% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% Project Gutenberg's An Introduction to Astronomy, by Forest Ray Moulton %
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
% with this eBook or online at www.gutenberg.org                          %
%                                                                         %
%                                                                         %
% Title: An Introduction to Astronomy                                     %
%                                                                         %
% Author: Forest Ray Moulton                                              %
%                                                                         %
% Release Date: April 24, 2010 [EBook #32000]                             %
% Most recently updated: June 11, 2021                         %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: UTF-8                                           %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO ASTRONOMY ***
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Project Gutenberg's An Introduction to Astronomy, by Forest Ray Moulton

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: An Introduction to Astronomy

Author: Forest Ray Moulton

Release Date: April 24, 2010 [EBook #32000]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO ASTRONOMY ***
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Produced by Brenda Lewis, Andrew D. Hwang, Bup, and the
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

\frontmatter

\pagenumbering{roman}
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\normalsize

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% AN INTRODUCTION TO ASTRONOMY
\HalfTitle

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\null\vfill
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\Input[1.5in]{002}{png}%[** Publisher's Device]
\medskip

\footnotesize THE MACMILLAN COMPANY

\scriptsize
NEW YORK · BOSTON · CHICAGO · DALLAS \\
ATLANTA · SAN FRANCISCO
\medskip

\footnotesize MACMILLAN \& CO., \textsc{Limited}

\scriptsize
LONDON · BOMBAY · CALCUTTA \\
MELBOURNE
\medskip

\footnotesize THE MACMILLAN CO. OF CANADA, \textsc{Ltd.}

\scriptsize
TORONTO
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\Caption[The Lick Observatory, Mount Hamilton, California.]{Fig}{1}
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\setlength{\TmpLen}{0.2in}%
{\bfseries\LARGE AN INTRODUCTION}\\[2\TmpLen]
\small TO\\[2\TmpLen]
{\bfseries\MyHuge ASTRONOMY}\\[4\TmpLen]
\small BY\\[2\TmpLen]
\large FOREST RAY MOULTON, \textsc{Ph.D.}\\[\TmpLen]
\scriptsize PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF CHICAGO \\[0.5\TmpLen]
RESEARCH ASSOCIATE OF THE CARNEGIE INSTITUTION \\[0.5\TmpLen]
OF WASHINGTON\\[6\TmpLen]
\normalsize\textit{NEW AND REVISED EDITION}\\[6\TmpLen]
\textgoth{New York} \\[\TmpLen]
THE MACMILLAN COMPANY \\[\TmpLen]
1916 \\[\TmpLen]
\textit{\scriptsize All rights reserved}
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\null\vfill
\begin{center}
\scshape\footnotesize Copyright, 1906 and 1916,\\[2\TmpLen]
\small By THE MACMILLAN COMPANY.\\[\TmpLen]
\rule{0.5in}{0.5pt}
\medskip
\normalfont\footnotesize

\parbox{4in}{%
\setlength{\parindent}{1em}%
\spaceskip0.5em plus 0.5em minus 0.25em% ** Explicit spacing
Set up and electrotyped. Published April, 1906. Reprinted
November, 1907; July, 1908; April, 1910; April, 1911; September,
1912; September, 1913: October, 1914.

New and revised edition November, 1916.}

\vfill

\textgoth{Norwood Press} \\
J.~S. Cushing Co.\ --- Berwick \& Smith Co. \\
Norwood, Mass., U.S.A.
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\Preface

\textsc{The} necessity for a new edition of ``An Introduction to
Astronomy'' has furnished an opportunity for entirely rewriting
it. As in the first edition, the aim has been to present
the great subject of astronomy so that it can be easily
comprehended even by a person who has not had extensive
scientific training. It has been assumed that the reader has
no intention of becoming an astronomer, but that he has an
interest in the wonderful universe which surrounds him, and
that he has arrived at such a stage of intellectual development
that he demands the reasons for whatever conclusions he is
asked to accept. The first two of these assumptions have
largely determined the subject matter which is presented;
the third has strongly influenced the method of presenting it.

While the aims have not changed materially since the first
edition was written, the details of the attempt to accomplish
them have undergone many, and in some cases important,
modifications. For example, the work on reference points and
lines has been deferred to \Chapref{IV}. If one is to know the
sky, and not simply know about it, a knowledge of the coördinate
systems is indispensable, but they always present some
difficulties when they are encountered at the beginning of the
subject. It is believed that the present treatment prepares
so thoroughly for their study and leads so naturally to them
that their mastery will not be found difficult. The chapter on
telescopes has been regretfully omitted because it was not
necessary for understanding the remainder of the work, and
because the space it occupied was needed for treating more
vital parts of the subject. The numerous discoveries in the
sidereal universe during the last ten years have made it necessary
greatly to enlarge the last chapter.
%% -----File: 007.png---Folio vi-------

As now arranged, the first chapters are devoted to a discussion
of the earth and its motions. They present splendid
examples of the characteristics and methods of science, and
amply illustrate the care with which scientific theories are
established. The conclusions which are set forth are bound up
with the development of science from the dawn of recorded
history to the recent experiments on the rigidity and the elasticity
of the earth. They show how closely various sciences
are interlocked, and how much an understanding of the earth
depends upon its relations to the sky. They lead naturally to
a more formal treatment of the celestial sphere and a study of
the constellations. A familiarity with the brighter stars and
the more conspicuous constellations is regarded as important.
One who has become thoroughly acquainted with them will
always experience a thrill when he looks up at night into a
cloudless sky.

The chapter on the sun has been postponed until after the
treatment of the moon, planets, and comets. The reason is
that the discussion of the sun necessitates the introduction of
many new and difficult topics, such as the conservation of energy,
the disintegration of radioactive elements, and the principles
of spectrum analysis. Then follows the evolution of
the solar system. In this chapter new and more serious demands
are made on the reasoning powers and the imagination.
Its study in a measure develops a point of view and prepares
the way for the consideration, in the last chapter, of the transcendental
and absorbingly interesting problems respecting
the organization and evolution of the sidereal universe.

Lists of problems have been given at the ends of the principal
divisions of the chapters. They cannot be correctly
answered without a real comprehension of the principles which
they involve, and in very many cases, especially in the later
chapters, they lead to important supplementary results. It is
strongly recommended that they be given careful consideration.

The author is indebted to Mr.~Albert Barnett for the new
star maps and the many drawings with which the book is illustrated,
with the exception of Figs.\ \Fref{23}~and~\Fref{30}, which were
%% -----File: 008.png---Folio vii-------
kindly furnished by Mr.\ George Otis. He is indebted to
Professor David Eugene Smith for photographs of Newton,
Kepler, Herschel, Adams, and Leverrier. He is indebted to
the Lick, Lowell, Solar, and Yerkes observatories for a large
amount of illustrative material which was very generously
furnished. He is under deeper obligations to his colleague,
Professor W.~D. MacMillan, than this brief acknowledgment
can express for assistance on the manuscript, on the proofs,
and in preparing the many problems which appear in the book.

\null\hfill F.~R. MOULTON.\hspace*{\parindent}
\medskip

\settowidth{\TmpLen}{\small\textsc{The University of Chicago},}%
\parbox{\TmpLen}{%
  \centering\small\textsc{The University of Chicago}, \\
  September~25, 1916.%
}
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\fancyhf{}

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%% -----File: 010.png---Folio ix-------


\ToC{Contents}

%[** TN: Some headings set in all-caps, others titlecase. This is deliberate.]
\ToCChap{I}{Preliminary Considerations}

\ToCLine{1}{Science}% 1
\ToCLine{2}{The value of science}% 2
\ToCLine{3}{The origin of science}% 4
\ToCLine{4}{The methods of science}% 6
\ToCLine{5}{The imperfections of science}% 10
\ToCLine{6}{Great contributions of astronomy to science}% 14
\ToCLine{7}{The present value of astronomy}% 16
\ToCLine{8}{The scope of astronomy}% 19


\ToCChap{II}{THE EARTH}

\ToCSection{I}{The Shape of the Earth}

\ToCLine{9}{Astronomical problems respecting the earth}% 26
\ToCRange{10}{, }{11}{Proofs of the earth's sphericity}% 27
%[** TN: Need to set this manually]
\noindent\makebox[\linewidth][c]{%
  \ToCBox{12}, 14, 15.\hspace*{0.5em}Proofs of the earth's oblateness%
  \MyDotFill\pageref{art:12}--\pageref{art:15}}\\
%
\ToCLine{13}{Size and shape of the earth}% 33
\ToCLine{16}{The theoretical shape of the earth}% 38
\ToCLine{17}{Different kinds of latitude}% 39
%[** TN: Line does not match the heading in the main text]
\ToCLine{18}{Historical sketch on the shape of the earth}% 40

\ToCSection{II}{The Mass of the Earth and the Condition of its Interior}

\ToCLine{19}{The principle by which mass is determined}% 43
\ToCLine{20}{The mass and density of the earth}% 45
\ToCRange{21}{--}{23}{Methods of determining the density of the earth}% 46
%% -----File: 011.png---Folio x-------
\ToCLine{24}{Temperature and pressure in the earth's interior}% 51
\ToCRange{25}{, }{26}{Proofs of the earth's rigidity and elasticity}% 52
\ToCLine{27}{Historical sketch on the mass and rigidity of the earth}% 62

\ToCSection{III}{The Earth's Atmosphere}

\ToCLine{28}{Composition and mass of the earth's atmosphere}% 64
\ToCRange{29}{--}{31}{Methods of determining height of the atmosphere}% 65
\ToCLine{32}{The kinetic theory of gases}% 68
\ToCLine{33}{The escape of atmospheres}% 69
\ToCLine{34}{Effects of the atmosphere on climate}% 71
\ToCLine{35}{Importance of the constitution of the atmosphere}% 72
\ToCLine{36}{Rôle of the atmosphere in life processes}% 74
\ToCLine{37}{Refraction of light by the atmosphere}% 74
\ToCLine{38}{The twinkling of the stars}% 76


\ToCChap{III}{THE MOTIONS OF THE EARTH}

\ToCSection{I}{The Rotation of the Earth}

\ToCLine{39}{The relative rotation of the earth}% 77
\ToCLine{40}{The laws of motion}% 79
\ToCRange{41}{--}{43}{Proofs of the earth's rotation}% 82
\ToCLine{44}{Consequences of the earth's rotation}% 85
\ToCLine{45}{Uniformity of the earth's rotation}% 87
\ToCLine{46}{The variation of latitude}% 89
\ToCLine{47}{The precession of the equinoxes and nutation}% 92

\ToCSection{II}{The Revolution of the Earth}

\ToCLine{48}{Relative motion of the earth with respect to the sun}% 96
\ToCRange{49}{--}{52}{Proofs of the revolution of the earth}% 98
\ToCLine{53}{Shape of the earth's orbit}% 102
\ToCLine{54}{Motion of the earth in its orbit}% 103
\ToCLine{55}{Inclination of the earth's orbit}% 105
\ToCLine{56}{The cause of the seasons}% 107
\ToCLine{57}{Relation of altitude of pole to latitude of observer}% 108
\ToCLine{58}{The sun's diurnal circles}% 109
\ToCLine{59}{Hours of sunlight in different latitudes}% 111
\ToCLine{60}{The lag of the seasons}% 112
\ToCLine{61}{Effect of eccentricity of earth's orbit on seasons}% 113
\ToCLine{62}{Historical sketch of the motions of the earth}% 115
%% -----File: 012.png---Folio xi-------


\ToCChap{IV}{Reference Points and Lines}

\ToCLine{63}{Object and character of reference points and lines}% 121
\ToCLine{64}{The geographical system}% 122
\ToCLine{65}{The horizon system}% 123
\ToCLine{66}{The equator system}% 125
\ToCLine{67}{The ecliptic system}% 127
\ToCLine{68}{Comparison of systems of coördinates}% 127
\ToCRange{69}{, }{70}{Finding the altitude and azimuth}% 130
\ToCRange{71}{, }{72}{Finding the right ascension and declination}% 133
\ToCLine{73}{Other problems of position}% 135

\ToCChap{V}{The Constellations}

\ToCLine{74}{Origin of the constellations}% 138
\ToCLine{75}{Naming the stars}% 138
\ToCLine{76}{Star catalogues}% 141
\ToCLine{77}{The magnitudes of the stars}% 142
\ToCLine{78}{The first-magnitude stars}% 143
\ToCLine{79}{Number of stars in first six magnitudes}% 145
\ToCLine{80}{Motions of the stars}% 145
\ToCLine{81}{The Milky Way, or Galaxy}% 146
\ToCLine{82}{The constellations and their positions (Maps)}% 148
\ToCLine{83}{Finding the pole star}% 149
\ToCLine{84}{Units for estimating angular distances}% 150
% [** TN: Need to set this manually]
\pagebreak[3]%
\label{toc:85}%
\ifthenelse{\not\equal{\pageref{toc:85}}{\ToCAnchor}}{%
  \renewcommand{\ToCAnchor}{\pageref{toc:85}}%
  \noindent\makebox[\linewidth][c]{\scriptsize ARTS.\hfill PAGE}\\
}{}%
\raisebox{3\baselineskip}{\rule{0pt}{12pt}\ToCBox{85}--101.\hspace*{0.5em}}%
\settowidth{\TmpLen}{85--101.\hspace*{0.5em}999--999}%
\parbox[b]{\linewidth-\TmpLen}{%
Ursa Major, Cassiopeia, Locating the equinoxes, Lyra,
  Hercules, Scorpius, Corona Borealis, Boötes, Leo, Andromeda,
  Perseus, Auriga, Taurus, Orion, Canis Major,
  Canis Minor, Gemini\MyDotFill\hspace*{0.5em}}%
\PadTo[r]{\text{99--999}}{\text{\pageref{art:85}--\pageref{art:101}}}\\ % 150
%
\ToCLine{102}{On becoming familiar with the stars}% 167


\ToCChap{VI}{Time}

\ToCLine{103}{Definitions of equal intervals of time}% 169
\ToCLine{104}{The practical measure of time}% 170
\ToCLine{105}{Sidereal time}% 171
%% -----File: 013.png---Folio xii-------
\ToCLine{106}{Solar time}% 172
\ToCLine{107}{Variations in length of solar days}% 172
\ToCLine{108}{Mean solar time}% 175
\ToCLine{109}{The equation of time}% 176
\ToCLine{110}{Standard time}% 177
\ToCLine{111}{Distribution of time}% 179
\ToCLine{112}{Civil and astronomical days}% 181
\ToCLine{113}{Place of change of date}% 181
\ToCRange{114}{--}{116}{Sidereal, anomalistic, and tropical years}% 183
\ToCLine{117}{The calendar}% 184
\ToCLine{118}{Finding the day of week on any date}% 185


\ToCChap{VII}{The Moon}

\ToCLine{119}{The moon's apparent motion among the stars}% 188
\ToCLine{120}{The moon's synodical and sidereal periods}% 189
\ToCLine{121}{The phases of the moon}% 190
\ToCLine{122}{The diurnal circles of the moon}% 192
\ToCLine{123}{The distance of the moon}% 194
\ToCLine{124}{The dimensions of the moon}% 196
\ToCRange{125}{, }{126}{The moon's orbit with respect to earth and sun}% 197
\ToCLine{127}{The mass of the moon}% 198
\ToCLine{128}{The rotation of the moon}% 200
\ToCLine{129}{The librations of the moon}% 201
\ToCLine{130}{The density and surface gravity of the moon}% 202
\ToCLine{131}{The question of the moon's atmosphere}% 203
\ToCLine{132}{Light and heat received from the moon}% 204
\ToCLine{133}{The temperature of the moon}% 205
\ToCRange{134}{--}{138}{The surface of the moon}% 207
\ToCLine{139}{Effects of the moon on the earth}% 217
\ToCRange{140}{--}{142}{Eclipses of the moon and sun}% 218


\ToCChap{VIII}{THE SOLAR SYSTEM}

\ToCSection{I}{The Law of Gravitation}

\ToCLine{143}{The members of the solar system}% 226
\ToCLine{144}{Relative dimensions of the planetary orbits}% 227
%% -----File: 014.png---Folio xiii-------
\ToCLine{145}{Kepler's laws of motion}% 229
\ToCRange{146}{, }{147}{The law of gravitation}% 230
\ToCLine{148}{The conic sections}% 234
\ToCLine{149}{The question of other laws of force}% 236
\ToCLine{150}{Perturbations}% 237
\ToCLine{151}{The discovery of Neptune}% 238
\ToCLine{152}{The problem of three bodies}% 241
\ToCLine{153}{Cause of the tides}% 242
\ToCLine{154}{Masses of celestial bodies}% 244
\ToCLine{155}{Surface gravity of celestial bodies}% 245

\ToCSection{II}{Orbits, Dimensions, and Masses of the Planets}

\ToCLine{156}{Finding the dimensions of the solar system}% 246
\ToCLine{157}{Elements of the orbits of the planets (Table)}% 248
\ToCLine{158}{Dimensions and masses of the planets (Table)}% 252
\ToCLine{159}{Times for observing the planets}% 255
\ToCLine{160}{The planetoids}% 257
\ToCLine{161}{The question of undiscovered planets}% 261
\ToCLine{162}{The zodiacal light and the gegenschein}% 262


\ToCChap{IX}{THE PLANETS}

\ToCSection{I}{Mercury and Venus}

\ToCLine{163}{Phases of Mercury and Venus}% 266
\ToCLine{164}{Albedoes and atmospheres of Mercury and Venus}% 268
\ToCLine{165}{Surface markings and rotation of Mercury}% 269
\ToCLine{166}{The seasons of Mercury}% 270
\ToCLine{167}{Surface markings and rotation of Venus}% 271
\ToCLine{168}{The seasons of Venus}% 272

\ToCSection{II}{Mars}

\ToCLine{169}{The satellites of Mars}% 273
\ToCLine{170}{The rotation of Mars}% 274
\ToCLine{171}{The albedo and atmosphere of Mars}% 276
\ToCLine{172}{The polar caps and temperature of Mars}% 277
\ToCLine{173}{The canals of Mars}% 283
\ToCLine{174}{Explanations of the canals of Mars}% 285
%% -----File: 015.png---Folio xiv-------

\ToCSection{III}{Jupiter}

\ToCRange{175}{, }{176}{Jupiter's satellite system}% 289
\ToCLine{177}{Discovery of the velocity of light}% 291
\ToCRange{178}{, }{179}{Surface markings and rotation of Jupiter}% 292
\ToCLine{180}{Physical condition and seasons of Jupiter}% 296

\ToCSection{IV}{Saturn}

\ToCLine{181}{Saturn's satellite system}% 297
\ToCRange{182}{--}{184}{Saturn's ring system}% 299-304
\ToCLine{185}{Surface markings and rotation of Saturn}% 305
\ToCLine{186}{Physical condition and seasons of Saturn}% 306

\ToCSection{V}{Uranus and Neptune}

\ToCLine{187}{Satellite systems of Uranus and Neptune}% 306
\ToCLine{188}{Atmospheres and albedoes of Uranus and Neptune}% 307
\ToCLine{189}{Periods of rotation of Uranus and Neptune}% 307
\ToCLine{190}{Physical conditions of Uranus and Neptune}% 308


\ToCChap{X}{COMETS AND METEORS}

\ToCSection{I}{Comets}

\ToCLine{191}{General appearance of comets}% 311
\ToCLine{192}{The orbits of comets}% 313
\ToCRange{193}{, }{194}{The dimensions and masses of comets}% 316, 317
\ToCLine{195}{Families of comets}% 318
\ToCLine{196}{The capture of comets}% 320
\ToCLine{197}{On the origin of comets}% 322
\ToCLine{198}{Theories of comets' tails}% 323
\ToCLine{199}{The disintegration of comets}% 327
\ToCLine{200}{Historical comets}% 328
\ToCLine{201}{Halley's comet}% 332

\ToCSection{II}{Meteors}

\ToCLine{202}{Meteors, or ``shooting stars''}% 337
\ToCLine{203}{The number of meteors}% 338
\ToCRange{204}{, }{205}{Meteoric showers}% 339
%% -----File: 016.png---Folio xv-------
\ToCLine{206}{Connection between comets and meteors}% 341
\ToCLine{207}{Effects of meteors on the solar system}% 343
\ToCLine{208}{Meteorites}% 343
\ToCLine{209}{Theories respecting the origin of meteors}% 345


\ToCChap{XI}{THE SUN}

\ToCSection{I}{The Sun's Heat}

\ToCLine{210}{The problem of the sun's heat}% 349
\ToCLine{211}{Amount of energy received from sun}% 349
\ToCLine{212}{Sources of energy used by man}% 351
\ToCLine{213}{Amount of energy radiated by sun}% 353
\ToCLine{214}{The temperature of the sun}% 354
\ToCLine{215}{Principle of the conservation of energy}% 355
\ToCRange{216}{, }{217}{Theories of the sun's heat}% 356-359
\ToCLine{218}{Past and future of sun on contraction theory}% 360
\ToCLine{219}{The age of the earth}% 360

\ToCSection{II}{Spectrum Analysis}

\ToCLine{220}{The nature of light}% 365
\ToCLine{221}{On the production of light}% 366
\ToCLine{222}{Spectroscopes and the spectrum}% 369
\ToCRange{223}{--}{226}{The laws of spectrum analysis}% 371-375

\ToCSection{III}{The Constitution of the Sun}

\ToCLine{227}{Outline of the sun's constitution}% 378
\ToCLine{228}{The photosphere}% 379
\ToCRange{229}{--}{231}{Sunspots, distribution, periodicity, and motions}% 381-384
\ToCLine{232}{The rotation of the sun}% 388
\ToCLine{233}{The reversing layer}% 390
\ToCLine{234}{Chemical constitution of reversing layer}% 392
\ToCRange{235}{, }{236}{The chromosphere and prominences}% 394, 395
\ToCLine{237}{The spectroheliograph}% 398
\ToCLine{238}{The corona}% 401
\ToCLine{239}{The eleven-year cycle}% 404
%% -----File: 017.png---Folio xvi-------


\ToCChap{XII}{EVOLUTION OF THE SOLAR SYSTEM}

\ToCSection{I}{General Considerations on Evolution}

\ToCLine{240}{Essence of the doctrine of evolution}% 407
\ToCLine{241}{Value of a theory of evolution}% 408
\ToCLine{242}{Outline of growth of doctrine of evolution}% 410

%[** TN: Line does not exactly match the heading in the main text]
\ToCSection{II}{Data of Problem of Evolution of Solar System}

\ToCLine{243}{General evidences of orderly development}% 413
\ToCLine{244}{Distribution of mass in the solar system}% 414
\ToCLine{245}{Distribution of moment of momentum}% 416
\ToCLine{246}{The energy of the solar system}% 419

\ToCSection{III}{The Planetesimal Theory}

\ToCLine{247}{Outline of the planetesimal theory}% 421
\ToCLine{248}{Examples of planetesimal organization}% 422
\ToCLine{249}{Suggested origin of spiral nebulæ}% 424
\ToCLine{250}{The origin of planets}% 431
\ToCLine{251}{The planes of the planetary orbits}% 433
\ToCLine{252}{The eccentricities of the planetary orbits}% 434
\ToCLine{253}{The rotation of the sun}% 436
\ToCLine{254}{The rotation of the planets}% 437
\ToCLine{255}{The origin of satellites}% 440
\ToCLine{256}{The rings of Saturn}% 441
\ToCRange{257}{, }{258}{The planetoids and zodiacal light}% 442
\ToCLine{259}{The comets}% 442
\ToCLine{260}{The future of the solar system}% 443

\ToCSection{IV}{Historical Cosmogonies}

\ToCLine{261}{The hypothesis of Kant}% 446
\ToCLine{262}{The hypothesis of Laplace}% 449
\ToCRange{263}{, }{264}{Tidal forces and tidal evolution}% 452, 454
\ToCLine{265}{Effects of tides on motions of the moon}% 456
\ToCLine{266}{Effects of tides on motions of the earth}% 456
\ToCLine{267}{Tidal evolution of the planets}% 460
%% -----File: 018.png---Folio xvii-------


\ToCChap{XIII}{THE SIDEREAL UNIVERSE} %[** TN: Adding heading]

\ToCSection{I}{The Apparent Distribution of the Stars}

\ToCLine{268}{On the problems of the sidereal universe}% 463
\ToCLine{269}{Number of stars of various magnitudes}% 464
\ToCLine{270}{Apparent distribution of the stars}% 470
\ToCLine{271}{Form and structure of the Milky Way}% 473

\ToCSection{II}{Distances and Motions of the Stars}

\ToCLine{272}{Direct parallaxes of nearest stars}% 476
\ToCLine{273}{Distances of stars from proper motions and radial velocities}% 481
\ToCLine{274}{Motion of sun with respect to stars}% 482
\ToCLine{275}{Distances of stars from motion of sun}% 484
\ToCLine{276}{Kapteyn's results on distances of stars}% 486
\ToCLine{277}{Distances of moving groups of stars}% 487
\ToCLine{278}{Star streams}% 490
\ToCLine{279}{On the dynamics of the stellar system}% 491
\ToCLine{280}{Runaway stars}% 498
\ToCLine{281}{Globular clusters}% 500

\ToCSection{III}{The Stars}

\ToCLine{282}{Double stars}% 505
\ToCRange{283}{, }{284}{Orbits and masses of binary stars}% 507
\ToCRange{285}{, }{286}{Spectroscopic binary stars}% 510
\ToCRange{287}{--}{293}{Variable stars of various types}% 515
\ToCLine{294}{Temporary stars}% 523
\ToCLine{295}{The spectra of the stars}% 527
\ToCLine{296}{Phenomena associated with spectral types}% 530
\ToCLine{297}{On the evolution of the stars}% 532
\ToCLine{298}{Tacit assumptions of theories of stellar evolution}% 534
\ToCLine{299}{Origin and evolution of binary stars}% 543
\ToCLine{300}{On the infinity of the physical universe in space and in time}% 548

\ToCSection{IV}{The Nebulæ}

\ToCLine{301}{Irregular nebulæ}% 550
\ToCLine{302}{Spiral nebulæ}% 554
\ToCLine{303}{Ring nebulæ}% 560
\ToCLine{304}{Planetary nebulæ}% 560
%% -----File: 019.png---Folio xviii-------
% [Blank Page]
%% -----File: 020.png---Folio xix-------


\ToC{List of Tables}

\noindent\makebox[\linewidth][c]{\scriptsize\qquad NO.\hfill PAGE}\\
\LoTLine{I}{The first-magnitude stars}% 144
\LoTLine{II}{Numbers of stars in first six magnitudes}% 145
\LoTLine{III}{The constellations}% 147
\LoTLine{IV}{Elements of the orbits of the planets}% 249
\LoTLine{V}{Data on sun, moon, and planets}% 254
\LoTLine{VI}{Dates of eastern elongation and opposition}% 256
\LoTLine{VII}{Jupiter's satellite System}% 290
\LoTLine{VIII}{Saturn's satellite system}% 298
\LoTLine{IX}{Saturn's ring system}% 300
\LoTLine{X}{Rotation of the sun in different latitudes}% 389
\LoTLine{XI}{Elements found in the sun}% 393
\LoTLine{XII}{Distribution of moment of momentum in solar system}% 417
\LoTLine{XIII}{Distances of ejection for various initial velocities}% 428
\LoTLine{XIV}{Numbers of stars in magnitudes 5 to 17}% 466
\LoTLine{XV}{Distribution of the stars with respect to the Galaxy}% 471
\LoTLine{XVI}{Table of nineteen nearest stars}% 478
\LoTLine{XVII}{Distances of stars of magnitudes 1 to 15}% 486
\LoTLine{XVIII}{Binary stars whose masses are known}% 509

%% -----File: 021.png---Folio xx-------
% [Blank Page]
%% -----File: 022.png---Folio xxi-------


\ToC{List of Photographic Illustrations}

\LoPLine[{\hyperref[Fig:frontispiece]{\textit{frontispiece}}}]{1}{The Lick Observatory, Mt.\ Hamilton, Cal.}%
\LoPLine[{\hyperref[Fig:2]{\textit{facing} 1}}]{2}{The Yerkes Observatory, Williams Bay, Wis.}%
%[** Typo: "Figure 3" in original]
\LoPLine{4}{The moon at $8.5$ days (Ritchey; Yerkes Observatory)}% 20
\LoPLine{24}{Orion star trails (Barnard; Yerkes Observatory)}% 77
\LoPLine{25}{Circumpolar star trails (Ritchey)}% 78
\LoPLine{54}{The $40$-inch telescope of the Yerkes Observatory}% 138
\LoPLine{55}{The Big Dipper (Hughes; Yerkes Observatory)}% 149
\LoPLine{57}{The sickle in Leo (Hughes; Yerkes Observatory)}% 157
\LoPLine{58}{Great Andromeda Nebula (Ritchey; Yerkes Observatory)}% 158
\LoPLine{59}{The Pleiades (Wallace; Yerkes Observatory)}% 161
\LoPLine{60}{Orion (Hughes; Yerkes Observatory)}% 163
\LoPLine{61}{Great Orion Nebula (Ritchey; Yerkes Observatory)}% 164
\LoPLine{68}{The earth-lit moon (Barnard; Yerkes Observatory)}% 192
\LoPLine{75}{Moon at $9\frac{3}{4}$ days (Ritchey; Yerkes Observatory)}% 208
\LoPLine{77}{The Crater Theophilus (Ritchey; Yerkes Observatory)}% 210
\LoPLine{78}{Great Crater Clavius (Ritchey; Yerkes Observatory)}% 212
\LoPLine{79}{The full moon (Wallace; Yerkes Observatory)}% 215
\LoPLine{86}{Johann Kepler (Collection of David Eugene Smith)}% 229
\LoPLine{87}{Isaac Newton (Collection of David Eugene Smith)}% 232
\LoPLine{90}{William Herschel (Collection of David Eugene Smith)}% 239
\LoPLine{91}{John Couch Adams (Collection of David Eugene Smith)}% 240
\LoPLine{92}{Joseph Leverrier (Collection of David Eugene Smith)}% 240
\LoPLine{99}{Trail of Planetoid Egeria (Parkhurst; Yerkes Observatory)}% 259
\LoPLine{103}{Mars (Barnard; Yerkes Observatory)}% 275
\LoPLine{108}{Mars (Mount Wilson Solar Observatory)}% 286
\LoPLine{113}{Jupiter (E.~C. Slipher; Lowell Observatory)}% 295
\LoPLine{117}{Saturn (Barnard; Yerkes Observatory)}% 301
\LoPLine{119}{Brooks' Comet (Barnard; Yerkes Observatory)}% 312
\LoPLine{124}{Delavan's Comet (Barnard; Yerkes Observatory)}% 325
\LoPLine{125}{Encke's Comet (Barnard; Yerkes Observatory)}% 329
\LoPLine{126}{Morehouse's Comet (Barnard; Yerkes Observatory)}% 333
\LoPLine{128}{Halley's Comet (Barnard; Yerkes Observatory)}% 335
\LoPLine{133}{Long Island, Kan., meteorite (Farrington)}% 344
\LoPLine{134}{Cañon Diablo, Ariz., meteorite (Farrington)}% 345
%% -----File: 023.png---Folio xxii-------
\LoPLine{135}{Durango, Mexico, meteorite (Farrington)}% 345
\LoPLine{136}{Tower telescope of the Mt.\ Wilson Solar Observatory}% 348
\LoPLine{141}{The Sun (Fox; Yerkes Observatory)}% 376
\LoPLine{144}{Sun spot, July~17, 1905 (Fox; Yerkes Observatory)}% 382
\LoPLine{146}{Sun spots with opposite polarities (Hale; Solar Observatory)}% 386
\LoPLine{147}{Solar Observatory of the Carnegie Institution,\break Mt.\ Wilson, Cal.}% 387
\LoPLine{149}{Solar prominence $80,000$ miles high (Solar Observatory)}% 396
\LoPLine{150}{Motion in solar prominences (Slocum; Yerkes Observatory)}% 397
\LoPLine{152}{Spectroheliogram of sun (Hale and Ellerman; Yerkes Observatory)}% 400
\LoPLine{153}{Spectroheliograms of sun spot (Hale and Ellerman; Solar Observatory)}% 401
\LoPLine{154}{The sun's corona (Barnard and Ritchey)}% 402
\LoPLine{157}{Eruptive prominences (Slocum; Yerkes Observatory)}% 426
\LoPLine{159}{Great spiral nebula M.~51 (Ritchey; Yerkes Observatory)}% 429
\LoPLine{160}{Great spiral nebula M.~33 (Ritchey; Yerkes Observatory)}% 430
\LoPLine{162}{Laplace (Collection of David Eugene Smith)}% 449
\LoPLine{165}{Milky Way in Aquila (Barnard; Yerkes Observatory)}% 462
\LoPLine{166}{Star clouds in Sagittarius (Barnard; Yerkes Observatory)}% 472
\LoPLine{167}{Region of Rho Ophiuchi (Barnard; Yerkes Observatory)}% 474
\LoPLine{171}{Hercules star cluster (Ritchey; Yerkes Observatory)}% 501
\LoPLine{173}{Spectra of Mizar (Frost; Yerkes Observatory)}% 511
\LoPLine{174}{Spectra of Mu Orionis (Frost; Yerkes Observatory)}% 513
\LoPLine{180}{Nova Persei (Ritchey; Yerkes Observatory)}% 525
\LoPLine{181}{The spectrum of Sirius (Yerkes Observatory)}% 527
\LoPLine{182}{The spectrum of Beta Geminorum (Yerkes Observatory)}% 528
\LoPLine{183}{The spectrum of Arcturus (Yerkes Observatory)}% 529
\LoPLine{184}{The Pleiades (Ritchey; Yerkes Observatory)}% 537
\LoPLine{187}{Nebula in Cygnus (Ritchey; Yerkes Observatory)}% 551
\LoPLine{188}{Bright and dark nebulæ (Barnard; Yerkes Observatory)}% 554
\LoPLine{189}{The Trifid Nebula (Crossley reflector; Lick Observatory)}% 555
\LoPLine{190}{Spiral nebula in Ursa Major (Ritchey; Yerkes Observatory)}% 556
\LoPLine{191}{Spiral nebula in Andromeda (Crossley reflector; Lick Observatory)}% 557
\LoPLine{192}{Great nebula in Andromeda (Ritchey; Yerkes Observatory)}% 559
\LoPLine{193}{Ring nebula in Lyra (Sullivan; Yerkes Observatory)}% 560
\LoPLine{194}{Planetary nebula (Crossley reflector; Lick Observatory)}% 561

%% -----File: 024.png---Folio xxiii-------
\clearpage
\fancyhf{}

%AN INTRODUCTION TO ASTRONOMY
\HalfTitle

%% -----File: 025.png---Folio xxiv-------

\thispagestyle{empty}
\begin{sidewaysfigure}
\centering\Input{025}{jpg} %[Illustration: Fig. 2]
\Caption[The Yerkes Observatory of the University of Chicago, Williams Bay, Wisconsin.]{Fig}{2}
\end{sidewaysfigure}

%% -----File: 026.png---Folio 1-------

\mainmatter
\phantomsection
\pdfbookmark[-1]{Main Matter}{Main Matter}

% [** TN: Printed by the \Chapter macro]
%AN INTRODUCTION TO
%ASTRONOMY

\Chapter{I}{Preliminary Considerations}

\Article{1}{Science.}---The progress of mankind has been marked
\index{Science}%
by a number of great intellectual movements. At one time
the ideas of men were expanding with the knowledge which
they were obtaining from the voyages of Columbus, Magellan,
\index[xnames]{Columbus}%
\index[xnames]{Magellan}%
and the long list of hardy explorers who first visited the
remote parts of the earth. At another, millions of men laid
down their lives in order that they might obtain toleration
in religious beliefs. At another, the struggle was for political
freedom. It is to be noted with satisfaction that those
movements which have involved the great mass of people,
from the highest to the lowest, have led to results which
have not been lost.

The present age is known as the age of science. Never
before have so many men been actively engaged in the
pursuit of science, and never before have its results contributed
so enormously to the ordinary affairs of life. If all
its present-day applications were suddenly and for a considerable
time removed, the results would be disastrous.
With the stopping of trains and steamboats the food supply
in cities would soon fail, and there would be no fuel with
which to heat the buildings. Water could no longer be
pumped, and devastating fires might follow. If people escaped
to the country, they would perish in large numbers
%% -----File: 027.png---Folio 2-------
because without modern machinery not enough food could
be raised to supply the population. In fact, the more the
subject is considered, the more clearly it is seen that at the
present time the lives of civilized men are in a thousand ways
directly dependent on the things produced by science.

Astronomy is a science. That is, it is one of those subjects,
such as physics, chemistry, geology, and biology,
which have made the present age in very many respects
altogether different from any earlier one. Indeed, it is the
oldest science and the parent of a number of the others, and,
in many respects, it is the most perfect one. For these reasons
it illustrates most simply and clearly the characteristics
of science. Consequently, when one enters on the study
of astronomy he not only begins an acquaintance with a
subject which has always been noted for its lofty and unselfish
ideals, but, at the same time, he becomes familiar with
the characteristics of the scientific movement.

\Article{2}{The Value of Science.}---The importance of science
\index{Science!value of}%
in changing the relations of men to the physical universe
about them is easy to discern and is generally more or less
recognized. That the present conditions of life are better
than those which prevailed in earlier times proves the value
of science, and the more it is considered from this point of
view, the more valuable it is found to be.

The changes in the mode of living of man which science
has brought about, will probably in the course of time give
rise to marked alterations in his physique; for, the better
food supply, shelter, clothing, and sanitation which have
recently been introduced as a consequence of scientific discoveries,
correspond in a measure to the means by which the
best breeds of domestic animals have been developed, and
without which they degenerate toward the wild stock from
which they have been derived. And probably, also, as the
factors which cause changes in living organisms become
better known through scientific investigations, man will
consciously direct his own evolution.
%% -----File: 028.png---Folio 3-------

But there is another less speculative respect in which
science is important and in which its importance will enormously
increase. It has a profound influence on the minds
of those who devote themselves to it, and the number of
those who are interested in it is rapidly increasing. In the
first place, it exalts truth and honestly seeks it, wherever the
search may lead. In the second place, its subject matter
often gives a breadth of vision which is not otherwise obtained.
For example, the complexity and adaptability of
living beings, the irresistible forces which elevate the mountains,
or the majestic motions of the stars open an intellectual
horizon far beyond that which belongs to the ordinary affairs
of life. The conscious and deliberate search for truth
and the contemplation of the wonders of nature change the
mental habits of a man. They tend to make him honest
with himself, just in his judgment, and serene in the midst
of petty annoyances. In short, the study of science makes
character, as is splendidly illustrated in the lives of many
celebrated scientific men. It would undoubtedly be of very
great benefit to the world if every one could have the discipline
of the sincere and honest search for the truth which
is given by scientific study, and the broadening influence of
an acquaintance with scientific theories.

There is an important possible indirect effect of science
on the intellectual development of mankind which should
not be overlooked. One of the results of scientific discoveries
has been the greatly increased productivity of the human
race. All of the necessities of life and many of its luxuries
can now be supplied by the expenditure of much less time
than was formerly required to produce the bare means of
existence. This leaves more leisure for intellectual pursuits.
Aside from its direct effects, this is, when considered in its
broad aspects, the most important benefit conferred by
science, because, in the final analysis, intellectual experiences
are the only things in which men have an interest. As an
illustration, any one would prefer a normal conscious life
%% -----File: 029.png---Folio 4-------
for one year rather than an existence of five hundred years
with the certainty that he would be completely unconscious
during the whole time.

It is often supposed that science and the fine arts, whose
importance every one recognizes, are the antitheses of each
other. The arts are believed to be warm and human,---science,
cold and austere. This is very far from being the
case. While science is exacting in its demands for precision,
it is not insensible to the beauties of its subject. In
all branches of science there are wonderful harmonies which
appeal strongly to those who fully comprehend them. Many
of the great scientists have expressed themselves in their
writings as being deeply moved by the æsthetic side of their
subject. Many of them have had more than ordinary taste
for art. Mathematicians are noted for being gifted in music,
and there are numerous examples of scientific men who
were fond of painting, sculpture, or poetry. But even if
the common opinion that science and art are opposites were
correct, yet science would contribute indirectly to art through
the leisure it furnishes men.

\Article{3}{The Origin of Science.}---It is doubtful if any important
\index{Science!origin of}%
scientific idea ever sprang suddenly into the mind of a
single man. The great intellectual movements in the world
have had long periods of preparation, and often many men
were groping for the same truth, without exactly seizing it,
before it was fully comprehended.

The foundation on which all science rests is the principle
that the universe is orderly, and that all phenomena succeed
one another in harmony with invariable laws. Consequently,
science was impossible until the truth of this principle was
perceived, at least as applied to a limited part of nature.

The phenomena of ordinary observation, as, for example,
the wea\-ther, depend on such a multitude of factors that it
was not easy for men in their primitive state to discover
that they occur in harmony with fixed laws. This was the
age of superstition, when nature was supposed to be controlled
%% -----File: 030.png---Folio 5-------
by a great number of capricious gods whose favor
could be won by childish ceremonies. Enormous experience
was required to dispel such errors and to convince men that
the universe is one vast organization whose changes take
place in conformity with laws which they can in no way
alter.

The actual dawn of science was in prehistoric times,
probably in the civilizations that flourished in the valleys
of the Nile and the Euphrates. In the very earliest records
of these people that have come down to modern times it is
found that they were acquainted with many astronomical
phenomena and had coherent ideas with respect to the motions
of the sun, moon, planets, and stars. It is perfectly
clear from their writings that it was from their observations
of the heavenly bodies that they first obtained the idea that
the universe is not a chaos. Day and night were seen to
succeed each other regularly, the moon was found to pass
through its phases systematically, the seasons followed one
another in order, and in fact the more conspicuous celestial
phenomena were observed to occur in an orderly sequence.
It is to the glory of astronomy that it first led men to the
conclusion that law reigns in the universe.

The ancient Greeks, at a period four or five hundred
years preceding the Christian era, definitely undertook to
find from systematic observation how celestial phenomena
follow one another. They determined very accurately the
number of days in the year, the period of the moon's revolution,
and the paths of the sun and the moon among the
stars; they correctly explained the cause of eclipses and
learned how to predict them with a considerable degree of
accuracy; they undertook to measure the distances to the
heavenly bodies, and to work out a complete system that
would represent their motions. The idea was current among
the Greek philosophers that the earth was spherical, that it
turned on its axis, and, among some of them, that it revolved
around the sun. They had true science in the modern
%% -----File: 031.png---Folio 6-------
acceptance of the term, but it was largely confined to the
relations among celestial phenomena. The conception that
the heavens are orderly, which they definitely formulated and
acted on with remarkable success, has been extended, especially
in the last two centuries, so as to include the whole
universe. The extension was first made to the inanimate
world and then to the more complicated phenomena associated
with living beings. Every increase in carefully
recorded experience has confirmed and strengthened the
belief that nature is perfectly orderly, until now every one
who has had an opportunity of becoming familiar with any
science is firmly convinced of the truth of this principle,
which is the basis of all science.

