
nearly toward or from the sun; others are square to the line joining sun
and star; while the vast majority occupy some position intermediate
between these two. Now, our relation to these real motions of the stars
is well illustrated in Fig. 112, where the observer finds in some of the
shooting stars a tremendous proper motion across the sky, but sees
nothing of their rapid approach to him, while others appear to stand
motionless, although, in fact, they are moving quite as rapidly as are
their fellows. The fixed star resembles the shooting star in this
respect, that its proper motion is only that part of its real motion
which lies at right angles to the line of sight, and this needs to be
supplemented by that other part of the motion which lies parallel to the
line of sight, in order to give us any knowledge of the star's real

[Illustration: FIG. 123.--Motion of Polaris in the line of sight as
determined by the spectroscope. FROST.]

193. MOTION IN THE LINE OF SIGHT.--It is only within the last 25 years
that anything whatever has been accomplished in determining these
stellar motions of approach or recession, but within that time much
progress has been made by applying the Doppler principle (§ 89) to the
study of stellar spectra, and at the present time nearly every great
telescope in the world is engaged upon work of this kind. The shifting
of the lines of the spectrum toward the violet or toward the red end of
the spectrum indicates with certainty the approach or recession of the
star, but this shifting, which must be determined by comparing the
star's spectrum with that of some artificial light showing corresponding
lines, is so small in amount that its accurate measurement is a matter
of extreme difficulty, as may be seen from Fig. 123. This cut shows
along its central line a part of the spectrum of Polaris, between wave
lengths 4,450 and 4,600 tenth meters, while above and below are the
corresponding parts of the spectrum of an electric spark whose light
passed through the same spectroscope and was photographed upon the same
plate with that of Polaris. This comparison spectrum is, as it should
be, a discontinuous or bright-line one, while the spectrum of the star
is a continuous one, broken only by dark gaps or lines, many of which
have no corresponding lines in the comparison spectrum. But a certain
number of lines in the two spectra do correspond, save that the dark
line is always pushed a very little toward the direction of shorter wave
lengths, showing that this star is approaching the earth. This spectrum
was photographed for the express purpose of determining the star's
motion in the line of sight, and with it there should be compared Figs.
124 and 125, which show in the upper part of each a photograph obtained
without comparison spectra by allowing the star's light to pass through
some prisms placed just in front of the telescope. The lower section of
each figure shows an enlargement of the original photograph, bringing
out its details in a way not visible to the unaided eye. In the enlarged
spectrum of β Aurigæ a rate of motion equal to that of the
earth in its orbit would be represented by a shifting of 0.03 of a
millimeter in the position of the broad, hazy lines.

[Illustration: FIG. 124.--Spectrum of β Aurigæ.--PICKERING.]

Despite the difficulty of dealing with such small quantities as the
above, very satisfactory results are now obtained, and from them it is
known that the velocities of stars in the line of sight are of the same
order of magnitude as the velocities of the planets in their orbits,
ranging all the way from 0 to 60 miles per second--more than 200,000
miles per hour--which latter velocity, according to Campbell, is the
rate at which μ Cassiopeiæ is approaching the sun.

The student should not fail to note one important difference between
proper motions and the motions determined spectroscopically: the latter
are given directly in miles per second, or per hour, while the former
are expressed in angular measure, seconds of arc, and there can be no
direct comparison between the two until by means of the known distances
of the stars their proper motions are converted from angular into linear
measure. We are brought thus to the very heart of the matter; parallax,
proper motion, and motion in the line of sight are intimately related
quantities, all of which are essential to a knowledge of the real
motions of the stars.

[Illustration: FIG. 125.--Spectrum of Pollux.--PICKERING.]

194. STAR DRIFT.--An illustration of how they may be made to work
together is furnished by some of the stars--which make up the Great
Dipper--β, γ, ε, and ζ Ursæ Majoris, whose proper motions have
long been known to point in nearly the same direction across the sky and
to be nearly equal in amount. More recently it has been found that these
stars are all moving toward the sun with approximately the same
velocity--18 miles per second. One other star of the Dipper, δ Ursæ
Majoris, shares in the common proper motion, but its velocity in the
line of sight has not yet been determined with the spectroscope. These
similar motions make it probable that the stars are really traveling
together through space along parallel lines; and on the supposition
that such is the case it is quite possible to write out a set of
equations which shall involve their known proper motions and motions in
the line of sight, together with their unknown distances and the unknown
direction and velocity of their real motion along their orbits. Solving
these equations for the values of the unknown quantities, it is found
that the five stars probably lie in a plane which is turned nearly
edgewise toward us, and that in this plane they are moving about twice
as fast as the earth moves around the sun, and are at a distance from us
represented by a parallax of less than 0.02"--i. e., six times as great
as the outermost circle in Fig. 122. A most extraordinary system of
stars which, although separated from each other by distances as great as
the whole breadth of Fig. 122, yet move along in parallel paths which it
is difficult to regard as the result of chance, and for which it is
equally difficult to frame an explanation.

[Illustration: FIG. 126.--The Great Dipper, past, present, and future.]

The stars α and η of the Great Dipper do not share in this motion, and
must ultimately part company with the other five, to the complete
destruction of the Dipper's shape. Fig. 126 illustrates this change of
shape, the upper part of the figure (_a_) showing these seven stars as
they were grouped at a remote epoch in the past, while the lower
section (_c_) shows their position for an equally remote epoch in the
future. There is no resemblance to a dipper in either of these
configurations, but it should be observed that in each of them the stars
α and η keep their relative position unaltered, and the other five stars
also keep together, the entire change of appearance being due to the
changing positions of these two groups with respect to each other.

This phenomenon of groups of stars moving together is called _star
drift_, and quite a number of cases of it are found in different parts
of the sky. The Pleiades are perhaps the most conspicuous one, for here
some sixty or more stars are found traveling together along similar
paths. Repeated careful measurements of the relative positions of stars
in this cluster show that one of the lucid stars and four or five of the
telescopic ones do not share in this motion, and therefore are not to be
considered as members of the group, but rather as isolated stars which,
for a time, chance to be nearly on line with the Pleiades, and probably
farther off, since their proper motions are smaller.

To rightly appreciate the extreme slowness with which proper motions
alter the constellations, the student should bear in mind that the
changes shown in passing from one section of Fig. 126 to the next
represent the effect of the present proper motions of the stars
accumulated for a period of 200,000 years. Will the stars continue to
move in straight paths for so long a time?

195. THE SUN'S WAY.--Another and even more interesting application of
proper motions and motions in the line of sight is the determination
from them of the sun's orbit among the stars. The principle involved is
simple enough. If the sun moves with respect to the stars and carries
the earth and the other planets year after year into new regions of
space, our changing point of view must displace in some measure every
star in the sky save those which happen to be exactly on the line of the
sun's motion, and even these will show its effect by their apparent
motion of approach or recession along the line of sight. So far as their
own orbital motions are concerned, there is no reason to suppose that
more stars move north than south, or that more go east than west; and
when we find in their proper motions a distinct tendency to radiate from
a point somewhere near the bright star Vega and to converge toward a
point on the opposite side of the sky, we infer that this does not come
from any general drift of the stars in that direction, but that it marks
the course of the sun among them. That it is moving along a straight
line pointing toward Vega, and that at least a part of the velocities
which the spectroscope shows in the line of sight, comes from the motion
of the sun and earth. Working along these lines, Kapteyn finds that the
sun is moving through space with a velocity of 11 miles per second,
which is decidedly below the average rate of stellar motion--19 miles
per second.

196. DISTANCE OF SIRIAN AND SOLAR STARS.--By combining this rate of
motion of the sun with the average proper motions of the stars of
different magnitudes, it is possible to obtain some idea of the average
distance from us of a first-magnitude star or a sixth-magnitude star,
which, while it gives no information about the actual distance of any
particular star, does show that on the whole the fainter stars are more
remote. But here a broad distinction must be drawn. By far the larger
part of the stars belong to one of two well-marked classes, called
respectively Sirian and solar stars, which are readily distinguished
from each other by the kind of spectrum they furnish. Thus β
Aurigæ belongs to the Sirian class, as does every other star which has a
spectrum like that of Fig. 124, while Pollux is a solar star presenting
in Fig. 125 a spectrum like that of the sun, as do the other stars of
this class.

Two thirds of the sun's near neighbors, shown in Fig. 122, have spectra
of the solar type, and in general stars of this class are nearer to us
than are the stars with spectra unlike that of the sun. The average
distance of a solar star of the first magnitude is very approximately
represented by the outer circle in Fig. 122, 2,000,000 times the
distance of the sun from the earth; while the corresponding distance for
a Sirian star of the first magnitude is represented by the number

A third-magnitude star is on the average twice as far away as one of the
first magnitude, a fifth-magnitude star four times as far off, etc.,
each additional two magnitudes doubling the average distance of the
stars, at least down to the eighth magnitude and possibly farther,
although beyond this limit we have no certain knowledge. Put in another
way, the naked eye sees many Sirian stars which _may_ have "gone out"
and ceased to shine centuries ago, for the light by which we now see
them left those stars before the discovery of America by Columbus. For
the student of mathematical tastes we note that the results of Kapteyn's
investigation of the mean distances (_D_) of the stars of magnitude
(_m_) may be put into two equations:

    For Solar Stars, D = 23 × 2^{m/2}

    For Sirian Stars, D = 52 × 2^{m/2}

where the coefficients 23 and 52 are expressed in light years. How long
a time is required for light to come from an average solar star of the
sixth magnitude?

197. CONSEQUENCES OF STELLAR DISTANCE.--The amount of light which comes
to us from any luminous body varies inversely as the square of its
distance, and since many of the stars are changing their distance from
us quite rapidly, it must be that with the lapse of time they will grow
brighter or fainter by reason of this altered distance. But the
distances themselves are so great that the most rapid known motion in
the line of sight would require more than 1,000 years (probably several
thousand) to produce any perceptible change in brilliancy.

The law in accordance with which this change of brilliancy takes place
is that the distance must be increased or diminished tenfold in order to
produce a change of five magnitudes in the brightness of the object, and
we may apply this law to determine the sun's rank among the stars. If it
were removed to the distance of an average first-, or second-, or
third-magnitude star, how would its light compare with that of the
stars? The average distance of a third-magnitude star of the solar type
is, as we have seen above, 4,000,000 times the sun's distance from the
earth, and since 4,000,000 = 10^{6.6}, we find that at this distance the
sun's stellar magnitude would be altered by 6.6 × 5 magnitudes, and
would therefore be -26.5 + 33.0 = 6.5--i. e., the sun if removed to the
average distance of the third-magnitude stars of its type would be
reduced to the very limit of naked-eye visibility. It must therefore be
relatively small and feeble as compared with the brightness of the
average star. It is only its close proximity to us that makes the sun
look brighter than the stars.

The fixed stars may have planets circling around them, but an
application of the same principles will show how hopeless is the
prospect of ever seeing them in a telescope. If the sun's nearest
neighbor, α Centauri, were attended by a planet like Jupiter, this
planet would furnish to us no more light than does a star of the
twenty-second magnitude--i. e., it would be absolutely invisible, and
would remain invisible in the most powerful telescope yet built, even
though its bulk were increased to equal that of the sun. Let the student
make the computation leading to this result, assuming the stellar
magnitude of Jupiter to be -1.7.

198. DOUBLE STARS.--In the constellation Taurus, not far from Aldebaran,
is the fourth-magnitude star θ Tauri, which can readily be seen to
consist of two stars close together. The star α Capricorni is plainly
double, and a sharp eye can detect that one of the faint stars which
with Vega make a small equilateral triangle, is also a double star.
Look for them in the sky.

In the strict language of astronomy the term double star would not be
applied to the first two of these objects, since it is usually
restricted to those stars whose angular distance from each other is so
small that in the telescope they appear much as do the stars named above
to the naked eye--i. e., their angular separation is measured by a few
seconds or fractions of a single second, instead of the six minutes
which separate the component stars of θ Tauri or α Capricorni. There are
found in the sky many thousands of these close double stars, of which
some are only optically double--i. e., two stars nearly on line with the
earth but at very different distances from it--while more of them are
really what they seem, stars near each other, and in many cases near
enough to influence each other's motion. These are called _binary_
systems, and in cases of this kind the principles of celestial mechanics
set forth in Chapter IV hold true, and we may expect to find each
component of a double star moving in a conic section of some kind,
having its focus at the common center of gravity of the two stars.
We are thus presented with problems of orbital motion quite similar
to those which occur in the solar system, and careful telescopic
observations are required year after year to fix the relative positions
of the two stars--i. e., their angular separation, which it is customary
to call their _distance_, and their direction one from the other, which
is called _position angle_.

199. ORBITS OF DOUBLE STARS.--The sun's nearest neighbor, α Centauri, is
such a double star, whose position angle and distance have been measured
by successive generations of astronomers for more than a century, and
Fig. 127 shows the result of plotting their observations. Each black dot
that lies on or near the circumference of the long ellipse stands for
an observed direction and distance of the fainter of the two stars from
the brighter one, which is represented by the small circle at the
intersection of the lines inside the ellipse. It appears from the figure
that during this time the one star has gone completely around the other,
as a planet goes around the sun, and the true orbit must therefore be
an ellipse having one of its foci at the center of gravity of the two
stars. The other star moves in an ellipse of precisely similar shape,
but probably smaller size, since the dimensions of the two orbits are
inversely proportional to the masses of the two bodies, but it is
customary to neglect this motion of the larger star and to give to the
smaller one an orbit whose diameter is equal to the sum of the diameters
of the two real orbits. This practice, which has been followed in Fig.
127, gives correctly the relative positions of the two stars, and makes
one orbit do the work of two.

[Illustration: FIG. 127.--The orbit of α Centauri.--SEE.]

In Fig. 127 the bright star does not fall anywhere near the focus of the
ellipse marked out by the smaller one, and from this we infer that the
figure does not show the true shape of the orbit, which is certainly
distorted, foreshortened, by the fact that we look obliquely down upon
its plane. It is possible, however, by mathematical analysis, to find
just how much and in what direction that plane should be turned in order
to bring the focus of the ellipse up to the position of the principal
star, and thus give the true shape and size of the orbit. See Fig. 128
for a case in which the true orbit is turned exactly edgewise toward the
earth, and the small star, which really moves in an ellipse like that
shown in the figure, appears to oscillate to and fro along a straight
line drawn through the principal star, as shown at the left of the

In the case of α Centauri the true orbit proves to have a major
axis 47 times, and a minor axis 40 times, as great as the distance of
the earth from the sun. The orbit, in fact, is intermediate in size
between the orbits of Uranus and Neptune, and the periodic time of the
star in this orbit is 81 years, a little less than the period of Uranus.

[Illustration: FIG. 128.--Apparent orbit and real orbit of the double
star 42 Comæ Berenicis.--SEE.]