\Article{4}{The Methods of Science.}---Science is concerned with
\index{Science!methods of}%
the relations among phenomena, and it must therefore rest
ultimately upon observations and experiments. Since its
ideal is exactness, the observations and experiments must
be made with all possible precision and the results must be
carefully recorded. These principles seem perfectly obvious,
yet the world has often ignored them. One of the chief
faults of the scientists of ancient times was that they indulged
in too many arguments, more or less metaphysical in character,
and made too few appeals to what would now seem obvious
observation or experiment. A great English philosopher,
Roger Bacon (1214--1294), made a powerful argument
\index[xnames]{Bacon, Roger}%
in favor of founding science and philosophy on experience.

It must not be supposed that the failure to rely on observations
and experiment, and especially to record the results
of experience, are faults that the world has outgrown. On
the contrary, they are still almost universally prevalent
among men. For example, there are many persons who believe
in dreams or premonitions because once in a thousand
cases a dream or a premonition comes true. If they had
written down in every case what was expected and what
actually happened, the absurdity of their theory would
have been evident. The whole mass of superstitions with
%% -----File: 032.png---Folio 7-------
which mankind has burdened itself survives only because
the results of actual experience are ignored.

In scientific work great precision in making observations
and experiments is generally of the highest importance.
Every science furnishes examples of cases where the data
seemed to have been obtained with greater exactness than
was really necessary, and where later the extra accuracy
led to important discoveries. In this way the foundation
of the theory of the motion of the planets was laid. Tycho
\index[xnames]{Tycho Brahe}%
Brahe was an observer not only of extraordinary industry,
but one who did all his work with the most painstaking care.
Kepler, who had been his pupil and knew of the excellence
\index[xnames]{Kepler}%
of his measurements, was a computer who sought to bring
theory and observation into exact harmony. He found it
impossible by means of the epicycles and eccentrics, which
his predecessors had used, to represent exactly the observation
of Tycho Brahe. In spite of the fact that the discrepancies
were small and might easily have been ascribed to
errors of observation, Kepler had absolute confidence in
his master, and by repeated trials and an enormous amount
of labor he finally arrived at the true laws of planetary
motion (\Artref{145}). These laws, in the hands of the genius
Newton, led directly to the law of gravitation and to the
\index[xnames]{Newton}%
explanation of all the many peculiarities of the motions of
the moon and planets (\Artref{146}).

Observations alone, however carefully they may have
been made and recorded, do not constitute science. First,
the phenomena must be related, and then, what they have
in common must be perceived. It might seem that it would
be a simple matter to note in what respects phenomena are
similar, but experience has shown that only a very few have
the ability to discover relations that are not already known.
If this were not true, there would not be so many examples
of new inventions and discoveries depending on very simple
things that have long been within the range of experience of
every one. After the common element in the observed
%% -----File: 033.png---Folio 8-------
phenomena has been discovered the next step is to infer, by
the process known as induction, that the same thing is true
\index{Induction}%
in all similar cases. Then comes the most difficult thing of
all. The vital relationships of the one class of phenomena
with other classes of phenomena must be discovered, and
the several classes must be organized into a coherent whole.

An illustration will make the process clearer than an
\index{Laws!of motion}%
extended argument. Obviously, all men have observed
moving bodies all their lives, yet the fact that a moving body,
subject to no exterior force, proceeds in a straight line with
uniform speed was not known until about the time of Galileo
\index[xnames]{Galileo}%
(1564--1642) and Newton (1643--1727). When the result
\index[xnames]{Newton}%
is once enunciated it is easy to recall many confirmatory
experiences, and it now seems remarkable that so simple
a fact should have remained so long undiscovered. It was
also noted by Newton that when a body is acted on by a
force it has an acceleration (acceleration is the rate of
\index{Acceleration, definition of}%
change of velocity) in the direction in which the force acts,
and that the acceleration is proportional to the magnitude
of the force. Dense bodies left free in the air fall toward
the earth with accelerated velocity, and they are therefore
subject to a force toward the earth. Newton observed these
things in a large number of cases, and he inferred by induction
that they are universally true. He focused particularly
on the fact that every body is subject to a force directed
toward the earth.

If taken alone, the fact that bodies are subject to forces
toward the earth is not so very important; but Newton
used it in connection with many other phenomena. For
example, he knew that the moon is revolving around the
earth in an approximately circular orbit. At~$P$, in \Figref{3},
it is moving in the direction~$PQ$ around the earth,~$E$. But
it actually moves from $P$ to~$R$. That is, it has fallen toward
the earth through the distance~$QR$. Newton perceived that
this motion is analogous to that of a body falling near the
surface of the earth, or rather to the motion of a body which
%% -----File: 034.png---Folio 9-------
has been started in a horizontal direction from~$p$ near the
surface of the earth. For, if the body were started horizontally,
it would continue in the straight line~$pq$, instead of
curving downward to~$r$, if it were not acted upon by a force
directed toward the %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{1.5in}
\Input[1.5in]{034}{png}
\Caption[The motion
of the moon from $P$
to~$R$ around~$E$ is similar
to that of a body
projected horizontally
from~$p$.]{Fig}{3}
\end{wrapfigure}
earth. Newton also
\index[xnames]{Newton}%
knew Kepler's laws of planetary motion.
\index[xnames]{Kepler}%
By combining with wonderful insight a
number of classes of phenomena which
before his time had been supposed to be
unrelated, he finally arrived at the law of
\index{Law!of gravitation}%
\index{Gravitation!law of}%
gravitation---``Every particle of matter
in the universe attracts every other particle
with a force which is directly proportional
to the product of their masses
and inversely proportional to the square
of their distance apart.'' Thus, by perceiving
the essentials in many kinds of
phenomena and by an almost unparalleled
stroke of genius in combining them,
he discovered one of the relations which
every particle of matter in the universe
has to all the others. By means of the laws of motion
(\Artref{40}) and the law of gravitation, the whole problem of
the motions of bodies was systematized.

There is still another method employed in science which
is often very important. After general principles have
been discovered they can be used as the basis for deducing
\index{Deduction}%
particular conclusions. The value of the particular conclusions
may consist in leading to the accomplishment of some
desired end. For example, since a moving body tends to
continue in a straight line, those who build railways place
the outside rails on curves higher than those on the inside
so that trains will not leave the track. Or, the
knowledge of the laws of projectiles enables gunners to hit
invisible objects whose positions are known.

The value of particular conclusions may consist in enabling
%% -----File: 035.png---Folio 10-------
men to adjust themselves to phenomena over which
they have no control. For example, in many harbors large
boats can enter or depart only when the tide is high, and the
knowledge of the times when the tides will be high is valuable
to navigators. After the laws of meteorology have become
more perfectly known, so that approaching storms, or
frosts, or drouths, or hot waves can be accurately foretold
a considerable time in advance, the present enormous losses
due to these causes will be avoided.

The knowledge of general laws may lead to information
regarding things which are altogether inaccessible to observation
or experiment. For example, it is very important
for the geologist to know whether the interior of the earth
is solid or liquid; and, if it is solid, whether it is elastic or
viscous. Although at first thought it seems impossible to
obtain reliable information on this subject, yet by a number
of indirect processes (Arts.\ \hyperref[art:25]{25},~\hyperref[art:26]{26}) based on laws established
from observation, it has been possible to prove with certainty
that the earth, through and through, is about as
rigid as steel, and that it is highly elastic.

Another important use of the deductive process in science
\index{Deduction}%
is in drawing consequences of a theory which must be fulfilled
in experience if the theory is correct, and which may
fail if it is false. It is, indeed, the most efficient means of
testing a theory. Some of the most noteworthy examples
of its application have been in connection with the law of
gravitation. Time after time mathematicians, using this
law as a basis for their deductions, have predicted phenomena
that had not been observed, and time after time their
predictions have been fulfilled. This is one of the reasons
why the truth of the law of gravitation is regarded as having
been firmly established.

\Article{5}{The Imperfections of Science.}---One of the characteristics
\index{Science!imperfections of}%
of science is its perfect candor and fairness. It
would not be in harmony with its spirit to attempt to lead
one to suppose that it does not have sources of weakness.
%% -----File: 036.png---Folio 11-------
Besides, if its possible imperfections are analyzed, they can
be more easily avoided, and the real nature of the final
conclusions will be better understood.

It must be observed, in the first place, that science consists
of men's theories regarding what is true in the universe
about them. These theories are based on observation and
experiment and are subject to the errors and incompleteness
of the data on which they are founded. The fact that it
is not easy to record exactly what one may have attempted
to observe is illustrated by the divergence in the accounts
of different witnesses of anything except the most trivial
occurrence. Since men are far from being perfect, errors
in the observations cannot be entirely avoided, but in good
science every possible means is taken for eliminating them.

In addition to this source of error, there is another more
insidious one that depends upon the fact that observational
data are often collected for the purpose of testing a specific
theory. If the theory in question is due to the one who is
making the observations or experiments, it is especially
difficult for him to secure data uninfluenced by his bias in
its favor. And even if the observer is not the author of the
theory to which the observations relate, he is very apt to be
prejudiced either in its favor or against it.

Even if the data on which science is based were always
correct, they would not be absolutely exhaustive, and the
inductions to general principles from them would be subject
to corresponding uncertainties. Similarly, the general
principles, derived from various classes of phenomena, which
are used in formulating a complete scientific theory, do not
include all the principles which are involved in the particular
domain of the theory. Consequently it may be imperfect
for this reason also.

The sources of error in scientific theories which have been
enumerated are fundamental and will always exist. The
best that can be done is to recognize their existence and to
minimize their effects by all possible means. The fact that
%% -----File: 037.png---Folio 12-------
science is subject to imperfections does not mean that it is
of little value or that less effort should be put forth in its
cultivation. Wood and stone and brick and glass have
never been made into a perfect house; yet houses have been
very useful and men will continue to build them.

There are many examples of scientific theories which it
\index{Scientific theories}%
has been found necessary to modify or even to abandon.
These changes have not been more numerous than they
have been in other domains of human activities, but they
have been, perhaps, more frankly confessed. Indeed, there
are plenty of examples where scientists have taken evident
satisfaction in the alterations they have introduced. The
fact that scientific theories have often been found to be
imperfect and occasionally positively wrong, have led some
persons who have not given the question serious consideration
to suppose that the conclusions of science are worthy of no
particular respect, and that, in spite of the pretensions of
scientists, they are actually not far removed from the level
of superstitions. The respect which scientific theories
deserve and the gulf that separates them from superstitions
will be evident from a statement of their real nature.

Suppose a person were so situated that he could look
out from an upper window over a garden. He could make
a drawing of what he saw that would show exactly the relative
positions of the walks, shrubs, and flowers. If he were color
blind, the drawing could be made in pencil so as to satisfy
perfectly all his observations. But suppose some one else
who was not color blind should examine the drawing. He
would legitimately complain that it was not correct because
the colors were not shown. If the colors were correctly given,
both observers would be completely satisfied. Now suppose
a third person should look at the drawing and should then
go down and examine the garden in detail. He would find
that the various objects in it not only have positions but
also various heights. He would at once note that the
heights were not represented in the drawing, and a little
%% -----File: 038.png---Folio 13-------
reflection would convince him that the three-dimensional
garden could not be completely represented in a two-dimensional
drawing. He would claim that that method of trying
to give a correct idea of what was in the garden was fundamentally
wrong, and he might suggest a model of suitable
material in three dimensions. Suppose the three-dimensional
model were made satisfying the third observer. It is
important to note that it would correctly represent all the
relative positions observed by the first one and all the colors
observed by the second one, as well as the additional information
obtained by the third one.

A scientific theory is founded on the work of one or more
persons having only limited opportunities for observation
and experiment. It is a picture in the imagination, not on
paper, of the portion of the universe under consideration. It
represents all the observed relations, and it is assumed that
it will represent the relations that might be observed in
all similar circumstances. Suppose some new facts are
discovered which are not covered by the theory, just as the
second observer in the garden saw colors not seen by the
first. It will be necessary to change the scientific theory so
as to include them. Perhaps it can be done simply by adding
to the theory. But if the new facts correspond to the things
discovered by the third observer in the garden, it will be
necessary to abandon the old theory and to construct an
entirely new one. The new one must preserve all the relations
represented by the old one, and it must represent the
new ones as well.

In the light of this discussion it may be asked in what
sense scientific theories are true. The answer is that they
are all true to the extent that they picture nature. The
relations are the important things. When firmly established
they are a permanent acquisition; however the mode of
representing them may change, they remain. A scientific
theory is a convenient and very useful way of describing the
relations on which it is based. It correctly represents
%% -----File: 039.png---Folio 14-------
them, and in this respect differs from a superstition which
\index{Superstition}%
is not completely in harmony with its own data. It implies
many additional things and leads to their investigation. If
the implications are found to hold true in experience, the
theory is strengthened; if not, it must be modified. Hence,
there should be no reproach in the fact that a scientific
theory must be altered or abandoned. The necessity for
such a procedure means that new information has been
obtained, not that the old was false.\footnote
  {The comparison of scientific theories with the picture of the objects
  seen in the garden is for the purpose of making clear one of their particular
  features. It must be remembered that in most respects the comparison
  with so trivial a thing is very imperfect and unfair to science.}

\Article{6}{Great Contributions of Astronomy to Science.}---As
\index{Scientific theories!contributions to, by astronomy}%
was explained in \Artref{3}, science started in astronomy. Many
astronomical phenomena are so simple that it was possible
for primitive people to get the idea from observing them
that the universe is orderly and that they could discover its
laws. In other sciences there are so many varying factors
that the uniformity in a succession of events would not be
discovered by those who were not deliberately looking for it.
It is sufficient to consider the excessive complexities of the
weather or of the developments of plants or animals, to see
how hopeless would be the problem which a people without
a start on science would face if they were cut off from
celestial phenomena. It is certain that if the sky had always
been covered by clouds so that men could not have
observed the regular motions of the sun, moon, and stars,
the dawn of science would have been very much delayed.
It is entirely possible, if not probable, that without the help
of astronomy the science of the human race would yet be in
a very primitive state.

Astronomy has made positive and important contributions
to science within historical times. Spherical trigonometry
was invented and developed because of its uses in
determining the relations among the stars on the vault of
the heavens. Very many things in calculus and still higher
%% -----File: 040.png---Folio 15-------
branches of mathematics were suggested by astronomical
problems. The mathematical processes developed for astronomical
applications are, of course, available for use in
other fields. But the great science of mathematics does
not exist alone for its applications, and to have stimulated
its growth is an important contribution. While many
parts of mathematics did not have their origin in astronomical
problems, it is certain that had it not been for these
problems mathematical science would be very different from
what it now is.

The science of dynamics is based on the laws of motion.
These laws were first completely formulated by Newton,
\index[xnames]{Newton}%
who discovered them and proved their correctness by considering
the revolutions of the moon and planets, which
describe their orbits under the ideal condition of motion in a
vacuum without any friction. The immense importance
of mechanics in modern life is a measure of the value of this
contribution of astronomy to science.

The science of geography owes much to astronomy, both
directly and indirectly. A great period of exploration followed
the voyages of Columbus. It took courage of the
\index[xnames]{Columbus}%
highest order to sail for many weeks over an unknown ocean
in the frail boats of his time. He had good reasons for thinking
he could reach India, to the eastward, by sailing westward
from Spain. His reasons were of an astronomical
nature. He had seen the sun rise from the ocean in the
east, travel across the sky and set in the west; he had observed
that the moon and stars have similar motions; and
he inferred from these things that the earth was of finite extent
and that the heavenly bodies moved around it. This
led him to believe it could be circumnavigated. Relying
upon the conclusions that he drew from his observations of
the motions of the heavenly bodies, he maintained control
of his mutinous sailors during their perilous voyage across
the Atlantic, and made a discovery that has been of immense
consequence to the human race.
%% -----File: 041.png---Folio 16-------

One of the most important influences in modern scientific
thought is the doctrine of evolution. It has not only largely
\index{Evolution}%
given direction to investigations and speculations in biology
and geology, but it has also been an important factor in the
interpretation of history, social changes, and even religion.
The first clear ideas of the orderly development of the universe
were obtained by contemplating the relatively simple
celestial phenomena, and the doctrine of evolution was current
in astronomical literature more than half a century
before it appeared in the writings of Darwin, Spencer, and
\index[xnames]{Darwin, Charles}%
\index[xnames]{Spencer}%
their contemporaries. In fact, it was carried directly from
astronomy over into geology, and from geology into the
biological sciences (\Artref{242}).

\Article{7}{The Present Value of Astronomy.}---From what has
been said it will be admitted that astronomy has been of
great importance in the development of science, but it is
commonly believed that at the present time it is of little
practical value to mankind. While its uses are by no
means so numerous as those of physics and chemistry, it
is nevertheless quite indispensable in a number of human
activities.

Safe navigation of the seas is absolutely dependent upon
astronomy. In all long voyages the captains of vessels
frequently determine their positions by observations of the
celestial bodies. Sailors use the nautical mile, or knot,
\index{Mile, nautical}%
which approximately equals one and one sixth ordinary
miles. The reason they employ the nautical mile is that this
is the distance which corresponds to a change of one minute
of arc in the apparent positions of the heavenly bodies.
That is, if, for simplicity, the sun were over a meridian, its
altitude as observed from two vessels a nautical mile apart
on that meridian would differ by one minute of arc.

Navigation is not only dependent on simple observations
of the sun, moon, and stars, but the mathematical theory
of the motions of these bodies is involved. The subject is
so difficult and intricate that for a long time England and
%% -----File: 042.png---Folio 17-------
France offered substantial cash prizes for accurate tables of
the positions of the moon for the use of their sailors.

Just as a sea captain determines his position by astronomical
observations, so also are geographical positions
located. For example, explorers of the polar regions find
how near they have approached to the pole by observations
of the altitude of the sun. International boundary lines in
many cases are defined by latitudes and longitudes, instead
of being determined by natural barriers, as rivers, and in all
such cases they are located by astronomical observations.

It might be supposed that even though astronomy is essential
to navigation and geography, it has no value in the
ordinary activities of life. Here, again, first impressions are
erroneous. It is obvious that railway trains must be run according
to accurate time schedules in order to avoid confusion
and wrecks. There are also many other things in which accurate
time is important. Now, time is determined by observations
of the stars. The millions of clocks and watches in use in
the world are all ultimately corrected and controlled by
comparing them with the apparent diurnal motions of the
stars. For example, in the United States, observations are
made by the astronomers of the Naval Observatory, at
\index{Naval Observatory}%
Washington, on every clear night, and from these observations
their clocks are corrected. These clocks are in electrical
connection with more than $30,000$ other clocks in
various parts of the country. Every day time signals are
sent out from Washington and these $30,000$ clocks are
automatically corrected, and all other timepieces are
directly or indirectly compared with them.

It might be inquired whether some other means might
not be devised of measuring time accurately. It might be
supposed that a clock could be made that would run so
accurately as to serve all practical purposes. The fact is,
however, no clock ever was made which ran accurately for
any considerable length of time. No two clocks have been
made which ran exactly alike. In order to obtain a satisfactory
%% -----File: 043.png---Folio 18-------
measure of time it is necessary to secure the ideal
conditions under which the earth rotates and the heavenly
bodies move, and there is no prospect that it ever will be
possible to use anything else, as the fundamental basis, than
the apparent motions of the stars.

Astronomy is, and will continue to be, of great importance
in connection with other sciences. It supplies most of the
fundamental facts on which meteorology depends. It is
of great value to geology because it furnishes the geologist
information respecting the origin and pre-geologic history
of the earth, it determines for him the size and shape of the
earth, it measures the mass of the earth, and it proves important
facts respecting the condition of the earth's interior.
It is valuable in physics and chemistry because the universe
is a great laboratory which, with modern instruments, can
be brought to a considerable extent within reach of the
investigator. For example, the sun is at a higher temperature
than can be produced by any known means on the
earth. The material of which it is composed is in an incandescent
state, and the study of the light received from it has
proved the existence, in a number of instances, of chemical
elements which had not been known on the earth. In fact,
their discovery in the sun led to their detection on the earth.
It seems probable that similar discoveries will be made many
times in the future. The sun's corona and the nebulæ
contain material which seems to be in a more primitive state
than any known on the earth, and the revelations afforded
by these objects are having a great influence on physical
theories respecting the ultimate structure of matter.

Astronomy is of greatest value to mankind, however, in
an intellectual way. It furnishes men with an idea of the
wonderful universe in which they live and of their position
in it. Its effects on them are analogous to those which are
produced by travel on the earth. If a man visits various
countries, he learns many things which he does not and cannot
apply on his return home, but which, nevertheless,
%% -----File: 044.png---Folio 19-------
make him a broader and better man. Similarly, though
what one may learn about the millions of worlds which
occupy the almost infinite space within reach of the great
telescopes of modern times cannot be directly applied in the
ordinary affairs of life, yet the contemplation of such things,
in which there is never anything that is low or mean or sordid,
makes on him a profound impression. It strongly modifies
the particular philosophy which he has more or less definitely
formulated in his consciousness, and in harmony with which
he orders his life.

\Article{8}{The Scope of Astronomy.}---The popular conception
\index{Scope of astronomy}%
of astronomy is that it deals in some vague and speculative
way with the stars. Since it is obviously impossible to
visit them, it is supposed that all conclusions respecting them,
except the few facts revealed directly by telescopes, are pure
guesses. Many people suppose that astronomers ordinarily
engage in the harmless and useless pastime of gazing at the
stars with the hope of discovering a new one. Many of those
who do not have this view suppose that astronomers control
the weather, can tell fortunes, and are very shrewd to have
discovered the names of so many stars. As is true of most
conclusions that are not based on evidence, these conceptions
of astronomy and astronomers are absurd.

Astronomy contains a great mass of firmly established
facts. Astronomers demand as much evidence in support
of their theories as is required by other scientists. They
have actually measured the distances to the moon, sun, and
many of the stars. They have discovered the laws of their
motions and have determined the masses of the principal
members of the solar system. The precision attained in
much of their work is beyond that realized in most other
sciences, and their greatest interest is in measurable things
and not in vague speculations.

A more extended preliminary statement of the scope of
astronomy is necessary in order that its study may be entered
on without misunderstandings. Besides, the relations among
%% -----File: 045.png---Folio 20-------
the facts with which a science deals are very important,
and a preliminary outline of the subject will make it easier
to place in their proper position in an organized whole all
the various things which may be set forth in the detailed
discussions.

The most accessible and best-known astronomical object
is the earth. Those facts respecting it that are determined
entirely or in large
part by astronomical
means are properly
regarded as belonging
to astronomy.
Among them are the
shape and size of
the earth, its average
density, the condition
of its interior, the
height of its atmosphere,
its rotation on
its axis and revolution
around the sun,
and the climatic conditions
of its surface
so far as they are
determined by its relation
to the sun.

The nearest celestial
body is the
moon. Astronomers
have found by fundamentally the same methods as those
which surveyors employ that its distance from the earth
\index{Distance!of moon}%
\index{Moon!distance of}%
averages about $240,000$ miles, that its diameter is about
$2160$ miles, and that its mass is about one eightieth that of
the earth. The earth holds the moon in its orbit by its gravitational
control, and the moon in turn causes the tides on the
earth. It is found that there is neither atmosphere nor water
%% -----File: 046.png---Folio 21-------
on the moon, and the telescope shows that its surface is
covered with mountains and circular depressions, many of
great size, which are called craters.

\begin{wrapfigure}{\WLoc}{3in}%[Illustration: Moved down]
\Input[3in]{045}{jpg}
\Caption[The moon $1.5$~days after the first
quarter. \textit{Photographed with the 40-inch
telescope of the Yerkes Observatory.}]{Fig}{4}
\end{wrapfigure}
The earth is one of the eight planets which revolve around
the sun in nearly circular orbits. Three of them are smaller
than the earth and four are larger. The smallest, Mercury,
has a volume about one twentieth that of the earth, and the
largest, Jupiter, has a volume about one thousand times that
of the earth. The great sun, whose mass is seven hundred
times that of all of the planets combined, holds them in their
orbits and lights and warms them with its abundant rays.
Those nearest the sun are heated much more than the earth,
but remote Neptune gets only one nine-hundredth as much
light and heat per unit area as is received by the earth.
Some of the planets have no moons and others have several.
The conditions on one or two of them seem to be perhaps
favorable for the development of life, while the others certainly
cannot be the abode of such life as flourishes on the
earth.

In addition to the planets, over eight hundred small
planets, or planetoids, and a great number of comets circulate
around the sun in obedience to the same law of gravitation.
The orbits of nearly all the small planets lie between
the orbits of Mars and Jupiter; the orbits of the comets are
generally very elongated and are unrelated to the other
members of the system. The phenomena presented by the
comets, for example the behavior of their tails, raise many
interesting and puzzling questions.

The dominant member of the solar system is the sun.
Its volume is more than a million times that of the earth,
its temperature is far higher than any that can be produced
on the earth, even in the most efficient electrical furnaces,
and its surface is disturbed by the most violent storms.
Often masses of this highly heated material, in volumes
greater than the whole earth, move along or spout up from
its surface at the rate of several hundreds miles a minute.
%% -----File: 047.png---Folio 22-------
The spectroscope shows that the sun contains many of the
elements, particularly the metals, of which the earth is composed.
The consideration of the possible sources of the
sun's heat leads to the conclusion that it has supplied the
earth with radiant energy for many millions of years, and
that the supply will not fail for at least a number of million
years in the future.

The stars that seem to fill the sky on a clear night are
suns, many of which are much larger and more brilliant than
our own sun. They appear to be relatively faint points of
light because of their enormous distances from us. The
nearest of them is so remote that more than four years are
\index{Light!velocity of}%
\index{Velocity!of light}%
required for its light to come to the solar system, though
light travels at the rate of $186,330$ miles per second; and
others, still within the range of large telescopes, are certainly
a thousand times more distant. At these vast distances
such a tiny object as the earth would be entirely invisible
even though astronomers possessed telescopes ten thousand
times as powerful as those now in use. Sometimes stars
appear to be close together, as in the case of the Pleiades, but
\index{Pleiades}%
their apparent proximity is due to their immense distances
from the observer. There are doubtless regions of space
from which the sun would seem to be a small star forming a
close group with a number of others. There are visible
to the unaided eye in all the sky only about $5000$ stars, but
the great photographic telescopes with which modern
observatories are equipped show several hundreds of millions
of them. It might be supposed that telescopes with twice
the light-gathering power would show proportionately more
stars, and so on indefinitely, but this is certainly not true,
for there is evidence that points to the conclusion that they
do not extend indefinitely, at least with the frequency with
which they occur in the region around the sun. The visible
stars are not uniformly scattered throughout the space which
they occupy, but form a great disk-like aggregation lying in
the plane of the Milky Way.\index{Milky Way}%
%% -----File: 048.png---Folio 23-------

Many stars, instead of being single isolated masses, like
the sun, are found on examination with highly magnifying
telescopes to consist of two suns revolving around their
common center of gravity. In most cases the distances
between the two members of a double star is several times
as great as the distance from the earth to the sun. The
existence of double stars which may be much closer together
than those which are visible through telescopes has also
been shown by means of instruments called spectroscopes.
It has been found that a considerable fraction, probably
one fourth, of all the nearer stars are double stars. There
are also triple and quadruple stars; and in some cases
thousands of suns, all invisible to the unaided eye, occupy
a part of the sky apparently smaller than the moon. Even
in such cases the distances between the stars are enormous,
and such clusters, as they are called, constitute larger and
more wonderful aggregations of matter than any one ever
dreamed existed before they were revealed by modern
instruments.

While the sun is the center around which the planets and
\index{Stars!velocities of}%
\index{Velocity!of sun}%
\index{Velocity!of stars}%
comets revolve, it is not fixed with respect to the other
stars. Observations with both the telescope and the spectroscope
prove that it is moving, with respect to the brighter
stars, approximately in the direction of the brilliant Vega
\index{Vega}%
in the constellation Lyra. It is found by use of the spectroscope
\index{Lyra}%
that the rate of motion is about $400,000,000$ miles
per year. The other stars are also in motion with an average
velocity of about $600,000,000$ miles per year, though some of
them move much more slowly than this and some of them
many times faster. One might think that the great speed of
the sun would in a century or two so change its relations to
the stars that the appearance of the sky would be entirely
altered. But the stars are so remote that in comparison the
distance traveled by the sun in a year is negligible. When
those who built the pyramids turned their eyes to the sky
\index{Pyramids}%
at night they saw the stars grouped in constellations almost
%% -----File: 049.png---Folio 24-------
exactly as they are seen at present. During the time covered
by observations accurate enough to show the motion
of the sun it has moved sensibly in a straight line, though in
the course of time the direction of its path will doubtless be
changed by the attractions of the other stars. Similarly,
the other stars are moving in sensibly straight lines in every
direction, but not altogether at random, for it has been found
that there is a general tendency for them to move in two or
more roughly parallel streams.

In addition to learning what the universe is at present,
one of the most important and interesting objects of astronomy
is to find out through what great series of changes it
has gone in its past evolution, and what will take place in it
in the future. As a special problem, the astronomer tries
to discover how the earth originated, how long it has been
in existence, particularly in a state adapted to the abode of
life, and what reasonably may be expected for the future.
These great problems of cosmogony have been of deep interest
to mankind from the dawn of civilization; with increasing
knowledge of the wonders of the universe and of the laws
by which alone such questions can be answered, they have
become more and more absorbingly attractive.


\Section{I}{QUESTIONS}

1. Enumerate as many ways as possible in which science is
beneficial to men.

2. What is the fundamental basis on which science rests, and
what are its chief characteristics?

3. What is induction? Give examples. Can a science be developed
without inductions? Are inductions always true?

4. What is deduction? Give examples. Can a science be developed
without deductions? Are deductions always true?

5. In what respects may science be imperfect? How may its imperfections
be most largely eliminated? Are any human activities
perfect?

6. Name some superstition and show in what respects it differs
from scientific conclusions.

7. Why did science originate in astronomy?

%% -----File: 050.png---Folio 25-------

8. Are conclusions in astronomy firmly established, as they are
in other sciences?

9. In what fundamental respects do scientific laws differ from
civil laws?

10. What advantages may be derived from a preliminary outline
of the scope of astronomy? Would they hold in the case of a subject
not a science?

11. What questions respecting the earth are properly regarded
as belonging to astronomy? To what other sciences do they respectively
belong? Is there any science which has no common
ground with some other science?

12. What arts are used in astronomy? Does astronomy contribute
to any art?

13. What references to astronomy in the sacred or classical literatures
do you know?

14. Has astronomy exerted any influence on philosophy and
religion? Have they modified astronomy?

\normalsize
%% -----File: 051.png---Folio 26-------


\Chapter{II}{The Earth}

\Section{I}{The Shape of the Earth}

\Article{9}{Astronomical Problems respecting the Earth.}---The
earth is one of the objects belonging to the field of astronomical
investigations. In the consideration of it astronomy
has its closest contact with some of the other sciences, particularly
with geology and meteorology. Those problems
respecting the earth that can be solved for other planets also,
or that are essential for the investigation of other astronomical
questions, are properly considered as belonging to the
field of astronomy.

The astronomical problems respecting the earth can be
divided into two general classes. The first class consists of
those which can be treated, at least to a large extent, without
regarding the earth as a member of a family of planets
or considering its relations to them and the sun. Such problems
are its shape and size, its mass, its density, its interior
temperature and rigidity, and the constitution, mass, height,
and effects of its atmosphere. These problems will be treated
in this chapter. The second class consists of the problems
involved in the relations of the earth to other bodies, particularly
its rotation, revolution around the sun, and the consequences
of these motions. The treatment of these problems
will be reserved for the next chapter.

It would be an easy matter simply to state the astronomical
facts respecting the earth, but in science it is necessary
not only to say what things are true but also to give the
reasons for believing that they are true. Therefore one or
more proofs will be given for the conclusions astronomers
have reached respecting the earth. As a matter of logic
%% -----File: 052.png---Folio 27-------
one complete proof is sufficient, but it must be remembered
that a scientific doctrine consists of, and rests on, a great
number of theories whose truth may be more or less in question,
and consequently a number of proofs is always desirable.
If they agree, their agreement confirms belief in the
accuracy of all of them. It will not be regarded as a burden
to follow carefully these proofs; in fact, one who has arrived
at a mature stage of intellectual development instinctively
demands the reasons we have for believing that our conclusions
are sound.

\Article{10}{The Simplest and most Conclusive Proof of the
Earth's Sphericity.}\footnote
  {The earth is not exactly round, but the departure from sphericity
  will be neglected for the moment.}---Among the proofs that the earth is
\index{Earth!sphericity of}%
\index{Sphericity of earth}%
found, the simplest and most conclusive is that \textit{the plane of
the horizon, or the direction of the plumb line, changes by an
angle which is directly proportional
to the distance the observer travels
along the surface of the earth,
whatever the direction and distance
of travel}.

\begin{wrapfigure}{\WLoc}{2.25in}%[Illustration:]
\Input[2.25in]{052}{png}
\Caption[The change in the direction
of the plumb line is
proportional to the distance
traveled along the surface of
the earth.]{Fig}{5}
\end{wrapfigure}

It will be shown first that if
the earth were a true sphere the
statement would be true. For
simplicity, suppose the observer
travels along a meridian. If the
statement is true for this case,
it will be true for all others,
because a sphere has the same
curvature in every direction.
Suppose the observer starts from~$O_1$,
\Figref{5}, and travels northward
to~$O_2$. The length of the arc~$O_1O_2$
is proportional to the angle~$a$
which it subtends at the center of the sphere. The planes
of the horizon of $O_1$ and~$O_2$ are respectively $O_1H_1$ and~$O_2H_2$.
%% -----File: 053.png---Folio 28-------
These lines are respectively perpendicular to $CO_1$ and~$CO_2$.
Therefore the angle between them equals the angle~$a$. That
is, the distance traveled is proportional to the change of
direction of the plane of the horizon.

The plumb lines at $O_1$ and~$O_2$ are respectively $O_1Z_1$ and
$O_2Z_2$, and the angle between these lines is~$a$. Hence the distance
traveled is proportional to the change in the direction
of the plumb line.

It will be shown now that if the surface of the earth were
not a true sphere the change in the direction of the plane of
the horizon would not be proportional to the distance traveled
on the surface. Suppose
\Figref{6} represents a plane
section through the non-spherical
earth along
whose surface the observer
travels. Since the
earth is not a sphere, the
curvature of its surface
will be different at different
places. Suppose that
$O_1O_2$ is one of the flatter
regions and $O_3O_4$ is one
of the more convex ones.
In the neighborhood of $O_1O_2$ the direction of the plumb line
changes slowly, while in the neighborhood of $O_3O_4$ its direction
changes more rapidly. The large arc~$O_1O_2$ subtends an
angle at~$C_1$ made by the respective perpendiculars to the
surface which exactly equals the angle at~$C_3$ subtended by
the smaller arc~$O_3O_4$. Therefore in this case the change in
direction of the plumb line is \emph{not} proportional to the distance
traveled, for the same angular change corresponds to
two different distances. The same result is true for the
plane of the horizon because it is always perpendicular to
the plumb line.

\begin{wrapfigure}{\WLoc}{2.75in}%[Illustration:]
\Input[2.75in]{053}{png}
\Caption[If the earth were not spherical,
equal angles would be subtended by arcs
of different lengths.]{Fig}{6}
\end{wrapfigure}

Since the conditions of the statement would be satisfied
%% -----File: 054.png---Folio 29-------
in case the earth were spherical, and only in case it were
spherical, the next question is what the observations show.
Except for irregularities of the surface, which are not under
consideration here, and the oblateness, which will be discussed
in \Artref{12}, the observations prove absolutely that the
change in direction of the plumb line is proportional to the
arc traversed.

Two practical problems are involved in carrying out the
proof which has just been described. The first is that of
measuring the distance between two points along the surface
\begin{figure}[hbt]%[Illustration:]
\Input{054}{png}
\Caption[The base line $A_{1}A_{2}$ is measured directly and the other distances
are obtained by triangulation.]{Fig}{7}
\index{Triangulation}%
\end{figure}%
of the earth, and the second is that of determining the
change in the direction of the plumb line. The first is a
refined problem of surveying; the second is solved by
observations of the stars.

All long distances on the surface of the earth are determined
by a process known as triangulation. It is much
more convenient than direct measurement and also much
more accurate. A fairly level stretch of country, $A_1$ and~$A_2$
in \Figref{7}, a few miles long is selected, and the distance
between the two points, which must be visible from each
other, is measured with the greatest possible accuracy.
%% -----File: 055.png---Folio 30-------
This line is called the \textit{base line}. Then a point~$A_{3}$ is taken
\index{Base line}%
which can be seen from both $A_{1}$ and~$A_{2}$. A telescope is set
up at $A_{1}$ and pointed at~$A_{2}$. It has a circle parallel to the
surface of the earth on which the degrees are marked. The
position of the telescope with respect to this circle is recorded.
Then the telescope is turned until it points toward~$A_{3}$.
The difference of its position with respect to the circle when
pointed at $A_{2}$ and at $A_{3}$ is the angle $A_{2}A_{1}A_{3}$. Similarly,
the telescope is set up at $A_{2}$ and the angle $A_{1}A_{2}A_{3}$ is measured.
Then in the triangle $A_{1}A_{2}A_{3}$ two angles and the included
side are known. By plane geometry, two triangles
that have two angles and the included side of one respectively
equal to two angles and the included side of the other are
exactly alike in size and shape. This simply means that
when two angles and the included side of the triangle are
given, the triangle is uniquely defined. The remaining parts
can be computed by trigonometry. In the present case
suppose the distance $A_{2}A_{3}$ is computed.

Now suppose a fourth point~$A_{4}$ is taken so that it is
visible from both $A_{2}$ and~$A_{3}$. Then, after the angles at $A_{2}$
and $A_{3}$ in the triangle $A_{2}A_{3}A_{4}$ have been measured, the line
$A_{3}A_{4}$ can be computed. This process evidently can be continued,
step by step, to any desired distance.

Suppose $A_{1}$ is regarded as the original point from which
measurements are to be made. Not only have various distances
been determined, but also their directions with respect
to the north-south line are known. Consequently, it is
known how far north and how far east $A_{2}$ is from~$A_{1}$. The
next step gives how far south and how far east $A_{3}$ is from~$A_{2}$.
By combining the two results it is known how far south and
how far east $A_{3}$ is from~$A_{1}$, and so on for succeeding points.

The convenience in triangulation results partly from the
long distances that can be measured, especially in rough
country. It is sometimes advisable to go to the trouble of
erecting towers in order to make it possible to use stations
separated by long distances. The accuracy arises, at least
%% -----File: 056.png---Folio 31-------
in part, from the fact that the angles are measured by instruments
which magnify them. The fact that the stations
are not all on the same level, and the curvature of the earth,
introduce little difficulties in the computations that must
be carefully overcome.

The direction of the plumb line at the station~$A_{1}$, for
example, is determined by noting the point among the stars
at which it points. The plumb line at~$A_{2}$ will point to a
different place among the stars. The difference in the two
places among the stars gives the difference in the directions
of the plumb lines at the two stations. The stars apparently
move across the sky from east to west during the night and
are not in the same positions at the same time of the day
on different nights. Hence, there are here also certain circumstances
to which careful attention must be given in
order to get accurate results.

\Article{11}{Other Proofs of the Earth's Sphericity.}---There are
many reasons given for believing that the earth is not a
plane, and that it is, indeed, some sort of a convex figure;
but most of them do not prove that it is actually spherical.
It will be sufficient to mention them.

(\textit{a})~The earth has been circumnavigated, but so far as
this fact alone is concerned it might be the shape of a cucumber.
(\textit{b})~Vessels disappear below the horizon hulls first
and masts last, but this only proves the convexity of the
surface. (\textit{c})~The horizon appears to be a circle when viewed
from an elevation above the surface of the water. This is
theoretically good but observationally it is not very exact.
(\textit{d})~The shadow of the earth on the moon at the time of a
lunar eclipse is always an arc of a circle, but this proof is
very inconclusive, in spite of the fact that it is often mentioned,
because the shadow has no very definite edge and
its radius is large compared to that of the moon.

\Article{12}{Proof of the Oblateness of the Earth by Arcs of
Latitude.}---The latitude of a place on the earth is determined
\index{Earth!oblateness of}%
\index{Oblateness of earth}%
by observations of the direction of the plumb line
%% -----File: 057.png---Folio 32-------
with respect to the stars. This is the reason that a sea captain
refers to the heavenly bodies in order to find his location
on the ocean. It is found by actual observations of the
stars and measurements of arcs that the length of a degree
of arc is longer the farther it is from the earth's equator.
This proves that
\begin{wrapfigure}[17]{\WLoc}{2.25in}%[Illustration: Break]
\Input[2.25in]{057}{png}
\Caption[The length of a degree
of latitude is least at the equator
and greatest at the poles.]{Fig}{8}
\end{wrapfigure}
the earth is less curved at the poles than
it is at the equator. A body which is thus flattened at the
poles and bulged at the equator is called \textit{oblate}.
\index{Oblate figure}%

In order to see that in the case of an oblate body a degree
of latitude is longer near the poles than it is at the equator,
consider \Figref{8}. In this figure $E$~represents a plane section
of the body through its poles.
The curvature at the equator is
the same as the curvature of the
circle~$C_1$, and a degree of latitude
on~$E$ at its equator equals a
degree of latitude on~$C_1$. The
curvature of~$E$ at its pole is the
same as the curvature of the
circle~$C_2$, and a degree of latitude
on~$E$ at its pole equals a
degree of latitude on~$C_2$. Since
$C_2$ is greater than~$C_1$, a degree
of latitude near the pole of the
oblate body is greater than a degree of latitude near its
equator.