200. MASSES OF DOUBLE STARS.--If we apply to this orbit Kepler's Third
Law in the form given it at page 179, we shall find--

    a^3 / T^2 = (23.5)^3 / (81)^2 = k (M + m),

where _M_ and _m_ represent the masses of the two stars. We have already
seen that _k_, the gravitation constant, is equal to 1 when the masses
are measured in terms of the sun's mass taken as unity, and when _T_ and
_a_ are expressed in years and radii of the earth's orbit respectively,
and with this value of _k_ we may readily find from the above equation,
_M_ + _m_ = 2.5--i. e., the combined mass of the two components of
α Centauri is equal to rather more than twice the mass of the sun. It is
not every double star to which this process of weighing can be applied.
The major axis of the orbit, _a_, is found from the observations in
angular measure, 35" in this case, and it is only when the parallax of
the star is known that this can be converted into the required linear
units, radii of the earth's orbit, by dividing the angular major axis by
the parallax; 47 = 35" ÷ 0.75".

Our list of distances (§ 189) contains four double stars whose periodic
times and major axes have been fairly well determined, and we find in
the accompanying table the information which they give about the masses
of double stars and the size of the orbits in which they move:

  STAR.                | Major | Minor | Periodic | Mass.
                       | axis. | axis. | time.    |
  α Centauri           |  47   |  40   |   81 y.  |   2
  70 Ophiuchi          |  56   |  48   |   88     |   3
  Procyon              |  34   |  31   |   40     |   3
  Sirius               |  43   |  34   |   52     |   4

The orbit of Uranus, diameter = 38, and Neptune, diameter = 60, are of
much the same size as these double-star orbits; but the planetary orbits
are nearly circular, while in every case the double stars show a
substantial difference between the long and short diameters of their
orbits. This is a characteristic feature of most double-star orbits, and
seems to stand in some relation to their periodic times, for, on the
average, the longer the time required by a star to make its orbital
revolution the more eccentric is its orbit likely to prove.

Another element of the orbits of double stars, which stands in even
closer relation to the periodic time, is the major axis; the smaller the
long diameter of the orbit the more rapid is the motion and the shorter
the periodic time, so that astronomers in search of interesting
double-star orbits devote themselves by preference to those stars whose
distance apart is so small that they can barely be distinguished one
from the other in the telescope.

Although the half-dozen stars contained in the table all have orbits of
much the same size and with much the same periodic time as those in
which Uranus and Neptune move, this is by no means true of all the
double stars, many of which have periods running up into the hundreds if
not thousands of years, while a few complete their orbital revolutions
in periods comparable with, or even shorter than, that of Jupiter.

201. DARK STARS.--Procyon, the next to the last star of the preceding
table, calls for some special mention, as the determination of its mass
and orbit stands upon a rather different basis from that of the other
stars. More than half a century ago it was discovered that its proper
motion was not straight and uniform after the fashion of ordinary stars,
but presented a series of loops like those marked out by a bright point
on the rim of a swiftly running bicycle wheel. The hub may move straight
forward with uniform velocity, but the point near the tire goes up and
down, and, while sharing in the forward motion of the hub, runs
sometimes ahead of it, sometimes behind, and such seemed to be the
motion of Procyon and of Sirius as well. Bessel, who discovered it, did
not hesitate to apply the laws of motion, and to affirm that this
visible change of the star's motion pointed to the presence of an unseen
companion, which produced upon the motions of Sirius and Procyon just
such effects as the visible companions produce in the motions of double
stars. A new kind of star, dark instead of bright, was added to the
astronomer's domain, and its discoverer boldly suggested the possible
existence of many more. "That countless stars are visible is clearly no
argument against the existence of as many more invisible ones." "There
is no reason to think radiance a necessary property of celestial
bodies." But most astronomers were incredulous, and it was not until
1862 that, in the testing of a new and powerful telescope just built, a
dark star was brought to light and the companion of Sirius actually
seen. The visual discovery of the dark companion of Procyon is of still
more recent date (November, 1896), when it was detected with the great
telescope of the Lick Observatory. This discovery is so recent that the
orbit is still very uncertain, being based almost wholly upon the
variations in the proper motion of the star, and while the periodic time
must be very nearly correct, the mass of the stars and dimensions of the
orbit may require considerable correction.

The companion of Sirius is about ten magnitudes and that of Procyon
about twelve magnitudes fainter than the star itself. How much more
light does the bright star give than its faint companion? Despite the
tremendous difference of brightness represented by the answer to this
question, the mass of Sirius is only about twice as great as that of its
companion, and for Procyon the ratio does not exceed five or six.

The visual discovery of the companions to Sirius and Procyon removes
them from the list of dark stars, but others still remain unseen,
although their existence is indicated by variable proper motions or by
variable orbital motion, as in the case of ζ Cancri, where one
of the components of a triple star moves around the other two in a
series of loops whose presence indicates a disturbing body which has
never yet been seen.

202. MULTIPLE STARS.--Combinations of three, four, or more stars close
to each other, like ζ Cancri, are called multiple stars, and
while they are far from being as common as are double stars, there is a
considerable number of them in the sky, 100 or more as against the more
than 10,000 double stars that are known. That their relative motions are
subject to the law of gravitation admits of no serious doubt, but
mathematical analysis breaks down in face of the difficulties here
presented, and no astronomer has ever been able to determine what will
be the general character of the motions in such a system.

[Illustration: FIG. 129.--Illustrating the motion of a spectroscopic

203. SPECTROSCOPIC BINARIES.--In the year 1890 Professor Pickering, of
the Harvard Observatory, announced the discovery of a new class of
double stars, invisible as such in even the most powerful telescope,
and producing no perturbations such as have been considered above, but
showing in their spectrum that two or more bodies must be present in the
source of light which to the eye is indistinguishable from a single
star. In Fig. 129 we suppose _A_ and _B_ to be the two components of a
double star, each moving in its own orbit about their common center of
gravity, _C_, whose distance from the earth is several million times
greater than the distance between the stars themselves. Under such
circumstances no telescope could distinguish between the two stars,
which would appear fused into one; but the smaller the orbit the more
rapid would be their motion in it, and if this orbit were turned
edgewise toward the earth, as is supposed in the figure, whenever the
stars were in the relative position there shown, _A_ would be rapidly
approaching the earth by reason of its orbital motion, while _B_ would
move away from it, so that in accordance with the Doppler principle the
lines composing their respective spectra would be shifted in opposite
directions, thus producing a doubling of the lines, each single line
breaking up into two, like the double-sodium line _D_, only not spaced
so far apart. When the stars have moved a quarter way round their orbit
to the points _A´_, _B´_, their velocities are turned at right angles to
the line of sight and the spectrum returns to the normal type with
single lines, only to break up again when after another quarter
revolution their velocities are again parallel with the line of sight.
The interval of time between consecutive doublings of the lines in the
spectrum thus furnishes half the time of a revolution in the orbit. The
distance between the components of a double line shows by means of the
Doppler principle how fast the stars are traveling, and this in
connection with the periodic times fixes the size of the orbit, provided
we assume that it is turned exactly edgewise to the earth. This
assumption may not be quite true, but even though the orbit should
deviate considerably from this position, it will still present the
phenomenon of the double lines whose displacement will now show
something less than the true velocities of the stars in their orbits,
since the spectroscope measures only that component of the whole
velocity which is directed toward the earth, and it is important to note
that the real orbits and masses of these _spectroscopic binaries_, as
they are called, will usually be somewhat larger than those indicated by
the spectroscope, since it is only in exceptional cases that the orbit
will be turned exactly edgewise to us.

The bright star Capella is an excellent illustration of these
spectroscopic binaries. At intervals of a little less than a month the
lines of its spectrum are alternately single and double, their maximum
separation corresponding to a velocity in the line of sight amounting to
37 miles per second. Each component of a doubled line appears to be
shifted an equal amount from the position occupied by the line when it
is single, thus indicating equal velocities and equal masses for the two
component stars whose periodic time in their orbit is 104 days. From
this periodic time, together with the velocity of the star's motion, let
the student show that the diameter of the orbit--i. e., the distance of
the stars from each other--is approximately 53,000,000 miles, and that
their combined mass is a little less than that of α Centauri, provided
that their orbit plane is turned exactly edgewise toward the earth.

There are at the present time (1901) 34 spectroscopic binaries known,
including among them such stars as Polaris, Capella, Algol, Spica, β
Aurigæ, ζ Ursæ Majoris, etc., and their number is rapidly increasing,
about one star out of every seven whose motion in the line of sight
is determined proving to be a binary or, as in the case of Polaris,
possibly triple. On account of smaller distance apart their periodic
times are much shorter than those of the ordinary double stars, and
range from a few days up to several months--more than two years in the
case of η Pegasi, which has the longest known period of any star of this

Spectroscopic binaries agree with ordinary double stars in having masses
rather greater than that of the sun, but there is as yet no assured case
of a mass ten times as great as that of the sun.

204. VARIABLE STARS.--Attention has already been drawn (§ 23) to the
fact that some stars shine with a changing brightness--e. g., Algol, the
most famous of these _variable stars_, at its maximum of brightness
furnishes three times as much light as when at its minimum, and other
variable stars show an even greater range. The star ο Ceti has
been named Mira (Latin, _the wonderful_), from its extraordinary range
of brightness, more than six-hundred-fold. For the greater part of the
time this star is invisible to the naked eye, but during some three
months in every year it brightens up sufficiently to be seen, rising
quite rapidly to its maximum brilliancy, which is sometimes that of a
second-magnitude star, but more frequently only third or even fourth
magnitude, and, after shining for a few weeks with nearly maximum
brilliancy, falling off to become invisible for a time and then return
to its maximum brightness after an interval of eleven months from the
preceding maximum. In 1901 it should reach its greatest brilliancy about
midsummer, and a month earlier than this for each succeeding year. Find
it by means of the star map, and by comparing its brightness from night
to night with neighboring stars of about the same magnitude see how it
changes with respect to them.

The interval of time from maximum to maximum of brightness--331.6 days
for Mira--is called the star's period, and within its period a star
regularly variable runs through all its changes of brilliancy, much as
the weather runs through its cycle of changes in the period of a year.
But, as there are wet years and dry ones, hot years and cold, so also
with variable stars, many of them show differences more or less
pronounced between different periods, and one such difference has
already been noted in the case of Mira; its maximum brilliancy is
different in different years. So, too, the length of the period
fluctuates in many cases, as does every other circumstance connected
with it, and predictions of what such a variable star will do are
notoriously unreliable.

205. THE ALGOL VARIABLES.--On the other hand, some variable stars
present an almost perfect regularity, repeating their changes time after
time with a precision like that of clockwork. Algol is one type of these
regular variables, having a period of 68.8154 hours, during six sevenths
of which time it shines with unchanging luster as a star of the 2.3
magnitude, but during the remaining 9 hours of each period it runs down
to the 3.5 magnitude, and comes back again, as is shown by a curve in
Fig. 130. The horizontal scale here represents hours, reckoned from the
time of the star's minimum brightness, and the vertical scale shows
stellar magnitudes. Such a diagram is called the star's light curve, and
we may read from it that at any time between 5h. and 32h. after the time
of minimum the star's magnitude is 2.32; at 2h. after a minimum the
magnitude is 2.88, etc. What is the magnitude an hour and a half before
the time of minimum? What is the magnitude 43 days after a minimum?

[Illustration: FIG. 130.--The light curve of Algol.]

The arrows shown in Fig. 130 are a feature not usually found with light
curves, but in this case each one represents a spectroscopic
determination of the motion of Algol in the line of sight. These
observations extended over a period of more than two years, but they are
plotted in the figure with reference to the number of hours each one
preceded or followed a minimum of the star's light, and each arrow shows
not only the direction of the star's motion along the line of sight, the
arrows pointing down denoting approach of the star toward the earth, but
also its velocity, each square of the ruling corresponding to 10
kilometers (6.2 miles per second). The differences of velocity shown by
adjacent arrows come mainly from errors of observation and furnish some
idea of how consistent among themselves such observations are, but there
can be no doubt that before minimum the star is moving away from the
earth, and after minimum is approaching it. It is evident from these
observations that in Algol we have to do with a spectroscopic binary,
one of whose components is a dark star which, once in each revolution,
partially eclipses the bright star and produces thus the variations in
its light. By combining the spectroscopic observations with the
variations in the star's light, Vogel finds that the bright star, Algol,
itself has a diameter somewhat greater than that of the sun, but is of
low density, so that its mass is less than half that of the sun, while
the dark star is a very little smaller than the sun and has about a
quarter of its mass. The distance between the two stars, dark and
bright, is 3,200,000 miles. Fig. 129, which is drawn to scale, shows the
relative positions and sizes of these stars as well as the orbits in
which they move.

The mere fact already noted that close binary systems exist in
considerable numbers is sufficient to make it probable that a certain
proportion of these stars would have their orbit planes turned so nearly
edgewise toward the earth as to produce eclipses, and corresponding to
this probability there are already known no less than 15 stars of the
Algol type of eclipse variables, and only a beginning has been made in
the search for them.

[Illustration: FIG. 131.--The light curve of β Lyræ.]

206. VARIABLES OF THE β LYRÆ TYPE.--In addition to these there
is a certain further number of binary variables in which both components
are bright and where the variation of brightness follows a very
different course. Capella would be such a variable if its orbit plane
were directed exactly toward the earth, and the fact that its light is
not variable shows conclusively that such is not the position of the
orbit. Fig. 131 represents the light curve of one of the best-known
variable systems of this second type, that of β Lyræ, whose
period is 12 days 21.8 hours, and the student should read from the curve
the magnitude of the star for different times during this interval.
According to Myers, this light curve and the spectroscopic observations
of the star point to the existence of a binary star of very remarkable
character, such as is shown, together with its orbit and a scale of
miles, in Fig. 132. Note the tide which each of these stars raises in
the other, thus changing their shapes from spheres into ellipsoids. The
astonishing dimensions of these stars are in part compensated by their
very low density, which is less than that of air, so that their masses
are respectively only 10 times and 21 times that of the sun! But these
dimensions and masses perhaps require confirmation, since they depend
upon spectroscopic observations of doubtful interpretation. In Fig. 132
what relative positions must the stars occupy in their orbit in order
that their combined light should give β Lyræ its maximum
brightness? What position will furnish a minimum brightness?

[Illustration: FIG. 132.--The system of β Lyræ.--MYERS.]