%[Illustration: Place opposite figure 8.]
\ifthenelse{\boolean{ForPrinting}}{%
  \begin{wrapfigure}[17]{i}{2.75in}%
  }{%
  \begin{wrapfigure}[17]{l}{2.75in}%
}
\Input[2.75in]{058}{png}
\Caption[Perpendiculars to the surface of
an oblate body, showing that equal arcs
subtend largest angles at its equator and
smallest at its poles.]{Fig}{9}
\end{wrapfigure}
A false argument is sometimes made which leads to the
opposite conclusion. Lines are drawn from the center of
the oblate body dividing the quadrant into a number of
equal angles. Then it is observed that the arc intercepted
between the two lines nearest the equator is longer than
that intercepted between the two lines nearest the pole.
The error of this argument lies in the fact that, with the
exception of those drawn to the equator and poles, these
lines are not perpendicular to the surface. \Figureref{9} shows
an oblate body with a number of lines drawn perpendicular
%% -----File: 058.png---Folio 33-------
to its surface. Instead of their all passing through the
center of the body, they are tangent to the curve~$AB$. The
line~$AE$ equals the radius
of a circle having the
same curvature as the
oblate body at~$E$, and
$BP$ is the radius of the
circle having the curvature
at~$P$.

\Article{13}{Size and Shape
of the Earth.}---The size
\index{Earth!dimensions of}%
\index{Shape of earth}%
and shape of the earth
can both be determined
from measurements of
arcs. If the earth were
spherical, a degree of arc
would have the same length everywhere on its surface, and
its circumference would be $360$~times the length of one degree.
Since the earth is oblate, the matter is not quite so
simple. But from the lengths of arcs in different latitudes
both the size and the shape of the earth can be computed.

It is sufficiently accurate for ordinary purposes to state
that the diameter of the earth is about $8000$ miles, and that
the difference between the equatorial and polar diameters is
$27$~miles.

The dimensions of the earth have been computed with
great accuracy by Hayford, who found for the equatorial
\index[xnames]{Hayford}%
diameter $7926.57$ miles, and for the polar diameter $7899.98$
miles. The error in these results cannot exceed a thousand
feet. The equatorial circumference is $24,901.7$ miles, and the
length of one degree of longitude at the equator is $69.17$
miles. The lengths of degrees of latitude at the equator
and at the poles are respectively $69.40$ and $68.71$ miles.
The total area of the earth is about $196,400,000$ square miles.
The volume of the earth is equal to the volume of a sphere
whose radius is $3958.9$~miles.
%% -----File: 059.png---Folio 34-------

\Article{14}{Newton's Proof of the Oblateness of the Earth.}---The
\index{Earth!oblateness of}%
\index{Oblateness of earth}%
\index[xnames]{Newton}%
first proof that the earth is oblate was due to Newton.
He based his demonstration on the laws of motion, the law
of gravitation, and the rotation of the earth. It is therefore
much more complicated than that depending on the lengths
of degrees of latitude, which is purely geometrical. It has
the advantage, however, of not requiring any measurements
of arcs.

Suppose the earth, \Figref{10}, rotates around the axis~$PP'$.
Imagine that a tube filled with water exists reaching from
the pole~$P$ to the center~$C$,
and then to the surface
on the equator at~$Q$.
The water in this tube
exerts a pressure toward
the center because of the
attraction of the earth
for it. Consider a unit
volume in the part~$CP$
at any distance~$D$ from
the center; the pressure
it exerts toward the
center equals the earth's
attraction for it because
it is subject to no other
forces. Suppose for the moment that the earth is a sphere,
as it would be if it were not rotating on its axis, and consider
a unit volume in the part~$CQ$ at the distance~$D$ from
the center. Because of the symmetry of the sphere it
will be subject to an attraction equal to that on the corresponding
unit %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{059}{png}
\Caption[Because of the earth's rotation
around~$PP'$ the column~$CQ$ must be
longer than~$PC$.]{Fig}{10}
\end{wrapfigure}
in~$CP$. But, in addition to the earth's attraction,
this mass of water is subject to the centrifugal force
due to the earth's rotation, which to some extent counter-balances
the attraction. Therefore, the pressure it exerts
toward the center is less than that exerted by the corresponding
unit in~$CP$. If the earth were spherical, all units
%% -----File: 060.png---Folio 35-------
in the two columns could be paired in this way. The result
would be that the pressure exerted by~$PC$ would be greater
than that exerted by~$QC$; but such a condition would not
be one of equilibrium, and water would flow out of the
mouth of the tube from the center to the equator. In
order that the two columns of water shall be in equilibrium
the equatorial column must be longer than the polar.

Newton computed the amount~$RQ$ by which the one tube
\index[xnames]{Newton}%
must be longer than the other in order that for a body having
the mass, dimensions, and rate of rotation of the earth,
there should be equilibrium. This gave him the oblateness
of the earth. In spite of the fact that his data were
not very exact, he obtained results which agree very well
with those furnished by modern measurements of arcs.

The objection at once arises that the tubes did not
actually exist and that they could not possibly be constructed,
and therefore that the conclusion was as insecure as those
usually are which rest on imaginary conditions. But the
fears aroused by these objections are dissipated by a little
more consideration of the subject. It is not necessary that
the tubes should run in straight lines from the surface to
the center in order that the principle should apply. They
might bend in any manner and the results would be the same,
just as the level to which the water rises in the spout of a
teakettle does not depend on its shape. Suppose the tubes
are deformed into a single one connecting $P$ and~$Q$ along
the surface of the earth. The principles still hold; but the
ocean connection of pole and equator may be considered as
being a tube. Hence the earth must be oblate or the ocean
would flow from the poles toward the equator.

\Article{15}{Pendulum Proof of the Oblateness of the Earth.}---It
\index{Earth!oblateness of}%
seems strange at first that the shape of the earth can be
determined by means of the pendulum. Evidently the
method cannot rest on such simple geometrical principles as
were sufficient in using the lengths of arcs. It will be found
that it involves the laws of motion and the law of gravitation.
%% -----File: 061.png---Folio 36-------

The time of oscillation of a pendulum depends on the intensity
of the force acting on the bob and on the distance
from the point of support to the bob. It is shown in analytic
mechanics that the formula for a complete oscillation is
\[
t = 2\pi \sqrt{l/g},
\]
where $t$ is the time, $\pi = 3.1416$, $l$ is the length of the pendulum,
and $g$ is the resultant acceleration\footnote
  {Force equals mass times acceleration. On a large pendulum the force of
gravity is greater but the acceleration is the same.}
produced by all
the forces to which the pendulum is subject. If $l$ is determined
by measurement and $t$ is found by observations, the
resultant acceleration is given by
\[
g = \frac{4\pi^2 l}{t^2}.
\]
Consequently, the pendulum furnishes a means of finding
the gravity~$g$ at any place.

In order to treat the problem of determining the shape
of the earth from a knowledge of~$g$ at various places on its
surface, suppose first that it is a homogeneous sphere. If
this were its shape, its attraction would be equal for all points
on its surface. But the gravity~$g$ would not be the same
at all places, because it is the resultant of the earth's attraction
and the centrifugal acceleration due to the earth's
rotation. The gravity~$g$ would be the greatest at the poles,
where there is no centrifugal acceleration, and least at the
equator, where the attraction is exactly opposed by the
centrifugal acceleration. Moreover, the value of~$g$ would
vary from the poles to the equator in a perfectly definite
manner which could easily be determined from theoretical
considerations.

Now suppose the earth is oblate. It can be shown mathematically
that the attraction of an oblate body for a particle
at its pole is greater than that of a sphere of equal volume
and density for a particle on its surface, and that at its
equator the attraction is less. Therefore at the pole, where
%% -----File: 062.png---Folio 37-------
there is no centrifugal acceleration, $g$ is greater on an oblate
body than it is on an equal sphere. On the other hand, at
the equator $g$ is less on the oblate body than on the sphere
both because the attraction of the former is less, and also
because its equator is farther from its axis so that the centrifugal
acceleration is greater. That is, the manner in
which $g$ varies from pole to equator depends upon the oblateness
of the earth, and it can be computed when the oblateness
is given. Conversely, when $g$ has been found by experiment,
the shape of the earth can be computed.

Very extensive determinations of~$g$ by means of the pendulum,
taken in connection with the mathematical theory,
not only prove that the earth is oblate, but give a degree of
flattening agreeing closely with that obtained from the
measurement of arcs.

The question arises why $g$ is determined by means of the
pendulum. Its variations cannot be found by using balance
scales, because the forces on both the body to be weighed and
the counter weights vary in the same proportion. However,
the variations in~$g$ can be determined with some approximation
by employing the spring balance. The choice between
the spring balance and the pendulum is to be settled on the
basis of convenience and accuracy. It is obvious that spring
balances are very convenient, but they are not very accurate.
On the other hand, the pendulum is capable of furnishing
the variation of~$g$ with almost indefinite precision by the
period in which it vibrates. Suppose the pendulum is
moved from one place to another where $g$ differs by one
hundred-thousandth of its value. This small difference could
not be detected by the use of spring balances, however many
times the attempt might be made. It follows from the
formula that the time of a swing of the pendulum would be
changed by about one two-hundred-thousandth of its value.
If the time of a complete oscillation were a second, for example,
the difference could not be detected in a second; but
the deviation for the following second would be equal to
%% -----File: 063.png---Folio 38-------
that in the first, and the difference would be doubled. The
effect would accumulate, second after second, and in a day
of $86,400$ seconds it would amount to nearly one half of a
second, a quantity which is easily measured. In ten days
the difference would amount to about $4.3$~seconds. The
important point in the pendulum method is that the effects
of the quantities to be measured accumulate until they become
observable.

\Article{16}{The Theoretical Shape of the Earth.}---The oblateness
\index{Shape of earth}%
of the earth is not an accident; its shape depends on its
size, mass, distribution of density, and rate of rotation. If
\begin{figure}[hbt]%[Illustration:]
\centering
\begin{minipage}[b]{1.875in}
\Input[1.75in]{063a}{png}
\Caption[Oblate spheroid.]{Fig}{11}
\end{minipage}
\hfil
\begin{minipage}[b]{2.3in}
\Input[2.25in]{063b}{png}
\Caption[Prolate spheroid.]{Fig}{12}
\end{minipage}
\index{Spheroid, oblate and prolate}%
\end{figure}%
it were homogeneous, its shape could be theoretically determined
without great difficulty. It has been found from
mathematical discussions that if a homogeneous fluid body
is slowly rotating it may have either of two forms of equilibrium,
one of which is nearly spherical while the other is
very much flattened like a discus. These figures are not
simply oblate, but they are figures known as spheroids. A
spheroid is a solid generated by the rotation of an ellipse
(\Artref{53}) about one of its diameters. \Figureref{11} is an \textit{oblate}
spheroid generated by the rotation of the ellipse $PQP'Q'$
about its shortest diameter~$PP'$. Its equator is its largest
circumference. \Figureref{12} is a \textit{prolate} spheroid generated
by the rotation of the ellipse $PQP'Q'$ about its longest diameter~$PP'$.
The equator of this figure is its smallest circumference.
The oblate and prolate spheroids are fundamentally
different in shape.
%% -----File: 064.png---Folio 39-------

Of the two oblate spheroids which theory shows are
figures of equilibrium for slow rotation, that which is the
more nearly spherical is stable, while the other is unstable.
That is, if the former were disturbed a little, it would
retake its spheroidal form, while if the latter were deformed
a little, it would take an entirely different shape, or might
even break all to pieces. In spite of the fact that the earth
is neither a fluid nor homogeneous, its shape is almost
exactly that of the more nearly spherical oblate spheroid
corresponding to its density and rate of rotation. This fact
might tempt one to the conclusion that it was formerly in a
fluid state. But this conclusion is not necessarily sound,
because, in such an enormous body, the strains which would
result from appreciable departure from the figure of equilibrium
would be so great that they could not be withstood
by the strongest material known. Besides this, if the conditions
for equilibrium were not exactly satisfied by the
solid parts of the earth, the water and atmosphere would
move and make compensation.

The sun, moon, and planets are bodies whose forms can
likewise be compared with the results furnished by theory.
Their figures agree closely with the theoretical forms. The
only appreciable disagreements are in the case of Jupiter
and Saturn, both of which are more nearly spherical than
\index{Saturn!shape of}%
the corresponding homogeneous bodies would be. The
reason for this is that these planets are very rare in their
outer parts and relatively dense at their centers. It is
probable that they are even more stable than the corresponding
homogeneous figures.

\Article{17}{Different Kinds of Latitude.}---It was seen in \Artref{12}
that perpendiculars to the water-level surface of the
earth, except on the equator and at the poles, do not pass
through the center of the earth. This leads to the definition
of different kinds of latitude.

The geometrically simplest latitude is that defined by a
line from the center of the earth to the point on its surface
%% -----File: 065.png---Folio 40-------
occupied by the observer. Thus, in \Figref{13}, $PC$ is the earth's
axis of rotation, $QC$ is in the plane of its equator, and $O$ is
the position of the observer. The angle~$l$ is called the \textit{geocentric
latitude}.  % [** TN: "latitude" unitalicized in original.]
\index{Latitude!geocentric}%

The observer at~$O$ cannot see the center of the earth and
cannot locate it by any kind of observation made at his
station alone. Consequently, he cannot directly determine~$l$.
All he has is the perpendicular
\begin{wrapfigure}{\WLoc}{2.75in}%[Illustration:]
\Input[2.75in]{065}{png}
\Caption[Geocentric and astronomical
latitudes.]{Fig}{13}
\end{wrapfigure}
to the surface defined
by his plumb line
which strikes the line~$CQ$
at~$A$. The angle~$l_1$ between
this line and $CQ$ is
his \textit{astronomical latitude}.
\index{Latitude!astronomical}%
The difference between
the geocentric and astronomical
latitudes varies
from zero at the poles
and equator to about~$11'$
in latitude~$45°$.

Sometimes the plumb line has an abnormal direction
because of the attractions of neighboring mountains, or
because of local excesses or deficiencies of matter under the
surface. The astronomical latitude, when corrected for these
anomalies, is called the \textit{geographical latitude}. The astronomical
\index{Latitude!geographical}%
and geographical latitudes rarely differ by more than
a few seconds of arc.

\Article{18}{Historical Sketch of Measurements of the Earth.}---While
the earth was generally supposed to be flat down to
the time of Columbus, yet there were several Greek philosophers
\index[xnames]{Columbus}%
who believed that it was a sphere. The earliest philosopher
who is known certainly to have maintained that
the earth is spherical was Pythagoras, author of the famous
\index[xnames]{Pythagoras}%
Pythagorean proposition of geometry, who lived from about
569 to 490~\BC. He was followed in this conclusion, among
others, by Eudoxus (407--356~\BC), by Aristotle (384--322~\BC),
\index[xnames]{Aristotle}%
\index[xnames]{Eudoxus}%
%% -----File: 066.png---Folio 41-------
the most famous philosopher of antiquity if not of all
time, and by Aristarchus of Samos (310--250~\BC). But
\index[xnames]{Aristarchus}%
none of these men seems to have had so clear convictions as
Eratosthenes (275--194~\BC), who not only believed in the
\index[xnames]{Eratosthenes}%
earth's sphericity but undertook to determine its dimensions.
He had noticed that the altitude of the pole star was less
when he was in Egypt than when he was farther north in
Greece, and he correctly interpreted this as meaning that
in traveling northward he journeyed around the curved surface
of the earth. By very crude means he undertook to
measure the length of a degree in Egypt, and in spite of the
fact that he had neither accurate instruments for obtaining
the distances on the surface of the earth, nor telescopes
with which to determine the changes of the direction of
the plumb line with respect to the stars, he secured results
that were not surpassed in accuracy until less than 300
years ago.

After the decline of the Greek civilization and science, no
progress was made in proving the earth is spherical until the
voyage of Columbus in 1492. His ideas regarding the size
\index[xnames]{Columbus}%
of the earth were very erroneous, as is shown by the fact
that he supposed he had reached India by crossing the Atlantic
Ocean. The great explorations and geographical discoveries
that quickly followed the voyages of Columbus convinced
men that the earth is at least globular and gave them
some idea of its dimensions.

There were no serious attempts made to obtain accurate
knowledge of the shape and size of the earth until about the
middle of the seventeenth century. The first results of any
considerable degree of accuracy were obtained in 1671 by
Picard from a measurement of an arc in France.
\index[xnames]{Picard}%

In spite of the fact that Newton proved in 1686 that the
\index[xnames]{Newton}%
earth is oblate, the conclusion was by no means universally
accepted. Imperfections in the measures of the French led
Cassini to maintain until about 1745 that the earth is prolate.
\index[xnames]{Cassini, J.}%
But the French were taking hold of the question in
%% -----File: 067.png---Folio 42-------
earnest and they finally agreed with the conclusion of Newton.
\index[xnames]{Newton}%
They extended the arc that Picard had started from
\index[xnames]{Picard}%
the Pyrenees to Dunkirk, an angular distance of~$9°$. The
results were published in 1720. They sent an expedition to
Peru, on the equator, in 1735, under Bouguer, Condamine,
\index[xnames]{Bouguer}%
\index[xnames]{Condamine}%
and Godin. By 1745 these men had measured an arc of~$3°$.
\index[xnames]{Godin}%
In the meantime an expedition to Lapland, near the Arctic
circle, had measured an arc of~$1°$. On comparing these
measurements it was found that a degree of latitude is
greater the farther it is from the equator.

In the last century all the principal governments of the
world have carried out very extensive and accurate surveys
of their possessions. The English have not only triangulated
the British Isles but they have done an enormous amount of
work in India and Africa. The Coast and Geodetic Survey
in the United States has triangulated with unsurpassed precision
a great part of the country. They have run a level
from the Atlantic Ocean to the Pacific. The names most
often encountered in this connection are Clarke of England,
\index[xnames]{Clarke}%
Helmert of Germany, and Hayford of the United States.
\index[xnames]{Hayford}%
\index[xnames]{Helmert}%
Hayford has taken up an idea first thrown out by the English
in connection with their work in India along the borders
of the Himalaya Mountains, and by using an enormous
amount of observational data and making appalling computations
he has placed it on a firm basis. The observations
in India showed that under the Himalaya Mountains the
earth is not so dense as it is under the plains to the south.
Hayford has proved that the corresponding thing is true in
the United States, even in the case of very moderate elevations
and depressions. Moreover, deficiency in density
under the elevated places is just enough to offset the elevations,
so that the total weight of the material along every
radius from the surface of the earth to its center is almost
exactly the same. This theory is known as the theory of
isostasy, and the earth is said to be in almost perfect isostatic
\index{Isostasy}%
adjustment.
%% -----File: 068.png---Folio 43-------


\Section{II}{QUESTIONS}

1. In order to prove the sphericity of the earth by measurement
of arcs, would it be sufficient to measure only along meridians?
(Consider the anchor ring.)

2. Do the errors in triangulation accumulate with the length of
the distance measured? Do the errors in the astronomical determination
of the angular length of the arc increase with its length?

3. How accurately must a base line of five miles be measured in
order that it may not introduce an error in the determination of the
earth's circumference of more than $1000$~feet?

4. Which of the reasons given in \Artref{11} actually prove, so far
as they go, that the earth is spherical? What other reasons are
there for believing it is spherical?

5. The acceleration~$g$ in mid-latitudes is about $32.2$~feet per
second; how long would a pendulum have to be to swing in $1$, $2$, $3$, $4$
seconds?

6. Draw to scale a meridian section of a figure having the earth's
oblateness.

7. Newton\DPtypo{\,}{'}s proof of the earth's oblateness depends on the
knowledge that the earth rotates; what proofs of it do not depend
upon this knowledge?

8. Suppose time can be measured with an error not exceeding
one tenth of a second; how accurately can $g$ be determined by the
pendulum in $10$~days?

9. Suppose the solid part of the earth were spherical and perfectly
rigid; what would be the distribution of land and water over
the surface?

10. Is the astronomical latitude greater than, or equal to, the
geocentric latitude for all points on the earth's surface?

11. What distance on the earth's surface corresponds to a degree
of arc, a minute of arc, a second of arc?

12. Which of the proofs of the earth's sphericity depend upon
modern discoveries and measurements?

\normalsize


\Section{II}{The Mass of the Earth and the Condition of its Interior}

\Article{19}{The Principle by which Mass is Determined.}---It is
important to understand clearly the principles which are at
the foundation of any subject in which one may be interested,
and this applies in the present problem. The ordinary
method of determining the mass of a body is to weigh it.
%% -----File: 069.png---Folio 44-------
That is the way in which the quantity of most commodities,
such as coal or ice or sugar, is found. The reason a body
has weight at the surface of the earth is that the earth
attracts it. It will be seen later (\Artref{40}) that the body
attracts the earth equally in the opposite direction. Consequently,
the real property of a body by which its mass is
determined is its attraction for some other body. The
underlying principle is that \textit{the mass of a body is proportional
to the attraction which it has for another body}.

Now consider the problem of finding the mass of the
earth, which must be solved by considering its attraction
for some other body. Its attraction for any given mass, for
example, a cubic inch of iron, can easily be measured. But
this does not give the mass of the earth compared to the
cubic inch of iron. It is necessary to compare the attraction
of the earth for the iron with the attraction of some other
fully known body, as a lead ball of given size, for the same
unit of iron. Since the amount of attraction of one body
for another depends upon their distance apart, it is necessary
to measure the distance from the lead ball to the attracted
body, and also to know the distance of the attracted
body from the center of the earth. For this reason the mass
of the earth could not be found until after its dimensions
had been ascertained. By comparing the attractions of the
earth and the lead ball for the attracted body, and making
proper adjustments for the distances of their respective
centers from it, the number of times the earth exceeds the
lead ball in mass can be determined.

Not only is the mass of the earth computed from its attraction,
but the same principle is the basis for determining
the mass of every other celestial body. The masses of
those planets that have satellites are easily found from their
attractions for their respective satellites, and when two
stars revolve around each other in known orbits their masses
are defined by their mutual attractions. There is no means
of determining the mass of a single star.
%% -----File: 070.png---Folio 45-------

\Article{20}{The Mass and Density of the Earth.}---By applications
\index{Density!of earth}%
\index{Earth!density of}%
\index{Earth!mass of}%
(Arts.\ \hyperref[art:21]{21},~\hyperref[art:22]{22}) of the principle in \Artref{19} the mass of
the earth has been found. If it were weighed a small
quantity at a time at the surface, its total weight in tons
would be $6 × 10^{21}$, or $6$~followed by $21$~ciphers. This
makes no appeal to the imagination because the numbers
are so extremely far beyond all experience. A much better
method is to give its density, which is obtained by dividing
its mass by its volume. With water at its greatest
density as a standard, the average density of the earth
is~$5.53$.

The average density of the earth to the depth of a mile
or two is in the neighborhood of~$2.75$. Therefore there are
much denser materials in the earth's interior; their greater
density may be due either to their composition or to the
great pressure to which they are subject. The density of
quartz (sand) is~$2.75$, limestone~$3.2$, cast iron~$7.1$, steel~$7.8$,
lead~$11.3$, mercury~$13.6$, gold~$19.3$, and platinum~$21.5$. It
follows that no considerable part of the earth can be composed
of such heavy substances as mercury, gold, and platinum,
but, so far as these considerations bear on the question,
it might be largely iron.

The distribution of density in the earth was worked out
over $100$ years ago by Laplace on the basis of a certain assumption
\index[xnames]{Laplace}%
regarding the compressibility of the matter of
which it is composed. The results of this computation
have been compared with all the phenomena on which the
disposition of the mass of the earth has an influence, and the
results have been very satisfactory. Hence, it is supposed
that this law represents approximately the way the density
of the earth increases from its surface to its center. According
to this law, taking the density of the surface as~$2.72$,
the densities at depths of $1000$, $2000$, $3000$ miles, and the
center of the earth are respectively $5.62$, $8.30$, $10.19$, $10.87$.
At no depth is the average density so great as that of the
heavier metals.
%% -----File: 071.png---Folio 46-------

\Article{21}{Determination of the Density of the Earth by Means
\index{Density!of earth}%
\index{Earth!density of}%
of the Torsion Balance.}---The whole difficulty in determining
\index{Torsion balance}%
the density of the earth is due to the fact that the
attractions of masses of moderate dimensions are so feeble
that they almost escape detection with the most sensitive
apparatus. The problem from an experimental point of
view reduces to that of devising a means of measuring extremely
minute forces. It has been solved most successfully
by the torsion balance.

The torsion balance consists essentially of two small balls,
$bb$ in \Figref{14}, connected by a rod which is suspended from
\begin{figure}[hbt]%[Illustration:]
\Input{071}{png}
\Caption[The torsion balance.]{Fig}{14}
\end{figure}%
the point~$O$ by a quartz fiber~$OA$. If the apparatus is left
for a considerable time in a sealed case so that it is not disturbed
by air currents, it comes to rest. Suppose the balls~$bb$
are at rest and that the large balls~$BB$ are carefully
brought near them on opposite sides of the connecting rod,
as shown in the figure. They exert slight attractions for the
small balls and gradually move them against the feeble
resistance of the quartz fiber to torsion (twisting) to the
position~$b'b''$. The resistance of the quartz fiber becomes
greater the more it is twisted, and finally exactly balances
the attraction of the large balls. The forces involved are so
small that several hours may be required for the balls to
reach their final positions of rest. But they will finally be
reached and the angle through which the rod has been turned
can be recorded.
%% -----File: 072.png---Folio 47-------

The next problem is to determine from the deflection
which the large balls have produced how great the force is
which they have exerted. This would be a simple matter if
it were known how much resistance the quartz fiber offers
to twisting, but the resistance is so exceedingly small that
it cannot be directly determined. However, it can be found
by a very interesting indirect method.

Suppose the large balls are removed and that the rod
connecting the small balls is twisted a little out of its position
of equilibrium. It will then turn back because of the
resistance offered to twisting by the quartz fiber, and will
rotate past the position of equilibrium almost as far as it
was originally displaced in the opposite direction. Then
it will return and vibrate back and forth until friction destroys
its motion. It is evident that the characteristics of
the oscillations are much like those of a vibrating pendulum.
The formula connecting the various quantities involved is
\[
t = 2\pi\sqrt{l/f},
\]
where $t$ is the time of a complete oscillation of the rod
joining $b$ and~$b$, $l$~is the distance from $A$ to~$b$, and $f$~is the
resistance of torsion. This equation differs from that for
the pendulum, \Artref{15}, only in that $g$ has been replaced by~$f$.
Now $l$~is measured, $t$~is observed, and $f$~is computed from the
equation with great exactness however small it may be.

Now that $f$ and~$g$ are known it is easy to compute the
mass of the earth by means of the law of gravitation (\Artref{146}).
Let $E$~represent the mass of the earth, $R$~its radius,
$2B$~the mass of the two large balls, and $r$~the distances from
$BB$ to $bb$ respectively. Then, since gravitation is proportional
to the attracting mass and inversely as the square of
its distance from the attracted body, it follows that
\[
\frac{E}{R^2} : \frac{2B}{r^2} = g : f.
\]
In this proportion the only unknown is~$E$, which can therefore
be computed.
%% -----File: 073.png---Folio 48-------

\Article{22}{Determination of the Density of the Earth by the
Mountain Method.}---The characteristic of the torsion
\index{Density!of earth}%
\index{Earth!density of}%
\index{Mountain method of determining density of earth}%
balance is that it is very delicate and adapted to measuring
very small forces; the characteristic of the mountain method
is that a very large mass is employed, and the forces are
larger. In the torsion balance the balls~$BB$ are brought
near those suspended by the quartz fiber and are removed
at will. A mountain cannot be moved, and the advantage
of using a large mass is at least partly counterbalanced by
this disadvantage. The necessity for moving the attracting
body (in this case
the mountain) is
obviated in a very
ingenious manner.

\begin{wrapfigure}{\WLoc}{3in}%[Illustration:]
\Input[3in]{073}{png}
\Caption[The mountain method of determining
the mass of the earth.]{Fig}{15}
\end{wrapfigure}

For simplicity let
the oblateness of
the earth be neglected
in explaining
the mountain
method. In \Figref{15},
$C$~is the center
of the earth, $M$~is
the mountain, and
$O_1$ and~$O_2$ are two
stations on opposite
sides of the mountain
at which plumb
lines are suspended.
If it were not for
the attraction of the
mountain they would hang in the directions $O_1C$ and~$O_2C$.
The angle between these lines at~$C$ depends upon the distance
between the stations $O_1$ and~$O_2$. The distance between these
stations, even though they are on opposite sides of the mountain,
can be obtained by triangulation. Then, since the size
of the earth is known, the angle at~$C$ can be computed.
%% -----File: 074.png---Folio 49-------

But the attraction of the mountain for the plumb bobs
causes the plumb lines to hang in the directions $O_1A$ and~$O_2A$.
The directions of these lines with respect to the stars
can easily be determined by observations, and the difference
in their directions as thus determined is the angle at~$A$.

What is desired is the deflections of the plumb line produced
by the attractions of the mountain. It follows from
elementary geometry that the sum of the two small deflections
$CO_1A$ and~$CO_2A$ equals the angle~$A$ minus the angle~$C$.
Suppose, for simplicity, that the mountain is symmetrical
and that the deflections are equal. Then each one
equals one half the difference of the angles $A$~and~$C$. Therefore
the desired quantities have been found.

When the deflection has been found it is easy to obtain
the relation of the force exerted by the mountain to that
due to the earth. Let \Figref{16} represent the
plumb line on a large scale. If it were not
for the %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{1in}
\Input[1in]{074}{png}
\Caption[The
deflection of a
plumb line.]{Fig}{16}
\end{wrapfigure}
mountain it would hang in the direction~$O_1B_1$;
it actually hangs in the direction~$O_1B'_1$.
The earth's attraction is in the direction~$O_1B_1$,
and that of the mountain is in the
direction~$B_1B_1'$. The two forces are in the
same ratio as $O_1B_1$ is to~$B_1B_1'$, for, by the law
of the composition of forces, only then would
the plumb line hang in the direction~$O_1B_1'$.

The problem of finding the mass of the earth compared
to that of the mountain now proceeds just like that of finding
the mass of the earth compared to the balls~$BB$ in the
torsion-balance method. The mountain plays the rôle of
the large balls. A mountain $5000$~feet high and broad
would cause nearly $800$~times as much deflection as that
produced by an iron ball a foot in diameter. The advantage
of the large deflection is offset by not having very accurate
means of measuring it, and also by the fact that it is necessary
to determine the mass of a more or less irregular shaped
mountain made up of materials which may lack much of
%% -----File: 075.png---Folio 50-------
being uniform in density. In spite of these drawbacks this
method was the first one to give fairly accurate results.

\Article{23}{Determination of the Density of the Earth by the
Pendulum Method.}---It was explained in \Artref{15} that the
\index{Density!of earth}%
\index{Earth!density of}%
pendulum furnishes a very accurate means of determining
the force of gravity. Its delicacy arises from the fact that
in using it the effects of the changes in the forces accumulate
indefinitely; no such favorable circumstances were present
in the methods of the torsion balance and the mountain.

Suppose a pendulum has been swung at the surface of the
earth so long that the period of its oscillation has been accurately
determined. Then suppose it is taken at the same
place down into a deep pit or mine. The force to which it
is subject will be changed for three different reasons. (\textit{a})~The
pendulum will be nearer the axis of rotation of the earth and
the centrifugal acceleration to which it is subject will be
diminished. The relative change in gravity due to this
cause can be accurately computed from the latitude of the
position and the depth of the pit or mine. (\textit{b})~The pendulum
will be nearer the center of the earth, and, so far as this
factor alone is concerned, the force to which it is subject
will be increased. Moreover, the relative change due to
this cause also can be computed. (\textit{c})~The pendulum will be
below a certain amount of material whose attraction will
now be in the opposite direction. This cannot be computed
directly because the amount of attraction due to a ton of
matter, for example, is unknown. This is what is to be
found out. But from the time of the oscillation of the pendulum
at the bottom of the pit or mine the whole force to
which it is subject can be computed. Then, on making correction
for the known changes (\textit{a}) and~(\textit{b}), the unknown
change~(\textit{c}) can be obtained simply by subtraction. From
the amount of force exerted by the known mass above the
pendulum, the density of the earth can be computed by
essentially the same process as that employed in the case
of the torsion-balance method and the mountain method.
%% -----File: 076.png---Folio 51-------

\Article{24}{Temperature and Pressure in the Earth's Interior.}---There
\index{Earth!pressure in}%
\index{Earth!temperature in}%
\index{Temperature!of earth}%
are many reasons for believing that the interior of the
earth is very hot. For example, volcanic phenomena prove
that at least in many localities the temperature is above the
melting point of rock at a comparatively short distance
below the earth's surface. Geysers and hot springs show
that the interior of the earth is hot at many other places.
Besides this, the temperature has been found to rise in deep
mines at the rate of about one degree Fahrenheit for a descent
of $100$~feet, the amount depending somewhat on the
locality.

Suppose the temperature should go on increasing at the
rate of one degree for every hundred feet from the surface
to the center of the earth. At a depth of ten miles it would
be over $500$~degrees, at $100$~miles over $5000$ degrees, at
$1000$ miles over $50,000$ degrees, and at the center of the
earth over $200,000$ degrees. While there is no probability
that the rate of increase of temperature which prevails
near the surface keeps up to great depths, yet it is reasonably
certain that at a depth of a few hundred miles it is
several thousand degrees. Since almost every substance
melts at a temperature below $5000$ degrees, it has been
supposed until recent times that the interior of the earth,
below the depth of $100$~miles, is liquid.

But the great pressure to which matter in the interior of
the earth is subject is a factor that cannot safely be neglected.
A cylinder one inch in cross section and $1728$~inches,
or $144$~feet, in height has a volume of one cubic foot.
If it is filled with water, the pressure on the bottom equals
the weight of a cubic foot of water, or $62.5$~pounds. The
pressure per square inch on the bottom of the column $144$~feet
high having the density~$2.75$, or that of the earth's
crust, is $172$~pounds. The pressure per square inch at the
depth of a mile is $6300$ pounds, or $3$~tons in round numbers.
The pressure is approximately proportional to the depth for
a considerable distance. Therefore, the pressure per square
%% -----File: 077.png---Folio 52-------
inch at the depth of $100$~miles is approximately $300$~tons,
and at $1000$ miles it is $3000$ tons. However, the pressure
is not strictly proportional to the depth, and more refined
means must be employed to find how great it is at the earth's
center. Moreover, the pressure at great depths depends
upon the distribution of mass in the earth. On the basis
of the Laplacian law of density, which probably is a good
approximation to the truth, the pressure per square inch at
the center of the earth is $3,000,000$ times the atmospheric
pressure at the earth's surface, or $22,500$~tons.

It is a familiar fact that pressure increases the boiling
points of liquids. It has been found recently by experiment
that pressure increases the melting points of solids. Therefore,
in view of the enormous pressures at moderate depths
in the earth, it is not safe to conclude that its interior is
molten without further evidence. The question cannot be
answered directly because, in the first place, there is no very
exact means of determining the temperature, and, in the
second place, it is not possible to make experiments at such
high pressures. There are, however, several methods of
proving that the earth is solid through and through, and
they will now be considered.

\Article{25}{Proof of the Rigidity and Elasticity of the Earth by
the Tide Experiment.}---Among the several lines of attack
\index{Earth!rigidity of}%
\index{Elasticity of earth}%
\index{Rigidity of earth}%
that have been made on the question of the rigidity of the
earth, the one depending on the tides generated in the earth
by the moon and sun has been most satisfactory; and of the
methods of this class, that devised by Michelson and carried
\index[xnames]{Michelson}%
out in collaboration with Gale, in 1913, has given by far
\index[xnames]{Gale}%
the most exact results. Besides, it has settled one very
important question, which no other method has been able
to answer, namely, that the earth is highly elastic instead of
being viscous. For these reasons the work of Michelson
and Gale will be treated first.

The important difference between a solid and a liquid is
that the former offers resistance to deforming forces while
%% -----File: 078.png---Folio 53-------
the latter does not. If a perfect solid existed, no force whatever
could deform it; if a perfect liquid existed, the only resistance
it would offer to deformation would be the inertia
of the parts moved. Neither perfect solids nor absolutely
perfect liquids are known. If a solid body has the property
of being deformed more and more by a continually applied
force, and if, on the application of the force being discontinued,
the body not only does not retake its original form
but does not even tend toward it, then it is said to be \textit{viscous}.
Putty is a good example of a material that is viscous. On
the other hand, if on the application of a continuous force
the body is deformed to a certain extent beyond which it
does not go, and if, on the removal of the force, it returns
absolutely to its original condition, it is said to be \textit{elastic}.
While there are no solid bodies which are either perfectly
viscous or perfectly elastic, the distinction is a clear and
important one, and the characteristics of a solid may be
described by stating how far it approaches one or the other
of these ideal states.

In order to find how the earth is deformed by forces it is
necessary to consider what forces there are acting on it.
The most obvious ones are the attractions of the sun and
moon. But it is not clear in the first place that these attractions
tend to deform the earth, and in the second place
that, even if they have such a tendency, the result is at
all appreciable. A ball of iron attracted by a magnet is not
sensibly deformed, and it seems that the earth should behave
similarly. But the earth is so large that one's intuitions
utterly fail in such considerations. The sun and
moon actually do tend to alter the shape of the earth, and
the amount of its deformation due to their attractions is
measurable. The forces are precisely those that produce
the tides in the ocean.

It will be sufficient at present to give a rough idea, correct
so far as it goes, of the reason that the moon and sun
raise tides in the earth, reserving for Arts.\ \hyperref[art:263]{263},~\hyperref[art:264]{264} a more
%% -----File: 079.png---Folio 54-------
complete treatment of the question. In \Figref{17} let $E$~represent
the center of the earth, the arrow the direction toward
the moon, and $A$~and~$B$ the points where the line from $E$ to
the moon pierces the earth's surface. The moon is $4000$
miles nearer to~$A$ than it is to~$E$, and $4000$ miles nearer to~$E$
than it is to~$B$. Therefore the attraction of the moon for
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{079}{png}
\Caption[The tidal bulges at $A$~and~$B$ on the earth produced by
the moon.]{Fig}{17}
\index{Tidal!bulges}%
\index{Tide-raising!acceleration}%
\end{figure}%
a unit mass at~$A$ is greater than it is for a unit mass at~$E$,
and greater for a unit mass at~$E$ than it is for one at~$B$.
Since the distance from the earth to the moon is $240,000$
miles, the distance of the moon from~$A$ is fifty-nine sixtieths
of its distance from~$E$. Since the attraction varies inversely
as the square of the distance, the force on~$A$ is about one
thirtieth greater than that on~$E$, and the difference between
the forces on $E$~and~$B$ is only slightly less.

It follows from the relation of the attraction of the moon
for masses at $A$,~$E$, and~$B$ that it tends to pull the nearer
material at~$A$ away from the center of the earth~$E$, and the
center of the earth away from the more remote material at~$B$.
Since the forces are known, it is possible to compute the
elongation the earth would suffer if it were a perfect fluid.
The result is two elevations, or tidal bulges, at $A$~and~$B$.
%% -----File: 080.png---Folio 55-------
The concentric lines shown in \Figref{17} are the lines of equal
elevation. A rather difficult mathematical discussion shows
that the radii $EA$~and~$EB$ would each be lengthened by
about four feet. Since the earth possesses at least some
degree of rigidity its actual tidal elongation is somewhat less
than four feet. When it is remembered that the uncertainty
in the diameter of the earth, in spite of the many years that
have been devoted to determining it, is still several hundred
feet, the problem of finding how much the earth's elongation,
as a consequence of the rapidly changing tidal forces,
falls short of four feet seems altogether hopeless of solution.
Nevertheless the problem has been solved.

Suppose a pipe half filled with water is fastened in a horizontal
position to the surface of the earth. The water in the
pipe is subject to the attraction of the moon. To fix the
ideas, suppose the pipe lies in the east-and-west direction
in the same latitude as the point~$A$, \Figref{17}. Suppose, first,
that the earth is absolutely rigid so that it is not deformed
by the moon, and consider what happens to the water in the
pipe as the rotation of the earth carries it past the point~$A$.
When the pipe is to the west of~$A$ the water rises in its
eastern end, and settles correspondingly in its western end,
because the moon tends to make an elevation on the earth
at~$A$. When the pipe is carried past~$A$ to the east the water
rises in its western end and settles in its eastern end. Since
the earth is not absolutely rigid the magnitudes of the tides
under the hypothesis that it is rigid cannot be experimentally
determined; but, since all the forces that are involved
are known, the heights the tides would be on a rigid earth
can be computed.

Suppose now that the earth yields perfectly to the disturbing
forces of the moon. Its surface is in this case always
the exact figure of equilibrium. Consider the pipe, which
is attached to this surface, when it is to the west of~$A$. The
water would be high in its eastern end if the shape of the
surface of the earth were unchanged. But the surface to
%% -----File: 081.png---Folio 56-------
the east of it is elevated and the pipe is raised with it. Moreover,
the elevation of the surface is, under the present
hypothesis, just that necessary for equilibrium. Therefore,
in this case there is no tide at all with respect to the pipe.

The actual earth is neither absolutely rigid nor perfectly
fluid. Consequently the tides in the pipe will actually be
neither their theoretical maximum nor zero. The amount
by which they fall short of the value they would have if the
earth were perfectly rigid depends upon the extent to which
it yields to the moon's forces, and is a measure of this yielding.
Therefore the problem of finding how much the earth
is deformed by the moon is reduced to computing how great
the tides in the pipe would be if the earth were absolutely
rigid, and then comparing these results with the actual tides
in the pipe as determined by direct experiment. After the
amount the earth yields has been determined in this way,
its rigidity can be found by the theory of the deformation
of solid bodies.

In the experiment of Michelson and Gale two pipes were
\index[xnames]{Gale}%
\index[xnames]{Michelson}%
\index{Tidal!experiments}%
used, one lying in the plane of the meridian and the other in
the east-and-west direction. In order to secure freedom
from vibrations due to trains and heavy wagons they were
placed on the grounds of the Yerkes Observatory, and to
avoid variations in temperature they were buried a number
of feet in the ground. Since the tidal forces are very small,
pipes $500$~feet long were used, and even then the maximum
tides were only about two thousandths of an inch.