207. VARIABLES OF LONG AND SHORT PERIODS.--It must not be supposed that
all variable stars are binaries which eclipse each other. By far the
larger part of them, like Mira, are not to be accounted for in this way,
and a distinction which is pretty well marked in the length of their
periods is significant in this connection. There is a considerable
number of variable stars with periods shorter than a month, and there
are many having periods longer than 6 months, but there are very few
having periods longer than 18 months, or intermediate between 1 month
and 6 months, so that it is quite customary to divide variable stars
into two classes--those of long period, 6 months or more, and those of
short period less than 6 months, and that this distinction corresponds
to some real difference in the stars themselves is further marked by the
fact that the long-period variables are prevailingly red in color, while
the short-period stars are almost without exception white or very pale
yellow. In fact, the longer the period the redder the star, although it
is not to be inferred that all red stars are variable; a considerable
percentage of them shine with constant light. The eclipse explanation of
variability holds good only for short-period variables, and possibly not
for all of them, while for the long-period variables there is no
explanation which commands the general assent of astronomers, although
unverified hypotheses are plenty.

The number of stars known to be variable is about 400, while a
considerable number of others are "suspected," and it would not be
surprising if a large fraction of all the stars should be found to
fluctuate a little in brightness. The sun's spots may suffice to make it
a variable star with a period of 11 years.

The discovery of new variables is of frequent occurrence, and may be
expected to become more frequent when the sky is systematically explored
for them by the ingenious device suggested by Pickering and illustrated
in Fig. 133. A given region of the sky--e. g., the Northern Crown--is
photographed repeatedly upon the same plate, which is shifted a little
at each new exposure, so that the stars shall fall at new places upon
it. The finally developed plate shows a row of images corresponding to
each star, and if the star's light is constant the images in any given
row will all be of the same size, as are most of those in Fig. 133; but
a variable star such as is shown by the arrowhead reveals its presence
by the broken aspect of its row of dots, a minimum brilliancy being
shown by smaller and a maximum by larger ones. In this particular case,
at two exposures the star was too faint to print its image upon the

[Illustration: FIG. 133.--Discovery of a variable star by means of

208. NEW STARS.--Next to the variable stars of very long or very
irregular period stand the so-called _new_ or _temporary stars_, which
appear for the most part suddenly, and after a brief time either vanish
altogether or sink to comparative insignificance. These were formerly
thought to be very remarkable and unusual occurrences--"the birth of a
new world"--and it is noteworthy that no new star is recorded to have
been seen from 1670 to 1848 A. D., for since that time there have been
no less than five of them visible to the naked eye and others
telescopic. In so far as these new stars are not ordinary variables
(Mira, first seen in 1596, was long counted as a new star), they are
commonly supposed due to chance encounters between stars or other cosmic
bodies moving with considerable velocities along orbits which approach
very close to each other. The actual collision of two dark bodies moving
with high velocities is clearly sufficient to produce a luminous
star--e. g., meteors--and even the close approach of two cooled-off
stars, might result in tidal actions which would rend open their crusts
and pour out the glowing matter from within so as to produce temporarily
a very great accession of brightness.

The most famous of all new stars is that which, according to Tycho
Brahe's report, appeared in the year 1572, and was so bright when at its
best as to be seen with the naked eye in broad daylight. It continued
visible, though with fading light, for about 16 months, and finally
disappeared to the naked eye, although there is some reason to suppose
that it can be identified with a ruddy star of the eleventh magnitude in
the constellation Cassiopeia, whose light still shows traces of

No modern temporary star approaches that of Tycho in splendor, but in
some respects the recent ones surpass it in interest, since it has been
possible to apply the spectroscope to the analysis of their light and to
find thereby a much more complex set of conditions in the star than
would have been suspected from its light changes alone.

One of the most extraordinary of new stars, and the most brilliant one
since that of Tycho, appeared suddenly in the constellation Perseus in
February, 1901, and for a short time equaled Capella in brightness. But
its light rapidly waned, with periodic fluctuations of brightness like
those of a variable star, and at the present time (September, 1902) it
is lost to the naked eye, although in the telescope it still shines like
a star of the ninth or tenth magnitude.

By the aid of powerful photographic apparatus, during the period of its
waning brilliancy a ring of faint nebulous matter was detected
surrounding the star and drifting around and away from it much as if a
series of nebulæ had been thrown off by the star at the time of its
sudden outburst of light. But the extraordinary velocity of this nebular
motion, nearly a billion miles per hour, makes such an explanation
almost incredible, and astronomers are more inclined to believe that the
ring was merely a reflection of the star's own light from a cloud of
meteoric matter, into which a rapidly moving dark star plunged and,
after the fashion of terrestrial meteors, was raised to brilliant
incandescence by the collision. If we assume this to be the true
explanation of these extraordinary phenomena, it is possible to show
from the known velocity with which light travels through space and from
the rate at which the nebula spread, that the distance of Nova Persei,
as the new star is called, corresponds to a parallax of about one
one-hundredth of a second, a result that is, in substance, confirmed by
direct telescopic measurements of its parallax.

Another modern temporary star is Nova Aurigæ, which appeared suddenly in
December, 1891, waned, and in the following April vanished, only to
reappear three months later for another season of renewed brightness.
The spectra of both these modern Novæ contain both dark and bright lines
displaced toward opposite ends of the spectrum, and suggesting the
Doppler effect that would be produced by two or more glowing bodies
having rapid and opposite motions in the line of sight. But the most
recent investigations cast discredit on this explanation and leave the
spectra of temporary stars still a subject of debate among astronomers,
with respect both to the motion they indicate and the intrinsic nature
of the stars themselves. The varying aspect of the spectra suggested at
one time the sun's chromosphere, at another time the conditions that are
present in nebulæ, etc.



209. STELLAR COLORS.--We have already seen that one star differs from
another in respect of color as well as brightness, and the diligent
student of the sky will not fail to observe for himself how the luster
of Sirius and Rigel is more nearly a pure white than is that of any
other stars in the heavens, while at the other end of the scale
α Orionis and Aldebaran are strongly ruddy, and Antares presents an
even deeper tone of red. Between these extremes the light of every star
shows a mixture of the rainbow hues, in which a very pale yellow is the
predominant color, shading off, as we have seen, to white at one end of
the scale and red at the other. There are no green stars, or blue stars,
or violet stars, save in one exceptional class of cases--viz., where the
two components of a double star are of very different brightness, it is
quite the usual thing for them to have different colors, and then,
almost without exception, the color of the fainter star lies nearer to
the violet end of the spectrum than does the color of the bright one,
and sometimes shows a distinctly blue or green hue. A fine type of such
double star is β Cygni, in which the components are respectively
yellow and blue, and the yellow star furnishes eight times as much light
as the blue one.

The exception which double stars thus make to the general rule of
stellar colors, yellow and red, but no color of shorter wave length, has
never been satisfactorily explained, but the rule itself presents no
difficulties. Each star is an incandescent body, giving off radiant
energy of every wave length within the limits of the visible spectrum,
and, indeed, far beyond these limits. If this radiant energy could come
unhindered to our eyes every star would appear white, but they are all
surrounded by atmospheres--analogous to the chromosphere and reversing
layer of the sun--which absorb a portion of their radiant energy and,
like the earth's atmosphere, take a heavier toll from the violet than
from the red end of the spectrum. The greater the absorption in the
star's atmosphere, therefore, the feebler and the ruddier will be its
light, and corresponding to this the red stars are as a class fainter
than the white ones.

210. CHEMISTRY OF THE STARS.--The spectroscope is pre-eminently the
instrument to deal with this absorption of light in the stellar
atmospheres, just as it deals with that absorption in the sun's
atmosphere to which are due the dark lines of the solar spectrum,
although the faintness of starlight, compared with that of the sun,
presents a serious obstacle to its use. Despite this difficulty most of
the lucid stars and many of the telescopic ones have been studied with
the spectroscope and found to be similar to the sun and the earth as
respects the material of which they are made. Such familiar chemical
elements as hydrogen and iron, carbon, sodium, and calcium are scattered
broadcast throughout the visible universe, and while it would be
unwarranted by the present state of knowledge to say that the stars
contain nothing not found in the earth and the sun, it is evident that
in a broad way their substance is like rather than unlike that composing
the solar system, and is subject to the same physical and chemical laws
which obtain here. Galileo and Newton extended to the heavens the
terrestrial sciences of mathematics and mechanics, but it remained to
the nineteenth century to show that the physics and chemistry of the sky
are like the physics and chemistry of the earth.

211. STELLAR SPECTRA.--When the spectra of great numbers of stars are
compared one with another, it is found that they bear some relation to
the colors of the stars, as, indeed, we should expect, since spectrum
and color are both produced by the stellar atmospheres, and it is found
useful to classify these spectra into three types, as follows:

_Type I. Sirian stars._--Speaking generally, the stars which are white
or very faintly tinged with yellow, furnish spectra like that of Sirius,
from which they take their name, or that of β Aurigæ (Fig.
124), which is a continuous spectrum, especially rich in energy of short
wave length--i. e., violet and ultraviolet light, and is crossed by a
relatively small number of heavy dark lines corresponding to the
spectrum of hydrogen. Sometimes, however, these lines are much fainter
than is here shown, and we find associated with them still other faint
ones pointing to the presence of other metallic substances in the star's
atmosphere. These metallic lines are not always present, and sometimes
even the hydrogen lines themselves are lacking, but the spectrum is
always rich in violet and ultraviolet light.

Since with increasing temperature a body emits a continually increasing
proportion of energy of short wave length (§ 118), the richness of these
spectra in such energy points to a very high temperature in these stars,
probably surpassing in some considerable measure that of the sun. Stars
with this type of spectrum are more numerous than all others combined,
but next to them in point of numbers stands--

_Type II. Solar stars._--To this type of spectrum belong the yellow
stars, which show spectra like that of the sun, or of Pollux (Fig. 125).
These are not so rich in violet light as are those of Type I, but in
complexity of spectrum and in the number of their absorption lines they
far surpass the Sirian stars. They are supposed to be at a lower
temperature than the Sirian stars, and a much larger number of chemical
elements seems present and active in the reversing layer of their
atmospheres. The strong resemblance which these spectra bear to that of
the sun, together with the fact that most of the sun's stellar neighbors
have spectra of this type, justify us in ranking both them and it as
members of one class, called _solar stars_.

_Type III. Red stars._--A small number of stars show spectra comparable
with that of α Herculis (Fig. 134), in which the blue and the violet
part of the spectrum is almost obliterated, and the remaining yellow and
red parts show not only dark lines, but also numerous broad dark bands,
sharp at one edge, and gradually fading out at the other. It is this
_selective absorption_, extinguishing the blue and leaving the red end
of the spectrum, which produces the ruddy color of these stars, while
the bands in their spectra "are characteristic of chemical combinations,
and their presence ... proves that at certain elevations in the
atmospheres of these stars the temperature has sunk so low that chemical
combinations can be formed and maintained" (Scheiner-Frost). One of the
chemical compounds here indicated is a hydrocarbon similar to that found
in comets. In the white and yellow stars the temperatures are so high
that the same chemical elements, although present, can not unite one
with another to form compound substances.

[Illustration: FIG. 134.--The spectrum of α Herculis.--ESPIN.]

Most of the variable stars are red and have spectra of the third type;
but this does not hold true for the eclipse variables like Algol, all of
which are white stars with spectra of the first type. The ordinary
variable star is therefore one with a dense atmosphere of relatively low
temperature and complex structure, which produces the prevailing red
color of these stars by absorbing the major part of their radiant
energy of short wave length while allowing the longer, red waves to
escape. Although their exact nature is not understood, there can be
little doubt that the fluctuation in the light of these stars is due to
processes taking place within the star itself, but whether above or
below its photosphere is still uncertain.

212. CLASSES OF STARS.--There is no hard-and-fast dividing line between
these types of stellar spectra, but the change from one to another is by
insensible gradations, like the transition from youth to manhood and
from manhood to old age, and along the line of transition are to be
found numberless peculiarities and varieties of spectra not enumerated
above--e. g., a few stars show not only dark absorption lines in their
spectra but bright lines as well, which, like those in Fig. 48, point to
the presence of incandescent vapors, even in the outer parts of their
atmospheres. Among the lucid stars about 75 per cent have spectra of the
first type, 23 per cent are of the second type, 1 per cent of the third
type, and the remaining 1 per cent are peculiar or of doubtful
classification. Among the telescopic stars it is probable that much the
same distribution holds, but in the present state of knowledge it is not
prudent to speak with entire confidence upon this point.

That the great number of stars whose spectra have been studied should
admit of a classification so simple as the above, is an impressive fact
which, when supplemented by the further fact of a gradual transition
from one type of spectrum to the next, leaves little room for doubt that
in the stars we have an innumerable throng of individuals belonging to
the same species but in different stages of development, and that the
sun is only one of these individuals, of something less than medium size
and in a stage of development which is not at all peculiar, since it is
shared by nearly a fourth of all the stars.

213. STAR CLUSTERS.--In previous chapters we have noted the Pleiades and
Præsepe as star clusters visible to the naked eye, and to them we may
add the Hyades, near Aldebaran, and the little constellation Coma
Berenices. But more impressive than any of these, although visible only
in a telescope, is the splendid cluster in Hercules, whose appearance in
a telescope of moderate size is shown in Fig. 135, while Fig. 136 is a
photograph of the same cluster taken with a very large reflecting
telescope. This is only a type of many telescopic clusters which are
scattered over the sky, and which are made up of stars packed so closely
together as to become indistinguishable, one from another, at the center
of the cluster. Within an area which could be covered by a third of the
full moon's face are crowded in this cluster more than five thousand
stars which are unquestionably close neighbors, but whose apparent
nearness to each other is doubtless due to their great distance from us.
It is quite probable that even at the center of this cluster, where more
than a thousand stars are included within a radius of 160", the actual
distances separating adjoining stars are much greater than that
separating earth and sun, but far less than that separating the sun from
its nearest stellar neighbor.

[Illustration: FIG 135.--Star cluster in Hercules.]

An interesting discovery of recent date, made by Professor Bailey in
photographing star clusters, is that some few of them, which are
especially rich in stars, contain an extraordinary number of variable
stars, mostly very faint and of short period. Two clusters, one in the
northern and one in the southern hemisphere, contain each more than a
hundred variables, and an even more extraordinary case is presented by
a cluster, called Messier 5, not far from the star α Serpentis,
which contains no less than sixty-three variables, all about of the
fourteenth magnitude, all having light periods which differ but little
from half a day, all having light curves of about the same shape, and
all having a range of brightness from maximum to minimum of about one
magnitude. An extraordinary set of coincidences which "points
unmistakably to a common origin and cause of variability."

[Illustration: FIG. 136.--Star cluster in Hercules.--KEELER.]

[Illustration: FIG. 137.--The Andromeda nebula as seen in a very small

[Illustration: FIG. 138.--The Andromeda nebula and Holmes's comet.
Photographed by BARNARD.]

[Illustration: FIG. 139.--A drawing of the Andromeda nebula.]