An ingenious method of measuring these small changes in
level was devised. The ends of the pipes were sealed with
plane glass windows through which their interiors could be
viewed. Sharp pointers, fastened to the pipe, were placed
just under the surface of the water near the windows. When
viewed from below the level of the water the pointer and its
reflected image could be seen. \Figureref{18} shows an end of
one of the pipes, $S$~is the surface of the water, $P$~is the pointer,
and $P'$~is its reflected image. The distances of $P$~and~$P'$
%% -----File: 082.png---Folio 57-------
from the surface~$S$ are equal. Now suppose the water rises;
since $P$ and $P'$ are equidistant from~$S$, the change in their
apparent distance is twice the change
in the water level. The distances
between $P$ and $P'$ were accurately
measured with the help of permanently
fixed microscopes, and the
variations in the water level were
determined within one per cent of
their whole amount.

\begin{wrapfigure}{\WLoc}{1.75in}%[Illustration:]
\Input[1.75in]{082}{png}
\Caption[End of pipe in
the Michelson-Gale tide
experiment.]{Fig}{18}
\end{wrapfigure}

In order to make clear the accuracy
of the results, the complicated nature
of the tides must be pointed out.
Consider the tidal bulges $A$ and~$B$, \Figref{17}, which give an idea
of what happened to the water in the pipes. For simplicity,
fix the attention on the east-and-west pipe, which in the experiment
was about $13°$~north of the highest latitude~$A$ ever
attains. The rotating earth carried it daily across the meridian
of~$A$ to the north of~$A$, and similarly across the meridian
of~$B$. When the relations were as represented in the diagram
there were considerable tides in the pipe before and
after it crossed the meridian at~$A$ because it was, so to speak,
well on the tidal bulge. On the other hand, when it crossed
the meridian of~$B$ about $12$~hours later, the tides were very
small because the bulge~$B$ was far south of the equator.
But the moon was not all the time north of the plane of the
earth's equator. Once each month it was $28°$~north and
once each month $28°$~south, and it varied from hour to hour
in a rather irregular manner. Moreover, its distance, on
which the magnitudes of the tidal forces depend, also changed
continuously. Then add to all these complexities the corresponding
ones due to the sun, which are unrelated to those
of the moon, and which mix up with them and make the
phenomena still more involved. Finally, consider the north-and-south
pipe and notice, by the help of \Figref{17}, that its
tides are altogether distinct in character from those in the
%% -----File: 083.png---Folio 58-------
east-and-west pipe. With all this in mind, remember that
the observations made every two hours of the day for a
period of several months agreed perfectly in all their characteristics
with the results given by theory. The only difference
was that the observed tides were reduced in a constant
ratio by the yielding of the earth.

The perfection of this domain of science is proved by the
satisfactory coördination in this experiment of a great many
distinct theories. The perfect agreement in their characteristics
of more than a thousand observed tides with their
computed values depended on the correctness of the laws of
motion, the truth of the law of gravitation, the size of the
earth, the distance of the moon and the theory of its motion,
the mass of the moon, the distance to the sun and the theory
of the earth's motion around it, the mass of the sun, the
theory of tides, the numerous observations, and the lengthy
calculations. How improbable that there would be perfect
harmony between observation and theory in so many cases
unless scientific conclusions respecting all these things are
correct!

The extent to which the earth yields to the forces of the
moon was obtained from the amount by which the observed
tides were less than their theoretical values for an unyielding
earth. It was found that in the east-and-west pipe the observed
tides were about $70$~per~cent of the computed, while
in the north-and-south pipe the observed tides were only
about $50$~per~cent of the computed. This led to the astonishing
conclusion, which, however, had been reached earlier by
Schweydar on the basis of much less certain observational
\index[xnames]{Schweydar}%
data, that the earth's resistance to deformation in the east-and-west
direction is greater than it is in the north-and-south
direction. Love has suggested that the difference may be
\index[xnames]{Love}%
due indirectly to the effects of the oceanic tides on the general
body of the earth.

On using the amount of the yielding of the earth established
by observations and the magnitude of the forces exerted
%% -----File: 084.png---Folio 59-------
by the moon and sun, it was found by the mathematical
processes which are necessary in treating such problems,
that the earth, taken as a whole, is as rigid as steel. That
\index{Earth!elasticity of}%
\index{Elasticity of earth}%
is, it resists deformation as much as it would if it were made
of solid steel having throughout the properties of ordinary
good steel.

The work of Michelson and Gale for the first time gave a
\index[xnames]{Gale}%
\index[xnames]{Michelson}%
reliable answer to the question whether the earth is viscous
or elastic. It had almost invariably been supposed that
the earth is viscous, because it was thought that even if
the enormous pressure keeps the highly heated material of
its interior in a solid state, yet it would be only stiff like
a solid is when its temperature approaches the melting point.
In fact, Sir George Darwin had built up an elaborate theory
\index[xnames]{Darwin, George H.}%
of tidal evolution (Arts.\ \hyperref[art:265]{265},~\hyperref[art:266]{266}), at the cost of a number
of years of work, on the hypothesis that the earth is viscous.
But the experiments of Michelson and Gale prove that it is
very elastic.

If the earth were viscous, it would yield somewhat slowly
to the forces of the moon and sun. Consequently, the tilting
of the surface, which carries the pipes, would lag behind the
forces which caused both the tilting and the tides in the
pipes. There is no appreciable lag of a water tide in the
pipe only $500$~feet long, and consequently the observed and
computed tides would not agree in phase. On the other
hand, if the earth were elastic, there would be agreement in
phase between the observed and computed tides. It is more
difficult practically to determine accurately the phase of the
tides than it is to measure their magnitudes, but the observations
showed that there is no appreciable difference in the
phases of the observed and computed tides. These results
force the conclusion that the elasticity of the earth, taken as
a whole, cannot be less than that of steel,---a result obviously
of great interest to geologists.

\Article{26}{Other Proofs of the Earth's Rigidity.}---(\textit{a})~There is
\index{Earth!rigidity of}%
\index{Rigidity of earth}%
a method of finding how much the earth yields to the forces
%% -----File: 085.png---Folio 60-------
of the moon and sun which is fundamentally equivalent to
that of measuring tides in a pipe. It depends on the fact
that the position of a pendulum depends upon all the forces
\index{Pendulum!horizontal}%
acting on it, and, if the earth were in equilibrium, the line
of its direction would always be perpendicular to the water-level
surface. Consequently, if the earth yielded perfectly
to the forces of the moon and sun, a pendulum would constantly
remain perpendicular to its water-level surface.
But if the earth did not yield perfectly, the pendulum would
undergo very minute oscillations with respect to the solid
part analogous to those of the water in the pipes. A modification
of the ordinary pendulum, known as the horizontal
pendulum, was found to be sensitive enough to show the
oscillations, giving the rigidity of the earth but no satisfactory
evidence regarding its elasticity.

\phantomsection\label{subart:26b}%
(\textit{b})~The principles at the basis of the method of employing
tides in pipes apply equally well to tides in the ocean.
Longer columns of water are available in this case, but there
is difficulty in obtaining the exact heights of the actual tides,
and very much greater difficulty in determining their theoretical
heights on a shelving and irregular coast where they
would necessarily be observed. In fact, it has not yet been
found possible to predict in advance with any considerable
degree of accuracy the height of tides where they have not
been observed. Yet, Lord Kelvin with rare judgment inferred
\index[xnames]{Kelvin}%
on this basis that the earth is very rigid.

(\textit{c})~Earthquakes are waves in the earth which start from
\index{Earthquakes}%
some restricted region and spread all over the earth, diminishing
in intensity as they proceed. Modern instruments,
depending primarily on some adaptation of the horizontal
pendulum, can detect important earthquakes to a distance
of thousands of miles from their origin. Earthquake waves
are of different types; some proceed through the surface
rocks around the earth in undulations like the waves in the
ocean, while others, compressional in character like waves of
sound in the air, radiate in straight lines from their sources.
%% -----File: 086.png---Folio 61-------

The speed of a wave depends upon the density and the
rigidity of the medium through which it travels. This principle
applies to earthquake waves, and when tested on those
which travel in undulations through the surface rocks there
is good agreement between observation and theory. Consider
its application to the compressional waves that go
through the earth. The time required for them to go from
the place of their origin to the place where they are observed
is given by the observations. The density of the earth is
known. If its rigidity were known, the time could be computed;
but the time being known, the rigidity can be computed.
While the results are subject to some uncertainties,
they agree with those found by other methods.

(\textit{d})~The attraction of the moon for the equatorial bulge
slowly changes the plane of the earth's equator (\Artref{47}).
The magnitude of the force that causes this change is known.
If the earth consisted of a crust not more than a few hundred
miles deep floating on a liquid interior, the forces would
cause the crust to slip on the liquid core, just as a vessel containing
water can be rotated without rotating the water. If
the crust of the earth alone were moved, it would be shifted
rapidly because the mass moved would not be great. But
the rate at which the plane of the earth's equator is moved,
as given by the observations, taken together with the forces
involved, proves that the whole earth moves. When the
effects of forces acting on such an enormous body are considered,
it is found that this fact means that the earth has a
considerable degree of rigidity.

(\textit{e})~Every one knows that a top may be spun so that its
axis remains stationary in a vertical direction, or so that it
wabbles. Similarly, a body rotating freely in space may
rotate steadily around a fixed axis, or its axis of rotation
may wabble. The period of the wabbling depends upon the
size, shape, mass, rate of rotation, and rigidity of the body.
In the case of the earth all these factors except the last may
be regarded as known. If it were known, the rate of wabbling
%% -----File: 087.png---Folio 62-------
could be computed; or, if the rate of wabbling were
found from observation, the rigidity could be computed.
It has recently been found that the earth's axis of rotation
wabbles slightly, and the rate of this motion proves that the
rigidity of the earth is about that of steel.
\index{Latitude!variation of}% [** TN: Moving up one page; see Art. 46]
\index{Variation!of latitude}%

\Article{27}{Historical Sketch on the Mass and Rigidity of the
Earth.}---The history of correct methods of attempting to
find the mass of the earth necessarily starts with Newton,
\index[xnames]{Newton}%
because the ideas respecting mass were not clearly formulated
before his time, and because the determination of mass
depends on the law of gravitation which he discovered. By
some general but inconclusive reasoning he arrived at the
conjecture that the earth is five or six times as dense as
water.

The first scientific attempt to determine the density of
the earth was made by Maskelyne, who used the mountain
\index[xnames]{Maskelyne}%
method, in 1774, in Scotland. He found $4.5$ for the density
of the earth. The torsion balance, devised by Michell, was
\index[xnames]{Michell}% [** TN: Mitchell in original]
first employed by Cavendish, in England, in 1798. His
result agreed closely with those obtained by later experimenters,
among whom may be mentioned Baily (1840) in
\index[xnames]{Baily}%
England, and Reich (1842) in Germany, Cornu (1872) in
\index[xnames]{Cornu}%
\index[xnames]{Reich}%
France, Wilsing (1887) in Germany, Boys (1893) in England,
\index[xnames]{Boys}%
\index[xnames]{Wilsing}%
and Braun (1897) in Austria. The pendulum method, using
\index[xnames]{Braun}%
either a mountain or a mine to secure difference in elevation,
has been employed a number of times.

Lord Kelvin (then Sir William Thomson) first gave in
\index[xnames]{Kelvin}%
1863 good reasons for believing the earth is rigid. His conclusion
was based on the height of the oceanic tides, as outlined
in \hyperref[subart:26b]{Art.~26~(\textit{b})}. The proof by means of the rate of
transmission of earthquake waves owes its possibility largely
to John Milne, an Englishman who long lived in Japan,
\index[xnames]{Milne}%
which is frequently disturbed by earthquakes. His interest
in the character of earthquakes stimulated him to the invention
of instruments, known as seismographs, for detecting
\index{Seismograph}%
and recording faint earth tremors. The change of the position
%% -----File: 088.png---Folio 63-------
of the plane of the earth's equator, known as the precession
of the equinoxes, has been known observationally
ever since the days of the ancient Greeks, and its cause was
understood by Newton, but it has not been used to prove
\index[xnames]{Newton}%
the rigidity of the earth because it takes place very slowly.
The wabbling of the axis of the earth was first established
observationally, in 1888, by Chandler of Cambridge, Mass.,
\index[xnames]{Chandler}%
and Küstner of Berlin. The theoretical applications of the
\index[xnames]{Kustner@{Küstner}}%
rigidity of the earth were made first by Newcomb of Washington,
\index[xnames]{Newcomb}%
and then more completely by S.~S. Hough of England.
\index[xnames]{Hough, S. S.}%
The first attempt at the determination of the rigidity
of the earth by the amount it yields to the tidal forces of the
moon and sun was made unsuccessfully in 1879 by George
and Horace Darwin, in England. Notable success has been
\index[xnames]{Darwin, George H.}%
\index[xnames]{Darwin, Horace}%
achieved only in the last $15$~years, and that by improvements
in the horizontal pendulum and by taking great care in
\index{Pendulum!horizontal}%
keeping the instruments from being disturbed. The names
that stand out are von Rebeur-Paschwitz, Ehlert, Kortozzi,
\index[xnames]{Ehlert}%
\index[xnames]{Kortozzi}%
\index[xnames]{Rebeur-Paschwitz}%
Schweydar, Hecker, and Orloff. The observations of
\index[xnames]{Hecker}%
\index[xnames]{Orloff}%
\index[xnames]{Schweydar}%
Hecker at Potsdam, Germany, were especially good, and
Schweydar made two exhaustive mathematical discussions
of the subject.


\Section{III}{QUESTIONS}

1. What is the difference between mass and weight? Does the
weight of a body depend on its position? Does the inertia of a
body depend on its position?

2. Can the mass of a small body be determined from its inertia?
Can the mass of the earth be determined in the same way?

3. What is the average weight of a cubic mile of the earth?

4. Discuss the relative advantages of the torsion-balance method
and mountain method in determining the density of the earth.
Which one has the greater advantages?

5. What is the pressure at the bottom of an ocean six miles
deep?

6. Discuss the character of the tides in east-and-west and north-and-south
pipes during a whole day when the moon is in the position
indicated in \Figref{17}, and when it is over the earth's equator.
%% -----File: 089.png---Folio 64-------

7. What are the advantages and disadvantages of a long pipe
in the tide experiment?

8. If a body is at~$A$, \Figref{17}, is its weight greater or less than
normal as determined by spring balances? By balance scales?
What are the facts, if it is at~$B$?

9. Enumerate the scientific theories and facts involved in the
tide experiment.

10. List the principles on which the several proofs of the earth's
rigidity depend. How many fundamentally different methods are
there of determining its rigidity?

\normalsize


\Section{III}{The Earth's Atmosphere}
\index{Atmosphere}%
\index{Atmosphere!composition of}%

\Article{28}{Composition and Mass of the Earth's Atmosphere.}---The
atmosphere is the gaseous envelope which surrounds
the earth. Its chief constituents are the elements nitrogen
\index{Nitrogen}%
and oxygen, but there are also minute quantities of argon,
\index{Oxygen}%
helium, neon, krypton, \DPtypo{xeon}{xenon}, and some other very rare constituents.
\index{Xenon}%
When measured by volume at the earth's surface,
$78$~per cent of the atmosphere is nitrogen, $21$~per cent
is oxygen, $0.94$~per cent is argon, and the remaining elements
occur in much smaller quantities.

Nitrogen, oxygen, etc., are elements; that is, they are
substances which are not broken up into more fundamental
units by any physical or any chemical changes. The thousands
of different materials that are found on the earth are
all made up of about $90$~elements, only about half of which
are of very frequent occurrence. The union of elements
into a chemical compound is a very fundamental matter, for
the compound may have properties very unlike those of any
of the elements of which it is composed. For example,
hydrogen, carbon, and nitrogen are in almost all food, but
hydrocyanic acid, which is composed of these elements alone,
\index{Hydrocyanic acid}%
is a deadly poison.

Besides the elements which have been enumerated, the
atmosphere contains some carbon dioxide, which is a compound
\index{Carbon dioxide}%
of carbon and oxygen, and water vapor, which is
a compound of oxygen and hydrogen. In volume three
%% -----File: 090.png---Folio 65-------
hundredths of one per cent of the earth's atmosphere is
carbon dioxide; but this compound is heavier than nitrogen
and oxygen, and by weight, $0.05$~per~cent of the atmosphere
is carbon dioxide. The amount of water vapor in the air
varies greatly with the position on the earth's surface and
with the time. There are also small quantities of dust, soot,
ammonia, and many other things which occur in variable
quantities and which are considered as impurities.

The pressure of the atmosphere at sea level is about $15$~pounds
per square inch and its density is about one eight-hundredth
that of water. This means that the weight of a
column of air reaching from the earth's surface to the limits
of the atmosphere and having a cross section of one square
inch weighs $15$~pounds. The total mass of the atmosphere
\index{Atmosphere!mass of}%
\index{Atmosphere!pressure of}%
\index{Mass!of atmosphere}%
can be obtained by multiplying the weight of one column
by the total area of the earth. In this way it is found
that the mass of the earth's atmosphere is nearly
$6,000,000,000,000,000$ tons, or approximately one millionth
the mass of the solid earth. The total mass of even the
carbon dioxide of the earth's atmosphere is approximately
$3,000,000,000,000$ tons.

\Article{29}{Determination of Height of Earth's Atmosphere from
Observations of Meteors.}---Meteors, or shooting stars as
\index{Meteors}%
\index{Meteors!height of}%
\index{Shooting stars}%
they are commonly called, are minute bodies, circulating in
interplanetary space, which become visible only when they
penetrate the earth's atmosphere and are made incandescent
by the resistance which they encounter. The great heat
developed is a consequence of their high velocities, which
ordinarily are in the neighborhood of $25$~miles per second.

Let $m$, \Figref{19}, represent the path of a meteor before it
encounters the atmosphere at~$A$. Until it reaches~$A$ it is
invisible, but at~$A$ it begins to glow and continues luminous
until it is entirely burned up at~$B$. Suppose it is observed
from the two stations $O_1$~and~$O_2$ which are at a known distance
apart. The observations at~$O_1$ give the angle~$AO_1O_2$,
and those at~$O_2$ give the angle~$AO_2O_1$. From these data the
%% -----File: 091.png---Folio 66-------
other parts of the triangle can be computed (compare \Artref{10}).
After the distance~$O_1A$ has been computed the perpendicular
height %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3.4in}
\Input[3.4in]{091}{png}
\Caption[Determination of the height of meteors.]{Fig}{19}
\end{wrapfigure}
of~$A$ from the surface
of the earth
can be computed
by using the angle~$AO_1O_2$.
Similarly,
the height of~$B$
above the surface
of the earth can
be determined.

Observations of meteors from two stations show that they
ordinarily become visible at a height of from~$60$ to $100$~miles.
Therefore, the atmosphere is sufficiently dense to
\index{Atmosphere!height of}%
a height of about $100$~miles to offer sensible resistance to
meteors. Meteors usually disappear by the time they have
descended to within thirty or forty miles of the earth's
surface.

\Article{30}{Determination of Height of Earth's Atmosphere from
Observations of Auroræ.}---Auroræ are almost certainly
\index{Aurorae@{Auroræ}}%
electrical phenomena of the very rare upper atmosphere,
though their nature is not yet very well understood. Their
altitude can be computed from simultaneous observations
made at different stations. The method is the same as that
in obtaining the height of a meteor.

The southern ends of auroral streamers are usually more
than $100$ miles in height, and they are sometimes found at
an altitude of~$500$ or $600$~miles. Their northern ends are
much lower. This means that the density required to make
meteors incandescent is considerably greater than that which
is sufficient for auroral phenomena.

\Article{31}{Determination of Height of Earth's Atmosphere from
the Duration of Twilight.}---Often after sunset, even to the
east of the observer, high clouds are brilliantly illuminated
by the rays of the sun which still fall on them. The higher
%% -----File: 092.png---Folio 67-------
the clouds are, the longer they are illuminated. Similarly,
the sun shines on the upper atmosphere for a considerable
time after it has set or before it rises, and gives the twilight.
The duration of twilight depends upon the height of the
atmosphere. While it is difficult to determine the instant
at which the twilight ceases to be visible, observations show
that under favorable weather conditions it does not disappear
until the sun is $18$~degrees below the horizon.

In order to see how the height of the atmosphere can be
determined from the duration of the twilight, consider \Figref{20}.
The sun's rays
\begin{wrapfigure}{\WLoc}{3.25in}%[Illustration:]
\Input[3.25in]{092}{png}
\Caption[Determination of the height of the
atmosphere from the duration of twilight.]{Fig}{20}
\index{Twilight, duration of}%
\end{wrapfigure}
come in from the
left in lines that
are sensibly parallel.
The observer
at~$O$ can
see the illuminated
atmosphere
at~$P$; but if the
atmosphere were
much shallower,
it would not be
visible to him. The region~$P$ is midway between~$O$ and the
sunset point. Since~$O$ is $18$~degrees from the sunset point, it
is possible to compute the height of the plane of the horizon
at~$P$ above the surface of the earth. It is found that $18$~degrees
corresponds to an altitude of $50$~miles. That is, the
atmosphere extends to a height of $50$~miles above the earth's
surface in quantities sufficient to produce twilight.

The results obtained by the various methods for determining
the height of the atmosphere disagree because its density
decreases with altitude, as is found by ascending in balloons,
and different densities are required to produce the different
phenomena. It will convey the correct idea for most applications
to state that the atmosphere does not extend in appreciable
quantities beyond $100$~miles above the earth's surface.
%% -----File: 093.png---Folio 68-------
At this altitude its density is of the order of one four-millionth
of that at the surface. When the whole earth is
considered it is found that the atmosphere forms a relatively
thin layer. If the earth is represented by a globe $8$~inches
in diameter, the thickness of the atmosphere on the same
scale is only about one tenth of an inch.

\Article{32}{The Kinetic Theory of Gases.}---It has been stated
\index{Gases!kinetic theory of}%
\index{Kinetic theory of gases}%
that every known substance on the earth is composed of
about~$90$ fundamental elements. A chemical combination
of atoms is called a molecule. A molecule of oxygen consists
\index{Atoms}%
\index{Molecules}%
of two atoms of oxygen, a molecule of water of two
atoms of hydrogen and one of oxygen, and similarly for all
substances. Some molecules contain only a few atoms and
others a great many; for example, a molecule of cane sugar
is composed of $12$~atoms of carbon, $22$~of hydrogen, and $11$~of
oxygen. As a rule the compounds developed in connection
with the life processes contain many atoms.

The molecules are all very minute, though their dimensions
\index{Molecules!size of}%
doubtless vary with the number and kind of atoms
they contain. Lord Kelvin devised a number of methods
\index[xnames]{Kelvin}%
of determining their size, or at least the distances between
their centers. In water, for example, there are in round
numbers $500,000,000$ in a line of them one inch long, or the
cube of this number in a cubic inch.

In solids the molecules are constrained to keep essentially
the same relations to one another, though they are capable
of making complicated small vibrations. In liquids the
molecules continually suffer restraints from neighboring
molecules, but their relative positions are not fixed and they
move around among one another, though not with perfect
freedom. In gases the molecules are perfectly free from one
another except when they collide. They move with great
speed and collide with extraordinary frequency; but, in spite
of the frequency of the collisions, the time during which
they are uninfluenced by their neighbors is very much greater
than that in which they are in effective contact.
%% -----File: 094.png---Folio 69-------

The pressure exerted by a gas is due to the impact of its
molecules on the walls of the retaining vessel. To make the
ideas definite, consider a cubic foot of atmosphere at sea-level
pressure. Its weight is about one and one fourth ounces,
but it exerts a pressure of $15$~pounds on each square inch of
each of its six surfaces, or a total pressure on the surface of
the cube of more than six tons. This implies that the molecules
move with enormous speed. They do not all move
with the same speed, but some travel slowly while others go
much faster than the average. Theoretically, at least, in
every gas there are molecules moving with every velocity,
however great, but the number of those having any given
velocity diminishes rapidly as its difference from the average
velocity increases. The average velocity of molecules
in common air at ordinary temperature and pressure is more
than $1600$~feet per second, and on the average each molecule
\index{Molecules!velocity of}%
\index{Velocity!of molecules}%
has $5,000,000,000$ collisions per second. Therefore the
average distance traveled between collisions is only about
$\frac{1}{250000}$~of an inch.

From the kinetic theory of gases it is possible to determine
\index{Gases!pressure of}%
how fast the density of the air diminishes with increase
of altitude. It is found that about one half of the earth's
atmosphere is within the first $3.5$~miles of its surface, that
one half of the remainder is contained in the next $3.5$~miles,
and so on until it is so rare that the kinetic theory no longer
applies without sensible modifications.

\Article{33}{The Escape of Atmospheres.}---Suppose a body is
\index{Escape of atmosphere}%
\index{Velocity!of escape}%
projected upward from the surface of the earth. The height
to which it rises depends upon the speed with which it is
started. The greater the initial speed, the higher it will rise,
and there is a certain definite initial velocity for which, neglecting
the resistance of the air, it will leave the earth and
never return. This is the velocity of escape, and for the
earth it is a little less than $7$~miles per second.

The molecules in the earth's atmosphere may be considered
as projectiles which dart in every direction. It has
%% -----File: 095.png---Folio 70-------
been seen that there is a small fraction of them which
move with a velocity as great as $7$~miles per second. Half of
these will move toward points in the sky and consequently
would escape from the earth if they did not encounter other
molecules. But in view of the great frequency of collisions
of molecules, it is evident that only a very small fraction of
those which move with high velocities can escape from the
earth. However, it seems certain that some molecules will
be lost in this way, and, so far as this factor is concerned,
the earth's atmosphere is being continually depleted. The
process is much more rapid in the case of bodies, such as
the moon, for example, whose masses and attractions are
much smaller, and for which, therefore, the velocity of
escape is lower.

It should not be inferred from this that the earth's atmosphere
is diminishing in amount even if possible replenishment
from the rocks and its interior is neglected. When a
molecule escapes from the earth it is still subject to the attraction
of the sun and goes around it in an orbit which crosses
that of the earth. Therefore the earth has a chance of
acquiring the molecule again by collision. The only exception
to this statement is when the molecule escapes with a
velocity so high that the sun's attraction cannot control it.
The velocity necessary in order that the molecule shall
escape both the earth and the sun depends upon its direction
of motion, but averages about $25$~miles per second and cannot
be less than $19$~miles per second. But besides the molecules
that have escaped from the earth there are doubtless many
others revolving around the sun near the orbit of the earth.
These also can be acquired by collision. The earth is so
old and there has been so much time for losing and acquiring
an atmosphere, molecule by molecule, that probably an
equilibrium has been reached in which the number of molecules
lost equals the number gained. The situation is
analogous to a large vessel of water placed in a sealed
room. The water evaporates until the air above it becomes
%% -----File: 096.png---Folio 71-------
so nearly saturated that the vessel acquires as many molecules
of water vapor by collisions as it loses by evaporation.

The doctrine of the escape of atmospheres implies that
bodies of small mass will have limited and perhaps inappreciable
atmospheres, and that those of large mass will have
extensive atmospheres. The implications of the theory are
exactly verified in experience. For example, the moon, with
\index{Mass!of moon}%
\index{Moon!mass of}%
a mass $\frac{1}{80}$~that of the earth and a velocity of escape of
about $1.5$~miles per second, has no sensible atmosphere. On
the other hand, Jupiter, with a mass $318$~times that of the
earth and a velocity of escape of $37$~miles per second, has
an enormous atmosphere. These examples are typical of
the facts furnished by all known celestial bodies.

\Article{34}{Effects of the Atmosphere on Climate.}---Aside from
\index{Atmosphere!climatic influences of}%
the heat received from the sun, the most important factor
affecting the earth's climate is its atmosphere. It tends to
equalize the temperature in three important ways. (\textit{a})~It
makes the temperature at any one place more uniform than
it would otherwise be, and (\textit{b})~it reduces to a large extent
the variations in temperature in different latitudes that
would otherwise exist. And (\textit{c})~it distributes water over the
surface of the earth.

(\textit{a})~Consider the day side of the earth. The rays of the
sun are partly absorbed by the atmosphere and the heating
of the earth's surface is thereby reduced. The amount
absorbed at sea level is possibly as much as $40$~per~cent.
Every one is familiar with the fact that on a mountain,
above a part of the atmosphere, sunlight is more intense than
it is at lower levels. But at night the effects are reversed.
The heat that the atmosphere has absorbed in the daytime
is radiated in every direction, and hence some of it strikes
the earth and warms it. Besides this, at night the earth
radiates the heat it has received in the daytime. The atmosphere
above reflects some of the radiated heat directly
back to the earth. Another portion of it is absorbed and
radiated in every direction, and consequently in part back
%% -----File: 097.png---Folio 72-------
to the earth. In short, the atmosphere acts as a sort of
blanket, keeping out part of the heat in the daytime, and
helping to retain at night that which has been received. Its
action is analogous to that of a glass with which the gardener
covers his hotbed. The results are that the variations in
temperature between night and day are reduced, and the
average temperature is raised.

(\textit{b})~The unequal heating of the earth's atmosphere in
various latitudes is the primary cause of the winds. The
warmer air moves toward the cooler regions, and the cold
air of the higher latitudes returns toward the equator. The
trade winds are examples of these movements. Their importance
will be understood when it is remembered that
wind velocities of $15$~or~$20$ miles an hour are not uncommon,
and that there is about $15$~pounds of air above every square
inch of the earth's surface.

One of the effects of the winds is the production of the
ocean currents which are often said to be dominant factors
in modifying climate, but which are, as a matter of fact,
relatively unimportant consequences of the air currents. A
south wind will often in the course of a few hours raise the
temperature of the air over thousands of square miles of
territory by $20$~degrees, or even more. In order to raise the
temperature of the atmosphere at constant pressure, over
one square mile through $20$~degrees by the combustion of coal
it would be necessary to burn ten thousand tons. This
illustration serves to give some sort of mental image of the
great influence of air currents on climatic conditions, and if
it were not for them, it is probable that both the equatorial
and polar regions would be uninhabitable by man.

\Article{35}{Importance of the Constitution of the Atmosphere.}---The
blanketing effect of the atmosphere depends to a considerable
extent on its constitution. Every one is familiar
with the fact that the early autumn frosts occur only when
the air is clear and has low humidity. The reason is that
water vapor is less transparent to the earth's radiations than
%% -----File: 098.png---Folio 73-------
are nitrogen and oxygen gas. On the other hand, there is
not so much difference in their absorption of the rays that
come from the sun. The reason is that the very hot sun's
rays are largely of short wave length (\Artref{211}); that is,
they are to a considerable extent in the blue end of the spectrum,
while the radiation from the cooler earth is almost
entirely composed of the much longer heat rays. Ordinary
glass has the same property, for it transmits the sun's rays
almost perfectly, while it is a pretty good screen for the rays
emitted by a stove or radiator.

The water-vapor content of the atmosphere varies and
cannot surpass a certain amount. But carbon dioxide has
the same absorbing properties as water vapor, and in spite
of the fact that it makes up only a very small part of the
earth's atmosphere, Arrhenius believes that it has important
\index[xnames]{Arrhenius}%
climatic effects. He concluded that if the quantity of it
in the air were doubled the climate would be appreciably
warmer, and that if half of it were removed the average
temperature of the earth would fall. Chamberlin has shown
\index[xnames]{Chamberlin}%
that there are reasons for believing that the amount of
carbon dioxide has varied in long oscillations, and he suggested
that this may be the explanation of the ice ages, with
\index{Glacial epoch}%
intervening warm epochs, which the middle latitudes have
experienced.

If the effect of carbon dioxide on the climate has been
\index{Carbon dioxide!effects on climate}%
\index{Carbon dioxide!production of}%
correctly estimated, its production by the recent enormous
consumption of coal raises the interesting question whether
man at last is not in this way seriously interfering with the
cosmic processes. At the present time about $1,000,000,000$ tons
of coal are mined and burned annually. In order to
burn $12$~pounds of coal $32$~pounds of oxygen are required,
and the result of the combustion is $12 + 32 = 44$~pounds
of carbon dioxide. Consequently, by the combustion of
coal there is now annually produced by man about
$3,670,000,000$ tons of carbon dioxide. On referring to the
total amount of carbon dioxide now in the air (\Artref{28}), it
%% -----File: 099.png---Folio 74-------
is seen that at the present rate of combustion of coal it will
be doubled in $800$~years. Consequently, there are grounds
for believing that modern industry may have sensible
climatic effects in a few centuries.

\Article{36}{Rôle of the Atmosphere in Life Processes.}---Oxygen
\index{Atmosphere!role@{rôle of in life processes}}%
is an indispensable element in the atmosphere for all higher
forms of animal life. It is taken into the blood stream
through the lungs and is %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.5in}
\Input[2.5in]{099}{png}
\Caption[The refraction of light.]{Fig}{21}
\index{Light!refraction of}%
\index{Refraction}%
\end{wrapfigure}
used in the tissues. Its proportion
in the atmosphere is probably not very important, for it
seems probable that if it had always been much more or
much less, animals would have become adapted to the different
condition. But if the earth's crust had contained
enough material which readily unites with oxygen, such as
hydrogen, silicon, or iron, to have exhausted the supply, it
seems certain that animals with warm, red blood could not
have developed. Such considerations are of high importance
in speculating on the question of the habitability of
other planets.

The higher forms of vegetable matter are largely composed
of carbon and water. The carbon is obtained from the carbon
dioxide in the atmosphere. The carbon and oxygen are
separated in the cells of the
plants, the carbon is retained, and
the oxygen is given back to the air.

\Article{37}{Refraction of Light by the
Atmosphere.}---When light passes
\index{Atmosphere!refraction by}%
from a rarer to a denser medium
it is bent toward the perpendicular
to the surface between the
two media, and in general the
greater the difference in the densities
of the two media, the greater
is the bending, which is called refraction. Thus, in \Figref{21},
the ray $l$ which strikes the surface of the denser medium
at $A$ is bent from the direction $AB$ toward the perpendicular
to the surface $AD$ and takes the direction~$AC$.
%% -----File: 100.png---Folio 75-------

Now consider a ray of light striking the earth's atmosphere
obliquely. The density of the air increases from its outer
borders to the surface
\begin{wrapfigure}{\WLoc}{2.75in}%[Illustration:]
\Input[2.75in]{100a}{png}
\Caption[Refraction of light by the earth's atmosphere.]{Fig}{22}
\end{wrapfigure}
of the earth. Consequently,
a ray of light is
bent more and more as it
proceeds down through
the air. Let $l$, \Figref{22},
represent a ray of light
coming from a star $S$ to
an observer at $O$. The
star is really in the direction
$OS''$, but it appears to be in the direction $OS'$ from
which the light comes when it strikes the observer's eye. The
angle between $OS''$ and $OS'$ is the angle of refraction. It is
zero for a star at the zenith and increases to a little over
one-half of a degree for one at the horizon. For this reason a
\begin{figure}[hbt]%[Illustration:]
\Input{100b}{png}
\Caption[The sun is apparently flattened by refraction when it is on the
horizon.]{Fig}{23}
\end{figure}%
celestial body apparently rises before it is actually above the
horizon, and is visible until after it has really set. If the
sun or moon is on the horizon, its bottom part is apparently
raised more than its top part by refraction, so that it seems
to be flattened in the vertical direction, as is shown in \Figref{23}.
%% -----File: 101.png---Folio 76-------

\Article{38}{The Twinkling of the Stars.}---The atmosphere is not
\index{Scintillation of stars}%
\index{Stars!twinkling of}%
\index{Twinkling of stars}%
only of variable density from its highest regions to the surface
of the earth, but it is always disturbed by waves which
cause the density at a given point to vary continually.
These variations in density cause constant small changes in
the refraction of light, and consequently alterations in the
direction from which the light appears to come. When
the source is a point of light, as a star, it twinkles or scintillates.
The twinkling of the stars is particularly noticeable
in winter time on nights when the air is cold and unsteady.
The variation in refraction is different for different colors,
and consequently when a star twinkles it flashes sometimes
blue or green and at other times red or yellow. Objects that
have disks, even though they are too small to be discerned
with the unaided eye, appear much steadier than stars because
the irregular refractions from various parts seldom agree in
direction, and consequently do not displace the whole object.


\Section{IV}{QUESTIONS}

1. What is the weight of the air in a room $16$~feet square and
$10$~feet high?

2. How many pounds of air pass per minute through a windmill
$12$~feet in diameter in a breeze of $20$~miles per~hour?

3. Compute the approximate total atmospheric pressure to which
a person is subject.

4. What is the density of the air, compared to its density at the
surface, at heights of $50$,~$100$, and $500$~miles, the density being determined
by the law given at the end of \Artref{32}? This gives an idea
of the density required for the phenomena of twilight, of meteors,
and of auroræ.

5. Draw a diagram showing the earth and its atmosphere to scale.

6. The earth's mass is slowly growing by the acquisition of
meteors; if there is nothing to offset this growth, will its atmosphere
have a tendency to increase or to decrease in amount?

7. If the earth's atmosphere increases or decreases, as the case
may be, what will be the effect on the mean temperature, the daily
range at any place, and the range over the earth's whole surface?

8. If the earth's surface were devoid of water, what would be the
effect on the mean temperature, the daily range at any place, and
the range over its whole surface?

\normalsize

%% -----File: 102.png---Folio 77-------


\Chapter{III}{The Motions of the Earth}
\index{Earth!rotation of}%

\Section{I}{The Rotation of the Earth}

\Article{39}{The Relative Rotation of the Earth.}---The most
casual observer of the heavens has noticed that not only
the sun and moon, %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.5in}
\Input[2.5in]{102}{jpg}
\Caption[Star trails of brighter stars in Orion (Barnard).]{Fig}{24}
\index{Yerkes Observatory}%
\end{wrapfigure}
but also the stars, rise in the east, pass
across the sky, and set in
the west. At least this is
true of those stars which
cross the meridian south
of the zenith. \Figureref{24}
is a photograph of Orion
\index{Orion}%
in which the telescope was
kept fixed while the stars
passed in front of it, and
the horizontal streaks are
the images traced out by
the stars on the photographic
plate.

The stars in the northern
heavens describe circles
around the north pole of
the sky as a center. Two hours of observation of the position
of the Big Dipper will show the character of the motion
\index{Big Dipper}%
very clearly. \Figureref{25} shows circumpolar star trails secured
by pointing a fixed telescope toward the pole star and giving
an exposure of a little over an hour. The conspicuous
streak a little below and to the left of the center is the
trail of the pole star, which therefore is not exactly at the
pole of the heavens. A comparison of this picture with
%% -----File: 103.png---Folio 78-------
the northern sky will show that most of the stars whose
trails are seen are quite invisible to the unaided eye.

Since all the heavenly bodies rise in the east (except those
so near the pole that they simply go around it), travel across
the sky, and set
in the west, to
reappear again in
the east, it follows
that either
they go around
the earth from
east to west, or
the earth turns
from west to
east. So far as
the simple motions
of the sun,
moon, and stars
are concerned
both hypotheses
are in perfect
harmony with
the observations,
and it is not possible
to decide
which of them is correct without additional data. All the
apparent motions prove is that there is a relative motion
of the earth with respect to the heavenly bodies.

It is often supposed that the ancients were unscientific,
if not stupid, because they believed that the earth was fixed
and that the sky went %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3.25in}
\Input[3.25in]{103}{jpg}
\Caption[Circumpolar star trails (Ritchey).]{Fig}{25}
\index{Circumpolar star trails}%
\end{wrapfigure}
around it, but it has been seen that
so far as their data bore on the question one theory was as
good as the other. In fact, not all of them thought that
the earth was fixed. The earliest philosopher who is known
to have believed in the rotation of the earth was Philolaus,
\index[xnames]{Philolaus}%
a Pythagorean, who lived in the fifth century~\BC. His
%% -----File: 104.png---Folio 79-------
ideas were more or less mystical, but they seem to have had
some influence, for they were quoted by Copernicus (1473--1543)
\index[xnames]{Copernicus}%
in his great work on the theory of the motions in the
solar system. Aristotle (384--322~\BC) recognized the fact
\index[xnames]{Aristotle}%
that the apparent motions of the stars can be explained
either by their revolution around the earth, or by the rotation
of the earth on its axis. Aristarchus of Samos (310--250~\BC)
\index[xnames]{Aristarchus}%
made the clearest statements regarding both the
rotation and the revolution of the earth of any philosopher
of antiquity. But Hipparchus (180--110~\BC), who was the
\index[xnames]{Hipparchus}%
greatest astronomer of antiquity, and whose discoveries
were very numerous and valuable, believed in the fixity of
the earth. He was followed in this opinion by Ptolemy
\index[xnames]{Ptolemy}%
(100--170~\AD) and every other astronomer of note down to
Copernicus, who believed the earth rotated and revolved
around the sun.

\Article{40}{The Laws of Motion.}---One method of attacking
the question of whether or not any particular body, such as
the earth, moves is to consider the laws of motion of bodies
in general, and then to answer it on the basis of, and in
harmony with, these laws. The laws of nature are in a
fundamental respect different from civil laws, and it is unfortunate
that the same term is used for both of them. A
civil law prescribes or forbids a mode of conduct, with penalties
if it is violated. It can be violated at pleasure if one
is willing to run the chance of suffering the penalty. On
the other hand, a law of nature does not prescribe or compel
anything, but is a description of the way all phenomena of
a certain class succeed one another.

The laws of motion are statements of the way bodies
actually move. They were first given by Newton in 1686,
\index[xnames]{Newton}%
although they were to some extent understood by his predecessor
Galileo. Newton called them \textit{axioms} although they
\index[xnames]{Galileo}%
are by no means self-evident, as is proved by the fact that
for thousands of years they were quite unknown. The laws,
essentially as Newton gave them, are:
%% -----File: 105.png---Folio 80-------

\index{Laws!of motion}%
\textsc{Law I\@.} \textit{Every body continues in its state of rest, or of uniform
motion in a straight line, unless it is compelled to change
that state by an exterior force acting upon it.}

\textsc{Law II\@.} \textit{The rate of change of motion of a body is directly
proportional to the force applied to it and inversely proportional
to its mass, and the change of motion takes place in the
direction of the line in which the force acts.}

\textsc{Law III\@.} \textit{To every action there is an equal and oppositely
directed reaction; or, the mutual actions of two bodies are always
equal and oppositely directed.}

The importance of the laws of motion can be seen from the
fact that every astronomical and terrestrial phenomenon
involving the motion of matter is interpreted by using them
as a basis. They are, for example, the foundation of all
mechanics. A little reflection will lead to the conclusion
that there are few, if indeed any, phenomena that do not in
some way, directly or indirectly, depend upon the motion
of matter.

The first law states the important fact that if a body is at
rest it will never begin to move unless some force acts upon
it, and that if it is in motion it will forever move with uniform
speed in a straight line unless some exterior force acts upon
it. In two respects this law is contradictory to the ideas
generally maintained before the time of Newton. In the
\index[xnames]{Newton}%
first place, it had been supposed that bodies near the earth's
surface would descend, because it was natural for them to do
so, even though no forces were acting upon them. In the
second place, it had been supposed that a moving body would
stop unless some force were continually applied to keep it
going. These errors kept the predecessors of Newton from
getting any satisfactory theories regarding the motions of
the heavenly bodies.

The second law defines how the change of motion of a
body, in both direction and amount, depends upon the applied
force. It asserts what happens when any force is acting,
and this means that the statement is true whether or
%% -----File: 106.png---Folio 81-------
not there are other forces. In other words, the momentary
effects of forces can be considered independently of one
another. For example, if two forces, $PA$ and $PB$ in \Figref{26},
are acting on a body at $P$, it will move in the direction
$PA$ just as though $PB$
were not %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{106}{png}
\Caption[The parallelogram of forces.]{Fig}{26}
\index{Parallelogram of forces}%
\end{wrapfigure}
acting on it,
and it will move in the
direction $PB$ just as
though $PA$ were not
acting on it. The result
is that when they are
both acting it will go from $P$ to $C$ along $PC$. Since $PACB$
is a parallelogram, this is called the parallelogram law of
the composition of forces.

The first two laws refer to the motion of a single body;
the third expresses the way in which two bodies act on each
other. It means essentially that if one body changes the
state of motion of another body, its own state of motion is
also changed reciprocally in a definite way. The term
``action'' in the law means the mass times the rate of change
of motion (acceleration) of the body. Hence the third law
might read that if two bodies act on each other, then the
product of the mass and acceleration in one is equal and
opposite to the product of the mass and acceleration in the
other. This is a complete statement of the way two bodies
act upon each other. But the second law states that the
product of the mass and acceleration of a body is proportional
to the force acting on it. Hence it follows that the
third law might read that if two bodies act on each other,
then the force exerted by the first on the second is equal
and opposite to the force exerted by the second on the first.
This statement is not obviously true because it seems to
contradict ordinary experience. For example, the law states
that if a strong man and a weak man are pulling on a rope
(weight of the rope being neglected) against each other, the
strong man cannot pull any more than the weak man. The
%% -----File: 107.png---Folio 82-------
reason is, of course, that the weak man does not give the
strong one an opportunity to use his full strength. If the
strong man is heavier than the weak one and pulls enough,
he will move the latter while he remains in his tracks. This
seems to contradict the statement of the law in terms of
the acceleration; but the contradiction disappears when it
is remembered that the men are subject not only to the forces
they exert on each other, but also to their friction with the
earth. If they were in canoes in open water, they would
both move, and, if the weights of the canoes were included,
their motions would be in harmony with the third law.

Since the laws of motion are to be used fundamentally in
considering the motion of the earth, the question of their
truth at once arises. When they are applied to the motions
of the heavenly bodies, everything becomes orderly. Besides
this, they have been illustrated millions of times in
ordinary experience on the earth and they have been tested
in laboratories, but nothing has been found to indicate they
are not in harmony with the actual motions of material bodies.
In fact, they are now supported by such an enormous mass
of experience that they are among the most trustworthy conclusions
men have reached.

\Article{41}{Rotation of the Earth Proved by Its Shape.}---The
\index{Earth!rotation of}%
\index{Rotation!of earth}%
shape of the earth can be determined without knowing whether
or not it rotates. The simple measurements of arcs (\Artref{12})
prove that the earth is oblate.

It can be shown that it follows from the laws of motion
and the law of gravitation that the earth would be spherical
if it were not rotating. Since it is not spherical, it must be
rotating. Moreover, it follows from the laws of motion
that if it is rotating it will be bulged at the equator. Hence
the oblateness of the earth proves that it rotates and determines
the position of its axis, but does not determine in
which direction it turns.

\Article{42}{Rotation of the Earth Proved by the Eastward Deviation
of Falling Bodies.}---Let~$OP$, \Figref{27}, represent a
\index{Deviation!of falling bodies}%
\index{Falling bodies, deviations of}%
%% -----File: 108.png---Folio 83-------
tower from whose top a ball is dropped. Suppose that while
the ball is falling to the foot of the tower the earth rotates
through the angle~$QEQ'$. The top of the tower is carried
from $P$~to~$P'$, and its foot from $O$~to~$O'$. The distance~$PP'$
is somewhat greater than the distance~$OO'$. Now consider
the falling body.
It tends to move
in the direction~$PP'$
in accordance
with the first
law of motion because,
at the time
it is dropped, it
is carried in this
direction by the
rotation of the
earth. Moreover,
$PP'$~is the distance
through which it would be carried if it were not
dropped. But the earth's attraction causes it to descend,
and the force acts at \emph{right angles} to the line~$PP'$. Therefore,
by the second law of motion, %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3.375in}
\Input[3.375in]{108}{png}
\Caption[The eastward deviation of falling bodies
proves the eastward rotation of the earth.]{Fig}{27}
\end{wrapfigure}
the attraction of the earth
does not have any influence on the motion in the direction~$PP'$.
Consequently, while it is descending it moves in a
horizontal direction a distance equal to~$PP'$ and strikes
the surface at~$O''$ to the east of the foot of the tower~$O'$.
The eastward deviation is the distance~$O'O''$. The small
diagram at the right shows the tower and the path of the
falling body on a larger scale.

The foregoing reasoning has been made on the assumption
that the earth rotates to the eastward. The question arises
whether the conclusions are in harmony with experience.
The experiment for determining the deviation of falling bodies
is complicated by air currents and the resistance of the air.
Furthermore, the eastward deviation is very small, being
only $1.2$ inches for a drop of $500$~feet in latitude~$40°$. In
%% -----File: 109.png---Folio 84-------
spite of these difficulties, the experiment for moderate heights
proves that the earth rotates to the eastward. Father Hagen,
\index[xnames]{Hagen}%
of Rome, has devised an apparatus, having analogies with
Atwood's machine in physics, which avoids most of the disturbances
to which a freely falling body is subject. The
largest free fall so far tried was in a vertical mine shaft, near
Houghton, Mich., more than $4000$~feet deep. In spite
of the fact that the diameter of the mine shaft was many
times the deviation for that distance, the experiment utterly
failed because the balls which were dropped never reached
the bottom. It is probable that when they had fallen far
enough to acquire high speed the air packed up in front of
them until they were suddenly deflected far enough from
their course to hit the walls and become imbedded.

\Article{43}{Rotation of the Earth Proved by Foucault's Pendulum.}---One
\index{Earth!rotation of}%
\index{Foucault's pendulum}%
\index{Pendulum!Foucault's}%
\index{Rotation!of earth}%
\index[xnames]{Foucault}%
of the most ingenious and convincing experiments
for proving the
rotation of the
earth was devised
in 1851 by the
French physicist
Foucault. It depends
upon the
fact that according
to the laws of
motion a %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3.25in}
\Input[3.25in]{109}{png}
\Caption[The Foucault pendulum.]{Fig}{28}
\end{wrapfigure}
freely
swinging pendulum
tends constantly
to move in
the same plane.

Suppose a pendulum suspended at $O$, \Figref{28}, is started
swinging in the meridian $OQ$. Let $OV$ be the tangent at $O$
drawn in the plane of the meridian. After a certain interval
the meridian $OQ$ will have rotated to the position $O'Q'$.
The line $O'V'$ is drawn parallel to the line $OV$. Consequently
the pendulum will be swinging in the plane $EO'V'$.
%% -----File: 110.png---Folio 85-------
The tangent to the meridian at~$O'$ is~$O'V$. Consequently,
the angle between this line and the plane in which the
pendulum will be swinging is~$V'O'V$, which equals~$OVO'$.
That is, the angle at~$V$ between the meridian tangents equals
the apparent deviation of the plane of the pendulum from the
meridian. For points in the northern hemisphere the deviation
is from a north-and-south direction toward a northeast-and-southwest
direction. The angle around the cone at~$V$
equals the total deviation in one rotation of the earth. If
$O$~is at the earth's pole, the daily deviation is $360$~degrees.
If $O$~is on the earth's equator, the point~$V$ is infinitely far
away and the deviation is zero.

Foucault suspended a heavy iron ball by a steel wire about
\index[xnames]{Foucault}%
$200$~feet long, and the deviation became evident in a few
minutes. The experiment is very simple and has been repeated
in many places. It proves that the earth rotates
eastward, and the rate of deviation of the pendulum proves
that the relative motion of the earth with respect to the
stars is due entirely to its rotation and not at all to the
motions of the stars around it.

\Article{44}{Consequences of the Earth's Rotation.}---An important
\index{Deviation!of air currents}%
\index{Earth!rotation of}%
\index{Rotation!of earth}%
consequence of the earth's rotation is the direction of
air currents at
considerable distances
from the
equator in both
northern and
southern latitudes.
Suppose
the unequal heating
of the atmosphere
causes a
\begin{wrapfigure}{\WLoc}{3in}%[Illustration:]
\Input[3in]{110}{png}
\Caption[The deviation of air currents.]{Fig}{29}
\end{wrapfigure}
certain portion of
it to move northward
from~$O$, \Figref{29}, with such a velocity that if the
earth were not rotating, it would arrive at~$A$ in a certain
%% -----File: 111.png---Folio 86-------
interval of time. Suppose that in this interval of time the
meridian~$OQ$ rotates to the position~$O'Q'$. Hence the mass
of air under consideration actually had the velocities $OA$~and~$OO'$
when it started from~$O$, the former with respect to the
surface of the earth and the latter because of the rotation of
the earth. By the laws of motion these motions, being at
right angles to each other, are mutually independent, and
the air will move over both distances during the interval of
time and arrive at the point~$A''$, which is east of~$A'$. Consequently,
the mass of air that started straight northward
with respect to the surface of the earth along the meridian~$OA$
will have deviated eastward by the amount~$A'A''$.

The deviation for northward motion in the northern
hemisphere is toward the east; for southward motion, it
is toward the west. In both cases it is toward the right.
For similar reasons, in the southern hemisphere the deviation
is toward the left.

The deviations in the directions of air currents are evidently
greater the higher the latitude, because near the poles
a given distance along the earth's surface corresponds to
an almost equal change in the distance from the axis of
rotation, while at the equator there is no change in the distance
from the earth's axis. It might be supposed that in
middle latitudes a moderate northward or southward displacement
of the air would cause no appreciable change in
its direction of motion. But a point on the equator moves
eastward at the rate of over $1000$~miles an hour, at latitude
$60$~degrees the eastward velocity is half as great, and at the
pole it is zero. If it were not for friction with the earth's
surface, a mass of air moving from latitude $40$~degrees to
latitude $45$~degrees, a distance less than $350$~miles, would
acquire an eastward velocity with respect to the surface of
the earth of over $40$~miles an hour. The prevailing winds
of the northern hemisphere in middle latitudes are to the
northeast, and the eastward component has been found to
be strong for the very high currents.
%% -----File: 112.png---Folio 87-------

Obviously the same principles apply to water currents
and to air currents. Consequently water currents, such as
rivers, tend to deviate toward the right in the northern
\index{Deviation!of rivers}%
hemisphere. It has been found by examining the Mississippi
and Yukon rivers that the former to some extent,
and the latter to a much greater extent, on the whole scour
their right-hand banks.

All the proofs of the earth's rotation so far given depend
upon the laws of motion. There is one independent reason
for believing the earth rotates, though it falls a little short
of proof. It has been found by observations involving
only geometrical principles that the sun, moon, and planets
are comparable to the earth in size, some being larger and
others smaller. Direct observations with the telescope show
that a number of these bodies rotate on their axes, the remainder
being either very remote or otherwise unfavorably
situated for observation. The conclusion by analogy is
that the earth also rotates.

\Article{45}{The Uniformity of the Earth's Rotation.}---It follows
\index{Uniformity of earth's rotation}%
from the laws of motion, and in particular from the first
law, that if the earth were subject to no external forces and
were invariable in size, shape, and distribution of mass, it
would rotate on its axis with absolute uniformity. Since
the earth is a fundamental means of measuring time its
rotation cannot be tested by clocks. Its rotation might be
compared with other celestial phenomena, but then the
question of their uniformity would arise. The only recourse
is to make an examination of the possible forces and
changes in the earth which are capable of altering the rate
of its rotation.

The earth is subject to the attractions of the sun, moon,
and planets. But these attractions do not change its rate
of rotation because the forces pulling on opposite sides
balance, just as the earth's attraction for a rotating wheel
whose plane is vertical neither retards nor accelerates its
motion.
%% -----File: 113.png---Folio 88-------

The earth is struck by millions of small meteors daily
\index{Meteoric showers!matter, resistance of}%
\index{Meteors!effects of on earth's rotation}%
coming in from all sides. They virtually act as a resisting
medium and slightly retard its rotation, just as a top spinning
in the air is retarded by the molecules impinging on it.
But the mass of the earth is so large and the meteors are so
small that, at their present rate of infall, the length of the
day cannot be changed by this cause so much as a second in
\index{Day!invariability of}%
$100,000,000$ years.

The moon and the sun generate tides in the water around
\index{Tides!effects of, on day}%
the earth and the waves beat in upon the shores and are
gradually destroyed by friction. The energy of the waves
is transformed into heat. This means that something else
has lost energy, and a mathematical treatment of the subject
shows that the earth has suffered the loss. Consequently
its rotation is diminished. But as great and irresistible
as the tides may be, their energies are insignificant
compared to that of the rotating earth, and according to the
work of MacMillan the day is not increasing in length from
\index[xnames]{MacMillan}%
this cause more than one second in $500,000$ years.

Before discussing the effects of a change in the size of the
earth or in the distribution of its mass, it is necessary to
explain a very important property of the motion of rotating
bodies. It can be shown from the laws of motion that if
a body is not subject to any exterior forces, its total quantity
of rotation always remains the same no matter what changes
may take place in the body itself. The quantity of rotation
of a body, or \textit{moment of momentum}, as it is technically called
\index{Moment of momentum}%
in mechanics, is the sum of the rotations of all its parts.
The rotation of a single part, or particle, is the product of
its mass, its distance from the axis of rotation passing
through the center of gravity of the body, and the speed
with which it is moving at right angles to the line joining it
to the axis of rotation. It can be shown that in the case
of a body rotating as a solid, the quantity of rotation is
proportional to the product of the square of the radius and
the angular velocity of rotation, the angular velocity of
%% -----File: 114.png---Folio 89-------
rotation being the angle through which the body turns in
a unit of time.

Now apply this principle of the conservation of the moment
of momentum to the earth. If it should lose heat and
shrink so that its radius were diminished in length, then the
angular velocity of rotation would increase, for the product
of the square of the radius and the rate of rotation must
be constant. On the other hand, if the radio\DPtypo{-}{}active substances
in the earth should cause its temperature to rise and
its radius to expand, then the rate of rotation would decrease.
Neither of these causes can make a sensible change
in the rotation in $1,000,000$ years. Similarly, if a river
rising in low latitudes should carry sediment to higher latitudes
and deposit it nearer the earth's axis, then the rate
of rotation of the earth would be increased. While such
factors are theoretically effective in producing changes in
the rotation of the earth, from a practical point of view
they are altogether negligible.

It follows from this discussion that there are some influences
tending to decrease the rate of the earth's rotation,
and others tending to increase it, but that they are all so
small as to have altogether inappreciable effects even in a
period as long as $100,000$ years.

\Article{46}{The Variation of Latitude.}---It was mentioned in
\index{Latitude!variation of}%
\index{Variation!of latitude}%
connection with the discussion of the rigidity of the earth
(Arts.\ \hyperref[art:25]{25},~\hyperref[art:26]{26}), that its axis of rotation is not exactly fixed.
This does not mean that the direction of the axis changes,
but that the position of the earth itself changes so that its
axis of rotation continually pierces different parts of its
surface. That is, the poles of the earth are not fixed points
on its surface. Since the earth's equator is $90$~degrees from
its poles, the position of the equator also continually changes.
Therefore the latitude of any fixed point on the surface of
the earth undergoes continual variation. The fact was
discovered by very accurate determinations of latitude, and
for this reason is known as the variation of latitude.
%% -----File: 115.png---Folio 90-------

The pole wanders from its mean position not more than
$30$~feet, corresponding to a change of latitude of $0.3$~of a
second of arc. This is such a small quantity that it can be
measured only by the most refined means, and accounts
\begin{figure}[hbt]%[Illustration:]
\Input{115}{jpg}
\Caption[The position of the pole from 1906 to~1913.]{Fig}{30}
\end{figure}%
for the failure to discover it until the work of Chandler and
\index[xnames]{Chandler}%
Küstner about 1885.
\index[xnames]{Kustner@{Küstner}}%

In 1891 Chandler took up the problem of finding from
the observations how the pole actually moves. The variation
in its position is very complicated, \Figref{30} showing it
%% -----File: 116.png---Folio 91-------
from 1906 to 1913. Chandler found that this complicated
\index[xnames]{Chandler}%
motion is the result of two simpler ones. The first is a
yearly motion in an ellipse (\Artref{53}) whose longest radius is
$14$~feet and shortest radius $4$~feet; and the second is a
motion in a circle of radius $15$~feet, which is described in
about $428$~days. More recent discussions, based on observations
secured by the coöperation of the astronomers of several
countries, have modified these results to some extent and
have added other minor terms.

The problem is to account for the variation of latitude
and for the different periods. Unless a freely rotating oblate
rigid body is started turning exactly around its shortest
axis, it will undergo an oscillation with respect to its axis
of rotation in a period which depends upon its figure, mass,
and speed of rotation. Hence it might be supposed that
the earth in some way originally started rotating in this
manner. But since the earth is not perfectly rigid and unyielding,
friction would in the course of time destroy the
wabbling. In view of the fact that the earth is certainly
many millions of years old, it seems that friction should
long ago have reduced its rotation to sensible uniformity
around a fixed axis, and this is true unless it is very elastic
instead of being somewhat viscous. The tide experiment
(\Artref{25}) proves that the earth is very elastic and suggests
that perhaps the earth's present irregularities of rotation
have been inherited from greater ones produced at the time
of its origin, possibly by the falling together of scattered
meteoric masses. But the fact that the earth has two different
variations of latitude of almost equal magnitude is
opposed to this conclusion. The one which has the period
of a year is probably produced by meteorological causes, as
Jeffreys infers from a quantitative examination of the question.
\index[xnames]{Jeffreys}%
The one whose period is $428$~days, the natural period
of variation of latitude for a body having the dynamical
properties of the earth, is probably the consequence of the
other. In order to understand their relations consider a
%% -----File: 117.png---Folio 92-------
pendulum which naturally oscillates in seconds. Suppose it
starts from rest and is disturbed by a small periodic force
whose period is two thirds of a second. Presently it will be
moving, not like an undisturbed pendulum, but with one
oscillation in two thirds of a second, and with another
oscillation having an approximately equal magnitude, in its
natural period, or one second.

Euler showed about 1770 that if the earth were absolutely
\index[xnames]{Euler}%
rigid the natural period of oscillation of its pole would be $305$~days.
The increase of period to $428$~days is due to the fact
that the earth yields partially to disturbing forces (\Artref{25}).

Many parts of the earth have experienced wide variations
in climate during geological ages, and it has often been suggested
that these great changes in temperature were produced
by the wandering of its poles. There are no known
forces which could produce any greater variations in latitude
than those which have been considered, and there is not the
slightest probability that the earth's poles ever have been
far from their present position on the surface of the earth.

\Article{47}{Precession of the Equinoxes and Nutation.}---There
\index{Equinoxes!precession of}%
\index{Precession of equinoxes}%
is one more phenomenon to be considered in connection
with the rotation of the earth. In the variation of latitude
the poles of the earth are slightly displaced on its surface;
now the changes in the direction of its axis with respect to
the stars are under consideration.

The axis of the earth can be changed in direction only by
forces exterior to itself. The only important exterior forces
to which the earth is subject are the attractions of the moon
and sun. If the earth were a sphere, these bodies would
have no effect upon its axis of rotation, but its oblateness
gives rise to very important consequences.

Let~$O$, \Figref{31}, represent a point on the equator of the
oblate earth, and suppose the moon~$M$ is in the plane of the
meridian which passes through~$O$. The point~$O$ is moving
in the direction~$OA$ as a consequence of the earth's rotation.
The attraction of the moon for a particle at~$O$ is in the direction~$OM$.
%% -----File: 118.png---Folio 93-------
By the resolution of forces (the inverse of the
parallelogram of forces law) the force along~$OM$ can be resolved
in two others, one along~$OE$ and the other along the
line~$OB$ perpendicular to~$OE$. The former of these two
forces has no effect on the rotation; the latter tends to move
\begin{figure}[hbt]%[Illustration:]
\Input{118}{png}
\Caption[The attraction of the moon for the earth's equatorial bulge
causes the precession of the equinoxes.]{Fig}{31}
\end{figure}%
the particle in the direction~$OB$, and this tendency, combined
with the velocity~$OA$, causes it to move in the direction~$OC$
(the change is greatly exaggerated). Therefore the direction
of motion of~$O$ is changed; that is, the plane of the
equator is changed.

The moon, however, attracts every particle in the equatorial
bulge of the earth, and its effects vary with the position of
the particles. It can be shown by a mathematical discussion
that cannot be taken up here that the combined effect
on the entire bulge is to change the plane of the equator. It
is evident from \Figref{31} that the effect vanishes when the
moon is in the plane of the earth's equator. Therefore it
is natural to take the plane of the moon's orbit as a plane of
reference. These two planes intersect in a certain line whose
position changes as the plane of the earth's equator is shifted.
The plane of the earth's equator shifts in such a way that
the angle between it and the plane of the moon's orbit is
constant, while the line of intersection of the two planes rotates
%% -----File: 119.png---Folio 94-------
in the direction opposite to that in which the earth
turns on its axis.

The plane in which the sun moves is called the \textit{plane of
the ecliptic}, and the moon is always near this plane. For
\index{Ecliptic}%
the moment neglect its departure from the plane of the
ecliptic. Then the moon, and the sun similarly, cause the
line of the intersection of the plane of the earth's equator
and the plane of the ecliptic, called the \textit{line of the equinoxes},
\index{Equinoxes}%
to rotate in the direction opposite to that of the rotation of
the earth. This is the precession of the equinoxes, four
fifths of which is due to the moon and one fifth of which is
due to the sun. Since the axis of the earth is perpendicular
to the plane of its equator, the point in the sky toward which
the axis is directed describes a circle among the stars.

The mass of the earth is so great, the equatorial bulge is
relatively so small, and the forces due to the moon and sun
are so feeble that the precession is very slow, amounting only
to $50.2$~seconds of arc per year, from which it follows that
the line of the equinoxes will make a complete rotation only
after more than $25,800$ years have passed.

The precession of the equinoxes was discovered by Hipparchus
\index{Equinoxes!precession of}%
\index{Precession of equinoxes}%
\index[xnames]{Hipparchus}%
about 120~\BC\ from a comparison of his observations
with those made by earlier astronomers, but the cause
of it was not known until it was explained by Newton, in
\index[xnames]{Newton}%
1686, in his Principia. The theoretical results obtained for
the precession are in perfect harmony with the observations,
and the weight of this statement will be appreciated when
it is remembered that the calculations depend upon the size
of the earth, its density, the distribution of mass in it, the
laws of motion, the rate of rotation of the earth and its oblateness,
the distances to the moon and sun, their apparent motions
with respect to the earth, and the law of gravitation.

The moon does not move exactly in the plane of the
ecliptic, but deviates from it as much as $5$~degrees, and
consequently the precession which it produces is not exactly
with respect to the ecliptic. This circumstance would not
%% -----File: 120.png---Folio 95-------
be particularly important if it were not for the further fact
that the plane of the moon's orbit has a sort of precession
with respect to the ecliptic, completing a cycle in $18.6$~years.
This introduces a variation in the character of the precession
which is periodic with the same period of $18.6$~years. This
variation in the precession, which at its maximum amounts
to $9.2$~seconds of arc, is called the \textit{nutation}. It was discovered
\index{Nutation}%
by the great English astronomer Bradley from observations
\index[xnames]{Bradley}%
made during the period from 1727 to 1747. The
cause of it was first explained by D'Alembert, a famous
\index[xnames]{Dalembert@{D'Alembert}}%
French mathematician.


\Section{V}{QUESTIONS}

1. Which of the proofs of the rotation of the earth depend upon
the laws of motion?

2. Give three practical illustrations (one a train moving around
a curve) of the first law of motion.

3. Give three illustrations of the second law of motion.

4. Why is the kick in a heavy gun, for a given charge, less than
in a light gun?

5. If a man fixed on the shore pulls a boat by a rope, do the
interactions not violate the third law of motion?

6. For a body falling from a given height, in what latitude will
the eastward deviation be the greatest?

7. For what latitude will the rotation of the Foucault pendulum
be most rapid, and where would the experiment fail entirely?

8. In what latitude will the easterly (or westerly) deviation of
wind or water currents be most pronounced?

9. Is it easier to stop a large or small wheel of the same mass
rotating at the same rate?

10. If a wheel rotating without friction should diminish in size,
would its rate of rotation be affected?

11. Are boundaries that are defined by latitudes affected by the
wabbling of the earth's axis? By the precession of the equinoxes?

12. Would the precession be faster or slower if the earth were
more oblate? If the moon were nearer? If the earth were denser?

\normalsize

%% -----File: 121.png---Folio 96-------


\Section{II}{The Revolution of the Earth}
\index{Earth!revolution of}%
\index{Revolution of earth}%

\Article{48}{Relative Motion of the Earth with Respect to the
Sun.}---The diurnal motion of the sun is so obvious that the
\index{Motion!of sun}%
\index{Sun!apparent motion of}%
most careless observer fully understands it. But it is not
so well known that the sun has an apparent eastward motion
among the stars analogous to that of the moon, which every
one has noticed. The reason that people are not so familiar
with the apparent motion of the sun is that stars cannot
be observed in its neighborhood without telescopic aid, and,
besides, it moves slowly. However, the fact that it apparently
moves can be established without the use of optical
instruments; indeed, it was known in very ancient times.
\begin{figure}[hbt]%[Illustration:]
\Input{121}{png}
\Caption[The hypothesis that the sun revolves around the earth explains
the apparent eastward motion of the sun with respect to the stars.]{Fig}{32}
\end{figure}%
Suppose on a given date certain stars are seen directly south
on the meridian at 8~o'clock at night. The sun is therefore
$120°$ west of the star; or, what is equivalent, the stars in
question are $120°$ east of the sun. A month later at 8~o'clock
 at night the observed stars will be found to be $30°$
west of the meridian. Since at that time in the evening the
sun is $120°$ west of the meridian, the stars are $120° - 30°
= 90°$ east of the sun. That is, during a month the sun
apparently has moved $30°$ eastward with respect to the stars.

The question arises whether or not the sun's apparent
%% -----File: 122.png---Folio 97-------
motion eastward is produced by its actual motion around
the earth. It will be shown that the hypothesis that it
actually moves around the earth satisfies all the data so far
mentioned. Suppose $E$, \Figref{32}, represents the earth,
assumed fixed, and $S_1$ the position of the sun at a certain
time. As seen from the earth it will appear to be on the sky
among the stars at~$S_1'$. Suppose that at the end of $25$~days
the sun has moved forward in a path around the earth to
the position~$S_2$; it will then appear to be among the stars at~$S'_2$.
That is, it will appear to have moved eastward among
the stars in perfect accordance with the observations of its
apparent motion.

It will now be shown that the same observations can be
satisfied completely by the hypothesis that the earth revolves
\begin{figure}[hbt]%[Illustration:]
\Input{122}{png}
\Caption[The hypothesis that the earth revolves around the sun explains
the apparent eastward motion of the sun with respect to the stars.]{Fig}{33}
\end{figure}%
around the sun. Let~$S$, \Figref{33}, represent the sun,
assumed fixed, and suppose $E_1$ is the position of the earth at
a certain time. The sun will appear to be among the stars at~$S_1'$.
Suppose that at the end of $25$~days the earth has moved
forward in a path around the sun to~$E_2$; the sun will then
appear to be among the stars at~$S_2'$. That is, it will appear
to have moved eastward among the stars in perfect accordance
with the observations of its apparent motion. It is
noted that the assumed actual motion of the earth is in the
same direction as the sun's apparent motion; or, to explain
%% -----File: 123.png---Folio 98-------
the apparent motion of the sun by the motion of the earth,
the earth must be supposed to move eastward in its orbit.

Since all the data satisfy two distinct and mutually contradictory
hypotheses, new data must be employed in order
to determine which of them is correct. The ancients had
no facts by which they could disprove one of these hypotheses
and establish the truth of the other.

\Article{49}{Revolution of the Earth Proved from the Laws of
Motion.}---The first actual proof that the earth revolves
\index{Revolution of earth}%
around the sun was based on the laws of motion in 1686,
though the fact was generally believed by astronomers
somewhat earlier (\Artref{62}). It must be confessed at once,
however, that the statement requires a slight correction because
the sun and earth actually revolve around the center
of gravity of the two bodies, which is very near the center of
the sun because of the sun's relatively enormous mass.

It can be shown by measurements that have no connection
with the motion of the sun or earth that the volume of
the sun is more than a million times that of the earth. Hence,
unless it is extraordinarily rare, its mass is much greater
than that of the earth. In view of the fact that it is opaque,
the only sensible conclusion is that it has an appreciable
density. Hence, in the motion of the earth and sun around
their common center of gravity, the sun is nearly fixed while
the earth moves in an enormous orbit.

\Article{50}{Revolution of the Earth Proved by the Aberration of
Light.}---The second proof that the earth revolves was
made in 1728 when Bradley discovered what is known as
\index[xnames]{Bradley}%
the \textit{aberration of light}. This proof has the advantage of
depending neither on an assumption regarding the density
of the sun nor on the laws of motion.

Suppose rain falls vertically and that one stands still in it;
then it appears to him that it comes straight down. Suppose
he walks rapidly through it; then it appears to fall somewhat
obliquely, striking him in the face. Suppose he rides through
it rapidly; then it appears to descend more obliquely.
%% -----File: 124.png---Folio 99-------

In order to get at the matter qualitatively suppose~$T_1$,
\Figref{34}, is a tube at rest which is to be placed in such a
position that drops of rain shall descend through it without
striking the sides. Clearly it must be vertical. Suppose $T_2$
is a tube which is being carried to the right with moderate
speed. It is evident that the tube must be tilted slightly
in the direction of motion. Suppose
the tube~$T_3$ is being transported still
\begin{wrapfigure}{\WLoc}{1.75in}%[Illustration:]
\Input[1.75in]{124}{png}
\Caption[Explanation of
the aberration of light.]{Fig}{34}
\end{wrapfigure}
more rapidly; it must be given a
greater deviation from the vertical.
The distance~$A_3C_3$ is the distance the
tube moves while the drop descends
its length. Hence $A_3C_3$~is~to~$B_3C_3$ as
the velocity of the tube is to the velocity
\index{Light!velocity of}%
\index{Velocity!of light}%
of the drops. From the given
velocity of the rain and the velocity
of the tube at right angles to the
direction of the rain, the angle of the deviation from the
vertical, namely~$A_3B_3C_3$, can be computed.

Now suppose light from a distant star is considered instead
of falling rain, and let the tube represent a telescope.
All the relations will be qualitatively as in the preceding
case because the velocity of light is not infinite. In fact,
it has been found by experiments on the earth, which in no
way depend upon astronomical observations or theory, that
light travels in a vacuum at the rate of $186,330$ miles per
second. Hence, if the earth moves, stars should appear
displaced in the direction of its motion, the amount of the
displacement depending upon the velocity of the earth and
the velocity of light. Bradley observed such displacements,
at one time of the year in one direction and six months later,
when the earth was on the other side of its orbit, in the
opposite direction. The maximum displacement of a star
for this reason is $20.47$~seconds of arc which, at the present
time, is very easy to observe because measurements of position
are now accurate to one hundredth of this amount.
%% -----File: 125.png---Folio 100-------
Moreover, it is a quantity which does not depend on the
brightness or the distance of the star, and it can be checked
by observing as many stars as may be desired.

The aberration of light not only proves the revolution of
the earth, but its amount enables the astronomer to compute
the speed with which the earth moves. The result is accurate
to within about one tenth of one per cent. Since
the earth's period around the sun is known, this result gives
the circumference of the earth's orbit, from which the distance
from the earth to the sun can be computed. The distance
of the sun as found in this way agrees very closely
with that found by other methods.

There is, similarly, a small aberration due to the earth's
rotation, which, for a point on the earth's equator, amounts
at its maximum to $0.31$~second of arc.

\Article{51}{Revolution of the Earth Proved by the Parallax of
the Stars.}---The most direct method of testing whether or
\index{Parallax!of stars, definition of}%
\index{Revolution of earth}%
not the earth moves is to find whether the direction of a
star is the same when observed at different times of the
year. This was the first method tried, but for a long time
it failed because the stars are exceedingly remote. Even
with all the resources of modern instrumental equipment
fewer than $100$~stars are known which are so near that their
\begin{figure}[hbt]%[Illustration:]
\Input{125}{png}
\Caption[The parallax of~$A$ is the angle~$E_1AE_2$.]{Fig}{35}
\end{figure}%
differences in direction at different times of the year can be
measured with any considerable accuracy. Yet the observations
succeed in a considerable number of cases and really
prove the motion of the earth by purely geometrical means.

The angular difference in direction of a star as seen from
two points on the earth's orbit, which, in the direction perpendicular
to the line to the star, are separated from each
%% -----File: 126.png---Folio 101-------
other by the distance from the earth to the sun, is the \textit{parallax}
of the star. In \Figref{35} let $S$ represent the sun, $A$~a~star,
and $E_1$~and~$E_2$ two positions of the earth such that the
line~$E_1E_2$ is perpendicular to~$SA$ and such that $E_1E_2$~equals~$E_1S$.
Let $E_2B$ be parallel to~$E_1A$. Then, by definition,
the angle~$AE_2B$ is the parallax of~$A$. This angle equals~$E_1AE_2$.
Therefore an alternative definition of the parallax
of a star is that it is the angle subtended by the radius of
the earth's orbit as seen from the star.

It is obvious that the parallax is smaller the more remote
the star. The nearest known star, Alpha Centauri, in the
\index{Alpha Centauri}%
southern heavens, has a parallax of only $0.75$~second of arc,
from which it can be shown that its distance is $275,000$ times
as great as that from the earth to the sun, or about
$25,600,000,000,000$ miles. Suppose a point of light is seen
first with one eye and then with the other. If its distance
from the observer is about $11$~miles, then its difference in
direction as seen with the two eyes is $0.75$~second of arc, the
parallax of Alpha Centauri. This gives an idea of the
difficulties that must be overcome in order to measure the
distance of even the nearest star, especially when it is recalled
that the observations must be extended over several
months. The first success with this method was obtained
by Henderson about 1840.
\index[xnames]{Henderson}%

\Article{52}{Revolution of the Earth Proved by the Spectroscope.}---The
\index{Revolution of earth}%
\index{Spectroscope}%
spectroscope is an instrument of modern invention
which, among other things, enables the astronomer to
determine whether he and the source of light he may be examining
are relatively approaching toward, or receding from,
each other. Moreover, it enables him to measure the
speed of relative approach or recession irrespective of their
distance apart. (\Artref{226}.)

Consider the observation of a star~$A$, \Figref{36}, in the plane
of the earth's orbit when the earth is at~$E_1$, and again when
it is at~$E_2$. In the first position the earth is moving toward
the star at the rate of $18.5$~miles per second, and in the second
%% -----File: 127.png---Folio 102-------
position it is moving away from the star at the same rate.
Since in the case of many stars the %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{127}{png}
\Caption[Motion of the earth toward and from a star.]{Fig}{36}
\end{wrapfigure}
motion can be determined
to within one tenth of a mile per second, the observational
difficulties are not serious. If the star is not in the plane
of the earth's orbit, a correction
must be made in
order to find what fraction
of the earth's motion is
toward or from the star.
The method is independent
of the distance of the star and can be applied to all
stars which are bright enough except those whose directions
from the sun are nearly perpendicular to the plane of the
earth's orbit.

Since 1890 the spectroscope has been so highly perfected
that the spectroscopic proof of the earth's revolution has been
made with thousands of stars. This method gives the
earth's speed, and therefore the circumference of its orbit
and its distance from the sun. It should be stated, however,
that the motion of the earth was long ago so firmly established
that it has not been considered necessary to use the
spectroscope to give additional proof of it. Rather, it has
been used to determine how the stars move individually
(\Artref{273}) and how the sun moves with respect to them as a
whole (\Artref{274}). In order to obtain the motion of a star
with respect to the sun it is sufficient to observe it when
the earth is at~$E$, \Figref{36}. Then correction for the earth's
motion can be applied to the observations made when the
earth is at $E_1$~or~$E_2$.

\Article{53}{Shape of the Earth's Orbit.}---It has been tacitly
\index{Shape of earth's orbit}%
assumed so far that the earth's orbit is a circle with the sun
at the center. If this assumption were true, the apparent
diameter of the sun would be the same all the year because
the earth's distance from it would be constant. On the
other hand, if the sun were not at the center of the circle, or
if the orbit were not a circle, the apparent size of the sun
%% -----File: 128.png---Folio 103-------
would vary with changes in the earth's distance from it. It
is clear that the shape of the earth's orbit can easily be
\index{Earth's orbit}%
established by observation of the apparent diameter and
position of the sun.

It is found from the changes in the apparent diameter of
the sun that the earth's orbit is not exactly a circle. These
changes and the apparent motion of the sun together prove
that the earth moves around it in an elliptical orbit which
differs only a little from a circle. An ellipse is a plane curve
\index{Ellipse, definition of}%
such that the sum of the distances from two fixed points in
its interior, known as \textit{foci}, to any point on its circumference
\index{Foci}%
is always the same.

In \Figref{37}, $E$~represents an ellipse and $F$~and~$F'$ its two
foci. The definition of an ellipse suggests a convenient way
of drawing one. Two
\begin{wrapfigure}{\WLoc}{2.75in}%[Illustration:]
\Input[2.75in]{128}{png}
\Caption[An ellipse.]{Fig}{37}
\end{wrapfigure}
pins are put in drawing
paper at a convenient
distance apart and a
loop of thread somewhat
longer than twice
this distance is placed
over them. Then a
pencil~$P$ is placed inside the thread and the curve is drawn,
keeping the thread taut. The curve obtained in this way is
obviously an ellipse because the length of the thread is
constant, and this means that the sum of the distances
from $F$~and~$F'$ to the pencil~$P$ is the same for all points of
the curve.

\Article{54}{Motion of the Earth in Its Orbit.}---The earth moves
\index{Motion!of earth}%
in its orbit around the sun in such a way that the line drawn
from the sun to the earth sweeps over, or describes, equal
areas in equal intervals of time. Thus, in \Figref{38}, if the
three shaded areas are equal, the intervals of time required
for the earth to move over the corresponding arcs of its orbit
are also equal. This implies that the earth moves fastest
when it is at~$P$, the point nearest the sun, and slowest when
%% -----File: 129.png---Folio 104-------
it is at~$A$, the point farthest from the sun. The former is
called the \textit{perihelion point}, and the latter the \textit{aphelion point}.
\index{Aphelion point}%
\index{Perihelion point!definition of}%

It is obvious that an ellipse may be very nearly round or
much elongated. The extent of the elongation is defined
by what is known as the eccentricity, which is the ratio $CS$~divided~by~$CP$.
\index{Eccentricity}%
If the line~$CS$ is very short for a given line~$CP$,
the eccentricity is small and the ellipse is nearly circular.
In fact, a circle may be considered as being an ellipse whose
eccentricity is zero.

The eccentricity of the earth's orbit is very slight, being
\index{Earth's orbit}%
\index{Eccentricity!of earth's orbit}%
only $0.01677$. %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.375in}
\Input[2.375in]{129}{png}
\Caption[The earth moves so that
the line from the sun to the earth
sweeps over equal areas in equal intervals
of time.]{Fig}{38}
\index{Areas, law of}%
\index{Law!of areas}%
\end{wrapfigure}
That is, the distance~$CS$, \Figref{38}, in the
case of the earth's orbit is
about $\frac{1}{60}$~of~$CP$. Hence,
if the earth's orbit were
drawn to scale, its elongation
would be so slight that
it would not be obvious by
simple inspection.

The question arises as to
what occupies the second
focus of the elliptical orbit
of the earth. The answer
is that there is no body
there; nor is it absolutely
fixed in position because the earth's orbit is continually
modified to a very slight extent by the attractions of the
other planets.

It is easy to see how the earth might revolve around the
sun in a circle if it were started with the right velocity.
But it is not so easy to understand how it can revolve in an
elliptical orbit with the sun at one of the foci. While the
matter cannot be fully explained without some rather formidable
mathematical considerations, it can, at least, be
made plausible by a little reflection. Suppose a body is at~$P$,
\Figref{38}, and moving in the direction~$PT$. If its speed is
exactly such that its centrifugal acceleration balances the
%% -----File: 130.png---Folio 105-------
attraction of the sun, it will revolve around the sun in a
circle.

But suppose the initial velocity is a little greater than that
required for motion in a circular orbit. In this case the sun's
attraction does not fully counterbalance the centrifugal
acceleration, and the distance of the body from the sun
increases. Consider the situation when the body has
moved around in its orbit to the point~$Q$. At this point the
centrifugal acceleration is still greater than the attraction
of the sun, and the distance of the body from the sun is
increasing. It will be observed that the sun's attraction no
longer acts at right angles to the direction of motion of the
body, but that it tends to diminish its speed. It can be
shown by a suitable mathematical discussion, which must be
omitted here, that the diminution of the speed of the body
more than offsets the decreasing attraction of the sun due to
the increasing distance of the body, and that in elliptical
orbits a time comes in which the attraction and the centrifugal
acceleration balance. Suppose this takes place
when the body is at~$R$. Since its speed is still being diminished
by the attraction of the sun from that point on, the
attraction will more than counterbalance the centrifugal
acceleration. Eventually at~$A$ the distance of the body from
the sun will cease to increase. That is, it will again be moving
at right angles to a line joining it to the sun; but its
velocity will be so low that the sun will pull it inside of a circular
orbit tangent at that point. It will then proceed
back to the point~$P$, its velocity increasing as it decreases in
distance while going from $P$~to~$A$. The motion out from the
sun and back again is analogous to that of a ball projected
obliquely upward from the surface of the earth; its speed
decreases to its highest point, and then increases again as it
it descends.

\Article{55}{Inclination of the Earth's Orbit.}---The plane of the
\index{Ecliptic!obliquity of}%
\index{Inclination of earth's orbit}%
\index{Obliquity of ecliptic}%
earth's orbit is called the \textit{plane of the ecliptic}, and the line in
which this plane intersects the sky is called the \textit{ecliptic}. In
%% -----File: 131.png---Folio 106-------
\Figref{39} it is the circle~$RAR'V$. The plane of the earth's
equator cuts the sky in a circle which is called the \textit{celestial
equator}. In the figure it is~$QAQ'V$. The angle between the
\index{Equator}%
plane of the equator and the plane of the ecliptic is $23.5$~degrees.
This angle is called the \textit{inclination or obliquity of
the ecliptic}.