[Illustration: FIG. 140.--A photograph of the Andromeda

214. NEBULÆ.--Returning to Fig. 136, we note that its background has a
hazy appearance, and that at its center the stars can no longer be
distinguished, but blend one with another so as to appear like a bright
cloud. The outer part of the cluster is _resolved_ into stars, while in
the picture the inner portion is not so resolved, although in the
original photographic plate the individual stars can be distinguished to
the very center of the cluster. In many cases, however, this is not
possible, and we have an _irresolvable cluster_ which it is customary to
call a _nebula_ (Latin, _little cloud_).

The most conspicuous example of this in the northern heavens is the
great nebula in Andromeda (R. A. 0^{h} 37^{m}, Dec. + 41°), which may be
seen with the naked eye as a faint patch of foggy light. Look for it.
This appears in an opera glass or very small telescope not unlike Fig.
137, which is reproduced from a sketch. Fig. 138 is from a photograph of
the same object showing essentially the same shape as in the preceding
figure, but bringing out more detail. Note the two small nebulæ
adjoining the large one, and at the bottom of the picture an object
which might easily be taken for another nebula but which is in fact a
tailless comet that chanced to be passing that part of the sky when the
picture was taken. Fig. 139 is from another drawing of this nebula,
although it is hardly to be recognized as a representation of the same
thing; but its characteristic feature, the two dark streaks near the
center of the picture, is justified in part by Fig. 140, which is from a
photograph made with a large reflecting telescope.

[Illustration: FIG. 141.--Types of nebulæ.]

A comparison of these several representations of the same thing will
serve to illustrate the vagueness of its outlines, and how much the
impressions to be derived from nebulæ depend upon the telescopes
employed and upon the observer's own prepossessions. The differences
among the pictures can not be due to any change in the nebula itself,
for half a century ago it was sketched much as shown in the latest of
them (Fig. 140).

[Illustration: FIG. 142.--The Trifid nebula.--KEELER.]

215. TYPICAL NEBULÆ.--Some of the fantastic forms which nebulæ present
in the telescope are shown on a small scale in Fig. 141, but in recent
years astronomers have learned to place little reliance upon drawings
such as these, which are now almost entirely supplanted by photographs
made with long exposures in powerful telescopes. One of the most
exquisite of these modern photographs is that of the Trifid nebula in
Sagittarius (Fig. 142). Note especially the dark lanes that give to this
nebula its name, Trifid, and which run through its brightest parts,
breaking it into seemingly independent sections. The area of the sky
shown in this cut is about 15 per cent less than that covered by the
full moon.

[Illustration: FIG. 143.--A nebula in Cygnus.--KEELER.]

Fig. 143 shows a very different type of nebula, found in the
constellation Cygnus, which appears made up of filaments closely
intertwined, and stretches across the sky for a distance considerably
greater than the moon's diameter.

[Illustration: FIG. 144.--Spiral nebula in Canes Venatici.--KEELER.]

A much smaller but equally striking nebula is that in the constellation
Canes Venatici (Fig. 144), which shows a most extraordinary spiral
structure, as if the stars composing it were flowing in along curved
lines toward a center of condensation. The diameter of the circular part
of this nebula, omitting the projection toward the bottom of the
picture, is about five minutes of arc, a sixth part of the diameter of
the moon, and its thickness is probably very small compared with its
breadth, perhaps not much exceeding the width of the spiral streams
which compose it. Note how the bright stars that appear within the area
of this nebula fall on the streams of nebulous matter as if they were
part of them. This characteristic grouping of the stars, which is
followed in many other nebulæ, shows that they are really part and
parcel of the nebula and not merely on line with it. Fig. 145 shows how
a great nebula is associated with the star ρ Ophiuchi.

[Illustration: FIG. 145.--Great nebula about the star ρ

Probably the most impressive of all nebulæ is the great one in Orion
(Fig. 146), whose position is shown on the star map between Rigel and
ζ Orionis. Look for it with an opera glass or even with the
unaided eye. This is sometimes called an _amorphous_--i. e.,
shapeless--nebula, because it presents no definite form which the eye
can grasp and little trace of structure or organization. It is "without
form and void" at least in its central portions, although on its edges
curved filaments may be traced streaming away from the brighter parts
of the central region. This nebula, as shown in Fig. 146, covers an area
about equal to that of the full moon, without counting as any part of
this the companion nebula shown at one side, but photographs made with
suitable exposures show that faint outlying parts of the nebula extend
in curved lines over the larger part of the constellation Orion. Indeed,
over a large part of the entire sky the background is faintly covered
with nebulous light whose brighter portions, if each were counted as a
separate nebula, would carry the total number of such objects well into
the hundreds of thousands.

[Illustration: FIG. 146.--The Orion nebula.]

The Pleiades (Plate IV) present a case of a resolvable star cluster
projected against such a nebulous background whose varying intensity
should be noted in the figure. A part of this nebulous matter is shown
in wisps extending from one star to the next, after the fashion of a
bridge, and leaving little doubt that the nebula is actually a part of
the cluster and not merely a background for it.


Fig. 147 shows a series of so-called double nebulæ perhaps comparable
with double stars, although the most recent photographic work seems to
indicate that they are really faint spiral nebulæ in which only the
brightest parts are shown by the telescope.

According to Keeler, the spiral is the prevailing type of nebulæ, and
while Fig. 144 presents the most perfect example of such a nebula, the
student should not fail to note that the Andromeda nebula (Fig. 140)
shows distinct traces of a spiral structure, only here we do not see its
true shape, the nebula being turned nearly edgewise toward us so that
its presumably circular outline is foreshortened into a narrow ellipse.

[Illustration: FIG. 147.--Double nebulæ. HERSCHEL.]

Another type of nebula of some consequence presents in the telescope
round disks like those of Uranus or Neptune, and this appearance has
given them the name _planetary nebulæ_. The comet in Fig. 138, if
smaller, would represent fairly well the nebulæ of this type. Sometimes
a planetary nebula has a star at its center, and sometimes it appears
hollow, like a smoke ring, and is then called a ring nebula. The most
famous of these is in the constellation Lyra, not far from Vega.

216. SPECTRA OF NEBULÆ.--A star cluster, like the one in Hercules,
shows, of course, stellar spectra, and even when irresolvable the
spectrum is a continuous one, testifying to the presence of stars,
although they stand too close together to be separately seen. But in a
certain number of nebulæ the spectrum is altogether different, a
discontinuous one containing only a few bright lines, showing that here
the nebular light comes from glowing gases which are subject to no
considerable pressure. The planetary nebulæ all have spectra of this
kind and make up about half of all the known gaseous nebulæ. It is
worthy of note that a century ago Sir William Herschel had observed a
green shimmer in the light of certain nebulæ which led him to believe
that they were "not of a starry nature," a conclusion which has been
abundantly confirmed by the spectroscope. The green shimmer is, in fact,
caused by a line in the green part of the spectrum that is always
present and is always the brightest part of the spectrum of gaseous

In faint nebulæ this line constitutes the whole of their visible
spectrum, but in brighter ones two or three other and fainter lines are
usually associated with it, and a very bright nebula, like that in
Orion, may show a considerable number of extra lines, but for the most
part they can not be identified in the spectrum of any terrestrial
substances. An exception to this is found in the hydrogen lines, which
are well marked in most spectra of gaseous nebulæ, and there are
indications of one or two other known substances.

217. DENSITY OF NEBULÆ.--It is known from laboratory experiments that
diminishing the pressure to which an incandescent gas is subject,
diminishes the number of lines contained in its spectrum, and we may
surmise from the very simple character and few lines of these nebular
spectra that the gas which produces them has a very small density. But
this is far from showing that the nebula itself is correspondingly
attenuated, for we must not assume that this shining gas is all that
exists in the nebula; so far as telescope or camera are concerned, there
may be associated with it any amount of dark matter which can not be
seen because it sends to us no light. It is easy to think in this
connection of meteoric dust or the stuff of which comets are made, for
these seem to be scattered broadcast on every side of the solar system
and may, perchance, extend out to the region of the nebulæ.

But, whatever may be associated in the nebula with the glowing gas which
we see, the total amount of matter, invisible as well as visible, must
be very small, or rather its average density must be very small, for the
space occupied by such a nebula as that of Orion is so great that if the
average density of its matter were equal to that of air the resulting
mass by its attraction would exert a sensible effect upon the motion of
the sun through space. The brighter parts of this nebula as seen from
the earth subtend an angle of about half a degree, and while we know
nothing of its distance from us, it is easy to see that the farther it
is away the greater must be its real dimensions, and that this increase
of bulk and mass with increasing distance will just compensate the
diminishing intensity of gravity at great distances, so that for a given
angular diameter--e. g., half a degree--the force with which this nebula
attracts the sun depends upon its density but not at all upon its
distance. Now, the nebula must attract the sun in some degree, and must
tend to move it and the planets in an orbit about the attracting center
so that year after year we should see the nebula from slightly different
points of view, and this changed point of view should produce a change
in the apparent direction of the nebula from us--i. e., a proper motion,
whose amount would depend upon the attracting force, and therefore upon
the density of the attracting matter. Observations of the Orion nebula
show that its proper motion is wholly inappreciable, certainly far less
than half a second of arc per year, and corresponding to this amount of
proper motion the mean density of the nebula must be some millions of
times (10^{10} according to Ranyard) less than that of air at sea
level--i. e., the average density throughout the nebula is comparable
with that of those upper parts of the earth's atmosphere in which
meteors first become visible.

218. MOTION OF NEBULÆ.--The extreme minuteness of their proper motions
is a characteristic feature of all nebulæ. Indeed, there is hardly a
known case of sensible proper motion of one of these bodies, although a
dozen or more of them show velocities in the line of sight ranging in
amount from +30 to -40 miles per second, the plus sign indicating an
increasing distance. While a part of these velocities may be only
apparent and due to the motion of earth and sun through space, a part at
least is real motion of the nebulæ themselves. These seem to move
through the celestial spaces in much the same way and with the same
velocities as do the stars, and their smaller proper motions across the
line of sight (angular motions) are an index of their great distance
from us. No one has ever succeeded in measuring the parallax of a nebula
or star cluster.

[Illustration: FIG. 148.--A part of the Milky Way.]

The law of gravitation presumably holds sway within these bodies, and
the fact that their several parts and the stars which are involved
within them, although attracted by each other, have shown little or no
change of position during the past century, is further evidence of
their low density and feeble attraction. In a few cases, however, there
seem to be in progress within a nebula changes of brightness, so that
what was formerly a faint part has become a brighter one, or _vice
versa_; but, on the whole, even these changes are very small.

[Illustration: FIG. 149.--The Milky Way near θ Ophiuchi.--BARNARD.]

219. THE MILKY WAY.--Closely related to nebulæ and star clusters is
another feature of the sky, the _galaxy_ or _Milky Way_, with whose
appearance to the unaided eye the student should become familiar by
direct study of the thing itself. Figs. 148 and 149 are from photographs
of two small parts of it, and serve to bring out the small stars of
which it is composed. Every star shown in these pictures is invisible to
the naked eye, although their combined light is easily seen. The general
course of the galaxy across the heavens is shown in the star maps, but
these contain no indication of the wealth of detail which even the naked
eye may detect in it. Bright and faint parts, dark rifts which cut it
into segments, here and there a hole as if the ribbon of light had been
shot away--such are some of the features to be found by attentive

[Illustration: FIG. 150.--The Milky Way near β Cygni.--BARNARD.]

Speaking generally, the course of the Milky Way is a great circle
completely girdling the sky and having its north pole in the
constellation Coma Berenices. The width of this stream of light is very
different in different parts of the heavens, amounting where it is
widest, in Lyra and Cygnus, to something more than 30°, although its
boundaries are too vague and ill defined to permit much accuracy of
measurement. Observe the very bright part between β and γ Cygni, nearly
opposite Vega, and note how even an opera glass will partially resolve
the nebulous light into a great number of stars, which are here rather
brighter than in other parts of its course. But the resolution into
stars is only partial, and there still remains a background of
unresolved shimmer. Fig. 150 is a photograph of a small part of this
region in which, although each fleck of light represents a separate
star, the galaxy is not completely resolved. Compare with this region,
rich in stars, the nearly empty space between the branches of the galaxy
a little west of Altair. Another hole in the Milky Way may be found a
little north and east of α Cygni, and between the extremes of abundance
and poverty here noted there may be found every gradation of nebulous

The Milky Way is not so simple in its structure as might at first be
thought, but a clear and moonless night is required to bring out its
details. The nature of these details, the structure of the galaxy, its
shape and extent, the arrangement of its parts, and their relation to
stars and nebulæ in general, have been subjects of much speculation by
astronomers and others who have sought to trace out in this way what is
called the _construction of the heavens_.

220. DISTRIBUTION OF THE STARS.--How far out into space do the stars
extend? Are they limited or infinite in number? Do they form a system of
mutually related parts, or are they bunched promiscuously, each for
itself, without reference to the others? Here is what has been well
called "the most important problem of stellar astronomy, the acquisition
of well-founded ideas about the distribution of the stars." While many
of the ideas upon this subject which have been advanced by eminent
astronomers and which are still current in the books are certainly
wrong, and few of their speculations along this line are demonstrably
true, the theme itself is of such grandeur and permanent interest as to
demand at least a brief consideration. But before proceeding to its
speculative side we need to collect facts upon which to build, and
these, however inadequate, are in the main simple and not far to seek.

Parallaxes, proper motions, motions in the line of sight, while
pertinent to the problem of stellar distribution, are of small avail,
since they are far too scanty in number and relate only to limited
classes of stars, usually the very bright ones or those nearest to the
sun. Almost the sole available data are contained in the brightness of
the stars and the way in which they seem scattered in the sky. The most
casual survey of the heavens is enough to show that the stars are not
evenly sprinkled upon it. The lucid stars are abundant in some regions,
few in others, and the laborious star gauges, actual counting of the
stars in sample regions of the sky, which have been made by the
Herschels, Celoria, and others, suffice to show that this lack of
uniformity in distribution is even more markedly true of the telescopic

The rate of increase in the number of stars from one magnitude to the
next, as shown in § 187, is proof of another kind of irregularity in
their distribution. It is not difficult to show, mathematically, that if
in distant regions of space the stars were on the average as numerous
and as bright as they are in the regions nearer to the sun, then the
stars of any particular magnitude ought to be four times as numerous as
those of the next brighter magnitude--e. g., four times as many
sixth-magnitude stars as there are fifth-magnitude ones. But, as we have
already seen in § 187, by actual count there are only three times as
many, and from the discrepancy between these numbers, an actual
threefold increase instead of a fourfold one, we must conclude that on
the whole the stars near the sun are either bigger or brighter or more
numerous than in the remoter depths of space.

221. THE STELLAR SYSTEM.--But the arrangement of the stars is not
altogether lawless and chaotic; there are traces of order and system,
and among these the Milky Way is the dominant feature. Telescope and
photographic plate alike show that it is made up of stars which,
although quite irregularly scattered along its course, are on the
average some twenty times as numerous in the galaxy as at its poles,
and which thin out as we recede from it on either side, at first rapidly
and then more slowly. This tendency to cluster along the Milky Way is
much more pronounced among the very faint telescopic stars than among
the brighter ones, for the lucid stars and the telescopic ones down to
the tenth or eleventh magnitude, while very plainly showing the
clustering tendency, are not more than three times as numerous in the
galaxy as in the constellations most remote from it. It is remarkable as
showing the condensation of the brightest stars that one half of all the
stars in the sky which are brighter than the second magnitude are
included within a belt extending 12° on either side of the center line
of the galaxy.

In addition to this general condensation of stars toward the Milky Way,
there are peculiarities in the distribution of certain classes of stars
which are worth attention. Planetary nebulæ and new stars are seldom, if
ever, found far from the Milky Way, and stars with bright lines in their
spectra especially affect this region of the sky. Stars with spectra of
the first type--Sirian stars--are much more strongly condensed toward
the Milky Way than are stars of the solar type, and in consequence of
this the Milky Way is peculiarly rich in light of short wave lengths.
Resolvable star clusters are so much more numerous in the galaxy than
elsewhere, that its course across the sky would be plainly indicated by
their grouping upon a map showing nothing but clusters of this kind.

On the other hand, nebulæ as a class show a distinct aversion for the
galaxy, and are found most abundantly in those parts of the sky farthest
from it, much as if they represented raw material which was lacking
along the Milky Way, because already worked up to make the stars which
are there so numerous.

222. RELATION OF THE SUN TO THE MILKY WAY.--The fact that the galaxy is
a _great circle_ of the sky, but only of moderate width, shows that it
is a widely extended and comparatively thin stratum of stars within
which the solar system lies, a member of the galactic system, and
probably not very far from its center. This position, however, is not to
be looked upon as a permanent one, since the sun's motion, which lies
nearly in the plane of the Milky Way, is ceaselessly altering its
relation to the center of that system, and may ultimately carry us
outside its limits.

The Milky Way itself is commonly thought to be a ring, or series of
rings, like the coils of the great spiral nebula in Andromeda, and
separated from us by a space far greater than the thickness of the ring
itself. Note in Figs. 149 and 150 how the background is made up of
bright and dark parts curiously interlaced, and presenting much the
appearance of a thin sheet of cloud through which we look to barren
space beyond. While, mathematically, this appearance can not be
considered as proof that the galaxy is in fact a distant ring, rather
than a sheet of starry matter stretching continuously from the nearer
stellar neighbors of the sun into the remotest depths of space,
nevertheless, most students of the question hold it to be such a ring of
stars, which are relatively close together while its center is
comparatively vacant, although even here are some hundreds of thousands
of stars which on the whole have a tendency to cluster near its plane
and to crowd together a little more densely than elsewhere in the region
where the sun is placed.

223. DIMENSIONS OF THE GALAXY.--The dimensions of this stellar system
are wholly unknown, but there can be no doubt that it extends farther in
the plane of the Milky Way than at right angles to that plane, for stars
of the fifteenth and sixteenth magnitudes are common in the galaxy, and
testify by their feeble light to their great distance from the earth,
while near the poles of the Milky Way there seem to be few stars fainter
than the twelfth magnitude. Herschel, with his telescope of 18 inches
aperture, could count in the Milky Way more than a dozen times as many
stars per square degree as could Celoria with a telescope of 4 inches
aperture; but around the poles of the galaxy the two telescopes showed
practically the same number of stars, indicating that here even the
smaller telescope reached to the limits of the stellar system. Very
recently, indeed, the telescope with which Fig. 140 was photographed
seems to have reached the farthest limit of the Milky Way, for on a
photographic plate of one of its richest regions Roberts finds it
completely resolved into stars which stand out upon a black background
with no trace of nebulous light between them.

224. BEYOND THE MILKY WAY.--Each additional step into the depths of
space brings us into a region of which less is known, and what lies
beyond the Milky Way is largely a matter of conjecture. We shrink from
thinking it an infinite void, endless emptiness, and our intellectual
sympathies go out to Lambert's speculation of a universe filled with
stellar systems, of which ours, bounded by the galaxy, is only one.
There is, indeed, little direct evidence that other such systems exist,
but the Andromeda nebula is not altogether unlike a galaxy with a
central cloud of stars, and in the southern hemisphere, invisible in our
latitudes, are two remarkable stellar bodies like the Milky Way in
appearance, but cut off from all apparent connection with it, much as we
might expect to find independent stellar systems, if such there be.

These two bodies are known as the Magellanic clouds, and individually
bear the names of Major and Minor Nubecula. According to Sir John
Herschel, "the Nubecula Major, like the Minor, consists partly of large
tracts and ill-defined patches of irresolvable nebula, and of nebulosity
in every stage of resolution up to perfectly resolved stars like the
Milky Way, as also of regular and irregular nebulæ ... of globular
clusters in every stage of resolvability, and of clustering groups
sufficiently insulated and condensed to come under the designation of
clusters of stars." Its outlines are vague and somewhat uncertain, but
surely include an area of more than 40 square degrees--i. e., as much as
the bowl of the Big Dipper--and within this area Herschel counted
several hundred nebulæ and clusters "which far exceeds anything that is
to be met with in any other region of the heavens." Although its
excessive complexity of detail baffled Herschel's attempts at artistic
delineation, it has yielded to the modern photographic processes, which
show the Nubecula Major to be an enormous spiral nebula made up of
subordinate stars, nebulæ, and clusters, as is the Milky Way.

Compared with the Andromeda nebula, its greater angular extent suggests
a smaller distance, although for the present all efforts at determining
the parallax of either seem hopeless. But the spiral form which is
common to both suggests that the Milky Way itself may be a gigantic
spiral nebula near whose center lies the sun, a humble member of a great
cluster of stars which is roughly globular in shape, but flattened at
the poles of the galaxy and completely encircled by its coils. However
plausible such a view may appear, it is for the present, at least, pure
hypothesis, although vigorously advocated by Easton, who bases his
argument upon the appearance of the galaxy itself.

225. ABSORPTION OF STARLIGHT.--We have had abundant occasion to learn
that at least within the confines of the solar system meteoric matter,
cosmic dust, is profusely scattered, and it appears not improbable that
the same is true, although in smaller degree, in even the remoter parts
of space. In this case the light which comes from the farther stars over
a path requiring many centuries to travel, must be in some measure
absorbed and enfeebled by the obstacles which it encounters on the way.
Unless celestial space is transparent to an improbable degree the
remoter stars do not show their true brightness; there is a certain
limit beyond which no star is able to send its light, and beyond which
the universe must be to us a blank. A lighthouse throws into the fog its
beams only to have them extinguished before a single mile is passed, and
though the celestial lights shine farther, a limit to their reach is
none the less certain if meteoric dust exists outside the solar system.
If there is such an absorption of light in space, as seems plausible,
the universe may well be limitless and the number of stellar systems
infinite, although the most attenuated of dust clouds suffices to
conceal from us and to shut off from our investigation all save a minor
fraction of it and them.



226. NATURE OF THE PROBLEM.--To use a common figure of speech, the
universe is alive. We have found it filled with an activity that
manifests itself not only in the motions of the heavenly bodies along
their orbits, but which extends to their minutest parts, the molecules
and atoms, whose vibrations furnish the radiant energy given off by sun
and stars. Some of these activities, such as the motions of the heavenly
bodies in their orbits, seem fitted to be of endless duration; while
others, like the radiation of light and heat, are surely temporary, and
sooner or later must come to an end and be replaced by something
different. The study of things as they are thus leads inevitably to
questions of what has been and what is to be. A sound science should
furnish some account of the universe of yesterday and to-morrow as well
as of to-day, and we need not shrink from such questions, although
answers to them must be vague and in great measure speculative.

The historian of America finds little difficulty with events of the
nineteenth century or even the eighteenth, but the sources of
information about America in the fifteenth century are much less
definite; the tenth century presents almost a blank, and the history of
American mankind in the first century of the Christian era is wholly
unknown. So, as we attempt to look into the past or the future of the
heavens, we must expect to find the mists of obscurity grow denser with
remoter periods until even the vaguest outlines of its development are
lost, and we are compelled to say, beyond this lies the unknown. Our
account of growth and decay in the universe, therefore, can not aspire
to cover the whole duration of things, but must be limited in its scope
to certain chapters whose epochs lie near to the time in which we live,
and even for these we need to bear constantly in mind the logical bases
of such an inquiry and the limitations which they impose upon us.

227. LOGICAL BASES AND LIMITATIONS.--The first of these bases is: An
adequate knowledge of the present universe. Our only hope of reading the
past and future lies in an understanding of the present; not necessarily
a complete knowledge of it, but one which is sound so far as it goes.
Our position is like that of a detective who is called upon to unravel a
mystery or crime, and who must commence with the traces that have been
left behind in its commission. The foot print, the blood stain, the
broken glass must be examined and compared, and fashioned into a theory
of how they came to be; and as a wrong understanding of these elements
is sure to vitiate the theories based upon them, so a false science of
the universe as it now is, will surely give a false account of what it
has been; while a correct but incomplete knowledge of the present does
not wholly bar an understanding of the past, but only puts us in the
position of the detective who correctly understands what he sees but
fails to take note of other facts which might greatly aid him.

The second basis of our inquiry is: The assumed permanence of natural
laws. The law of gravitation certainly held true a century ago as well
as a year ago, and for aught we know to the contrary it may have been a
law of the universe for untold millions of years; but that it has
prevailed for so long a time is a pure assumption, although a necessary
one for our purpose. So with those other laws of mathematics and
mechanics and physics and chemistry to which we must appeal; if there
was ever a time or place in which they did not hold true, that time and
place lie beyond the scope of our inquiry, and are in the domain
inaccessible to scientific research. It is for this reason that science
knows nothing and can know nothing of a creation or an end of the
universe, but considers only its orderly development within limited
periods of time. What kind of a past universe would, under the operation
of known laws, develop into the present one, is the question with which
we have to deal, and of it we may say with Helmholtz: "From the
standpoint of science this is no idle speculation but an inquiry
concerning the limitations of its methods and the scope of its known

To ferret out the processes by which the heavenly bodies have been
brought to their present condition we seek first of all for lines of
development now in progress which tend to change the existing order of
things into something different, and, having found these, to trace their
effects into both past and future. Any force, however small, or any
process, however slow, may produce great results if it works always and
ceaselessly in the same direction, and it is in these processes, whose
trend is never reversed, that we find a partial clew to both past and

228. THE SUN'S DEVELOPMENT.--The first of these to claim our attention
is the shrinking of the sun's diameter which, as we have seen in Chapter
X, is the means by which the solar output of radiant energy is
maintained from year to year. Its amount, only a few feet per annum, is
far too small to be measured with any telescope; but it is cumulative,
working century after century in the same direction, and, given time
enough, it will produce in the future, and must have produced in the
past, enormous transformations in the sun's bulk and equally significant
changes in its physical condition.

Thus, as we attempt to trace the sun's history into the past, the
farther back we go the greater shall we expect to find its diameter and
the greater the space (volume) through which its molecules are spread.
By reason of this expansion its density must have been less then than
now, and by going far enough back we may even reach a time at which the
density was comparable with what we find in the nebulæ of to-day. If our
ideas of the sun's present mechanism are sound, then, as a necessary
consequence of these, its past career must have been a process of
condensation in which its component particles were year by year packed
closer together by their own attraction for each other. As we have seen
in § 126, this condensation necessarily developed heat, a part of which
was radiated away as fast as produced, while the remainder was stored
up, and served to raise the temperature of the sun to what we find it
now. At the present time this temperature is a chief obstacle to further
shrinkage, and so powerfully opposes the gravitative forces as to
maintain nearly an equilibrium with them, thus causing a very slow rate
of further condensation. But it is not probable that this was always so.
In the early stages of the sun's history, when the temperature was low,
contraction of its bulk must have been more rapid, and attempts have
been made by the mathematicians to measure its rate of progress and to
determine how long a time has been consumed in the development of the
present sun from a primitive nebulous condition in which it filled a
space of greater diameter than Neptune's orbit. Of course, numerical
precision is not to be expected in results of this kind, but, from a
consideration of the greatest amount of heat that could be furnished by
the shrinkage of a mass equal to that of the sun, it seems that the
period of this development is to be measured in tens of millions or
possibly hundreds of millions of years, but almost certainly does not
reach a thousand millions.

229. THE SUN'S FUTURE.--The future duration of the sun as a source of
radiant energy is surely to be measured in far smaller numbers than
these. Its career as a dispenser of light and heat is much more than
half spent, for the shrinkage results in an ever-increasing density,
which makes its gaseous substance approximate more and more toward the
behavior of a liquid or solid, and we recall that these forms of matter
can not by any further condensation restore the heat whose loss through
radiation caused them to contract. They may continue to shrink, but
their temperature must fall, and when the sun's substance becomes too
dense to obey the laws of gaseous matter its surface must cool rapidly
as a consequence of the radiation into surrounding space, and must
congeal into a crust which, although at first incandescent, will
speedily become dark and opaque, cutting off the light of the central
portions, save as it may be rent from time to time by volcanic outbursts
of the still incandescent mass beneath. But such outbursts can be of
short duration only, and its final condition must be that of a dark
body, like the earth or moon, no longer available as a source of radiant
energy. Even before the formation of a solid crust it is quite possible
that the output of light and heat may be seriously diminished by the
formation of dense vapors completely enshrouding it, as is now the case
with Jupiter and Saturn. It is believed that these planets were formerly
incandescent, and at the present time are in a state of development
through which the earth has passed and toward which the sun is moving.
According to Newcomb, the future during which the sun can continue to
furnish light and heat at its present rate is not likely to exceed
10,000,000 years.

This idea of the sun as a developing body whose present state is only
temporary, furnishes a clew to some of the vexing problems of solar
physics. Thus the sun-spot period, the distribution of the spots in
latitude, and the peculiar law of rotation of the sun in different
latitudes, may be, and very probably are, results not of anything now
operating beneath its photosphere, but of something which happened to it
in the remote past--e. g., an unsymmetrical shrinkage or possibly a
collision with some other body. At sea the waves continue to toss long
after the storm which produced them has disappeared, and, according to
the mathematical researches of Wilsing, a profound agitation of the
sun's mass might well require tens of thousands, or even hundreds of
thousands of years to subside, and during this time its effects would be
visible, like the waves, as phenomena for which the actual condition of
things furnishes no apparent cause.