The point on the sky pierced by a line drawn perpendicular
to the plane of the ecliptic is called the \textit{pole of the ecliptic},
\index{Ecliptic!pole of}%
\index{Pole!of ecliptic}%
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{131}{png}
\Caption[The ecliptic, celestial equator, and celestial pole.]{Fig}{39}
\end{figure}%
and the point where the earth's axis, extended, pierces the
sky is called the pole of the equator or, simply, \textit{the celestial
pole}. The orbit of the earth is so very small in comparison
\index{Pole}%
with the distance to the sky that the motion of the earth in
its orbit has no sensible effects on the position of the celestial
pole and it may be regarded as a fixed point. In \Figref{39},
$P'$~is the pole of the ecliptic and $P$~is the pole of the
equator. The angle between these lines is the same as the
angle between the planes, or $23.5$~degrees.

Now consider the precession of the equinoxes (\Artref{47}).
%% -----File: 132.png---Folio 107-------
The pole of the ecliptic remains fixed. As a consequence of
the precession of the equinoxes the pole~$P$ describes a circle
around it with a radius of $23.5$~degrees, and the direction of
the motion is opposite to that of the direction of the motion of
the earth around the sun. Or, the points $A$~and~$V$, which are
the equinoxes, continually move backward along the ecliptic
in the direction opposite to that of the revolution of the earth.

\Article{56}{Cause of the Seasons.}---Let the upper part of the
\index{Seasons!cause of}%
earth~$E$, \Figref{39}, represent its north pole. When the earth
is at~$E_1$ its north pole is turned away from the sun so that
it is in continual darkness; but, on the other hand, the
south pole is continually illuminated. At this time of the
year the northern hemisphere has its winter and the southern
hemisphere its summer. The conditions are reversed
when the earth is at~$E_3$. When the earth is at~$E_2$ the plane
of its equator passes through the sun, and it is the spring
season in the northern hemisphere. Similarly, when the
earth is at~$E_4$ the equator also passes through the sun and
it is autumn in the northern hemisphere.

Consider a point in a medium northern latitude when the
earth is %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.25in}
\Input[2.25in]{132}{png}
\Caption[Effects of obliquity of sun's rays.]{Fig}{40}
\end{wrapfigure}
at~$E_1$, and the same position again when the earth is
at~$E_3$. At~$E_1$ the sun's rays,
when it is on the meridian,
strike the surface of the earth
at the point in question more
obliquely than when the earth
is at~$E_3$. Their intensity is,
therefore, less in the former
case than it is in the latter;
for, in the former, the rays
whose cross section is~$PQ$,
\Figref{40}, are spread out over
the distance~$AB$, while in the latter they extend over the
smaller distance~$A'B$. This fact, and the variations in the
number of hours of sunshine per day (\Artref{58}), cause the
changes in the seasons.
%% -----File: 133.png---Folio 108-------

\Article{57}{Relation of the Position of the Celestial Pole to the
Latitude of the Observer.}---In order to make clear the
\index{Altitude!of pole}%
\index{Equator!altitude of}%
\index{Pole!altitude of}%
climatic effects of certain additional factors, consider the
apparent position of the celestial pole as seen by an observer
in any latitude. Since the pole is the place where
the axis of the earth, extended, pierces the sky, it is obvious
that, if an observer were at a pole of the earth, the celestial
equator would be on his horizon and the celestial pole would
be at his zenith; while, if he were on the equator of the
earth, the celestial equator would pass through his zenith,
and the celestial poles would be on his horizon, north and
south.

Consider an observer at~$O$, \Figref{41}, in latitude $l$~degrees
north of the equator. The line~$P'P$ points toward the
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4.5in]{133}{png}
\Caption[The altitude of the celestial pole equals the latitude of the
observer.]{Fig}{41}
\end{figure}%
north pole of the sky. Since the sky is extremely far away
compared to the dimensions of the earth, the line from~$O$
to the celestial pole is essentially parallel to~$P'P$. The angle
between the plane of the horizon and the line to the pole
is called the altitude of the pole. Since $ON$ is perpendicular
%% -----File: 134.png---Folio 109-------
to~$EO$, and $P'P$~is perpendicular to~$EQ$, it follows that $a$~equals~$l$,
or \textit{the altitude of the pole equals the latitude of the
observer}.

Consider also the altitude of the equator where it crosses
\index{Altitude!of equator}%
the meridian directly south of the observer. It is represented
by~$b$ in the diagram. It easily follows that $b = 90°- l$,
or the altitude of the equator where it crosses the meridian
equals $90°$~minus the latitude of the observer.

\Article{58}{The Diurnal Circles of the Sun.}---It is evident from
\index{Diurnal circles}%
\Figref{39} that when the earth is in the position~$E_1$, the
sun is seen south of the celestial equator; when the earth is
at $E_2$~or~$E_4$, the sun appears to be on the celestial equator;
and when the earth is at~$E_3$, the sun is seen north of the celestial
equator. If the equator is taken as the line of reference
and the apparent motion of the sun is considered, its
\begin{figure}[hbt]%[Illustration:]
\Input{134}{png}
\Caption[Relation of ecliptic and celestial equator.]{Fig}{42}
\end{figure}%
position with respect to the equator is represented in \Figref{42}.
The sun appears to be at~$V$ when the earth is at~$E_2$,
\Figref{39}. The point~$V$ is called the \textit{vernal equinox}, and
\index{Equinoxes}%
\index{Equinoxes!autumnal}%
\index{Equinoxes!vernal}%
\index{Vernal equinox}%
the sun has this position on or within one day of March~21.
The sun is at~$S$, called the \textit{summer solstice}, when the earth is
\index{Solstices}%
at~$E_3$, \Figref{39}, and it is in this position about June~21.
The sun is at~$A$, called the \textit{autumnal equinox}, when the earth
\index{Autumnal equinox}%
is at~$E_4$, and it has this position about September~23. Finally,
the sun is at~$W$, which is called the \textit{winter solstice}, when the
earth is at~$E_1$. The angle between the ecliptic and the
equator at $V$~and~$A$ is~$23°.5$; and the perpendicular distance
between the equator and the ecliptic at $S$~and~$W$ is~$23°.5$.
From these relations and those given in \Artref{57} the
diurnal paths of the sun can readily be constructed.
%% -----File: 135.png---Folio 110-------

Suppose the observer is in north latitude~$40°$. Let~$O$,
\Figref{43}, represent his position, and suppose his horizon is~$\mathit{SWNE}$,
where the letters stand for the four cardinal points.
Then it follows from the relation of %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{135}{png}
\Caption[Diurnal circles of the sun.]{Fig}{43}
\end{wrapfigure}
the altitude of the pole
to the latitude of the observer
that~$NP$, where $P$~represents
the pole, is~$40°$.
Likewise~$SQ$, where $Q$~represents
the place at which the
equator crosses the meridian,
is~$50°$. The equator is everywhere
$90$~degrees from the
pole and in the figure is
represented by the circle~$QWQ'E$.

Suppose the sun is on the
equator at $V$~or~$A$, \Figref{42}.
Since it takes six months for it to move from $V$~to~$A$, its
motion in one day is very small and may be neglected in the
present discussion. Hence, without serious error, it may be
supposed that the sun is on the equator all day. When this
is the case, its apparent diurnal path, due to the rotation
of the earth, is~$EQWQ'$, \Figref{43}. It will be noticed that
it rises directly in the east and sets directly in the west,
being exactly half the time above the horizon and half the
time below it. This is true whatever the latitude of the
observer. But the height at which it crosses the meridian
depends, of course, upon the latitude of the observer, and is
greater the nearer he is to the earth's equator.

Suppose now that it is June~21 and that the sun is at the
summer solstice~$S$, \Figref{42}. It is then $23°.5$~north of the
equator and will have essentially this distance from the
equator all day. The diurnal path of the sun in this case
is~$E_1Q_1W_1Q_1'$, \Figref{43}, which is a circle parallel to, and $23°.5$~north
of, the equator. In this case the sun rises north of
the east point by the angle~$EE_1$, and sets an equal distance
%% -----File: 136.png---Folio 111-------
north of the west point. Moreover, it is more than half
the twenty-four hours above the horizon. The fact that its
altitude at noon is $23°.5$~greater than it is when the sun is
on the equator, and the longer time from sunrise to sunset,
are the reasons that the temperature is higher in the summer
than in the spring or autumn. It is obvious from \Figref{43}
that the length of the day from sunrise to sunset depends
upon the latitude of the observer, being greater the farther
he is from the earth's equator.

When the sun is at the winter solstice~$W$, \Figref{42}, its
diurnal path is~$E_2Q_2W_2Q_2'$. At this time of the year it rises
in the southeast, crosses the meridian at a low altitude, and
sets in the southwest. The time during which it is above the
horizon is less than that during which it is below the horizon,
and the difference in the two intervals depends upon the
latitude of the observer.

\Article{59}{Hours of Sunlight in Different Latitudes.}---It follows
\index{Sunlight in all latitudes}%
from \Figref{43} that when the sun is north of the celestial
equator, an observer north of the earth's equator receives
more than $12$~hours of sunlight per day; and when the sun
is south of the celestial equator, he receives less than $12$~hours
of sunlight per day. It might be suspected that the
excess at one time exactly balances the deficiency at the
other. This suspicion is strengthened by the obvious fact
that, a point at the equator receives $12$~hours of sunlight
per day every day in the year, and at the pole the sun
shines continuously for six months and is below the horizon
for six months, giving the same total number of hours of
sunshine in these two extreme positions on the earth. The
conclusion is correct, for it can be shown that the total
number of hours of sunshine in a year is the same at all
places on the earth's surface. This does not, of course,
mean that the same amount of sunshine is received at all
places, because at positions near the poles the sun's rays
always strike the surface very obliquely, while at positions
near the equator, for at least part of the time they strike
%% -----File: 137.png---Folio 112-------
the surface perpendicularly. The intensity of sunlight at
the earth's equator when the sun is at the zenith is $2.5$~times
its maximum intensity at the earth's poles; and the
amount received per unit area on the equator in a whole
year is about $2.5$~times that received at the poles.

If the obliquity of the ecliptic were zero, the sun would
pass every day through the zenith of an observer at the
earth's equator; but actually, it passes through the zenith
only twice a year. Consequently, the effect of the obliquity
of the ecliptic is to diminish the amount of heat received on
the earth's equator. Therefore some other places on the
earth, which are obviously the poles, must receive a larger
amount than they would if the equator and the ecliptic
were coincident. That is, the obliquity of the ecliptic
causes the climate to vary less in different latitudes than it
would if the obliquity were zero.

\Article{60}{Lag of the Seasons.}---From the astronomical point
\index{Seasons!lag of}%
\index{Seasons!length of}%
of view March~21 and September~23, the times at which the
sun passes the two equinoxes are corresponding seasons.
The middle of the summer is when the sun is at the summer
solstice, June~21, and the middle of the winter when it is at
the winter solstice, December~21. But from the climatic
standpoint March~21 and September~23 are not corresponding
seasons, and June~21 and December~21 are not the
middle of summer and winter respectively. The climatic
seasons lag behind the astronomical.

The cause of the lag of the seasons is very simple. On
June~21 any place on the earth's surface north of the Tropic
of Cancer is receiving the largest amount of heat it gets at
any time in the year. On account of the blanketing effect
of the atmosphere, less heat is radiated than is received;
hence the temperature continues to rise. But after that
date less and less heat is received as day succeeds day;
on the other hand, more is radiated daily, for the hotter a
body gets, the faster it radiates. In a few weeks the loss
equals, and then exceeds, that which is received, after which
%% -----File: 138.png---Folio 113-------
the temperature begins to fall. The same reasoning applies
for all the other seasons. This phenomenon is quite analogous
to the familiar fact that the maximum daily temperature
normally occurs somewhat after noon.

If there were no atmosphere and if the earth radiated heat
as fast as it was acquired, there would be no lag in the
seasons. In high altitudes, where the air is thin and dry,
this condition is nearly realized and the lag of the seasons is
small, though the phenomenon is very much disturbed by the
great air currents which do much to equalize temperatures.

\Article{61}{The Effect of the Eccentricity of the Earth's Orbit
on the Seasons.}---It is found from observations of the
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{138}{png}
\Caption[Because of the eccentricity of the earth's orbit, summers in the
northern hemisphere are longer than the winters.]{Fig}{44}
\end{figure}%
apparent diameter of the sun that the earth is at its perihelion
on or about January~3, and at its aphelion on or about
July~4. It follows from the way the earth describes its
orbit, as explained in \Artref{54}, that the time required for it
to move from $P$ to~$Q$, \Figref{44}, is exactly equal to that
required for it to move from $Q$ to~$P$. But the line joining
the vernal and autumnal equinoxes, which passes through
the sun, is nearly at right angles to the line joining the
perihelion and aphelion points, and is represented by~$VA$,
\Figref{44}. Since the area swept over by the radius from the
sun to the earth, while the earth is moving over the arc~$VQA$,
%% -----File: 139.png---Folio 114-------
is greater than the area described while it goes over
the arc~$APV$, it follows that the interval of time in the former
case is greater than that in the latter. That is, since
$V$~is the vernal equinox, the summer in the northern hemisphere
is longer than the winter. The difference in length
is greatly exaggerated in the figure, but it is found that the
interval from vernal equinox to autumnal equinox is actually
about $186.25$~days, while that from autumnal equinox to
vernal equinox is only $179$~days. The difference is, therefore,
about $7.25$~days.

Since the summers are longer than the winters in the
northern hemisphere while the reverse is true in the southern
hemisphere, it might be supposed that points in corresponding
latitudes receive more heat in the northern hemisphere
than in the southern hemisphere. But it will be
noticed from \Figref{44} that, although the summer is longer in
the northern hemisphere than it is in the southern, the earth
is then farther from the sun. It can be shown from a discussion
of the way in which the earth's distance from the
sun varies and from the rate at which it moves at different
points in its orbit, that the longer summer season in
the northern hemisphere is exactly counterbalanced by the
greater distance the earth is then from the sun. The result
is that points in corresponding latitudes north and south of
the equator receive in the whole year exactly the same
amount of light and heat from the sun.

There is, however, a difference in the seasons in the northern
and southern hemispheres which depends upon the eccentricity
of the earth's orbit. When the sun is north of
the celestial equator so that its rays strike the surface in
northern latitudes most nearly perpendicularly, a condition
that tends to produce high temperatures, the greater distance
of the sun reduces them somewhat. Therefore, the
temperature does not rise in the summer so high as it would
if the earth's orbit were circular. In the winter time, at
the same place, when the sun's rays strike the surface slantingly,
%% -----File: 140.png---Folio 115-------
the earth is nearer to the sun than the average, and
consequently the temperature does not fall so low as it would
if the eccentricity of the earth's orbit were zero. The result
is that the seasonal variations in the northern hemisphere
are less extreme than they would be if the earth's
orbit were circular; and, for the opposite reason, in the
southern hemisphere they are more extreme. This does
not mean that actually there are greater extremes in the
temperature south of the equator than there are north of the
equator. The larger proportion of water in the southern
hemisphere, which tends to make temperature conditions
uniform, may more than offset the effects of the eccentricity
of the earth's orbit.

The attractions of the other planets for the earth change
very slowly both the eccentricity and the direction of the
perihelion of the earth's orbit. It has been shown by
mathematical discussions of these influences that the relation
of the perihelion to the line of the equinoxes will be
\index{Equinoxes!precession of}%
\index{Precession of equinoxes}%
reversed in about $50,000$ years. In fact, there is a cyclical
change in these relations with a period of somewhat more
than $100,000$ years. It was suggested by James Croll that
\index[xnames]{Croll}%
the condition of long winter and short summer, such as
now prevails in the southern hemisphere, especially when the
eccentricity of the earth's orbit was greatest, produced the
glaciation which large portions of the earth's surface are
known to have experienced repeatedly in the past. This
theory has now been abandoned because, on other grounds,
it is extremely improbable.

\Article{62}{Historical Sketch of the Motions of the Earth.}---The
history of the theory of the motion of the earth is intimately
associated with that of the motions of the planets,
and the whole problem of the relations of the members of
the solar system to one another may well be considered
together.

The planets are readily found by observations, even
without telescopes, to be moving among the stars. Theories
%% -----File: 141.png---Folio 116-------
respecting the meanings of these motions date back to the
very dawn of history. Many of the simpler phenomena
of the sun, moon, and planets had been carefully observed
by the Chaldeans and Egyptians, but it remained for the
brilliant and imaginative Greeks to organize and generalize
experience and to develop theories. Thales is credited with
\index[xnames]{Thales}%
having introduced Egyptian astronomy into Greece more
than $600$~years before the Christian era. The Pythagoreans
followed a century later and made important contributions
to the philosophy of the science, but very few to its data.
Their success was due to the weakness of their method; for,
not being too much hampered by the facts of observation,
they gave free rein to their imaginations and introduced
numerous ideas into a budding science which, though often
erroneous, later led to the truth. They believed that the
earth was round, immovable, at the center of the universe,
and that the heavenly bodies moved around it on crystalline
spheres.

Following the Pythagoreans came Eudoxus (409--356~\BC),
\index[xnames]{Pythagoras}%
\index[xnames]{Eudoxus}%
Aristotle (384--322~\BC), and Aristarchus (310--250~\BC),
\index[xnames]{Aristarchus}%
\index[xnames]{Aristotle}%
who were much more scientific, in the modern sense of
the term, and who made serious attempts to secure perfect
agreement between the observations and theory. Aristarchus
was the first to show that the apparent motions of
the sun, moon, and stars could be explained by the theory
that the earth rotates on its axis and revolves around the
sun. Aristotle's objection was that if this theory were true
the stars would appear to be in different directions at different
times of the year; the reply of Aristarchus was that the
stars were infinitely remote, a valid answer to a sensible
criticism. Aristarchus was a member of the Alexandrian
school, founded by Alexander the Great, and to which the
\index[xnames]{Alexander the Great}%
geometer Euclid belonged. His astronomy had the formal
\index[xnames]{Euclid}%
perfection which would be natural in a school where geometry
was so splendidly systematized that it has required almost
no modification for $2000$~years.
%% -----File: 142.png---Folio 117-------

The rather formal astronomy which resulted from the
influence of the mathematics of Alexandria was succeeded
by an epoch in which the greatest care was taken to secure
observations of the highest possible precision. Hipparchus
\index[xnames]{Hipparchus}%
(180--110~\BC), who belonged to this period, is universally
conceded to have been the greatest astronomer of antiquity.
His observations in both extent and accuracy had never been
approached before his time, nor were they again equaled
until the time of the Arab, Albategnius (850--929~\AD).
\index[xnames]{Albategnius}%
He systematically and critically compared his observations
with those of his predecessors. He developed trigonometry
without which precise astronomical calculations cannot be
made. He developed an ingenious scheme of eccentrics
and epicycles (which will be explained presently) to represent
the motions of the heavenly bodies.

Ptolemy (100--170~\AD) was the first astronomer of note
\index[xnames]{Ptolemy}%
after Hipparchus, and the last important astronomer of the
Alexandrian period. From his time until that of Copernicus
\index[xnames]{Copernicus}%
(1473--1543) not a single important advance was made
in the science of astronomy. From Pythagoras to Ptolemy
was $700$~years, from Ptolemy to Copernicus was $1400$~years,
and from Copernicus to the present time is $400$~years. The
work of Ptolemy, which is preserved in the \textit{Almagest} (\textit{i.e.}\ The
\index{Almagest}%
Greatest Composition), was the crowning achievement
of the second period, and that of Copernicus was the first
of the modern period; or, perhaps it would be more accurate
to say that the work of Copernicus constituted the transition
from ancient to modern astronomy, which was really begun
by Kepler (1571--1630) and Galileo (1564--1642).
\index[xnames]{Galileo}%
\index[xnames]{Kepler}%

The most elaborate theory of ancient times for explaining
the motions of the heavenly bodies was due to Ptolemy.
He supposed that the earth was a fixed sphere situated at
the center of the universe. He supposed that the sun and
moon moved around the earth in circles. It does not seem
to have occurred to the ancients that the orbits of the heavenly
bodies could be anything but circles, which were supposed
%% -----File: 143.png---Folio 118-------
to be perfect curves. In order to explain the varying distances
of the sun and moon, which were proved by the variations
in their apparent diameters, he supposed that the
earth was somewhat out of the centers of the circles in which
the various bodies were supposed to move around it. It is
clear that such motion, called eccentric motion, would have
\index{Eccentric motion}%
considerable similarity to motion in an ellipse around a body
at one of its foci.

Another device used by Ptolemy for the purpose of explaining
\index{Ptolemaic theory}%
\index[xnames]{Ptolemy}%
the motions of the planets was the epicycle. In
\index{Epicycle}%
this system the body was supposed to travel with uniform
speed along a small circle, the epicycle, whose center moved
with uniform speed along a large circle, the deferent, around
\index{Deferent}%
the earth. By carefully adjusting the dimensions and inclinations
of the epicycle and the deferent, together with
the rates of motion along them, Ptolemy succeeded in getting
a very satisfactory theory for the motions of the sun, moon,
and planets so far as they were then known.

Copernicus was not a great, or even a skillful, observer,
\index{Copernican theory}%
\index[xnames]{Copernicus}%
but he devoted many years of his life to the study of the
apparent motions of the heavenly bodies with a view to
discovering their real motions. The invention of printing
about 1450 had made accessible the writings of the Greek
philosophers, and Copernicus gradually became convinced
that the suggestion that the sun is the center, and that the
earth both rotates on its axis and revolves around the sun,
explains in the simplest possible way all the observed phenomena.
It must be insisted that Copernicus had no rigorous
proof that the earth revolved, but the great merit of his work
consisted in the faithfulness and minute care with which
he showed that the heliocentric theory would satisfy the
observation as well as the geocentric theory, and that from
the standpoint of common sense it was much more plausible.

The immediate successor of Copernicus was Tycho Brahe
\index[xnames]{Tycho Brahe}%
(1546--1610), who rejected the heliocentric theory both for
theological reasons and because he could not observe any
%% -----File: 144.png---Folio 119-------
displacements of the stars due to the annual motion of the
earth. He contributed nothing of value to the theory of
astronomy, but he was an observer of tireless industry whose
work had never been equaled in quality or quantity. For
example, he determined the length of the year correctly to
within one second of time.

Between the time of Tycho Brahe and that of Newton
\index[xnames]{Newton}%
\index[xnames]{Tycho Brahe}%
(1643--1727), who finally laid the whole foundation for mechanics
and particularly the theory of motions of the planets,
there lived two great astronomers, Galileo (1564--1642) and
\index[xnames]{Galileo}%
Kepler (1571--1630), who by work in quite different directions
\index[xnames]{Kepler}%
led to the complete overthrow of the Ptolemaic theory
of eccentrics and epicycles. These two men had almost no
characteristics in common. Galileo was clear, penetrating,
brilliant; Kepler was mystical, slow, but endowed with unwearying
industry. Galileo, whose active mind turned in
many directions, invented the telescope and the pendulum
clock, to some extent anticipated Newton in laying the
foundation of dynamics, proved that light and heavy bodies
fall at the same rate, covered the field of mathematical and
physical science, and defended the heliocentric theory in a
matchless manner in his \textit{Dialogue on the Two Chief Systems
of the World}. Kepler confined his attention to devising a
\index{Dialogues of Galileo}%
\index{Galileo's Dialogues}%
theory to account for the apparent motions of sun and planets,
especially as measured by his preceptor, Tycho Brahe. With
an honesty and thoroughness that could not be surpassed,
he tested one theory after another and found them unsatisfactory.
Once he had reduced everything to harmony except
some of the observations of Mars by Tycho Brahe
(of course without a telescope), and there the discrepancy
was below the limits of error of all observers except Tycho
Brahe. Instead of ascribing the discrepancies to minute
errors by Tycho Brahe, he had implicit faith in the absolute
reliability of his master and passed on to the consideration
of new theories. In his books he set forth the complete
record of his successes and his failures with a childlike candor
%% -----File: 145.png---Folio 120-------
not found in any other writer. After nearly twenty years
of computation he found the three laws of planetary motion
(\Artref{145}) which paved the way for Newton. Astronomy
\index[xnames]{Newton}%
owes much to the thoroughness of Kepler.


\Section{VI}{QUESTIONS}

1. Note carefully the position of any conspicuous star at 8~\PM\
and verify the fact that in a month it will be $30°$~farther west at the
same time in the evening.

2. From which of the laws of motion does it follow that two
attracting bodies revolve around their common center of gravity?

3. What are the fundamental principles on which each of the
four proofs of the revolution of the earth depend? How many
really independent proofs of the revolution of the earth are there?

4. Which of the proofs of the revolution of the earth give also
the size of its orbit?

5. The aberration of light causes a star apparently to describe a
small curve near its true place; what is the character of the curve if
the star is at the pole of the ecliptic? If it is in the plane of the
earth's orbit?

6. Discuss the questions corresponding to question 5 for the
small curve described as a consequence of the parallax of a star.
Do aberration and parallax have their maxima and minima at the
same times, or are their phases such that they can be separated?

7. Discuss the climatic conditions if the day were twice as long
as it is at present.

8. If the eccentricity of the earth's orbit were zero, in what
respects would the seasons differ from those which we have now?

9. If the inclination of the equator to the ecliptic were zero, in
what respects would the seasons differ from those which we have now?

10. Suppose the inclination of the equator to the ecliptic were~$90°$;
describe the phenomena which would correspond to our day
and to our seasons.

11. Draw diagrams giving the diurnal circles of the sun when the
sun is at an equinox and both solstices, for an observer at the earth's
equator, in latitude $75°$~north, and at the north pole.

12. At what times of the year is the sun's motion northward or
southward slowest (see \Figref{42})? For what latitude will it then pass
through or near the zenith? This place will then have its highest
temperature. Compare the amount of heat it receives with that
received by the equator during an equal interval when the sun is
near the equinox. Which will have the higher temperature?

\normalsize

%% -----File: 146.png---Folio 121-------


\Chapter{IV}{Reference Points and Lines}
\index{Reference points and lines}%

\Article{63}{Object and Character of Reference Points and
Lines.}---One of the objects at which astronomers aim is a
knowledge of the motions of the heavenly bodies. In order
fully to determine their motions it is necessary to learn how
their apparent positions change with the time. Another
important problem of the astronomer is the measurement of
the distances of the celestial objects, for without a knowledge
of their distances, their dimensions and many other of their
properties cannot be determined. In order to measure the
distance of a celestial body it is necessary to find how its
apparent direction differs as seen from different points on
the earth's surface (\Artref{123}), or from different points in the
the earth's orbit (\Artref{51}). For both of these problems it is
obviously important to have a precise and convenient means
of describing the apparent positions of the heavenly bodies.

Not only are systems of reference points and lines important
for certain kinds of serious astronomical work, but they
are also indispensable to those who wish to get a reasonable
familiarity with the wonders of the sky. Any one who has
traveled and noticed the stars has found that their apparent
positions are different when viewed from different latitudes
on the earth. It can be verified by any one on a single clear
evening that the stars apparently move during the night.
And if the sky is examined at the same time of night on different
dates the stars will be found to occupy different places.
That is, there is considerable complexity in the apparent
motions of the stars, and any such vague directions as are
ordinarily made to suffice for describing positions on the earth
would be absolutely useless when applied to the heavens.
%% -----File: 147.png---Folio 122-------

Although the celestial bodies differ greatly in distance
from the earth, some being millions of times as far away as
others, they all seem to be at about the same distance on a
spherical surface, which is called the \textit{celestial sphere}. In
\index{Celestial sphere}%
fact, the ancients actually assumed that the stars are attached
to a crystalline sphere. The celestial sphere is not a
sphere at any particular large distance; it is an imaginary
surface beyond all the stars and on which they are all projected,
at such an enormous distance from the earth that
two lines drawn toward a point on it from any two points
on the earth, or from any two points on the earth's orbit,
are so nearly parallel that their convergence can never be
detected with any instrument. For short, it is said to be
an infinite sphere.

While the real problem giving rise to reference points and
lines is that of describing accurately and concisely the directions
of celestial objects from the observer, its solution is
equivalent to describing their apparent positions on the
celestial sphere. Since it is much easier to imagine a position
on a sphere than it is to think of the direction of lines radiating
from its center, the heavenly bodies are located in direction
by describing their projected positions on the celestial
sphere. Fortunately, a similar problem has been solved in
locating positions on the surface of the earth, and the astronomical
problem is treated similarly.

\Article{64}{The Geographical System.}---Every one is familiar
\index{Geographical system}%
with the method of locating a position on the surface of the
earth by giving its latitude and longitude. Therefore it will
be sufficient to point out here the essential elements of this
process.

The geographical lines that cover the earth are composed
of two distinct sets which have quite different properties.
The first set consists of the equator, which is a great circle,
and the parallels of latitude, which are small circles parallel
to the equator. If the equator is defined in any way, the
two associated poles, which are $90°$~from it, are also uniquely
%% -----File: 148.png---Folio 123-------
located. Or, if there is any natural way in which the poles
are defined, the equator is itself given. In the case of the
earth the poles are the points on its surface at the ends of
its axis of rotation, and these points consequently have
properties not possessed by any others. If they are regarded
as being defined in this way, the equator is defined as the
great circle $90°$~from them.

The second set of circles on the surface of the earth consists
of great circles, called meridians, passing through the
poles and cutting the equator at right angles. All the
meridians are similar to one another, and a convenient
one is chosen as a line from which to measure longitudes.
The distances from the fundamental meridian to the other
meridians are given in degrees and are most conveniently
measured in arcs along the equator.

The fundamental meridian generally used as a standard
is that one which passes through the observatory at Greenwich,
England. However, in many cases, other countries
use the meridians of their own national observatories. For
example, in the United States, the meridian of the Naval
Observatory at Washington is frequently employed.
\index{Naval Observatory}%

In order to locate uniquely a point on the surface of the
earth, it is sufficient to give its \textit{latitude}, which is the angular
\index{Latitude!astronomical}%
distance from the equator, and its \textit{longitude}, which is the
\index{Longitude}%
angular distance east or west of the standard meridian.
These distances are called the \textit{coördinates} of the point. It
\index{Coordinates@{Coördinates}}%
is customary to measure the longitude either east or west, as
may be necessary in order that it shall not be greater than~$180°$.
In many respects it would be simpler if longitude were
counted from the fundamental meridian in a single direction.

\Article{65}{The Horizon System.}---The horizon, which separates
\index{Horizon}%
the visible portion of the sky from that which is invisible, is
a curve that cannot escape attention. If it were a great
circle, it might be taken as the principal circle for a system of
coördinates on the sky. But on the land the contour of
the horizon is subject to the numerous irregularities of surface,
%% -----File: 149.png---Folio 124-------
and on the sea it is always viewed from at least some
small altitude above the surface of the water. For this
\index{Altitude}%
reason it is called the sensible horizon to distinguish it from
the astronomical horizon, which will be defined in the next
paragraph.

The direction defined by the plumb line at any place
is perfectly definite. The point where the plumb line, if
extended upward, pierces the celestial sphere is called the
\textit{zenith}, and the opposite point is called %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.5in}
\Input[2.5in]{149}{png}
\Caption[The horizon system.]{Fig}{45}
\end{wrapfigure}
the \textit{nadir}. These two
\index{Nadir}%
\index{Zenith}%
points will be taken as poles of the first set of coördinates in
the horizon system, and the horizon is defined as the great
circle on the celestial sphere $90°$~from the zenith. The small
circles parallel to the horizon are called \textit{altitude circles} or,
sometimes, almucantars.
\index{Almucantars}%

The second set of circles in the horizon system consists of
the great circles which pass through the zenith and the
nadir and cut the horizon at right angles. They are called
\textit{vertical circles}. The fundamental vertical circle from which
\index{Vertical circles}%
distances along the horizon are measured is that one which
passes through the pole of
the sky; that is, the point
where the axis of the earth,
prolonged, cuts the celestial
sphere, and it is called the
\textit{meridian}.
\index{Meridian}%

The coördinates of a point
in the horizon system are (\textit{a})~the
angular distance above or
below the horizon, which is
called \textit{altitude}, and (\textit{b})~the
angular distance west from
the south point along the
horizon to the place where the vertical circle through the
object crosses the horizon. This is called the \textit{azimuth} of
\index{Azimuth}%
the object.

In \Figref{45}, $O$~represents the position of the observer,
%% -----File: 150.png---Folio 125-------
$\mathit{SWNE}$~his horizon, and $Z$~his zenith. The point where the
earth's axis pierces the sky is perfectly definite and is represented
by~$P$ in the diagram. The vertical circle which passes
through $Z$~and~$P$ is the meridian. The points at which the
meridian cuts the horizon are the north and south points.
The north point, for positions in the northern hemisphere
of the earth, is the one nearest the pole~$P$. In this way the
cardinal points are uniquely defined.

Consider a star at~$A$. Its altitude is~$BA$, which, in this
case, is about~$40°$, and its azimuth is~$\mathit{SWNEB}$, which, in
this case, is about~$300°$. It is, of course, understood that
the object might be below the horizon and the azimuth
might be anything from zero to~$360°$. When the object is
above the horizon, its altitude is considered as being positive,
and when below, as being negative.

\Article{66}{The Equator System.}---The poles of the sky have
\index{Equator}%
been defined as the points where the earth's axis prolonged
intersects the celestial sphere. It might be supposed at
first that these would not be conspicuous points because the
earth's axis is a line which of course cannot be seen. But
the rotation of the earth causes an apparent motion of
the stars around the pole of the sky. Consequently, an
equally good definition of the poles is that they are the
common centers of the diurnal circles of the stars. That
pole which is visible from the position of an observer is a
point no less conspicuous than the zenith.

The celestial equator is a great circle $90°$~from the poles
of the sky. An alternative definition is that the celestial
equator is the great circle in which the plane of the earth's
equator intersects the celestial sphere. The small circles
parallel to the celestial equator are called \textit{declination circles}.

The second set of circles in the equatorial system consists
of those which pass through the poles and are perpendicular
to the celestial equator. They are called \textit{hour circles}. The
fundamental hour circle, called the \textit{equinoctial colure}, from
\index{Equinoctial colure}%
\index{Hour circle}%
which all others are measured, is that one which passes
%% -----File: 151.png---Folio 126-------
through the vernal equinox, that is, the place at which the
sun in its apparent annual motion around the sky crosses
the celestial equator from south to north.

The coördinates in the equator system are (a)~the angular
distance north or south of the celestial equator, which is called
declination, and (b)~the angular distance eastward from the
\index{Declination}%
vernal equinox along the equator to the point where the
hour circle through the object crosses the equator. This
distance is called right ascension. The direction eastward is
\index{Right ascension}%
defined as that in which the sun moves in its apparent
motion among the stars.

In \Figref{46}, let $O$ represent the position of the observer,
$\mathit{NESW}$~his horizon, $PNQ'SQ$~his meridian. Suppose the
star is at~$A$ and that %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.5in}
\Input[2.5in]{151}{png}
\Caption[The equator system.]{Fig}{46}
\end{wrapfigure}
the vernal
equinox is at~$V$. Then the
declination of the star is the
arc~$CA$ and its right ascension
is~$VQC$. In this case the declination
is about~$40°$ and the
right ascension is about~$75°$.
It is not customary to express
the right ascension in degrees,
but to give it in hours, where
an hour equals~$15°$. In the
present case the right ascension
of~$A$ is, therefore, about $5$~hours.

It is easy to understand why it is convenient to count
right ascension in hours. The sky has an apparent motion
westward because of the earth's actual rotation eastward,
and it makes a complete circuit of~$360°$ in $24$~hours. Therefore
it apparently moves westward $15°$ in one hour. It
follows that a simple method of finding the right ascension
of an object is to note when the vernal equinox crosses the
meridian and to measure the time which elapses before the
object is observed to cross the meridian. The interval of
time is its right ascension expressed in hours.
%% -----File: 152.png---Folio 127-------

\Article{67}{The Ecliptic System.}---The third system which is
\index{Ecliptic}%
employed in astronomy, but much less frequently than the
other two, is known as the ecliptic system because the fundamental
circle in its first set is the ecliptic. The \textit{ecliptic} is
the great circle on the celestial sphere traced out by the sun
in its apparent annual motion around the sky. The points
on the celestial sphere $90°$~from the ecliptic are the poles of
the ecliptic. The small circles parallel to the ecliptic are
called \textit{parallels of latitude}. The great circles which cross
\index{Latitude!celestial}%
the ecliptic at right angles are called \textit{longitude circles}.
\index{Longitude!celestial}%

The coördinates in the ecliptic system are the angular distance
north or south of the ecliptic, which is called \textit{latitude},
and the distance eastward
from the vernal equinox along
the ecliptic to the point where
\begin{wrapfigure}{\WLoc}{2.5in}%[Illustration:]
\Input[2.5in]{152}{png}
\Caption[The ecliptic system.]{Fig}{47}
\end{wrapfigure}
the longitude circle through
the object intersects the ecliptic,
which is called \textit{longitude}.

In \Figref{47}, $O$~represents the
position of the observer and
$QEQ'W$ the celestial equator.
Suppose that at the time in
question the vernal equinox is
at~$V$ and that the autumnal
equinox is at~$A$. Then, since the angle between the ecliptic
and the equator is~$23°.5$, the position of the ecliptic is~$AX'VX$.

\Article{68}{Comparison of the Systems of Coördinates.}---All
three of the systems of coördinates are geometrically like the
one used in geography; but there are important differences
in the way in which they arise and in the purposes for which
their use is convenient.

The horizon system depends upon the position of the
observer and the direction of his plumb line. It always
has the same relation to him, and if he travels he takes it
with him. The equator system is defined by the apparent
%% -----File: 153.png---Folio 128-------
rotation of the sky, which is due, of course, to the actual
rotation of the earth, and it is altogether independent of
the position of the observer. The ecliptic system is defined
by the apparent motion of the sun around the sky and also
is independent of the position of the observer.

Since the horizon system depends upon the position of
the observer, the altitude and azimuth of an object do not
really locate it unless the place of the observer is given.
Since the stars have diurnal motions across the sky, the time
of day must also be given; and since different stars cross the
meridian at different times on succeeding days, it follows
that the day of the year must also be given. The inconvenience
of the horizon system arises from the fact that its
circles are not fixed on the sky. Yet it is important for the
observer because the horizon is approximately the boundary
which separates the visible from the invisible portion of
the sky.

In the equator system the reference points and lines are
fixed with respect to the stars. This statement, however,
requires two slight corrections. In the first place, the
earth's equator, and therefore the celestial equator, is subject
to precession (\Artref{47}). In the second place, the stars have
very small motions with reference to one another which
become appreciable in work of extreme precision, generally
in the course of a few years. But in the present connection
these motions will be neglected and the equator coördinates
will be considered as being absolutely fixed with respect
to the stars. With this understanding the apparent position
of an object is fully defined if its right ascension and declination
are given. The reference points and lines of the ecliptic
system also have the desirable quality of being fixed with
respect to the stars.

From what has been said it might be inferred that the
equator and ecliptic systems are equally convenient, but
such is by no means the case. The equator always crosses
the meridian at an altitude which is equal to $90°$~minus the
%% -----File: 154.png---Folio 129-------
latitude of the observer (\Artref{57}) and always passes through
the east and west points of the horizon. Consequently, all
objects having the same declination cross the meridian at the
same altitude. Suppose, for example, that the observer is in
latitude $40°$~north. Then the equator crosses his meridian
at an altitude of~$50°$. If he observes that a star crosses
the meridian at an altitude of~$60°$, he knows that it is $10°$~north
of the celestial equator, or that its declination is~$10°$;
and by noting the time that has elapsed from the time of
the passage of the vernal equinox across the meridian to the
passage of the star, he has its right ascension. Nothing
could be simpler than getting the coördinates of an object in
the equator system.

Now consider the ecliptic system. Suppose~$V$, in \Figref{48},
represents the position of the vernal equinox on a certain
\begin{figure}[hbt]%[Illustration:]
\Input{154}{png}% [** TN: Side-by-side figures; special handling]
\caption{\footnotesize Equator and ecliptic.}
\label{Fig:48}%
\label{Fig:49}%
\end{figure}%
date and time of day. Then the pole of the ecliptic~$XVX'A$
is at~$R$ and the ecliptic crosses the meridian below the
equator. In this case the star might have north celestial
latitude and be on the meridian south of the equator. Twelve
hours later the vernal equinox has apparently rotated westward
with the sky to the point~$V$, \Figref{49}. The pole of the
ecliptic has gone around the pole~$P$ to the point~$R$, and the
ecliptic crosses the meridian north of the equator. It is
%% -----File: 155.png---Folio 130-------
clear from Figs.\ \Fref{48}~and~\Fref{49} that the position of the ecliptic
with respect to the horizon system changes continually with
the apparent rotation of the sky. It follows that for most
purposes the ecliptic system is not convenient. Its use
in astronomy is limited almost entirely to describing the
position of the sun, which is always on the ecliptic, and
the positions of the moon and planets, which are always
near it.

\Article{69}{Finding the Altitude and Azimuth when the Right
Ascension, Declination, and Time are Given.}---Suppose
the right ascension and declination of a star are given and
that its altitude and azimuth are desired. It is necessary
also to have given the latitude of the observer, the time of
day, and the time of year, because the altitude and azimuth
depend on these quantities. Most of the difficulty of the
problem arises from the fact that the vernal equinox has a
diurnal motion around the sky and that it is a point which
is not easily located. By computing the right ascension of
the sun at the date in question, direct use of the vernal
equinox may be avoided. It has been found convenient to
solve the problem in four distinct steps.

\textit{Step~1. The right ascension of the sun on the date in question.}---It
has been found by observation that the sun passes
the vernal equinox March~21. (The date may vary a day
because of the leap year, but it will be sufficiently accurate
for the present purposes to use March~21 for all cases.) In
a year the sun moves around the sky $24$~hours in right ascension,
or at the rate of two hours a month. Although the
rate of apparent motion of the sun is not perfectly uniform,
the variations from it are small and will be neglected in the
present connection. It follows from these facts that the
right ascension of the sun on any date may be found by
counting the number of months from March~21 to the date
in question and multiplying the result by two. For example,
October~6 is $6.5$~months from March~21, and the
right ascension of the sun on this date is, therefore, $13$~hours.
%% -----File: 156.png---Folio 131-------

\textit{Step~2. The right ascension of the meridian at the given
time of day on the date in question.}---Suppose the right
ascension of the sun has been determined by Step~1. Since
the sun moves $360°$ in $365$~days, or only one degree per day,
its motion during one day may be neglected. Suppose, for
example, that it is 8~o'clock at night. Then the sun is %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.5in}
\Input[2.5in]{156}{png}
\Caption[The right ascension of
the meridian.]{Fig}{50}
\end{wrapfigure}
$8$~hours
west of the meridian at
the position indicated in \Figref{50}.
Since right ascension is
counted eastward and the right
ascension of the sun is known,
the right ascension of~$Q$ may
be found by adding the number
of hours from the sun to~$Q$
to the right ascension of the
sun. If the right ascension of
the sun is $13$~hours and the
time of the day is 8~\PM, the
right ascension of the meridian
is $13 + 8 = 21$~hours. The general rule is, the right ascension
of the meridian is obtained by adding to the right ascension
of the sun the number of hours after noon.