230. THE NEBULAR HYPOTHESIS.--The theory of the sun's progressive
contraction as a necessary result of its radiation of energy is
comparatively modern, but more than a century ago philosophic students
of Nature had been led in quite a different way to the belief that in
the earlier stages of its career the sun must have been an enormously
extended body whose outer portions reached even beyond the orbit of the
remotest planet. Laplace, whose speculations upon this subject have had
a dominant influence during the nineteenth century, has left, in a
popular treatise upon astronomy, an admirable statement of the phenomena
of planetary motion, which suggest and lead up to the nebular theory of
the sun's development, and in presenting this theory we shall follow
substantially his line of thought, but with some freedom of translation
and many omissions.

He says: "To trace out the primitive source of the planetary movements,
we have the following five phenomena: (1) These movements all take place
in the same direction and nearly in the same plane. (2) The movements of
the satellites are in the same direction as those of the planets. (3)
The rotations of the planets and the sun are in the same direction as
the orbital motions and nearly in the same plane. (4) Planets and
satellites alike have nearly circular orbits. (5) The orbits of comets
are wholly unlike these by reason of their great eccentricities and
inclinations to the ecliptic." That these coincidences should be purely
the result of chance seemed to Laplace incredible, and, seeking a cause
for them, he continues: "Whatever its nature may be, since it has
produced or controlled the motions of the planets, it must have reached
out to all these bodies, and, in view of the prodigious distances which
separate them, the cause can have been nothing else than a fluid of
great extent which must have enveloped the sun like an atmosphere. A
consideration of the planetary motions leads us to think that ... the
sun's atmosphere formerly extended far beyond the orbits of all the
planets and has shrunk by degrees to its present dimensions." This is
not very different from the idea developed in § 228 from a consideration
of the sun's radiant energy; but in Laplace's day the possibility of
generating the sun's heat by contraction of its bulk was unknown, and he
was compelled to assume a very high temperature for the primitive
nebulous sun, while we now know that this is unnecessary. Whether the
primitive nebula was hot or cold the shrinkage would take place in much
the same way, and would finally result in a star or sun of very high
temperature, but its development would be slower if it were hot in the
beginning than if it were cold.

But again Laplace: "How did the sun's atmosphere determine the rotations
and revolutions of planets and satellites? If these bodies had been
deeply immersed in this atmosphere its resistance to their motion would
have made them fall into the sun, and we may therefore conjecture that
the planets were formed, one by one, at the outer limits of the solar
atmosphere by the condensation of zones of vapor which were cast off in
the plane of the sun's equator." Here he proceeds to show by an appeal
to dynamical principles that something of this kind must happen, and
that the matter sloughed off by the nebula in the form of a ring,
perhaps comparable to the rings of Saturn or the asteroid zone, would
ultimately condense into a planet, which in its turn might shrink and
cast off rings to produce satellites.

[Illustration: PIERRE SIMON LAPLACE (1749-1827).]

Planets and satellites would then all have similar motions, as noted at
the beginning of this section, since in every case this motion is an
inheritance from a common source, the rotation of the primitive
nebula about its own axis. "All the bodies which circle around a planet
having been thus formed from rings which its atmosphere successively
abandoned as rotation became more and more rapid, this rotation should
take place in less time than is required for the orbital revolution of
any of the bodies which have been cast off, and this holds true for the
sun as compared with the planets."

slower rate of motion was also supposed to hold true for Saturn's rings
as compared with the rotation of Saturn itself, but, as we have seen in
Chapter XI, this ring is made up of a great number of independent
particles which move at different rates of speed, and comparing, through
Kepler's Third Law, the motion of the inner edge of the ring with the
known periodic time of the satellites, we may find that these particles
must rotate about Saturn more rapidly than the planet turns upon its
axis. Similarly the inner satellite of Mars completes its revolution in
about one third of a Martian day, and we find in cases like this grounds
for objection to the nebular theory. Compare also Laplace's argument
with the peculiar rotations of Uranus, Neptune, and their satellites
(Chapter XI). Do these fortify or weaken his case?

Despite these objections and others equally serious that have been
raised, the nebular theory agrees with the facts of Nature at so many
points that astronomers upon the whole are strongly inclined to accept
its major outlines as being at least an approximation to the course of
development actually followed by the solar system; but at some
points--e. g., the formation of planets and satellites through the
casting off of nebulous rings--the objections are so many and strong as
to call for revision and possibly serious modification of the theory.

One proposed modification, much discussed in recent years, consists in
substituting for the primitive _gaseous_ nebula imagined by Laplace, a
very diffuse cloud of meteoric matter which in the course of its
development would become transformed into the gaseous state by rising
temperature. From this point of view much of the meteoric dust still
scattered throughout the solar system may be only the fragments left
over in fashioning the sun and planets. Chamberlin and Moulton, who have
recently given much attention to this subject, in dissenting from some
of Laplace's views, consider that the primitive nebulous condition must
have been one in which the matter of the system was "so brought together
as to give low mass, high momentum, and irregular distribution to the
outer part, and high mass, low momentum, and sphericity to the central
part," and they suggest a possible oblique collision of a small nebula
with the outer parts of a large one.

232. BODE'S LAW.--We should not leave the theory of Laplace without
noting the light it casts upon one point otherwise obscure--the meaning
of Bode's law (§ 134). This law, stated in mathematical form, makes a
geometrical series, and similar geometrical series apply to the
distances of the satellites of Jupiter and Saturn from these planets.
Now, Roche has shown by the application of physical laws to the
shrinkage of a gaseous body that its radius at any time may be expressed
by means of a certain mathematical formula very similar to Bode's law,
save that it involves the amount of time that has elapsed since the
beginning of the shrinking process. By comparing this formula with the
one corresponding to Bode's law he reaches the conclusion that the
peculiar spacing of the planets expressed by that law means that they
were formed at successive _equal_ intervals of time--i. e., that Mars is
as much older than the earth as the earth is older than Venus, etc. The
failure of Bode's law in the case of Neptune would then imply that the
interval of time between the formation of Neptune and Uranus was shorter
than that which has prevailed for the other planets. But too much
stress should not be placed upon this conclusion. So long as the manner
in which the planets came into being continues an open question,
conclusions about their time of birth must remain of doubtful validity.

233. TIDAL FRICTION BETWEEN EARTH AND MOON.--An important addition to
theories of development within the solar system has been worked out by
Prof. G. H. Darwin, who, starting with certain very simple assumptions
as to the present condition of things in earth and moon, derives from
these, by a strict process of mathematical reasoning, far-reaching
conclusions of great interest and importance. The key to these
conclusions lies in recognition of the fact that through the influence
of the tides (§ 42) there is now in progress and has been in progress
for a very long time, a gradual transfer of motion (moment of momentum)
from the earth to the moon. The earth's motion of rotation is being
slowly destroyed by the friction of the tides, as the motion of a
bicycle is destroyed by the friction of a brake, and, in consequence of
this slowing down, the moon is pushed farther and farther away from the
earth, so that it now moves in a larger orbit than it had some millions
of years ago.

Fig. 24 has been used to illustrate the action of the moon in raising
tides upon the earth, but in accordance with the third law of motion
(§ 36) this action must be accompanied by an equal and contrary reaction
whose nature may readily be seen from the same figure. The moon moves
about its orbit from west to east and the earth rotates about its axis
in the same direction, as shown by the curved arrow in the figure. The
tidal wave, _I_, therefore points a little _in advance_ of the moon's
position in its orbit and by its attraction must tend to pull the moon
ahead in its orbital motion a little faster than it would move if the
whole substance of the earth were placed inside the sphere represented
by the broken circle in the figure. It is true that the tidal wave at
_I´´_ pulls back and tends to neutralize the effect of the wave at _I_,
but on the whole the tidal wave nearer the moon has the stronger
influence, and the moon on the whole moves a very little faster, and by
virtue of this added impetus draws continually a little farther away
from the earth than it would if there were no tides.

moving the moon away from the earth is a cumulative one, going on
century after century, and with reference to it the moon's orbit must be
described not as a circle or ellipse, or any other curve which returns
into itself, but as a spiral, like the balance spring of a watch, each
of whose coils is a little larger than the preceding one, although this
excess is, to be sure, very small, because the tides themselves are
small and the tidal influence feeble when compared with the whole
attraction of the earth for the moon. But, given time enough, even this
small force may accomplish great results, and something like 100,000,000
years of past opportunity would have sufficed for the tidal forces to
move the moon from close proximity with the earth out to its present

For millions of years to come, if moon and earth endure so long, the
distance between them must go on increasing, although at an ever slower
rate, since the farther away the moon goes the smaller will be the tides
and the slower the working out of their results. On the other hand, when
the moon was nearer the earth than now, tidal influences must have been
greater and their effects more rapidly produced than at the present
time, particularly if, as seems probable, at some past epoch the earth
was hot and plastic like Jupiter and Saturn. Then, instead of tides in
the water of the sea, such as we now have, the whole substance of the
earth would respond to the moon's attraction in _bodily tides_ of
semi-fluid matter not only higher, but with greater internal friction of
their molecules one upon another, and correspondingly greater effect in
checking the earth's rotation.

But, whether the tide be a bodily one or confined to the waters of the
sea, so long as the moon causes it to flow there will be a certain
amount of friction which will affect the earth much as a brake affects a
revolving wheel, slowing down its motion, and producing thus a longer
day as well as a longer month on account of the moon's increased
distance. Slowing down the earth's rotation is the direct action of the
moon upon the earth. Pushing the moon away is the form in which the
earth's equal and contrary reaction manifests itself.

plastic the earth must have raised in it a bodily tide manifold greater
than the lunar tides upon the earth, and, as we have seen in Chapter IX,
this tide has long since worn out the greater part of the moon's
rotation and brought our satellite to the condition in which it presents
always the same face toward the earth.

These two processes, slowing down the rotation and pushing away the
disturbing body, are inseparable--one requires the other; and it is
worth noting in this connection that when for any reason the tide ceases
to flow, and the tidal wave takes up a permanent position, as it has in
the moon (§ 99), its work is ended, for when there is no motion of the
wave there can be no friction to further reduce the rate of rotation of
the one body, and no reaction to that friction to push away the other.
But this permanent and stationary tidal wave in the moon, or elsewhere,
means that the satellite presents always the same face toward its
planet, moving once about its orbit in the time required for one
revolution upon its axis, and the tide raised by the moon upon the earth
tends to produce here the result long since achieved in our satellite,
to make our day and month of equal length, and to make the earth turn
always the same side toward the moon. But the moon's tidal force is
small compared with that of the earth, and has a vastly greater momentum
to overcome, so that its work upon the earth is not yet complete.
According to Thomson and Tait, the moon must be pushed off another
hundred thousand miles, and the day lengthened out by tidal influence to
seven of our present weeks before the day and the lunar month are made
of equal length, and the moon thereby permanently hidden from one
hemisphere of the earth.

236. THE EARTH-MOON SYSTEM.--Retracing into the past the course of
development of the earth and moon, it is possible to reach back by means
of the mathematical theory of tidal friction to a time at which these
bodies were much nearer to each other than now, but it has not been
found possible to trace out the mode of their separation from one body
into two, as is supposed in the nebular theory. In the earliest part of
their history accessible to mathematical analysis they are distinct
bodies at some considerable distance from each other, with the earth
rotating about an axis more nearly perpendicular to the moon's orbit and
to the ecliptic than is now the case. Starting from such a condition,
the lunar tides, according to Darwin, have been instrumental in tipping
the earth's rotation axis into its present oblique position, and in
determining the eccentricity of the moon's orbit and its position with
respect to the ecliptic as well as the present length of day and month.

337. TIDAL FRICTION UPON THE PLANETS.--The satellites of the outer
planets are equally subject to influences of this kind, and there
appears to be independent evidence that some of them, at least, turn
always the same face toward their respective planets, indicating that
the work of tidal friction has here been accomplished. We saw in Chapter
XI that it is at present an open question whether the inner planets,
Venus and Mercury, do not always turn the same face toward the sun,
their day and year being of equal length. In addition to the direct
observational evidence upon this point, Schiaparelli has sought to show
by an appeal to tidal theory that such is probably the case, at least
for Mercury, since the tidal forces which tend to bring about this
result in that planet are about as great as the forces which have
certainly produced it in the case of the moon and Saturn's satellite,
Japetus. The same line of reasoning would show that every satellite in
the solar system, save possibly the newly discovered ninth satellite of
Saturn, must, as a consequence of tidal friction, turn always the same
face toward its planet.

238. THE SOLAR TIDE.--The sun also raises tides in the earth, and their
influence must be similar in character to that of the lunar tides,
checking the rotation of the earth and thrusting earth and sun apart,
although quantitatively these effects are small compared with those of
the moon. They must, however, continue so long as the solar tide lasts,
possibly until the day and year are made of equal length--i. e., they
may continue long after the lunar tidal influence has ceased to push
earth and moon apart. Should this be the case, a curious inverse effect
will be produced. The day being then longer than the month, the moon
will again raise a tide in the earth which will run around it _from west
to east_, opposite to the course of the present tide, thus tending to
accelerate the earth's rotation, and by its reaction to bring the moon
back toward the earth again, and ultimately to fall upon it.

We may note that an effect of this kind must be in progress now between
Mars and its inner satellite, Phobos, whose time of orbital revolution
is only one third of a Martian day. It seems probable that this
satellite is in the last stages of its existence as an independent body,
and must ultimately fall into Mars.

239. ROCHE'S LIMIT.--In looking forward to such a catastrophe, however,
due regard must be paid to a dynamical principle of a different
character. The moon can never be precipitated upon the earth entire,
since before it reaches us it will have been torn asunder by the excess
of the earth's attraction for the near side of its satellite over that
which it exerts upon the far side. As the result of Roche's mathematical
analysis we are able to assign a limiting distance between any planet
and its satellite within which the satellite, if it turns always the
same face toward the planet, can not come without being broken into
fragments. If we represent the radius of the planet by _r_, and the
quotient obtained by dividing the density of the planet by the density
of the satellite by _q_, then

    Roche's limit = 2.44 r ∛q.

Thus in the case of earth and moon we find from the densities given in
§ 95, _q_ = 1.65, and with _r_ = 3,963 miles we obtain 11,400 miles as
the nearest approach which the moon could make to the earth without
being broken up by the difference of the earth's attractions for its
opposite sides.

We must observe, however, that Roche's limit takes no account of
molecular forces, the adhesion of one molecule to another, by virtue of
which a stick or stone resists fracture, but is concerned only with the
gravitative forces by which the molecules are attracted toward the
moon's center and toward the earth. Within a stone or rock of moderate
size these gravitative forces are insignificant, and cohesion is the
chief factor in preserving its integrity, but in a large body like the
moon, the case is just reversed, cohesion plays a small part and
gravitation a large one in holding the body together. We may conclude,
therefore, that at a proper distance these forces are capable of
breaking up the moon, or any other large body, into fragments of a size
such that molecular cohesion instead of gravitation is the chief agent
in preserving them from further disintegration.