\textit{Step~3. The hour angle of the object.}---Wherever the object
\index{Hour angle}%
may be, a certain hour circle passes through it and crosses
the equator at some point. The distance from the meridian
along the equator to this point is called the hour angle of
the object. The hour angle is counted either east or west
as may be necessary in order that the resulting number
shall not exceed~$12$.

Suppose the right ascension of the meridian has been
found by Step~2. The hour angle of the star is the difference
between its right ascension, which is one of the quantities
given in the problem, and the right ascension of the meridian.
If the right ascension of the star is greater than that of the
meridian, its hour angle is east, and if it is less than that of the
meridian, its hour angle is west. There is one case which,
%% -----File: 157.png---Folio 132-------
in a way, is an exception to this statement. Suppose the
right ascension of the meridian is $22$~hours and the right
ascension of the star is $2$~hours. According to the rule the
star is $20$~hours west, which, of course, is the same as 4 hours
east. But its right ascension of $2$~hours may be considered
as being a right ascension of $26$~hours, just as 2~o'clock in the
afternoon can be equally well called $14$~o'clock. When its
right ascension is called $26$~hours, the rule leads directly to
the result that the hour angle is $4$~hours east.

\textit{Step~4. Application of the declination and estimation of
the altitude and azimuth.}---In order to make the last step
clear, consider a special %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.5in}
\Input[2.5in]{157}{png}
\Caption[Application of the declination
in finding the position of a star.]{Fig}{51}
\end{wrapfigure}
example. Suppose the hour angle
of the object has been found
by Step~3 to be $7$~hours east.
This locates the point~$C$, \Figref{51}.
Therefore the star is
somewhere on the hour circle~$PCP'$.
The given declination
determines where the
star is on the circle. Suppose,
for example, that the
object is $35°$~north. In order
to locate it, it is only necessary
to measure off $35°$~from~$C$ along the circle~$CP$.
Hence the star is at~$A$.
Now draw a vertical circle from~$Z$ through~$A$ to the horizon
at~$B$. The altitude is~$BA$ and the azimuth is~$\mathit{SWNB}$.
These distances can be computed by spherical trigonometry,
but they may be estimated closely enough for present
purposes. In this problem the altitude is about~$12°$ and the
azimuth is about~$230°$. Whatever the data may be which
are supplied by the problem, the method of procedure is
always that which has been given in the present case.

\Article{70}{Illustrative Example for Finding Altitude and
Azimuth.}---In order to illustrate fully the processes that
%% -----File: 158.png---Folio 133-------
have been explained in \Artref{69}, an actual problem will be
solved. Suppose the observer is in latitude $40°$~north. The
altitude of the pole~$P$, \Figref{52}, as seen from his position, will
be~$40°$, and the point~$Q$, where the equator crosses the
meridian, will have an altitude
of~$50°$. Suppose the date
on which the observation is
made is June~21 and the time
of day is 8~\PM. Suppose the
right ascension of the star in
question is approximately $16$~hours
and that its declination
is~$-16°$. The problem is to
find its apparent altitude and
azimuth.

{\stretchyspace%
The steps of the solution
will be made in their natural
order. (1)~Since June~21 is three months after March~21,
the right ascension of the sun on that date is $6$~hours.}
(2)~Since the time of day is 8~\PM, and the right ascension is
counted eastward, the right %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{158}{png}
\Caption[Finding the altitude and
azimuth.]{Fig}{52}
\end{wrapfigure}
ascension of the meridian is
$6 + 8 = 14$~hours. (3)~Since the right ascension of the star
is $16$~hours, its hour angle is $2$~hours east, and it is on the
hour circle~$PCP'$. (4)~Since its declination is~$-16°$, it is
$16°$~south from~$C$ toward~$P'$ and at the point~$A$. Now draw
a vertical circle from~$Z$ through~$A$, cutting the horizon at~$B$.
The altitude is~$BA$, which is about~$22°$. The azimuth is~$\mathit{SWNEB}$,
which is about~$320°$.

\Article{71}{Finding the Right Ascension and Declination when
the Altitude and Azimuth are Given.}---The problem of
finding the right ascension and declination when the altitude
and azimuth are given is the converse of that treated in
\Artref{69}. It can also be conveniently solved in four steps.

In the first step, the right ascension of the sun is obtained,
and in the second, the right ascension of the meridian is
found. These steps are, of course, the same as those given
%% -----File: 159.png---Folio 134-------
in \Artref{69}. The third step is to draw through the position
of the given object an hour circle which, from its definition,
reaches from one pole of the sky to the other and cuts
the equator at right angles. The fourth step is to estimate
the hour angle of the hour circle drawn in Step~3 and the
distance of the star from the equator measured along the
hour circle. Then the right ascension of the object is equal
to the right ascension of the meridian plus the hour angle
of the object if it is east, and minus the hour angle if it is
west; and the declination of the object is simply its distance
from the equator.

\Article{72}{Illustrative Example for Finding Right Ascension
and Declination.}---Suppose the date of the observation is
May~6 and that the time of day is 8~\PM. Suppose the
observer's latitude is $40°$~north. Suppose he sees a bright
star whose altitude is estimated to be~$35°$ and whose azimuth
is estimated to be~$60°$. Its
right ascension and declination
are required, and after they
have been obtained it can be
found from \Tableref{I}, p.~\pageref{Table:I}, %144
what star is observed.

\begin{wrapfigure}{\WLoc}{2.5in}%[Illustration:]
\Input[2.5in]{159}{png}
\Caption[Finding the right ascension
and declination.]{Fig}{53}
\end{wrapfigure}

The right ascension of the
sun on May~6 is $3$~hours and
the right ascension of the meridian
at 8~\PM\ is $11$~hours.
The star then is at the point~$A$,
\Figref{53}, where $BA = 35°$
and $SB = 60°$. The part of
the vertical circle~$BA$ is much
less foreshortened than~$AZ$ by the projection of the celestial
sphere on a plane, and this fact must be remembered in
connection with the drawing. The hour circle~$PAP'$ cuts
the equator at the point~$C$. The arc~$QC$ is much more foreshortened
by projection than~$CW$. Consequently, it is seen
that the hour angle of the star is $3.5$~hours west. Therefore
%% -----File: 160.png---Folio 135-------
its right ascension is $11 - 3.5 = 7.5$~hours approximately.
It is also seen that the star is approximately $5°$~north of the
equator. On referring to \Tableref{I}, it is found that this star
must be Procyon.

All problems of the same class can be solved in a similar
manner. But reliance should not be placed in the diagrams
alone, especially because of the distortion to which certain
of the lines are subject. The diagrams should be supplemented,
if not replaced, by actually pointing out on the
sky the various points and lines which are used. A little
practice with this method will enable one to solve either
the problem of finding the altitude and azimuth, or that
of obtaining the right ascension and declination, with an
error not exceeding $5°$~or~$10°$.

\Article{73}{Other Problems Connected with Position.}---There
are two other problems of some importance which naturally
arise. The first is that of finding the time of the year at
which a star of given right ascension will be on the meridian
at a time in the evening convenient for observation.

In order to make the problem concrete, suppose the time
in question is 8~\PM. The right ascension of the sun is then
$8$~hours less than the right ascension of the meridian. Since
the object is supposed to be on the meridian, the right ascension
of the sun will be $8$~hours less than that of the object.
To find the time of the year at which the sun has a given
right ascension, it is only necessary to count forward from
March~21 two hours for each month. For example, if the
object is Arcturus, whose right ascension is $14$~hours, the
right ascension of the sun is $14 - 8 = 6$~hours, and the date
is June~21.

The second problem is that of finding the time of day
at which an object whose right ascension is given will be on
the meridian or horizon on a given date. A problem of this
character will naturally arise in connection with the
announcement of the discovery of a comet or some other
object whose appearance in a given position would be conspicuous
%% -----File: 161.png---Folio 136-------
only for a short time. This problem is solved by
first finding the right ascension of the sun on the date, and
then taking the difference between this result and the right
ascension of the object. This gives the hour angle of the
sun at the required time. If the sun is west of the meridian,
its hour angle is the time of day; if it is east of the meridian,
its hour angle is the number of hours before noon.


\Section{VII}{QUESTIONS}

1. Make a table showing the correspondences of the points,
circles, and coördinates of the horizon, equator, and ecliptic systems
with those of the geographical system.

2. What are the altitude and azimuth of the zenith, the east
point, the north pole? What are the angular distances from the
zenith to the pole and to the point where the equator crosses the
meridian in terms of the latitude~$l$ of the observer?

3. Estimate the horizon coördinates of the sun at $10$~o'clock this
morning; at $10$~o'clock this evening.

4. Describe the complete diurnal motions of stars near the pole.
What part of the sky for an observer in latitude~$40°$ is always above
the horizon? Always below the horizon?

5. How long is required for the sky apparently to turn~$1°$?
Through what angle does it apparently turn in $1$~minute?

6. Are there positions on the earth from which the diurnal
motions of the stars are along parallels of altitude? Along vertical
circles?

7. Develop a rule for finding the hour angle of the vernal
equinox on any date at any time of day.

8. Find the altitude and azimuth of the vernal equinox at
9~\AM\ to-day.

9. Given: $\text{Rt.\ asc.} = 19$~hrs., $\text{declination} = +20°$, $\text{date} = \text{July~21}$,
$\text{time} = \text{8~\PM{}}$; find the altitude and azimuth.

10. Find the altitude and the azimuth (constructing a diagram)
of each of the stars given in \Tableref{I}, p.~\pageref{Table:I}, at 8~\PM\ to-day. %p. 144

11. If a star whose right ascension is $18$~hours is on the meridian
at 8~\PM, what is the date?

12. At what time of the day is a star whose right ascension is
$14$~hours on the meridian on May~21?

13. At what time of the day does a comet whose right ascension
is $4$~hours and declination is zero rise on Sept.~21?

14. The Leonid meteors have their radiant at right ascension
%% -----File: 162.png---Folio 137-------
$10$~hours and they appear on Nov.~14. At what time of the night
are they visible?

15. What is the right ascension of the point on the celestial sphere
toward which the earth is moving on June~21?

16. What are the altitude and azimuth of the point toward which
the earth is moving to-day at noon? At 6~\PM? At midnight?
At 6~\AM?

17. Observe some conspicuous star (avoid the planets), estimate
its altitude and azimuth, approximately determine its right ascension
and declination (\Artref{71}), and with these data identify it in
\Tableref{I}, p.~\pageref{Table:I}. %144

\normalsize

%% -----File: 163.png---Folio 138-------

\thispagestyle{empty}
\begin{figure}[hbtp]%[Illustration:]
\centering\Input{163}{jpg}
\Caption[The 40-inch telescope of the Yerkes Observatory.]{Fig}{54}
\index{Yerkes Observatory}% [** TN: Moved; typo "p. 139" in original]
\end{figure}

%% -----File: 164.png---Folio 139-------


\Chapter{V}{The Constellations}
\index{Constellations}%

\Article{74}{Origin of the Constellations.}---A moment's observation
of the sky on a clear and moonless night shows that
the stars are not scattered uniformly over its surface. Every
one is acquainted with such groups as the Big Dipper and
\index{Big Dipper}%
the Pleiades. This natural grouping of the stars was observed
\index{Pleiades}%
in prehistoric times by primitive and childlike peoples
who imagined the stars formed outlines of various living
creatures, and who often wove about them the most fantastic
romances.

The earliest list of constellations, still in existence, is that
of Ptolemy (about 140~\AD), who enumerated, described,
\index[xnames]{Ptolemy}%
and located $48$~of them. These constellations not only did
not entirely cover the part of the sky visible from Alexandria,
where Ptolemy lived, but they did not even occupy all of
the northern sky. In order to fill the gaps and to cover the
southern sky many other constellations were added from
time to time, though some of them have now been abandoned.
The lists of Argelander (1799--1875) in the northern
\index[xnames]{Argelander}%
heavens, and the more recent ones of Gould in the southern
\index[xnames]{Gould}%
heavens, contain $80$~constellations, and these are the ones
now generally recognized.

\Article{75}{Naming the Stars.}---The ancients gave proper
names to many of the stars, and identified the others by
describing their relations to the anatomy of the fictitious
creatures in which they were situated. For example, there
were Sirius, Altair, Vega, etc., with proper names, and
\index{Altair}%
\index{Sirius}%
\index{Vega}%
``The Star at the End of the Tail of the Little Bear'' (Polaris),
\index{Polaris}%
``The Star in the Eye of the Bull'' (Aldebaran), etc.,
\index{Aldebaran}%
designated by their positions.
%% -----File: 165.png---Folio 140-------

In modern times the names of $40$~or~$50$ of the most conspicuous
stars are frequently used by astronomers and
writers on astronomy; the remainder are designated by
letters and numbers. A system in very common use, that
introduced by Bayer in 1603, is to give to the stars in each
\index[xnames]{Bayer}%
constellation, in the order of their brightness, the names of
the letters of the Greek alphabet in their natural order.
In connection with the Greek letters, the genitive of the name
of the constellation is used. For example, the brightest
star in the whole sky is Sirius, in Canis Major. Its name
\index{Sirius}%
according to the system of Bayer is Alpha Canis Majoris.
The second brightest star in Perseus, whose common name
\index{Perseus}%
\begin{figure}[hbt]%[Illustration:]
\centering\Input[3.5in]{165}{png}
\Caption[The Big Dipper and the Pole Star.]{Fig}{55}
\index{Big Dipper}%
\index[xnames]{Hughes}% [** TN: Typo in orig; points to "p. 149"]
\end{figure}%
is Algol, in this system is called Beta Persei. After the
\index{Algol}%
Greek letters are exhausted the Roman letters are used, and
then follow numbers for the stars in the order of their brightness.
While this is the general rule, there are numerous
exceptions in naming the stars, for example, in the case of
the stars which constitute the Big Dipper (\Figref{55}).

About 1700, Flamsteed published a catalogue of stars in
\index[xnames]{Flamsteed}%
which he numbered those in each constellation according to
their right ascensions regardless of their brightness. In
modern catalogues the stars are usually given in the order
of their right ascension and no reference is made either to the
constellation to which they belong or to their apparent
brightness.
%% -----File: 166.png---Folio 141-------

\Article{76}{Star Catalogues.}---Star catalogues are lists of stars,
\index{Catalogues of stars}%
\index{Stars!catalogues of}%
usually all above a given brightness, in certain parts of the
sky, together with their right ascensions and declinations on
a given date. It is necessary to give the date, for the stars
slowly move with respect to one another, and the reference
points and lines to which their positions are referred are not
absolutely fixed. The most important variation in the position
of the reference points and lines is due to the precession
of the equinoxes (\Artref{47}).

The earliest known star catalogue is one of $1080$ stars by
Hipparchus for the epoch 125~\BC. Ptolemy revised it and
\index[xnames]{Hipparchus}%
\index[xnames]{Ptolemy}%
reduced the star places to the epoch 150~\AD. Tycho Brahe
\index[xnames]{Tycho Brahe}%
made a catalogue of $1005$ stars in 1580, about $30$~years before
the invention of the telescope. Since the invention of
the telescope and the revival of science in Europe, numerous
catalogues have been made, containing in some cases more
than $100,000$ stars. While the positions in all these catalogues
are very accurately given, compared even to the
work of Tycho Brahe, they are not accurate enough for
certain of the most refined work in modern times. To meet
these needs, a number of catalogues, containing a limited
number of stars whose positions have been determined
with the very greatest accuracy, have been made. The
most accurate of these is the Preliminary General Catalogue
of Boss, in which the positions of $6188$~stars are given.
\index[xnames]{Boss, Lewis}%

A project for photographing the whole heavens by international
\index{Photographic chart of sky}%
coöperation was formulated at Paris in 1887.
The plan provided that each plate should cover $4$~square
degrees of the sky, and that they should overlap so that the
whole sky would be photographed twice. The number of
plates required, therefore, is nearly~$22,000$. On every plate
a number of stars are photographed whose positions are
already known from direct observations. The positions of
the other stars on the plate can then be determined by measuring
with a suitable machine their distances and directions
from the known stars. This work can, of course, be
%% -----File: 167.png---Folio 142-------
carried out at leisure in an astronomical laboratory. On
these plates, most of which have already been secured, there
will be shown in all about $8,000,000$ different stars. In the
first catalogue based on them only about $1,300,000$ of the
brightest stars will be given.

The photographic catalogue was an indirect outgrowth
of pho\-to\-graphs of the great comet of 1882 taken by Gill
\index[xnames]{Gill}%
at the Cape of Good Hope. The number of star images
obtained on his plates at once showed the possibilities of
making catalogues of stars by the photographic method.
In 1889 he secured photographs of the whole southern sky
from declination $-19°$~south, and the enormous labor of
measuring the positions of the $350,000$ star images on these
plates was carried out by Kapteyn, of Groningen, Holland.
\index[xnames]{Kapteyn}%

\Article{77}{The Magnitudes of the Stars.}---The magnitude of
\index{Magnitudes of stars}%
a star depends upon the amount of light received from it
by the earth, and is not determined altogether by the amount
of light it radiates, for a small star near the earth might
give the observer more light than a much larger one farther
away. It is clear from this fact that the magnitude of a
star depends upon its actual brightness and also upon its
distance from the observer.

The stars which are visible to the unaided eye are divided
arbitrarily into $6$~groups, or magnitudes, depending upon
their apparent brightness. The $20$~brightest stars constitute
the first-magnitude group, and the faintest stars
which can be seen by the ordinary eye on a clear night are
of the sixth magnitude, the other four magnitudes being distributed
between them so that the ratio of the brightness
of one group to that of the next is the same for all consecutive
magnitudes. The definition of what shall be exactly the
first magnitude is somewhat arbitrary; but a first-magnitude
star has been taken to be approximately equal to the
average brightness of the first $20$~stars. The sixth-magnitude
stars are about $\frac{1}{100}$ as bright as the average of the first
group, and, in order to make the ratio from one magnitude
%% -----File: 168.png---Folio 143-------
to the other perfectly definite, it has been agreed that the
technical sixth-magnitude stars shall be those which are
\index{Stars!first-magnitude}%
exactly $\frac{1}{100}$ as bright as the technical first-magnitude stars.
\index{First-magnitude stars}%
The problem arises of finding what the ratio is for successive
magnitudes.

Let $r$ be the ratio of light received from a star of one
magnitude to that received from a star of the next fainter
magnitude. Then stars of the fifth magnitude are $r$~times
brighter than those of the sixth, and those of the fourth are
$r$~times brighter than those of the fifth, and they are therefore
$r^2$~times brighter than those of the sixth. By a repetition
of this process it is found that the first-magnitude stars
are $r^5$~times brighter than those of the sixth magnitude.
Therefore $r^5 = 100$, from which it is found that $r = 2.512$\,\ldots.

Since the amount of light received from different stars
varies almost continuously from the faintest to the brightest,
it is necessary to introduce fractional magnitudes. For
example, if a star is brighter than the second magnitude and
fainter than the first, its magnitude is between $1$ and~$2$.
A step of one tenth of a magnitude is such a ratio that,
when repeated ten times, it gives the value~$2.512$\,\ldots. It
is found by computation, which can easily be carried out by
logarithms, that a first-magnitude star is $1.097$~times as
bright as a star of magnitude~$1.1$. The ratio of brightness
of a star of magnitude~$1.1$ to that of a star of~$1.2$ is likewise
$1.097$; and, consequently, a star of magnitude~$1$ is
$1.097 × 1.097 = 1.202$ times as bright as a star of magnitude~$1.2$.

A star which is $2.512$~times as bright as a first-magnitude
star is of magnitude~$0$, and still brighter stars have negative
magnitudes. For example, Sirius, the brightest star in the
\index{Sirius}%
sky, has a magnitude of~$-1.58$, and the magnitude of the
full moon on the same system is about~$-12$, while that of
the sun is~$-26.7$.
\index{Sun!magnitude of}%

\Article{78}{The First-magnitude Stars.}---As first-magnitude
stars are conspicuous and relatively rare objects, they serve
%% -----File: 169.png---Folio 144-------
as guideposts in the study of the constellations. All of
those which are visible in the latitude of the observer should
be identified and learned. They will, of course, be recognized
partly by their relations to neighboring stars.

In \Tableref{I} the first column contains the names of the first-magnitude
stars; the second, the constellations in which
they are found; the third, their magnitudes according to the
Harvard determination; the fourth, their right ascensions;
the fifth, their declinations; the sixth, the dates on which
they cross the meridian at 8~\PM; and the seventh, the
velocity toward or from the earth in miles per second, the
negative sign indicating approach and the positive, recession.
Their apparent positions at any time can be determined
from their right ascensions and declinations by the principles
explained in \Artref{69}.
\begin{sidewaystable}[hpbt]
\begin{center}
%\TFontsize%
\settowidth{\ColOneLen}{\THF Alpha Crucis}%
\settowidth{\ColTwoLen}{\THF Constellation}%
\Caption{Table}{I}
\index{First-magnitude stars}%
\index{Harvard College Observatory}%
\index{Radial velocity}%
\index{Stars!first-magnitude}%
\begin{tabular}{|l|l|r<{\ }|>{\quad}r@{\Skip}l|r@{\ }l|>{\ }l@{}r<{\ }|r<{\quad}|}
\hline
\TEntry{\ColOneLen}{\THead Name} &
\TEntry{\ColTwoLen}{\THead Constellation} &
\settowidth{\TmpLen}{\THF Mag-}\TEntry{\TmpLen}{\THead Mag\-ni\-tude} &
\TCEntry{2}{c|}{\THF Right As-}{\THead Right As\-cension} &
\TCEntry{2}{c|}{\THF nation}{\THead Decli\-nation} &
\TCEntry{2}{c|}{\THF at 8~\PM.}{\THead\medskip On Me\-ridian\\ at 8~\PM.\medskip} &
\TCEntry{1}{|c|}{\THF Velocity}{\THead Radial\\ Velocity} \\
\hline
\Strut
\DTE{Sirius}         & \DTE{Canis Major}
\index{Sirius}%
  & $-1.6$ & $\Z6$\rlap{h} & $41$\rlap{m} & $-16$\rlap{$°$} & \rlap{$36'$}\Z\Z & Feb. & 28 & $-\Z5.6$ \\
\DTE{Canopus}        & \DTE{Carina}
\index{Canopus}%
  & $-0.9$ & $\Z6$ & $22$ & $-52$ & $39$ & Feb. & 23 &  $+12.7$ \\
\TEntry[b]{\ColOneLen}{\DTE{Alpha\\ \Skip Centauri}} & \DTE{Centaurus}
\index{Alpha Centauri}%
\index{Beta Centauri}%
  &  $0.1$ &  $14$ & $34$ & $-60$ & $29$ & June & 29 &  $-13.8$ \\
\DTE{Vega}           & \DTE{Lyra}
\index{Vega}%
  &  $0.1$ &  $18$ & $34$ & $+38$ & $42$ & Aug. & 30 & $-\Z8.5$ \\
\DTE{Capella}        & \DTE{Auriga}
\index{Capella}%
  &  $0.2$ & $\Z5$ & $10$ & $+45$ & $55$ & Feb. &  5 &  $+19.7$ \\
\DTE{Arcturus}       & \DTE{Boötes}
\index{Arcturus}%
  &  $0.2$ &  $14$ & $12$ & $+19$ & $37$ & June & 24 & $-\Z2.4$ \\
\DTE{Rigel}          & \DTE{Orion}
\index{Rigel}%
  &  $0.3$ & $\Z5$ & $11$ & $- 8$ & $17$ & Feb. &  5 &  $+13.6$ \\
\DTE{Procyon}        & \DTE{Canis Minor}
\index{Procyon}%
  &  $0.5$ & $\Z7$ & $35$ & $+ 5$ & $26$ & Mar. & 14 & $-\Z2.5$ \\
\DTE{Achernar}       & \DTE{Eridanus}
\index{Achernar}%
  &  $0.6$ & $\Z1$ & $35$ & $-57$ & $40$ & Dec. & 16 &  $+10.0$ \\
\TEntry[b]{\ColOneLen}{\DTE{Beta \\ \Skip Centauri}} & \DTE{Centaurus}
  &  $0.9$ &  $13$ & $58$ & $-59$ & $58$ & June & 21 &  \QMark  \\
\DTE{Betelgeuze}     & \DTE{Orion}
\index{Betelgeuze}%
  &  $0.9$ & $\Z5$ & $51$ & $+ 7$ & $24$ & Feb. & 15 &  $+13.0$ \\
\DTE{Altair}         & \DTE{Aquila}
\index{Altair}%
  &  $0.9$ &  $19$ & $47$ & $+ 8$ & $39$ & Sept.& 19 &  $-20.5$ \\
Alpha Crucis             & \DTE{Crux}
\index{Alpha Crucis}%
  &  $1.1$ &  $12$ & $22$ & $-62$ & $38$ & May  & 29 & $+\Z4.3$ \\
\DTE{Aldebaran}      & \DTE{Taurus}
\index{Aldebaran}%
  &  $1.1$ & $\Z4$ & $31$ & $+16$ & $21$ & Jan. & 26 &  $+34.2$ \\
\DTE{Pollux}         & \DTE{Gemini}
\index{Pollux}%
  &  $1.2$ & $\Z7$ & $40$ & $+28$ & $14$ & Mar. & 15 & $+\Z2.4$ \\
\DTE{Spica}          & \DTE{Virgo}
\index{Spica}%
  &  $1.2$ &  $13$ & $21$ & $-10$ & $44$ & June & 12 & $+\Z1.2$ \\
\DTE{Antares}        & \DTE{Scorpius}
\index{Antares}%
  &  $1.2$ &  $16$ & $24$ & $-26$ & $15$ & July & 27 & $-\Z1.9$ \\
\raisebox{\baselineskip}{\DTE{Fomalhaut}} &
\index{Fomalhaut}%
  \TEntry[b]{\ColTwoLen}{\DTE{Piscis \\ \Skip Australis}}
  &  $1.3$ &  $22$ & $53$ & $-30$ & $ 4$ & Nov. &  8 & $+\Z4.2$ \\
\DTE{Deneb}          & \DTE{Cygnus}
\index{Deneb}%
  &  $1.3$ &  $20$ & $39$ & $+44$ & $59$ & Oct. &  4 & $-\Z2.5$ \\
\DTE{Regulus}        & \DTE{Leo}
\index{Regulus}%
  &  $1.3$ &  $10$ &  $4$ & $+12$ & $23$ & Apr. & 23 & $-\Z5.0$ \\
\hline
\end{tabular}
\end{center}
\end{sidewaystable}
%% -----File: 170.png---Folio 145-------

\Article{79}{Number of Stars in the First Six Magnitudes.}---The
number of stars in each of the first six magnitudes is given
in \Tableref{II}. The sum of the numbers is~$5000$.
\begin{table}[hbt]
\begin{center}
\Caption{Table}{II}
\index{Number of stars}%
\index{Stars!number of}%
\begin{tabular}{|*{2}{l@{}p{1in}|}}
\hline
\Strut
First Magnitude  & \MyDotFill   $20$ &  Fourth Magnitude & \MyDotFill  $425$ \\
Second Magnitude & \MyDotFill   $65$ &  Fifth Magnitude  & \MyDotFill $1100$ \\
Third Magnitude  & \MyDotFill  $190$ &  Sixth Magnitude  & \MyDotFill $3200$ \\
\hline
\end{tabular}
\end{center}
\end{table}
There are, therefore, in the whole sky only about $5000$ stars
which are visible to the unaided eye. At any one time
only half the sky is above the horizon, and those stars which
are near the horizon are largely extinguished by the absorption
of light by the earth's atmosphere. Therefore one
never sees at one time more than about $2000$ stars, although
the general impression is that they are countless.

It is seen from the \Tableref{II} that the number of stars in
each magnitude is about three times as great as the number
in the preceding magnitude. This ratio holds approximately
down to the ninth magnitude, and in the first nine
magnitudes there are in all nearly $200,000$ stars. Since
a telescope $3$~inches in aperture will show objects as faint as
the ninth magnitude, it is seen what enormous aid is obtained
from optical instruments. Only a rough guess can
be made respecting the number of stars which are still
fainter, but there are probably more than $300,000,000$ of
them within the range of present visual and photographic
instruments.

\Article{80}{The Motions of the Stars.}---The stars have motions
\index{Motion!of stars}%
\index{Stars!motions of}%
with respect to one another which, in the course of immense
ages, appreciably change the outlines of the constellations,
but which have not made important alterations in the visible
sky during historic times. Nevertheless, they are so large
that they must be taken into account when using star catalogues
in work of precision.
%% -----File: 171.png---Folio 146-------

One result of the motions of the stars is that they drift
with respect to fixed reference points and lines. The yearly
change in the position of a star with respect to fixed reference
points and lines is called its proper motion. The largest
\index{Proper motion of stars}%
\index{Stars!proper motions of}%
known proper motion is that of an eighth-magnitude star
in the southern heavens, whose annual displacement on the
sky is about $8.7$~seconds of arc. The slight extent to which
the proper motions of the stars can change the appearance
of the constellations is shown by the fact that even this
star, whose proper motion is more than $100$~times the average
proper motion of the brighter stars, will not move over an
apparent distance as great as the diameter of the moon in
less than $220$~years.

Another component of the motion of a star is that which is
in the line joining it with the earth. This component can
be measured by the spectroscope (\Artref{222}), and is found
to range all the way from a velocity of approach of $40$~miles
per second to one of recession with the same speed; and
in some cases even higher velocities are encountered. In
the course of immense time the changes in the distances of
the stars will alter their magnitudes appreciably; but the
distances of the stars are so great that there is probably no
case in which the motion of a star toward or from the earth
will sensibly change its magnitude in $20,000$ years.

\Article{81}{The Milky Way, or Galaxy.}---The Milky Way is a
\index{Galaxy}%
\index{Milky Way}%
hazy band of light giving indications to the unaided eye of
being made up of faint stars; it is on the average about $20°$
in width and stretches in nearly a great circle entirely around
the sky. The telescope shows that it is made up of millions
of small stars which can be distinguished separately only
with optical aid. It is clear that because of its irregular
form and great width its position cannot be precisely described,
but in a general way its location is defined by the
fact that it intersects the celestial equator at two places
whose right ascensions are approximately $6$~hours $40$~minutes
and $18$~hours $40$~minutes, and it has an inclination to the
%% -----File: 172.png---Folio 147-------
equator of about~$62°$. Or, in other terms, the north pole
of the Milky Way is at right ascension about $12$~hours $40$~minutes
and at declination about~$+28°$. For a long distance
it is divided more or less completely into two parts, and at
one place in the southern heavens it is cut entirely across by
a dark streak. A very interesting feature for observers in
northern latitudes is a singular dark region north of the star
Deneb.

\Article{82}{The Constellations and Their Positions.}---The work
on reference points and lines in the preceding chapter together
with the discussions so far given in this chapter are
sufficient to prepare for the study of the constellations with
interest and profit, and the student should not stop short
of an actual acquaintance with all the first-magnitude stars
and the principal constellations that are visible in his latitude.
\Tableref{III} contains a list of the constellations and gives their
positions. The numbers at the top show the degrees of declination
between which the constellations lie, the numerals
at the left show their right ascensions, and the numbers
placed in connection with the names of the constellations
give the number of stars in them which are easily visible to
the unaided eye. The constellations which lie on the ecliptic,
or the so-called zodiacal constellations, are printed in italics.

The \hyperref[Maps]{following maps} show the constellations from the north
pole to $-50°$~declination. When \Mapref{I} is held up toward
the sky, facing north, with its center in the line joining the
eye with the north pole, and with the hour circle having the
right ascension of the meridian placed directly above
its center, it shows the circumpolar constellations in their
true relations to one another and to the horizon and pole.
The other maps are to be used, facing south, with their centers
held on a line joining the eye to the celestial equator,
and with the hour circle having the right ascension of the
meridian held in the plane of the eye and the meridian.
When they are placed in this way, they show the constellations
to the south of the observer in their true relationships. In
%% -----File: 173.png---Folio 148-------
\begin{sidewaystable}[hbtp]
\begin{center}
\Caption{Table}{III}
\index{Constellations!list of}%
\scriptsize%
\setlength{\TmpLen}{1in}%
\setlength{\unitlength}{0.5in}%
\begin{tabular}{|*{7}{p{\TmpLen}|}}
  \hline
  \rule[-12pt]{0pt}{36pt}
  \smash{%
    \begin{picture}(2,1)
      \put(0,1){\line(2,-1){2}}%
      \put(0.05,0.25){R.A.}
      \put(1.95,0.75){\makebox(0,0)[tr]{\textsc{Dec.}}}
    \end{picture}}
        & \multicolumn{1}{c|}{$+90°$ \textsc{to} $+50°$}
        & \multicolumn{1}{c|}{$+50°$ \textsc{to} $+25°$}
        & \multicolumn{1}{c|}{$+25°$ \textsc{to} $0°$}
        & \multicolumn{1}{c|}{$0°$ \textsc{to} $-25°$}
        & \multicolumn{1}{c|}{$-25°$ \textsc{to} $-50°$}
        & \multicolumn{1}{c|}{$-50°$ \textsc{to} $-90°$} \\ \hline
%
\DTE{I--II}
  & \TEntry{\TmpLen}{Cassiopeia, 46.}
  & \TEntry{\TmpLen}{Andromeda, 18\DPtypo{.}{;} \\
            Triangulum, 5.}
  & \TEntry{\TmpLen}{\textit{Pisces}, 18; \\
            \textit{Aries}, 17.}
  & \TEntry{\TmpLen}{Cetus, 37.}
  & \TEntry{\TmpLen}{\medskip Ph{\oe}nix, 32; \\
            Apparatus \\
            \Skip Sculptoris, 13.\medskip}
  & \TEntry{\TmpLen}{(Ph{\oe}nix); \\
            Hydrus, 18.} \\
%
\DTE{III--IV}
  &  \CDash
  &  Perseus, 46.
  & \textit{Taurus}, 58.
  &  Eridanus, 64.
  & (Eridanus.)
  & \TEntry{\TmpLen}{Horologium, 11;  \\
            Reticulum, 9.\medskip} \\
%
\DTE{V--VI}
  & \TEntry{\TmpLen}{Camelo-\\ \Skip pardalis, 36.}
  & Auriga, 35.
  & \TEntry{\TmpLen}{Orion, 58; \\
               \textit{Gemini}, 33.}
  & Lepus, 18.
  & Columba, 15.
  & \TEntry{\TmpLen}{Dorado, 16; \\
            Pictor, 14; \\
            Mons Mensa, 12.\medskip} \\
%
\DTE{VII--VIII}
  & \CDash
  & Lynx, 28.
  & \TEntry{\TmpLen}{Canis Minor, 8; \\
            \textit{Cancer}, 15.}
  & \TEntry{\TmpLen}{Canis Major, 27; \\
            Monoceros, 12.}
  &  Argo-Navis, 149.
  & \TEntry{\TmpLen}{(Argo-Navis, \\
            \Skip Puppis); \\
            Piscis Volans, 9.\medskip} \\
%
\DTE{IX--X}
  & \CDash
  & Leo Minor, 15.
  & \textit{Leo}, 47.
  & \TEntry{\TmpLen}{Hydra, 49; \\
            Sextans, 5.}
  &  \CDash
  & \TEntry{\TmpLen}{(Argo-Navis, \\
            \Skip Vela.)\medskip} \\
%
\DTE{XI--XII}
  & Ursa Major, 53.
  & \CDash
  & \TEntry{\TmpLen}{Coma \\
            \Skip Berenices, 20.}
  & \TEntry{\TmpLen}{Crater, 15; \\
            Corvus, 8.}
  & Centaurus, 56.
  & \TEntry{\TmpLen}{(Argo-Navis, \\
            \Skip Carina); \\
            Chameleon, 13.\medskip} \\
%
\DTE{XIII--XIV}
  & \CDash
  & \TEntry{\TmpLen}{Canes Venatici,\\ \Skip 15; \\
            Boötes, 36.}
  & \CDash
  & \textit{Virgo}, 39.
  & Lupus, 34.
  & \TEntry{\TmpLen}{(Centaurus); \\
            Crux, 13; \\
            Musca, 15.\medskip} \\
%
\DTE{XV--XVI}
  & Ursa Minor, 23.
  & \TEntry{\TmpLen}{Corona Borealis,\\ \Skip 19; \\
            Hercules, 65.\medskip}
  &  Serpens, 25.
  & \textit{Libra}, 23.
  & Norma, 14.
  & Circinus, 10. \\
%
\DTE{XVII--XVIII}
  & Draco, 80.
  & Lyra, 18.
  & \TEntry{\TmpLen}{Aquila, 37; \\
            Sagitta, 5.}
  & \TEntry{\TmpLen}{\textit{Scorpius}, 34; \\
            Ophiuchus, 46.}
  & Ara, 15\DPtypo{}{.}
  & \TEntry{\TmpLen}{Triangulum \\
            \Skip Australe, 11; \\
            Apus, 8.\medskip} \\
%
\DTE{XIX--XX}
  & \CDash
  & Cygnus, 67.
  & \TEntry{\TmpLen}{Vulpecula, 23; \\
            Delphinus, 10.}
  & \textit{Sagittarius}, 48.
  & \TEntry{\TmpLen}{Corona \\
            \Skip Australis, 8.}
  & \TEntry{\TmpLen}{Telescopium, 16; \\
            Pavo, 37; \\
            Octans, 22.\medskip} \\
%
\DTE{XXI--XXII}
  & Cepheus, 44.
  & Lacerta, 16.
  & Equuleus, 5.
  & \textit{Capricornus}, 22.
  & \TEntry{\TmpLen}{Piscis \\
            \Skip Australis, 16.}
  & \TEntry{\TmpLen}{Indus, 15; \\
            \Skip (Octans).\medskip} \\
%
\DTE{XXIII--XXIV}
  & \CDash
  & \CDash
  & Pegasus, 43.
  & \textit{Aquarius}, 36.
  & Grus, 30.
  & \TEntry{\TmpLen}{(Octans); \\
            Tucana, 22.\medskip} \\
\hline
\end{tabular}
\end{center}
\end{sidewaystable}
%% -----File: 174.p n g----------
\begin{figure}[p]
\begin{center}
\phantomsection\label{Maps}%
\Caption{Map}{I}
\makebox[0pt][c]{%
\Input[\MapWidth]{174}{jpg}} %[Illustration: Map I]
\end{center}
\end{figure}%
%
%% -----File: 175.p n g----------
\begin{figure}[p]
\begin{center}
\Caption{Map}{II}
\makebox[0pt][c]{%
\Input[\MapWidth]{175}{jpg}} %[Illustration: Map II]
\end{center}
\end{figure}%
%
%% -----File: 176.p n g----------
\begin{figure}[p]
\begin{center}
\Caption{Map}{III}
\makebox[0pt][c]{%
\Input[\MapWidth]{176}{jpg}} %[Illustration: Map III]
\end{center}
\end{figure}%
%
%% -----File: 177.p n g----------
\begin{figure}[p]
\begin{center}
\Caption{Map}{IV}
\makebox[0pt][c]{%
\Input[\MapWidth]{177}{jpg}} %[Illustration: Map IV]
\end{center}
\end{figure}%
%
%% -----File: 178.p n g----------
\begin{figure}[p]
\begin{center}
\Caption{Map}{V}
\makebox[0pt][c]{%
\Input[\MapWidth]{178}{jpg}} %[Illustration: Map V]
\end{center}
\end{figure}%
%
%% -----File: 179.png---Folio 149-------
order to apply the maps according to these instructions, it
is necessary to know the right ascension of the meridian for
the day and hour in question, and it can be computed with
sufficient approximation by the method of \Artref{69}.

\Article{83}{Finding the Pole Star.}---The first step to be taken
in finding the constellations, either from their right ascensions
and declinations or from star maps, is to determine
the north-and-south line. It is defined closely enough for
present purposes by the position of the pole star.

The Big Dipper is the best known and one of the most
\index{Big Dipper}%
conspicuous groups of stars in the northern heavens. It is
always above the
horizon for an observer
in latitude
$40°$~north, and, because
of its definite
shape, it can
never be mistaken
for any other group
of stars. It is
made up of $7$~stars
of the second magnitude
which form the outline of a great dipper in the sky.
\Figureref{56} is a photograph of this group of stars distinctly
showing the dipper. The stars Alpha and Beta are called
The Pointers because they are almost directly in a line with
\index{Pointers}%
the pole star Polaris. In order to find the pole star, start with
\index{Polaris}%
Beta, \Figref{55}, go through Alpha, and continue about five
times the distance from Beta to Alpha. At the point reached
there will be found the second-magnitude star Polaris with
no other one so bright anywhere in the neighborhood.

\begin{wrapfigure}{\WLoc}{3.375in}%[Illustration:]
\Input[3.375in]{179}{jpg}
\Caption[The Big Dipper.]{Fig}{56}
\end{wrapfigure}

Besides defining the north-and-south line and serving as
a guide for a study of the constellations in the northern
heavens, the pole star is an interesting object in several
other respects. It has a faint companion of the ninth magnitude,
distant from it about $18.5$~seconds of arc. This
%% -----File: 180.png---Folio 150-------
faint companion cannot be seen with the unaided eye because,
in order that two stars may be seen as separate objects
without a telescope, they must be distant from each other
at least $3$~minutes of arc, and, besides, they must not be too
bright or too faint. The brighter of the two components of
Polaris is also a double star, a fact which was discovered by
\index{Polaris}%
means of the spectroscope in 1899. Indeed, it has turned
out on more recent study at the Lick Observatory that the
\index{Lick Observatory}%
principal star of this system is really a triple sun.

\Article{84}{Units for Estimating Angular Distances.}---The distances
\index{Angular distances}%
between stars, as seen projected on the celestial
sphere, are always given in degrees. There is, in fact, no
definite content to the statement that two stars seem to be
a yard apart. In order to estimate angular distances, it is
important to have a few units of known length which can
always be seen.

It is $90°$ from the horizon to the zenith, and one would
suppose that it would be a simple matter to estimate half
of this distance. As a matter of fact, few people place the
zenith high enough. In order to test the accuracy with
which one locates it, he should face the north and fix his
attention on the star which he judges to be at the zenith,
and then, keeping it in view, turn slowly around until he
faces the south. The first trial is apt to furnish a surprise.