240. SATURN'S RINGS.--Saturn's rings are of peculiar interest in this
connection. The outer edge of the ring system lies just inside of
Roche's limit for this planet, and we have already seen that the rings
are composed of small fragments independent of each other. Whatever may
have been the process by which the nine satellites of Saturn came into
existence, we have in Roche's limit the explanation why the material of
the ring was not worked up into satellites; the forces exerted by Saturn
would tear into pieces any considerable satellite thus formed and
equally would prevent the formation of one from raw material.

Saturn's rings present the only case within the solar system where
matter is known to be revolving about a planet at a distance less than
Roche's limit, and it is an interesting question whether these rings can
remain as a permanent part of the planet's system or are only a
temporary feature. The drawings of Saturn made two centuries ago agree
among themselves in representing the rings as larger than they now
appear, and there is some reason to suppose that as a consequence of
mutual disturbances--collisions--their momentum is being slowly wasted
so that ultimately they must be precipitated into the planet. But the
direct evidence of such a progress that can be drawn from present data
is too scanty to justify positive conclusions in the matter. On the
other hand, Nolan suggests that in the outer parts of the ring small
satellites might be formed whose tidal influence upon Saturn would
suffice to push them away from the ring beyond Roche's limit, and that
the very small inner satellites of Saturn may have been thus formed at
the expense of the ring.

The inner satellite of Mars is very close to Roche's limit for that
planet, and, as we have seen above, must be approaching still nearer to
the danger line.

241. THE MOON'S DEVELOPMENT.--The fine series of photographs of the moon
obtained within the last few years at Paris, have been used by the
astronomers of that observatory for a minute study of the lunar
formations, much as geologists study the surface of the earth to
determine something about the manner in which it was formed. Their
conclusions are, in general, that at some past time the moon was a hot
and fluid body which, as it cooled and condensed, formed a solid crust
whose further shrinkage compressed the liquid nucleus and led to a long
series of fractures in the crust and outbursts of liquid matter, whose
latest and feeblest stages produced the lunar craters, while traces of
the earlier ones, connected with a general settling of the crust,
although nearly obliterated, are still preserved in certain large but
vague features of the lunar topography, such as the distribution of the
seas, etc. They find also in certain markings of the surface what they
consider convincing evidence of the existence in past times of a lunar
atmosphere. But this seems doubtful, since the force of gravity at the
moon's surface is so small that an atmosphere similar to that of the
earth, even though placed upon the moon, could not permanently endure,
but would be lost by the gradual escape of its molecules into the
surrounding space.

The molecules of a gas are quite independent one of another, and are in
a state of ceaseless agitation, each one darting to and fro, colliding
with its neighbors or with whatever else opposes its forward motion, and
traveling with velocities which, on the average, amount to a good many
hundreds of feet per second, although in the case of any individual
molecule they may be much less or much greater than the average value,
an occasional molecule having possibly a velocity several times as great
as the average. In the upper regions of our own atmosphere, if one of
these swiftly moving particles of oxygen or nitrogen were headed away
from the earth with a velocity of seven miles per second, the whole
attractive power of the earth would be insufficient to check its motion,
and it would therefore, unless stopped by some collision, escape from
the earth and return no more. But, since this velocity of seven miles
per second is more than thirty times as great as the average velocity of
the molecules of air, it must be very seldom indeed that one is found to
move so swiftly, and the loss of the earth's atmosphere by leakage of
this sort is insignificant. But upon the moon, or any other body where
the force of gravity is small, conditions are quite different, and in
our satellite a velocity of little more than one mile per second would
suffice to carry a molecule away from the outer limits of its
atmosphere. This velocity, only five times the average, would be
frequently attained, particularly in former times when the moon's
temperature was high, for then the average velocity of all the molecules
would be considerably increased, and the amount of leakage might become,
and probably would become, a serious matter, steadily depleting the
moon's atmosphere and leading finally to its present state of
exhaustion. It is possible that the moon may at one time have had an
atmosphere, but if so it could have been only a temporary possession,
and the same line of reasoning may be applied to the asteroids and to
most of the satellites of the solar system, and also, though in less
degree, to the smaller planets, Mercury and Mars.

242. STELLAR DEVELOPMENT.--We have already considered in this chapter
the line of development followed by one star, the sun, and treating this
as a typical case, it is commonly believed that the life history of a
star, in so far as it lies within our reach, begins with a condition in
which its matter is widely diffused, and presumably at a low
temperature. Contracting in bulk under the influence of its own
gravitative forces, the star's temperature rises to a maximum, and then
falls off in later stages until the body ceases to shine and passes over
to the list of dark stars whose existence can only be detected in
exceptional cases, such as are noted in Chapter XIII. The most
systematic development of this idea is due to Lockyer, who looks upon
all the celestial bodies--sun, moon and planets, stars, nebulæ, and
comets--as being only collections of meteoric matter in different stages
of development, and who has sought by means of their spectra to classify
these bodies and to determine their stage of advancement. While the
fundamental ideas involved in this "meteoritic hypothesis" are not
seriously controverted, the detailed application of its principles is
open to more question, and for the most part those astronomers who hold
that in the present state of knowledge stellar spectra furnish a key to
a star's age or degree of advancement do not venture beyond broad
general statements.

[Illustration: FIG. 151.--Types of stellar spectra substantially
according to SECCHI.]

243. STELLAR SPECTRA.--Thus the types of stellar spectra shown in Fig.
151 are supposed to illustrate successive stages in the development of
an average star. Type I corresponds to the period in which its
temperature is near the maximum; Type II belongs to a later stage in
which the temperature has commenced to fall; and Type III to the period
immediately preceding extinction.

While human life, or even the duration of the human race, is too short
to permit a single star to be followed through all the stages of its
career, an adequate picture of that development might be obtained by
examining many stars, each at a different stage of progress, and,
following this idea, numerous subdivisions of the types of stellar
spectra shown in Fig. 151 have been proposed in order to represent with
more detail the process of stellar growth and decay; but for the most
part these subdivisions and their interpretation are accepted by
astronomers with much reserve.

It is significant that there are comparatively few stars with spectra of
Type III, for this is what we should expect to find if the development
of a star through the last stages of its visible career occupied but a
small fraction of its total life. From the same point of view the great
number of stars with spectra of the first type would point to a long
duration of this stage of life. The period in which the sun belongs,
represented by Type II, probably has a duration intermediate between the
others. Since most of the variable stars, save those of the Algol class,
have spectra of the third type, we conclude that variability, with its
associated ruddy color and great atmospheric absorption of light, is a
sign of old age and approaching extinction. The Algol or eclipse
variables, on the other hand, having spectra of the first type, are
comparatively young stars, and, as we shall see a little later, the
shortness of their light periods in some measure confirms this
conclusion drawn from their spectra.

We have noted in § 196 that the sun's near neighbors are prevailingly
stars with spectra of the second type, while the Milky Way is mainly
composed of first-type stars, and from this we may now conclude that in
our particular part of the entire celestial space the stars are, as a
rule, somewhat further developed than is the case elsewhere.

244. DOUBLE STARS.--The double stars present special problems of
development growing out of the effects of tidal friction, which must
operate in them much as it does between earth and moon, tending steadily
to increase the distance between the components of such a star. So, too,
in such a system as is shown in Fig. 132, gravity must tend to make each
component of the double star shrink to smaller dimensions, and this
shrinkage must result in faster rotation and increased tidal friction,
which in turn must push the components apart, so that in view of the
small density and close proximity of those particular stars we may
fairly regard a star like β Lyræ as in the early stages of its
career and destined with increasing age to lose its variability of
light, since the eclipses which now take place must cease with
increasing distance between the components unless the orbit is turned
exactly edgewise toward the earth. Close proximity and the resulting
shortness of periodic time in a double star seem, therefore, to be
evidence of its youth, and since this shortness of periodic time is
characteristic of both Algol variables and spectroscopic binaries as a
class, we may set them down as being, upon the whole, stars in the early
stages of their career. On the other hand, it is generally true that the
larger the orbit, and the greater the periodic time in the orbit, the
farther is the star advanced in its development.

In his theory of tidal friction, Darwin has pointed out that whenever
the periodic time in the orbit is more than twice as long as the time
required for rotation about the axis, the effect of the tides is to
increase the eccentricity of the orbit, and, following this indication,
See has urged that with increasing distance between the components of a
double star their orbits about the common center of gravity must grow
more and more eccentric, so that we have in the shape of such orbits a
new index of stellar development; the more eccentric the orbit, the
farther advanced are the stars. It is important to note in this
connection that among the double stars whose orbits have been computed
there seems to run a general rule--the larger the orbit the greater is
its eccentricity--a relation which must hold true if tidal friction
operates as above supposed, and which, being found to hold true,
confirms in some degree the criteria of stellar age which are furnished
by the theory of tidal friction.

245. NEBULÆ.--The nebular hypothesis of Laplace has inclined astronomers
to look upon nebulæ in general as material destined to be worked up into
stars, but which is now in a very crude and undeveloped stage. Their
great bulk and small density seem also to indicate that gravitation has
not yet produced in them results at all comparable with what we see in
sun and stars. But even among nebulæ there are to be found very
different stages of development. The irregular nebula, shapeless and
void like that of Orion; the spiral, ring, and planetary nebulæ and the
star cluster, clearly differ in amount of progress toward their final
goal. But it is by no means sure that these several types are different
stages in one line of development; for example, the primitive nebula
which grows into a spiral may never become a ring or planetary nebula,
and _vice versa_. So too there is no reason to suppose that a star
cluster will ever break up into isolated stars such as those whose
relation to each other is shown in Fig. 122.

246. CLASSIFICATION.--Considering the heavenly bodies with respect to
their stage of development, and arranging them in due order, we should
probably find lowest down in the scale of progress the irregular nebulæ
of chaotic appearance such as that represented in Fig. 146. Above these
in point of development stand the spiral, ring, and planetary nebulæ,
although the exact sequence in which they should be arranged remains a
matter of doubt. Still higher up in the scale are star clusters whose
individual members, as well as isolated stars, are to be classified by
means of their spectra, as shown in Fig. 151, where the order of
development of each star is probably from Type I, through II, into III
and beyond, to extinction of its light and the cutting off of most of
its radiant energy. Jupiter and Saturn are to be regarded as stars which
have recently entered this dark stage. The earth is further developed
than these, but it is not so far along as are Mars and Mercury; while
the moon is to be looked upon as the most advanced heavenly body
accessible to our research, having reached a state of decrepitude which
may almost be called death--a stage typical of that toward which all the
others are moving.

Meteors and comets are to be regarded as fragments of celestial matter,
chips, too small to achieve by themselves much progress along the normal
lines of development, but destined sooner or later, by collision with
some larger body, to share thenceforth in its fortunes.

247. STABILITY OF THE UNIVERSE.--It was considered a great achievement
in the mathematical astronomy of a century ago when Laplace showed that
the mutual attractions of sun and planets might indeed produce endless
perturbations in the motions and positions of these bodies, but could
never bring about collisions among them or greatly alter their existing
orbits. But in the proof of this great theorem two influences were
neglected, either of which is fatal to its validity. One of these--tidal
friction--as we have already seen, tends to wreck the systems of
satellites, and the same effect must be produced upon the planets by any
other influence which tends to impede their orbital motion. It is the
inertia of the planet in its forward movement that balances the sun's
attraction, and any diminution of the planet's velocity will give this
attraction the upper hand and must ultimately precipitate the planet
into the sun. The meteoric matter with which the earth comes ceaselessly
into collision must have just this influence, although its effects are
very small, and something of the same kind may come from the medium
which transmits radiant energy through the interstellar spaces.

It seems incredible that the luminiferous ether, which is supposed to
pervade all space, should present absolutely no resistance to the motion
of stars and planets rushing through it with velocities which in many
cases exceed 50,000 miles per hour. If there is a resistance to this
motion, however small, we may extend to the whole visible universe the
words of Thomson and Tait, who say in their great Treatise on Natural
Philosophy, "We have no data in the present state of science for
estimating the relative importance of tidal friction and of the
resistance of the resisting medium through which the earth and moon
move; but, whatever it may be, there can be but one ultimate result for
such a system as that of the sun and planets, if continuing long enough
under existing laws and not disturbed by meeting with other moving
masses in space. That result is the falling together of all into one
mass, which, although rotating for a time, must in the end come to rest
relatively to the surrounding medium."

Compare with this the words of a great poet who in The Tempest puts into
the mouth of Prospero the lines:

    "The cloud-capp'd towers, the gorgeous palaces,
    The solemn temples, the great globe itself,
    Yea, all which it inherit, shall dissolve;
    And, like this insubstantial pageant faded,
    Leave not a rack behind."

248. THE FUTURE.--In spite of statements like these, it lies beyond the
scope of scientific research to affirm that the visible order of things
will ever come to naught, and the outcome of present tendencies, as
sketched above, may be profoundly modified in ages to come, by
influences of which we are now ignorant. We have already noted that the
farther our speculation extends into either past or future, the more
insecure are its conclusions, and the remoter consequences of present
laws are to be accepted with a corresponding reserve. But the one great
fact which stands out clear in this connection is that of _change_. The
old concept of a universe created in finished form and destined so to
abide until its final dissolution, has passed away from scientific
thought and is replaced by the idea of slow development. A universe
which is ever becoming something else and is never finished, as shadowed
forth by Goethe in the lines:

    "Thus work I at the roaring loom of Time,
    And weave for Deity a living robe sublime."



The Greek letters are so much used by astronomers in connection with the
names of the stars, and for other purposes, that the Greek alphabet is
printed below--not necessarily to be learned, but for convenient

    Greek.          Name.  English.

    Α  α            Alpha    a
    Β  β            Beta     b
    Γ  γ            Gamma    g
    Δ  δ            Delta    d
    Ε  ε or ϵ       Epsilon  ĕ
    Ζ  ζ            Zeta     z
    Η  η            Eta      ē
    Θ  ϑ or θ       Theta     th
    Ι  ι            Iota      i
    Κ  κ            Kappa     k
    Λ  λ            Lambda    l
    Μ  μ            Mu        m
    Ν  ν            Nu        n
    Ξ  ξ            Xi        x
    Ο  ο            Omicron   ŏ
    Π  π            Pi        p
    Ρ  ρ            Rho       r
    Σ  σ or ς       Sigma     s
    Τ  τ            Tau       t
    Υ  υ            Upsilon   u
    Φ  φ            Phi       ph
    Χ  χ            Chi       ch
    Ψ  ψ            Psi       ps
    Ω  ω            Omega     ō


The following brief bibliography, while making no pretense at
completeness, may serve as a useful guide to supplementary reading:

_General Treatises_

    YOUNG. _General Astronomy._ An admirable general survey of the
    entire field.

    NEWCOMB. _Popular Astronomy._ The second edition of a German
    translation of this work by Engelmann and Vogel is especially

    BALL. _Story of the Heavens._ Somewhat easier reading than
    either of the preceding.

    CHAMBERS. _Descriptive Astronomy._ An elaborate but elementary
    work in three volumes.

    LANGLEY. _The New Astronomy._ Treats mainly of the physical
    condition of the celestial bodies.

    PROCTOR and RANYARD. _Old and New Astronomy._

_Special Treatises_

    PROCTOR. _The Moon._ A general treatment of the subject.

    NASMYTH and CARPENTER. _The Moon._ An admirably illustrated but
    expensive work dealing mainly with the topography and physical
    conditions of the moon. There is a cheaper and very good edition
    in German.

    YOUNG. _The Sun._ International Scientific Series. The most
    recent and authoritative treatise on this subject.

    PROCTOR. _Other Worlds than Ours._ An account of planets,
    comets, etc.

    NEWTON. _Meteor._ Encyclopædia Britannica.

    AIRY. _Gravitation._ A non-mathematical exposition of the laws
    of planetary motion.

    STOKES. _On Light as a Means of Investigation._ Burnett
    Lectures. II. The basis of spectrum analysis.

    SCHELLEN. _Spectrum Analysis._

    THOMSON (Sir W., Lord KELVIN), _Popular Lectures, etc._ Lectures
    on the Tides, The Sun's Heat, etc.

    BALL. _Time and Tide._ An exposition of the researches of G. H.
    Darwin upon tidal friction.

    GORE. _The Visible Universe._ Deals with a class of problems
    inadequately treated in most popular astronomies.

    DARWIN. _The Tides._ An admirable elementary exposition.

    CLERKE. _The System of the Stars._ Stellar astronomy.

    NEWCOMB. Chapters on the Stars, in _Popular Science Monthly_ for

    CLERKE. _History of Astronomy during the Nineteenth Century._ An
    admirable work.

    WOLF. _Geschichte der Astronomie._ München, 1877. An excellent
    German work.


See § 20.

 NAME.           |  Magnitude.   | Right Ascension. | Declination.|
                 |               |                  |             |
                 |               |     h. m.        |      °      |
 β Ceti          |       2       |     0 38.6       |    - 18.5   |
 η Ceti          |       3       |     1  3.6       |    - 10.7   |
 α Ceti          |       3       |     2 57.1       |    +  3.7   |
 γ Eridani       |       3       |     3 53.4       |    - 13.8   |
 _Aldebaran_     |       1       |     4 30.2       |    + 16.3   |
                 |               |                  |             |
 _Rigel_         |       0       |     5  9.7       |    -  8.3   |
 κ Orionis       |       2       |     5 43.0       |    -  9.7   |
 β Canis Majoris |       2       |     6 18.3       |    - 17.9   |
 _Sirius_        |      -1       |     6 40.7       |    - 16.6   |
 _Procyon_       |       0       |     7 34.1       |    +  5.5   |
                 |               |                  |             |
 α Hydræ         |       2       |     9 22.7       |    -  8.2   |
 _Regulus_       |       1       |    10  3.0       |    + 12.5   |
 ν Hydræ         |       3       |    10 44.7       |    - 15.7   |
 ε Corvi         |       3       |    12  5.0       |    - 22.1   |
 γ Corvi         |       3       |    12 10.7       |    - 17.0   |
                 |               |                  |             |
 _Spica_         |       1       |    13 19.9       |    - 10.6   |
 ζ Virginis      |       3       |    13 29.6       |    -  0.1   |
 α Libræ         |       3       |    14 45.3       |    - 15.6   |
 β Libræ         |       3       |    15 11.6       |    -  9.0   |
 _Antares_       |       1       |    16 23.3       |    - 26.2   |
                 |               |                  |             |
 α Ophiuchi      |       2       |    17 30.3       |    + 12.6   |
 ε Sagittarii    |       2       |    18 17.5       |    - 34.4   |
 δ Aquilæ        |       3       |    19 20.5       |    +  2.9   |
 _Altair_        |       1       |    19 45.9       |    +  8.6   |
 β Aquarii       |       3       |    21 26.3       |    -  6.0   |
                 |               |                  |             |
 α Aquarii       |       3       |    22  0.6       |    -  0.8   |
 _Fomalhaut_     |       1       |    22 52.1       |    - 30.2   |


The references are to section numbers.

  Absorption of starlight, 225.

  Absorption spectra, 87.

  Accelerating force, 35.

  Adjustment of observations, 2.

  Albedo of moon, 97.
    of Venus, 148.

  Algol, 205.

  Altitudes, 4, 21.

  Andromeda nebula, 214.

  Angles, measurement of, 2.

  Angular diameter, 7.

  Annular eclipse, 64.

  Asteroids, 156.

  Atmosphere of the earth, 49.
    of the moon, 103.
    of Jupiter, 139.
    of Mars, 153.

  Aurora, 51.

  Azimuth, 5, 21.

  Biela's comet, 181.

  Bode's law, 134, 232.

  Bredichin's theory of comet tails, 180.

  Calendar, O. S. and N. S., 61.

  Capture of comets and meteors, 176.

  Canals of Mars, 154.

  Celestial mechanics, 32.

  Changes upon the moon, 108.

  Chemical constitution of sun, 116.
    of stars, 210.

  Chromosphere, the sun's, 124.

  Chronology, 59.

  Classification of stars, 212.

  Clocks and watches, 74.
    sidereal clock, 12.

  Collisions with comets, 183.

  Colors of stars, 209.

  Comets, general characteristics, 158-164.
    development of, 179, 181.
    groups, 177.
    orbits, 161.
    periodic, 176.
    spectra, 182.
    tails, 180.

  Comets and meteors, relation of, 175.

  Conic sections, 38.

  Constellations, 184.

  Corona, the sun's, 123.

  Craters, lunar, 105.

  Dark stars, 201.

  Day, 52, 62.

  Declination, 21.

  Development of comet, 179.
    of moon, 241.
    of nebulæ, 245.
    of stars, 242, 244.
    of sun, 228.
  of universe, 226.

  Distribution of stars and nebulæ, 220.

  Diurnal motion, 10, 15.

  Doppler principle, 89.

  Double nebulæ, 215.

  Double stars, 198.
    development of, 244.

  Driving clock, 80.

  Earth, atmosphere, 48.
    mass, 45.
    size and shape, 44.
    warming of the earth, 47.

  Eclipses, nature of, 63.
    annular eclipse, 64.
    eclipse limits, 68.
    eclipse maps, 70, 71.
    number of, in a year, 69.
    partial eclipse, 64.
    prediction of, 70, 71.
    recurrence of, 72.
    shadow cone, 64, 66.
    total eclipse, 64.
    uses of, 73.

  Eclipses of Jupiter's satellites, 141.

  Eclipse theory of variable stars, 205.

  Ecliptic, 26.
    obliquity of, 25.

  Ellipse, 33.

  Epochs for planetary motion, 30.

  Energy, radiant, 75.
    condensation of, 76.

  Epicycle, 32.

  Equation of time, 53.

  Equator, 16, 21.

  Equatorial mounting, 80.

  Equinoxes, 25.

  Ether, 75.

  Evening star, 31.

  Faculæ, 122.

  Falling bodies, law of, 35.

  Finding the stars, 14.

  Fraunhofer lines, 87.

  Galaxy, 219.

  Geography of the sky, 16.

  Graphical representation, 6.

  Grating, diffraction, 84.

  Gravitation, law of, 37.

  Harvest moon, 93.

  Heat of the sun, 118, 126.

  Helmholtz, contraction theory of the sun, 126, 228.

  Horizon, 4, 21.

  Hour angle, 21.

  Hour circle, 21.

  Hyperbola, 38.

  Japetus, satellite of Saturn, 145.

  Jupiter, 136.
    atmosphere, 139.
    belts, 137.
    invisible from fixed stars, 197.
    orbit of, 29.
    physical condition, 139.
    rotation and flattening, 138.
    satellites, 140.
    surface markings, 137.

  Kepler's laws, 33, 111.

  Latitude, determination of, 18.

  Leap year, 61.

  Lenses, 77.

  Leonid meteor shower, 172.
    perturbations of, 174.

  Librations of moon, 98.

  Life upon the planets, 157.

  Light curves, 205.

  Light, nature of, 75.

  Light year, 190.

  Limits of eclipses, 68.

  Longitude, 56.
    determination of, 58.

  Lunation, 60.

  Magnifying power of telescope, 79.

  Magnitude, stellar, 9, 186.

  Mars, atmosphere, temperature, 150.
    canals, 154.
    orbit, 30.
    polar caps, 152.
    rotation, 151.
    satellites, 155.
    surface markings, 150.

  Mass, determination of, 37.
    of comets, 164.
    of double stars, 200.
    of moon, 94.
    of planets, 40, 133.

  Measurements, accurate, 1.

  Mercury, 149.
    motion of its perihelion, 43.
    orbit of, 30.

  Meridian, 19, 21.

  Meteors, nature of, 165, 169.
    number of, 167.
    velocity, 170.

  Meteors and comets, relation of, 175.

  Meteor showers, radiant, 171.
    Leonids, capture of, 172, 173.
    perturbations, 174.

  Milky Way, 219.

  Mira, ο Ceti, 204.

  Mirrors, 77.

  Month, 60.

  Moon, 91.
    albedo, 97.
    atmosphere, 103.
    changes in, 108.
    density, surface gravity, 95.
    development of, 241.
    harvest moon, 93.
    influence upon the earth, 109, 233.
    librations, 98.
    map of, 101.
    mass and size, 94.
    motion, 24, 92.
    mountains and craters, 104.
    phases, 91, 92.
    physical condition, 100, 107.

  Month, 60.

  Morning star, 31.

  Motion in line of sight, 89, 193.

  Multiple stars, 202.

  Names of stars, 8.

  Nebulæ, 214.
    density, 217.
    development of, 245.
    motion, 218.
    spectra, 216.
    types and classes of, 215.

  Nebular hypothesis, 230.
    objections to, 231.

  Neptune, 146.
    discovery of, 41.

  Newton's laws of motion, 34.
    law of gravitation, 37, 43.

  Nodes, 39.
    relation to eclipses, 67, 71.

  Nucleus, of comet, 160.

  Objective, of telescope, 78.

  Obliquity of ecliptic, 25.

  Observations, of stars, 10.

  Occultation of stars, 103.

  Orbits, of comets, 161.
    of double stars, 199.
    of moon, 92.
    of planets, 28.

  Orion nebula, 215.

  Parabola, 35, 38, 161.

  Parabolic velocity, 38.

  Parallax, 114, 188.

  Penumbra, 64, 121.

  Perihelion, 38.

  Periodic comets, 176.

  Personal equation, 82.

  Perturbations, 39.
    of meteors, 174.

  Phases, of the moon, 91, 92.

  Photography, 81.
    of stars, 13.

  Photosphere, of sun, 121.

  Planets, 26, 133.
    distances from the sun, 134.
    how to find, 29.
    mass, density, size, 133.
    motion of, 27, 38.
    periodic times of, 30.

  Planetary nebulæ, 215.

  Pleiades, 16, 215.

  Plumb-line apparatus, 11, 18.

  Poles, 21.

  Precession, 46.

  Prisms, 84.

  Problem of three bodies, 39.

  Prominences, solar, 125.

  Proper motions, 191.

  Protractor, 2.

  Ptolemaic system, 32.

  Radiant energy, 75.

  Radiant, of meteor shower, 171.

  Radius victor, 33.

  Reference lines and circles, 17.

  Refraction, 50.

  Right ascension, 16, 20, 21.

  Roche's limit, 239.

  Rotation, of earth, 55.
    of Mars, 151.
    of moon, 99.
    of Jupiter, 138.
    of Saturn, 144.
    of sun, 120, 132.

  Saros, 72.

  Satellites, of Jupiter, 136, 140.
    of Mars, 155.
    of Saturn, 145.

  Saturn, 142.
    ball of, 144.
    orbit, 29.
    rings, 142.
    rotation, 144.
    satellites, 145.

  Seasons, on the earth, 47.
    on Mars, 151.

  Shadow cone, 64, 66.

  Sidereal time, 20, 54.

  Shooting stars, 158. (See Meteor.)

  Spectroscope, 84.

  Spectroscopic binaries, 203.

  Spectrum, 84, 87.
    of comets, 182.
    of nebulæ, 216.
    of stars, 211.
    types of, 88.

  Spectrum analysis, 85.

  Spiral nebulæ, 215.

  Standard time, 57.

  Stars, 8, 184.
    classes of, 212.
    clusters, 213.
    colors, 209.
    dark stars, 201.
    development of, 242.
    distances from the sun, 188, 196.
    distribution of, 220.
    double stars, 198, 203.
    drift, 194.
    magnitudes, 9, 196.
    number of, 185.
    spectra, 211.
    temporary, 208.
    variable, 204.

  Starlight, absorption of, 225.

  Star maps, construction of, 23.

  Stellar system, extent of, 223.

  Sun's apparent motion, 25.
    real motion, 195.

  Sun, 110.
    chemical composition, 116.
    chromosphere, 124.
    corona, 123.
    distance from the earth, 111.
    faculæ, 119, 122.
    gaseous constitution, 127.
    heat of, 117.
    mechanism of, 126.
    physical properties, 115-120.
    prominences, 125.
    rotation, 120, 132.
    surface of, 119.
    temperature, 118.

  Sun spots, 119, 121.
    period, 129, 131.
    zones, 130.

  Telescopes, 78.
    equatorial mounting for, 80.
    magnifying power of, 79.

  Temperature of Jupiter, 139.
    of Mars, 152.
    of Mercury, 149.
    of moon, 107.
    of sun, 118.

  Temporary stars, 208.

  Terminator, 91.

  Tenth meter, 75.

  Tidal friction, 233-238.

  Tides, 42.

  Time, sidereal, 20, 54.
    solar, 52.
    determination of, 20.
    equation of, 53.
    standard, 57.

  Triangulation, 3.

  Trifid nebula, 215.

  Twilight, 51.

  Twinkling, of stars, 48.

  Universe, development of, 226.
    stability of, 247.

  Uranus, 146.

  Variable stars, 204.

  Velocity, its relation to orbital motion, 38.

  Venus, 148.
    orbit of, 30.

  Vernal equinox, 21, 25.

  Vertical circle, 21.

  Wave front, 76.

  Wave lengths, 75, 86.

  Year, 25.
    leap year, 61.
    sidereal year, 59.
    tropical year, 60.

  Zenith, 21.

  Zodiac, 26.

  Zodiacal light, 168.



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