The altitude of the pole star is equal to the latitude of
the observer which, in the United States, is from $25°$ to~$50°$.
This unit is not so satisfactory as some others because it
depends upon the position of the observer and also because
it is more difficult to estimate from the horizon to a star
than it is between two stars. Another large unit which can
always be observed from northern latitudes is the distance
between Alpha Ursæ Majoris and Polaris, which is~$28°$.
For a smaller unit the distance between The Pointers in the
\index{Pointers}%
Big Dipper, which is $5°\,20'$, is convenient.

\Article{85}{Ursa Major (The Greater Bear).}---The Big Dipper,
\index{Ursa Major}%
to which reference has already been made, and which is one
%% -----File: 181.png---Folio 151-------
of the most conspicuous configurations in the northern
heavens, is in the eastern part\footnote
  {East and west on the sky must be understood to be measured along
  declination circles. Consequently, near the pole east may have any direction
  with respect to the horizon. Above the pole, east on the sky is toward
  the eastern part of the horizon, while below the pole it is toward the western
  part of the horizon. All statements of direction in descriptions of the
  constellations refer to directions on the sky unless otherwise indicated, and
  care must be taken not to understand them in any other sense.}
of the constellation Ursa
Major and serves to locate the position of this constellation.
The outline of the Bear extends north, south, and west of
the bowl of the Dipper for more than~$10°$; but all the stars
in this part of the sky are of the third magnitude or fainter.

According to the Greek legend, Zeus changed the nymph
\index[xnames]{Zeus}%
Callisto into a bear in order to protect her from the jealousy
\index[xnames]{Callisto}%
of his wife Hera. While the transformed Callisto was wandering
\index[xnames]{Hera}%
in the forest, she met her son Arcas, who was about to
\index[xnames]{Arcas}%
slay her when Zeus intervened and saved her by placing them
both among the stars, where they became the Greater and
the Smaller Bears. Hera was still unsatisfied and prevailed
\index{Big Dipper}%
\index{Ursa Major}%
on Oceanus and Thetis to cause them to pursue forever their
\index[xnames]{Thetis}%
courses around the pole without resting beneath the ocean
waves. Thus was explained the circumpolar motions of
those stars which are always above the horizon.

The Pawnee Indians call the stars of the bowl of the Dipper
a stretcher on which a sick man is being carried, and the
first one in the handle is the medicine man.

The star at the bend of the handle of the Dipper, called
Mizar by the Arabs, has a faint one near it which is known
\index{Mizar}%
as Alcor. Mizar is of the second magnitude, and Alcor is of
\index{Alcor}%
the fifth. Any one with reasonably good eyes can see the two
stars as distinct objects, without optical aid. It is probable
that this was the first double star that was discovered. The
distance of~$11'.5$ between them is so great, astronomically
speaking, that it is no longer regarded as a true double star.
It has been supposed by some writers that the word Alcor
is derived from an Arabic word meaning the test, and the
%% -----File: 182.png---Folio 152-------
Arabs are said to have tested their eyesight on it. The
Pawnee Indians call it the Medicine Man's Wife's Dog.

The star Mizar itself is a fine telescopic double, the first
\index{Mizar}%
one ever discovered; the two components are distant from
each other~$14''.6$ and can be seen separately with a $3$-inch
telescope. The distance from the earth to Mizar, according
to the work of Ludendorff, is $4,800,000$ times as far as from
\index[xnames]{Ludendorff}%
the earth to the sun, and about $75$~years are required for
light to come from it to us. The star appears to be faint
only because of its immense distance, for, as a matter of
fact, it radiates $115$~times as much light as is given out by
the sun. The actual distance even from Mizar to Alcor,
which is barely discernible with the unaided eye, is $16,000$
times as far as from the earth to the sun.

The first of a series of very important discoveries was made
by E.~C. Pickering, in~1889, by spectroscopic observations
\index[xnames]{Pickering, E. C.}%
of the brighter component of Mizar. It was found by
methods which will be discussed in Arts.\ \hyperref[art:285]{285}~and~\hyperref[art:286]{286} that
this star is itself a double in which the components are so
close together that they cannot be distinguished separately
with the aid of any existing telescope. Such a star is called
a spectroscopic binary. The complete discussion showed
that the brighter component of Mizar is composed of two
great suns whose combined mass is many times that of our
sun, and that they revolve about their common center of
gravity at a distance of $25,000,000$ miles from each other in
a period of $20.5$~days.

\Article{86}{Cassiopeia (The Woman in the Chair).}---To find
\index{Cassiopeia}%
Cassiopeia go from the middle of the handle of the Big Dipper
through Polaris and about $30°$~beyond. The constellation
will be recognized because the principal stars of which it
is composed, ranging in magnitude from the second to the
fourth, form a zigzag, or letter~$W$. When it is tilted in a
particular way as it moves around the pole in its diurnal
motion, it has some resemblance to the outline of a chair.
The brightest of the 7~stars in the~$W$ is the one at the bottom
%% -----File: 183.png---Folio 153-------
of its second part, and a $2$-inch telescope will show that it
is a double star whose colors are described as rose and blue.

One of the most interesting objects in this constellation is
the star Eta Cassiopeiæ, which is near the middle of the third
\index{Eta Cassiopeiae@{Eta Cassiopeiæ}}%
stroke of the~$W$ and about $2°$~from Alpha. It is a fine
double which can be separated with a $3$-inch telescope.
The two stars are not only apparently close together, but
actually form a physical system, revolving around their
common center of gravity in a period of about $200$~years.
If there are planets revolving around either of these stars,
their phenomena of night and day and their seasons must
be very complicated.

In 1572 a new star suddenly blazed forth in Cassiopeia
\index{Cassiopeia}%
and became brighter than any other one in the sky. It
caught the attention of Tycho Brahe, who was then a young
\index[xnames]{Tycho Brahe}%
man, and did much to stimulate his interest in astronomy.

\Article{87}{How to Locate the Equinoxes.}---It is advantageous
\index{Equinoxes!how to locate}%
to know how to locate the equinoxes when the positions of
objects are defined by their right ascensions and declinations.
To find the vernal equinox, draw a line from Polaris through
\index{Polaris}%
the most westerly star in the~$W$ of Cassiopeia, and continue
it~$90°$. The point where it crosses the equator is the vernal
equinox which, unfortunately, has no bright stars in its
neighborhood.

If the vernal equinox is below the horizon, the autumnal
equinox may be conveniently used. One or the other of
them is, of course, always above the horizon. To find the
autumnal equinox, draw a line from Polaris through Delta
Ursæ Majoris, or the star where the handle of the Big
Dipper joins the dipper, and continue it $90°$ to the equator.
\index{Big Dipper}%
The autumnal equinox is in Virgo. This constellation
\index{Virgo}%
contains the first-magnitude star Spica, which is about $10°$~south
\index{Spica}%
and $20°$~east of the autumnal equinox.

\Article{88}{Lyra (The Lyre, or Harp).}---Lyra is a small but
\index{Lyra}%
very interesting constellation whose right ascension is about
$18.7$~hours and whose declination is about $40°$~north. It is,
%% -----File: 184.png---Folio 154-------
therefore, about $50°$~from the pole, and its position can easily
be determined by using the directions for finding the vernal
and autumnal equinoxes. Or, its distance east or west of
the meridian can be determined by the methods of \Artref{69}.
With an approximate idea of its location, it can always
be found because it contains the brilliant bluish-white,
first-magnitude star Vega. If there should be any doubt in
\index{Vega}%
regard to the identification of Vega, it can always be dispelled
by the fact that this star, together with two fourth-magnitude
stars, Epsilon and Zeta Lyræ, form an equilateral
triangle whose sides are about $2°$~in length. There
are no other stars so near Vega, and there is no other configuration
of this character in the whole heavens.

As was stated in \Artref{47}, the attractions of the moon and
sun for the equatorial bulge of the earth cause a precession
of the earth's equator, and therefore a change in the location
of the pole of the sky. About $12,000$ years from now the
north pole will be very close to Vega. What a splendid
pole star it will make! It is approaching us at the rate of
$8.5$~miles per~second, but its distance is so enormous that
even this high velocity will make no appreciable change in
its brightness in the next $12,000$ years. The distance of
Vega is not very accurately known, but it is probably more
than $8,000,000$ times as far from the earth as the earth is
from the sun. At its enormous distance the sun would appear
without a telescope as a faint star nearly at the limits
of visibility. Another point of interest is that the sun with
all its planets is moving nearly in the direction of Vega at
the rate of about $400,000,000$ miles a year.

The star Epsilon Lyræ, which is about $2°$~northeast of
\index{Epsilon Lyrae@{Epsilon Lyræ}}%
Vega, is an object which should be carefully observed. It is
a double star in which the apparent distance between the
two components is~$207''$.\DPnote{** [sic]} They are barely distinguishable
as separate objects with the unaided eye even by persons
of perfect eyesight. It is a noteworthy fact that, so far as
is known, this star was not seen to be a double by the Arabs,
%% -----File: 185.png---Folio 155-------
the early Greeks, or any primitive peoples. A century ago
astronomers gave their ability to separate this pair without
the use of the telescope as proof of their having exceptionally
keen sight. Perhaps with the more exacting use to which
the eyes of the human race are being subjected, they are
actually improving instead of deteriorating as is commonly
supposed.

Although the angular distance between the two components
of Epsilon Lyræ seems small, astronomers regularly
\index{Beta Lyrae@{Beta Lyræ}}%
\index{Epsilon Lyrae@{Epsilon Lyræ}}%
measure one two-thou\-sandth of this angle. The discovery
of Neptune was based on the fact that in $60$~years it had
\index{Discovery of Uranus and Neptune}%
\index{Neptune!discovery of}%
pulled Uranus from its predicted place, as seen from the
earth, only a little more than half of the angular distance
between the components of this double star. When Epsilon
Lyræ is viewed through a telescope of $5$~or $6$~inches' aperture,
it presents a great surprise. The two components are found
to be so far apart in the telescope that they can hardly be
seen at the same time, and a little close attention shows that
each of them also is a double. That is, the faint object
Epsilon Lyræ is a magnificent system of four suns.

About $5°.5$~south of Vega and $3°$~east is the third-magnitude
star Beta Lyræ. It is a very remarkable variable
whose brightness changes by nearly a magnitude in a period
of $12$~days and $22$~hours. The variability of this star is due
to the fact that it is a double whose plane of motion passes
nearly through the earth so that twice in each complete
revolution one star eclipses the other. A detailed study of
the way in which the light of this star varies shows that the
components are stars whose average density is approximately
that of the earth's atmosphere at sea level.

About $2°.5$~southeast of Beta Lyræ is the third-magnitude
star Gamma Lyræ. On a line joining these two stars and
about one third of the distance from Beta is a ring, or annular,
nebula, the only one of the few that are known that
\index{Ring nebula in Lyra}%
can be seen with a small telescope. It takes a large telescope,
however, to show much of its detail.
%% -----File: 186.png---Folio 156-------

\Article{89}{Hercules (The Kneeling Hero).}---Hercules is a very
\index{Hercules}%
large constellation lying west and southwest of Lyra. It
\index{Lyra}%
contains no stars brighter than the third magnitude, but it
can be recognized from a trapezoidal figure of 5~stars which
are about $20°$~west of Vega. The base of the trapezoid, which
\index{Vega}%
is turned to the north and slightly to the east, is about $6°$~long
and contains two stars in the northeast corner which
are of the third and fourth magnitudes. The star in the
southeast corner is of the fourth magnitude, and the others
are of the third magnitude. On the west side of the trapezoid,
about one third of the distance from the north end, is
one of the finest star clusters in the whole heavens, known as
Messier~13. It is barely visible to the unaided eye on a
\index[xnames]{Messier}%
clear dark night, appearing as a little hazy star; but through
a good telescope it is seen to be a wonderful object, containing
more than $5000$~stars (\Figref{171}) which are probably comparable
to our own sun in dimensions and brilliancy. The
cluster was discovered by Halley (1656--1742), but derives
\index[xnames]{Halley}%
its present name from the French comet hunter Messier
(1730--1817), who did all of his work with an instrument of
only $2.5$~inches' aperture.

\Article{90}{Scorpius (The Scorpion).}---There are $12$~constellations,
\index{Scorpius}%
one for each month, which lie along the ecliptic and
constitute the zodiac. Scorpius is the ninth of these and the
most brilliant one of all. In fact, it is one of the finest group
of stars that can be seen from our latitude. It is $60°$~straight
south of Hercules and can always be easily recognized by
its fiery red first-magnitude star Antares, which, in light-giving
\index{Antares}%
power, is equal to at least $200$~suns such as ours.
The word Antares means opposed to, or rivaling, Mars,
the red planet associated with the god of war. Antares is
represented as occupying the position of the heart of a scorpion.
About $7''$~west of Antares is a faint green star of the
sixth magnitude which can be seen through a $5$-~or $6$-inch
telescope under good atmospheric conditions. About $5°$~northwest
of Antares is a very compact and fine cluster,
%% -----File: 187.png---Folio 157-------
Messier~80. Scorpius lies in one of the richest and most
\index{Scorpius}%
\index[xnames]{Messier}%
varied parts of the Milky Way.

According to the Greek legend, Scorpius is the monster
that killed Orion and frightened the horses of the sun so that
Phaëton was thrown from his chariot when he attempted to
drive them.

\Article{91}{Corona Borealis (The Northern Crown).}---Just west
\index{Corona Borealis}%
\index{Northern Crown}%
of the great Hercules lies the little constellation Corona
Borealis. It is easily recognized by the semicircle, or crown,
of stars of the fourth and fifth magnitudes which opens
toward the northeast. The Pawnee Indians called it the
camp circle, and it is not difficult to imagine that the stars
represent warriors sitting in a semicircle around a central
campfire.

\Article{92}{Boötes (The Hunter).}---Boötes is a large constellation
\index{Bootes@{Boötes}}%
lying west of Corona Borealis, in right ascension about
$14$~hours, and extending from near the equator to within
$35°$~of the pole. It always can be easily recognized by its
bright first-magnitude star Arcturus, which is about $20°$~southwest
\index{Arcturus}%
of Corona Borealis. This
star is a deep orange in %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{1.5in}
\Input[1.5in]{187}{jpg}
\Caption[The sickle in Leo,
as seen when it is on the
meridian.]{Fig}{57}
\index[xnames]{Hughes}%
\end{wrapfigure}
color and is
one of the finest stars in the northern
sky. It is so far away that $100$~years
are required for its light to
come to the earth, and in radiating
power it is equivalent to more than
$500$~suns like our own.

In mythology Boötes is represented
as leading his hunting dogs in their
pursuit of the bear across the sky.

\Article{93}{Leo (The Lion).}---Leo lies
\index{Leo}%
about $60°$~west of Arcturus and is the
sixth zodiacal constellation. It is
easily recognized by the fact that it contains 7~stars which
form the outline of a sickle. In the photograph, \Figref{57}, only
the 5~brightest stars are shown. The most southerly star of
%% -----File: 188.png---Folio 158-------
\begin{figure}[hbtp]
\centering\Input{188}{jpg} %[Illustration: Fig. 58]
\Caption[The Great Andromeda Nebula. \textit{Photographed by Ritchey with
the two-foot reflector of the Yerkes Observatory.}]{Fig}{58}
\index{Andromeda!Nebula}%
\index{Yerkes Observatory}%
\end{figure}%
%% -----File: 189.png---Folio 159-------
the sickle is Regulus, at the end of the handle. The blade
\index{Regulus}%
of the sickle opens out toward the southwest. One of the
most interesting things in connection with this constellation
is that the meteors of the shower which occurs about
November~14 seem to radiate from a point within the blade
of the sickle (\Artref{204}).

The star Regulus is at the heart of the Nemean lion which,
according to classic legends, was killed by Hercules as the
\index{Hercules}%
first of his twelve great labors.

\Article{94}{Andromeda (The Woman Chained).}---Andromeda
\index{Andromeda}%
is a large constellation just south of Cassiopeia. It contains
\index{Cassiopeia}%
no first-mag\-ni\-tude stars, but it can be recognized from a
line of 3~second-magnitude stars extending northeast and
southwest. The most interesting object in this constellation
is the Great Andromeda Nebula, \Figref{58}, the brightest
nebula in the sky. It is about $15°$~directly south of Alpha
Cassiopeiæ, and it can be seen without difficulty on a clear,
moonless night as a hazy patch of light. When viewed
through a telescope it fills a part of the sky nearly $2°$~long and
$1°$~wide. In its center is a star which is probably variable.
The analysis of its light with the spectroscope seems to indicate
that it is composed of solid or liquid material surrounded
by cooler gases. It has been suggested that, instead
of being a nebula, it may be an aggregation of millions
of suns comparable to the Galaxy, but so distant from us
\index{Galaxy}%
that it apparently covers an insignificant part of the sky.

\Article{95}{Perseus (The Champion).}---Perseus is a large constellation
\index{Perseus}%
in the Milky Way directly east of Andromeda.
Its brightest star, Alpha, is in the midst of a star field which
presents the finest spectacle through field glasses or a small
telescope in the whole sky. The second brightest star in
this constellation is the earliest known variable star, Algol
\index{Algol}%
(the Demon). Algol is about $9°$~south and a little west of
Alpha Persei, and varies in magnitude from $2.2$ to $3.4$ in a
period of $2.867$~days. That is, at its minimum it loses more
than two thirds of its light. There is also a remarkable
%% -----File: 190.png---Folio 160-------
double cluster in this constellation about $10°$~east of Alpha
Cassiopeiæ.

Algol, together with the little stars near it, is the Medusa's
\index{Algol}%
\index[xnames]{Medusa}%
head which Perseus is supposed to carry in his hand and which
\index{Perseus}%
he used in the rescue of Andromeda. He is said to have
\index{Andromeda}%
stirred up the dust in heaven in his haste, and it now appears
as the Milky Way.
\index{Milky Way}%

\Article{96}{Auriga (The Charioteer).}---The next constellation
\index{Auriga}%
east of Perseus is Auriga, which contains the great first-magnitude
star Capella. Capella is about $40°$~from the
\index{Capella}%
Big Dipper and nearly in a line from Delta through Alpha
\index{Big Dipper}%
Ursæ Majoris. It is also distinguished by the fact that
near it are 3~stars known as The Kids, the name Capella
meaning The She-goat. It is receding from us at the rate
of nearly $20$~miles per second and its distance is $2,600,000$
times that of the earth from the sun. It was found at the
Lick Observatory, in 1889, to be a spectroscopic binary with
\index{Lick Observatory}%
a period of $104.2$~days. The computations of Maunder
\index[xnames]{Maunder}%
show that it radiates about $200$~times as much light as is
given out by the sun.

\Article{97}{Taurus (The Bull).}---Taurus is southwest of Auriga
\index{Taurus}%
and contains two conspicuous groups of stars, the Pleiades
\index{Pleiades}%
and the Hyades, besides the brilliant red star Aldebaran.
\index{Hyades}%

Among the many mythical stories regarding this constellation
there is one which describes the bull as charging down
on Orion. According to a Greek legend, Zeus took the form
\index{Orion}%
\index[xnames]{Zeus}%
of a bull when he captured Europa, the daughter of Agenor.
\index[xnames]{Agenor}%
\index[xnames]{Europa}%
While playing in the meadows with her friends, she leaped
upon the back of a beautiful white bull, which was Zeus
himself in disguise. He dashed into the sea and bore her
away to Crete. Only his head and shoulders are visible in
the sky because, when he swims, the rest of his body is
covered with water.

The Pleiades group, \Figref{59}, consists of 7~stars in the
form of a little dipper about $30°$~southwest of Capella and
nearly $20°$~south of, and a little east of, Algol. Six of them,
%% -----File: 191.png---Folio 161-------
which are of the fourth magnitude, are easily visible without
optical aid; but the seventh, which is near the one at the
end of the handle in the dipper, is more difficult. There
seems to have been considerable difficulty in seeing the faintest
one in ancient times, for it was frequently spoken of as
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{191}{jpg}
\Caption[The Pleiades. \textit{Photographed by Wallace at the Yerkes Observatory.}]{Fig}{59}
\index{Pleiades}%
\index{Yerkes Observatory}%
\index[xnames]{Wallace, R. J.}%
\end{figure}%
having been lost. There is no difficulty now, however, for
people with good eyes to see it, while those with exceptionally
keen sight can see $10$~or $11$~stars.

No group of stars in all the sky seems to have attracted
greater popular attention than the Pleiades, nor to have
been mentioned more frequently, not only in the classic
writings of the ancients, but also in the stories of primitive
peoples. They were The Seven Sisters of the Greeks, The
Many Little Ones of the ancient Babylonians, The Hen and
Chickens of the peoples of many parts of Europe, The
Little Eyes of the savage tribes of the South Pacific Islands,
and The Seven Brothers of some of the tribes of North
American Indians. They cross the meridian at midnight
in November, and many primitive peoples began their year
%% -----File: 192.png---Folio 162-------
at that time. It is said that on the exact date, November~17,
no petition was ever presented in vain to the kings of
ancient Persia. These stars had an important relation to
the religious ceremonies of the Aztecs, and certain of the
Australian tribes held dances in their honor.

Besides the $7$~stars which make up the Pleiades as observed
\index{Pleiades}%
without a telescope, there are at least $100$~others in the group
which can be seen with a small instrument. While their
distance from the earth is not known, it can scarcely be less
than $10,000,000$ times that of the sun. It follows that these
stars are apparently small only because they are so remote.
A star among them equal to the sun in brilliancy would appear
to us as a telescopic object of the ninth magnitude.
The larger stars of the group are at least from $100$ to $200$~times
as great in light-giving power as the sun.

About $8°$~southeast of the Pleiades is the Hyades group, a
\index{Hyades}%
cluster of small stars scarcely less celebrated in mythology.
They have been found recently to constitute a cluster of
stars, occupying an enormous space, all of which move in the
same direction with almost exactly equal speeds (\Artref{277}).
The magnificent scale of this group of stars is quite beyond
imagination. Individually they range in luminosity from $5$
to $100$~times that of the sun, and the diameter of the space
which they occupy is more than $2,000,000$ times the distance
from the earth to the sun.

\Article{98}{Orion (The Warrior).}---Southeast of Taurus and
\index{Orion}%
\index{Taurus}%
directly south of Auriga is the constellation Orion, lying
across the equator between the fifth and sixth hours of right
ascension. This is the finest region of the whole sky for
observation without a telescope.

The legends regarding Orion are many and in their details
conflicting. But in all of them he was a giant and a mighty
hunter who, in the sky, stands facing the bull (Taurus) with
a club in his right hand and a lion's skin in his left.

About $7°$~north of the equator and $15°$~southeast of Aldebaran
is the ruddy Betelgeuze. About $20°$~southwest of
\index{Betelgeuze}%
%% -----File: 193.png---Folio 163-------
Betelgeuze is the first-magnitude star Rigel, a magnificent
\index{Rigel}%
object which is at least $2000$ times as luminous as the sun.
About midway between Betelgeuze and Rigel and almost
on the equator is a row of second-magnitude stars running
northwest and southeast, which constitute the Belt of Orion,
\index{Belt of Orion}%
\index{Orion}%
\Figref{60}. From its southern end another row of fainter
stars reaches off to the southwest, nearly in the direction of
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{193}{jpg}
\Caption[Orion. \textit{Photographed at the Yerkes Observatory \(Hughes\).}]{Fig}{60}
\index{Yerkes Observatory}%
\end{figure}%
Rigel. These stars constitute the Sword of Orion. The central
\index{Sword of Orion}%
one of them appears a little fuzzy without a telescope,
and with a telescope is found to be a magnificent nebula,
\Figref{61}. In fact, the Great Orion Nebula impresses many
\index{Orion nebula}%
observers as being the most magnificent object in the whole
heavens. It covers more than a square degree in the sky,
and the spectroscope shows it to be a mass of glowing gas
whose distance is probably several million times as great as
that to the sun, and whose diameter is probably as great as
%% -----File: 194.png---Folio 164-------
the distance from the earth to the nearest star. The stars
in this region of the sky are generally supposed by astronomers
to be in an early stage of their development; most of them
\begin{figure}[hbt]%[Illustration:]
\Input{194}{jpg}
\Caption[The Great Orion Nebula. \textit{Photographed by Ritchey with the
two-foot reflector of the Yerkes Observatory.}]{Fig}{61}
\index{Orion nebula}%
\index{Yerkes Observatory}%
\end{figure}%
are of great luminosity, and a considerable fraction of them
are variable or double.
%% -----File: 195.png---Folio 165-------

\Article{99}{Canis Major (The Greater Dog).}---The constellation
\index{Canis Major}%
Canis Major is southeast of Orion and is marked by Sirius,
\index{Sirius}%
the brightest star in the whole sky. Sirius is almost in a
line with the Belt of Orion and a little more than $20°$ from it.
\index{Belt of Orion}%
It is bluish white in color and is supposed to be in an early
stage of its evolution, though it has advanced somewhat from
the condition of the Orion stars. Sirius is comparatively
near to us, being the third star in distance from the sun.
Nevertheless, $8.4$~years are required for its light to come to
us, and its distance is $47,000,000,000,000$ miles. It is approaching
us at the rate of $5.6$~miles per second; or, rather, it
is overtaking the sun, for the solar system is moving in nearly
the opposite direction.

The history of Sirius during the last two centuries is very
interesting, and furnishes a good illustration of the value
of the deductive method in making discoveries. First,
Halley found, in 1718, that Sirius has a motion with respect
\index[xnames]{Halley}%
to fixed reference points and lines; then, a little more than
a century later, Bessel found that this motion is slightly
\index[xnames]{Bessel}%
variable. He inferred from this, on the basis of the laws of
motion, that Sirius and an unseen companion were traveling
around their common center of gravity which was moving
with uniform speed in a straight line. This companion
actually was discovered by Alvan~G. Clark, in 1862, while
\index[xnames]{Clark}%
adjusting the $18$-inch telescope now of the Dearborn Observatory,
\index{Dearborn Observatory}%
at Evanston,~Ill. The distance of the two stars
from each other is $1,800,000,000$ miles, and they complete
a revolution in $48.8$~years. The combined mass of the
two stars is about $3.4$~times that of the sun. The larger
star is only about twice as massive as its companion but is
$20,000$ times brighter; together they radiate $48$~times as
much light as is emitted by the sun.

\Article{100}{Canis Minor (The Lesser Dog).}---Canis Minor is
\index{Canis Minor}%
directly east of Orion and is of particular interest in the
present connection because of its first-magnitude star Procyon,
\index{Procyon}%
which is about $25°$~east and just a little south of Betelgeuze.
\index{Betelgeuze}%
%% -----File: 196.png---Folio 166-------
The history of this star is much the same as that of
Sirius, the fainter companion having been discovered in
\index{Sirius}%
1896 by Schaeberle at the Lick Observatory. The period of
\index{Lick Observatory}%
\index[xnames]{Schaeberle}%
revolution of Procyon and its companion is $39$~years, its
\index{Procyon}%
distance is a little greater than that of Sirius, its combined
mass is about $1.3$ that of the sun, and its luminosity is about
$10$~times that of the sun. If the orbits of such systems as
Sirius and Procyon and their fainter companions were edgewise
to the earth, the brighter components would be regularly
eclipsed and they would be variable stars of the Algol
\index{Algol}%
type (\Artref{288}), though with such long periods and short
times of eclipse that their variability would probably not be
discovered.

\Article{101}{Gemini (The Twins).}---Gemini is the fourth zodiacal
\index{Gemini}%
constellation and lies directly north of Canis Minor. It has
been known as ``The Twins'' from the most ancient times
because its two principal stars, Castor and Pollux, are
\index{Castor}%
\index{Pollux}%
almost alike and only $4°.5$~apart. These stars are about
$25°$~north of Procyon, and Castor is the more northerly of
the two. Castor is a double star which can be separated by
a small telescope. In 1900 Bélopolsky, of Pulkowa, found
\index{Pulkowa}%
\index[xnames]{Belopolsky@{Bélopolsky}}%
that its fainter companion is a spectroscopic binary with a
period of $2.9$~days. In 1906 Curtis, of the Lick Observatory,
\index[xnames]{Curtis}%
found that the brighter companion is also a spectroscopic
binary with a period of $9.2$~days. Thus this star, instead
of being a single object as it appears to be without telescopic
and spectroscopic aid, is a system of four suns. The two
pairs revolve about the common center of gravity of the four
stars in a long period which probably lies between $250$ and
$2000$~years.

Castor is called Alpha Geminorum, because probably in
ancient times it was a little brighter than, or at least as bright
as, Pollux. Now Pollux is a little brighter than Castor.

About $10°$~southeast of Pollux is the large open Præsepe
\index{Praesepe@{Præsepe}}%
(The Beehive) star cluster which can be seen on a clear,
\index{Beehive (Præsepe)}%
moonless night without a telescope.
%% -----File: 197.png---Folio 167-------

\Article{102}{On Becoming Familiar with the Stars.}---The discussion
of the constellations will be closed here, not because
all have been described, or, indeed, any one of them adequately,
but because enough has been said to show that the
sky is full of objects of interest which can be found and enjoyed
with very little optical aid. The reader is expected
to observe all the objects which have been described, so far
as the time of year and the instrumental help at his command
will permit. If he does this, the whole subject will
have a deeper and more lively interest, and it will be a pleasure
to make constant appeals to the sky to verify statements
and descriptions.

The general features of the constellations are very simple,
but the whole subject cannot be mastered in an evening.
One should go over it several times with no greater optical
aid than that furnished by a field glass.


\Section{VIII}{QUESTIONS}

1. Show why about $22,000$~plates will be required to photograph
the whole sky as described in \Artref{76}.

2. Find the brightness of the stars in \Tableref{I} compared to that
of a first-magnitude star.

3. Find the amount of light received from the sun compared to
that received from a first-magnitude star.

4. Take the amount of light received from a first-magnitude
star as unity, and compute the amount of light received from each
of the first six magnitudes (\Tableref{II}).

5. If the ratio of the number of stars from one magnitude to the
next continued the same as it is in \Tableref{II}, how many stars would
there be in the first $20$~magnitudes?

6. At what time of the year is the most northerly part of the
Milky Way on the meridian at 8~\PM? What are its altitude and
azimuth at that time?

7. What constellations are within two hours of the meridian at
8~\PM\ to-night? Identify them.

8. If Lyra is visible at a convenient hour, test your eyes on
Epsilon Lyræ.

9. If Leo is visible at a convenient hour, test your eyes by finding
which star in the sickle has a very faint star near it.

%% -----File: 198.png---Folio 168-------

10. If Andromeda is visible at a convenient hour, find the great nebula.

11. How many stars can you see in the bowl of the Big Dipper?

12. If Perseus is visible at a convenient hour, identify Algol and
verify its variability.

13. How many of the Pleiades can you see?

14. If Orion is visible at a convenient hour, identify the Belt and
Sword and notice that the great nebula looks like a fuzzy star.

\normalsize

%% -----File: 199.png---Folio 169-------


\Chapter{VI}{Time}

\Article{103}{Definitions of equal Intervals of Time.}---It is impossible
\index{Time!equal intervals of}%
to give a definition of time in terms which are simpler
and better understood than the word itself; but it is profitable
to consider what it is that determines the length of an
interval of time. The subject may be considered from the
standpoint of the intellectual experience of the individual,
which varies greatly from time to time and which may differ
much from that of another person, or it may be treated with
reference to independent physical phenomena.

Consider first the definition of the length of an interval
of time or, rather, the equality of two intervals of time,
from the psychological point of view. If a person has had a
number of intellectual experiences, he is not only conscious
that they were distinct, but he has them arranged in his memory
in a perfectly definite order. When he recalls them and
notes their distinctness, number, and order, he feels that they
have occurred in time; that is, he has the perception of time.
An interval in which a person has had many and acute
intellectual experiences seems long; and two intervals of
time are of equal length, psychologically, when the individual
has had in them an equal number of equally intense intellectual
experiences. For example, in youth when most of life's
experiences are new and wonderful, the months and the
years seem to pass slowly; on the other hand, with increasing
age when life reduces largely to routine, the years slip
away quickly. Or, to take an illustration within the range
of the experience of many who are still young, a month of
travel, or the first month in college, seems longer than a whole
year in the accustomed routine of preparatory school life.
%% -----File: 200.png---Folio 170-------
It follows from these considerations that the true measure of
the length of the life of an individual from the psychological
point of view, which is the one in which he has greatest interest
as a thinking being, is the number, variety, and intensity
of his intellectual experiences. A man whose life has been
full, who has become acquainted with the world's history,
who is familiar with the wonders of the universe, who has
read and experienced again the finest thoughts of the best
minds of all ages, who has seen many places and come into
contact with many men, and who has originated ideas and
initiated intellectual movements of his own, has lived a
long life, however few may have been the number of revolutions
of the earth around the sun since he was born.

But since men must deal with one another, it is important
\index{Time!equal intervals of}%
to have some definition of the equality of intervals of time
that will be independent of their varying intellectual life.
The definition, or at least its consequences, must be capable
of being applied at any time or place, and it must not disagree
too radically with the psychological definition. Such
a definition is given by the first law of motion (\Artref{40}), or
rather a part of it, which for present purposes will be reworded
as follows:

\textit{Two intervals of time are equal, by definition, if a moving
body which is subject to no forces passes over equal distances in
them.} It is established by experience that it makes no
difference what moving body is used or at what rate it moves,
for they all give the same result.

\Article{104}{The Practical Measure of Time.}---A difficulty
\index{Time!practical measure of}%
with the first law of motion and the resulting definition of
equal intervals of time arises from the fact that it is impossible
to find a body which is absolutely uninfluenced by
exterior forces. Therefore, instead of using the law itself,
one of its indirect consequences is employed. It follows
from this law, together with the other laws of motion, that
a solid, rotating sphere which is subject to no exterior forces
turns at a uniform rate. There is no rotating body which
%% -----File: 201.png---Folio 171-------
is not subject to at least the attraction of other bodies; but
the simple attraction of an exterior body has no influence
on the rate of rotation of a sphere which is perfectly solid.
Therefore the earth rotates at a uniform rate, according to
the definition of uniformity implied in the first law of motion,
except for the slight and altogether negligible modifying influences
which were enumerated in \Artref{45}, and hence can
be used for the measurement of time.

If the rotation of the earth is to be used in the measurement
of time, it is only necessary to determine in some way
the angle through which it turns in any interval under consideration.
This can be done by observations of the position
of the meridian with reference to the stars. Since the stars
are extremely far away and do not move appreciably with
respect to one another in so short an interval as a day, the
rotation of the earth can be measured by reference to any
of them. Let it be remembered that, though the rate of
the rotation of the earth is subject to some possible slight
modifications, its uniformity is far beyond that of any clock
ever made.

\Article{105}{Sidereal Time.}---Sidereal time is the time defined
\index{Day!sidereal}%
\index{Sidereal!day}%
\index{Sidereal!time}%
\index{Time!sidereal}%
by the rotation of the earth with respect to the stars. A
sidereal day is the interval between the passage of the
meridian, in its eastward motion, across a star and its next
succeeding passage across the same star. Since the earth
rotates at a uniform rate, all sidereal days are of the same
length. The sidereal day is divided into $24$~sidereal hours,
which are numbered from $1$ to~$24$, the hours are divided into
$60$~minutes, and the minutes into $60$~seconds. The sidereal
time of a given place on the earth is zero when its meridian
crosses the vernal equinox.

Since the definition of sidereal time depends upon the
meridian of the observer, it follows that all places on the
earth having the same longitude have the same sidereal
time, and that those having different longitudes have different
sidereal time. It follows from the uniformity of the
%% -----File: 202.png---Folio 172-------
earth's rotation that equal intervals of sidereal time are
equal according to the first law of motion.

\Article{106}{Solar Time.}---Solar time is defined by the rotation
\index{Day!solar}%
\index{Solar!days}%
\index{Solar!time}%
\index{Time!solar}%
of the earth with respect to the sun. A solar day is the
interval of time between the passage of a meridian across
the center of the sun and its next succeeding passage across
the center of the sun. Since the sun apparently moves
eastward among the stars, a solar day is longer than the
sidereal day. The sun makes an apparent revolution of the
heavens in $365$~days, and therefore, since the circuit of the
heavens is~$360°$, it moves eastward on the average a little
less than $1°$~a~day. The earth turns $15°$ in $1$~hour, and $1°$
in $4$~minutes, from which it follows that the solar day is
nearly $4$~minutes longer on the average than the sidereal day.

\Article{107}{Variations in the Lengths of Solar Days.}---If the
\index{Variation!in lengths of days}%
apparent %[Illustration: Break]
\begin{wrapfigure}[17]{\WLoc}{3.125in}
\Input[3.125in]{202}{png}
\Caption[Solar days are longer than sidereal days.]{Fig}{62}
\end{wrapfigure}
motion of the sun eastward among the stars were
uniform, each
solar day would
be longer than
the sidereal day
by the same
amount; and
since the sidereal
days are all of
equal length, the
solar days also
would all be of
equal length.
But the eastward
apparent
motion of the
sun is somewhat variable because of two principal reasons,
which will now be explained.

The earth moves in its elliptical orbit around the sun in
such a way that the law of areas is fulfilled. The angular
distance the sun appears to move eastward among the
%% -----File: 203.png---Folio 173-------
stars equals the angular distance the earth moves forward
in its orbit. This is made evident from \Figref{62}, in which
$E_1$~represents the position of the earth when it is noon at~$A$.
At the next noon at~$A$, solar time, the earth has moved forward
in its orbit through the angle $E_1SE_2$ (of course the distance
is greatly exaggerated). Suppose that when the earth
is at~$E_1$ the direction of a star is~$E_1S$. When the earth is at~$E_2$,
the same direction is~$E_2S'$. The sun has apparently
moved through the angle $S'E_2S$, which equals~$E_2SE_1$.

Since the earth moves in its orbit in accordance with the
\index{Day!longest and shortest}%
law of areas, its angular motion is fastest when it is nearest
\begin{figure}[hbt]%[Illustration:]
\Input{203}{png}
\Caption[Length of solar days. Broken line gives effects of eccentricity;
dotted line, the inclination; full line, the combined effects.]{Fig}{63}
\end{figure}%
the sun. Consequently, when the earth is at its perihelion
the sun's apparent motion eastward is fastest, and the solar
days, so far as this factor alone is concerned, are then the
longest. The earth is at its perihelion point about the first
of January and at its aphelion point about the first of July.
Consequently, the time from noon to noon, so far as it
depends upon the eccentricity of the earth's orbit, is longest
about the first of January and shortest about the first of
July. The lengths of the solar days, so far as they depend
upon the eccentricity of the earth's orbit, are shown by the
broken line in \Figref{63}.

The second important reason why the solar days vary
in length is that the sun moves eastward along the ecliptic
and not along the equator. For simplicity, neglect the
eccentricity of the earth's orbit and the lack of uniformity of
the angular motion of the sun along the ecliptic. Consider
%% -----File: 204.png---Folio 174-------
the time when the sun is near the vernal equinox. Since
the ecliptic intersects the equator at an angle of~$23°.5$, only
one component of the sun's motion is directly eastward.
However, the reduction is somewhat less than might be
imagined for so large an inclination and amounts to only
about $10$~per~cent. When the sun is near the autumnal
equinox the situation is the same except that, at this time,
one component of the sun's motion is toward the south.
At these two times in the year the sun's apparent motion
eastward is less than it would otherwise be, and, consequently,
the solar days are shorter than the average. At the
solstices, midway between these two periods, the sun is
moving approximately along the arcs of small circles $23°.5$
from the equator, and its angular motion eastward is correspondingly
faster than the average. Therefore, so far
as the inclination of the ecliptic is concerned, the solar days
are longest about December~21 and June~21, and shortest
about March~21 and September~23. The lengths of the
solar days, so far as they depend upon the inclination of
the ecliptic, are shown by the dotted curve in \Figref{63}.

Now consider the combined effects of the eccentricity of
the earth's orbit and the inclination of the ecliptic on the
lengths of the solar days. Of these two influences, the
inclination of the ecliptic is considerably the more important.
On the first of January they both make the solar day
longer than the average. At the vernal equinox the eccentricity
has only a slight effect on the length of the solar day,
while the obliquity of the ecliptic makes it shorter than the
average. On June~21 the effect of the eccentricity is to
make the solar day shorter than the average, while the effect
of the obliquity of the ecliptic is to make it longer than the
average. At the autumnal equinox the eccentricity has
only a slight importance and the obliquity of the ecliptic
makes the solar day shorter than the average.

The two influences together give the following result:
The longest day in the year, from noon to noon by the sun,
%% -----File: 205.png---Folio 175-------
is about December~22, after which the solar day decreases
continually in length until about the 26th of March; it
then increases in length until about June~21; then it decreases
in length until the shortest day in the year is reached on
September~17; and then it increases in length continually
until December~22. On December~22 the solar day is about
$4$~minutes and $26$~seconds of mean solar time (\Artref{108}) %[** TN: Square brackets in original]
longer than the sidereal; on March~26 it is $3$~minutes and
$38$~seconds longer; on June~21 it is $4$~minutes $9$~seconds
longer; and on September~17 it is $3$~minutes and $35$~seconds
longer. The combined results are shown by the full line in
\Figref{63}. The difference in length between the longest and
the shortest day in the year is, therefore, about $51$~seconds of
mean solar time. While this difference for most purposes
is not important in a single day, it accumulates and gives
rise to what is known as the equation of time (\Artref{109}).

It might seem that it would be sensible for astronomers to
neglect the differences in the lengths of the solar days,
especially as the change in length from one day to the next
is very small. Only an accurate clock would show the disparity
in their lengths, and their slight differences would be
of no importance in ordinary affairs. But if astronomers
should use the rotation of the earth with respect to the
sun as defining equal intervals of time, they would be
employing a varying standard and they would find apparent
irregularities in the revolution of the earth and in all other
celestial motions which they could not bring under any fixed
laws. This illustrates the extreme sensitiveness of astronomical
theories to even slight errors.

\Article{108}{Mean Solar Time.}---Since the ordinary activities
\index{Day!mean solar}%
\index{Mean solar time}%
\index{Time!mean solar}%
of mankind are dependent largely upon the period of daylight,
it is desirable for practical purposes to have a unit of
time based in some way upon the rotation of the earth with
respect to the sun. On the other hand, it is undesirable to
have a unit of variable length. Consequently, the \textit{mean
solar day}, which has the average length of all the solar days
%% -----File: 206.png---Folio 176-------
of the year, is introduced. In sidereal time its length is
$24$~hours, $3$~minutes, and $56.555$~seconds.

The mean solar day is divided into 24~mean solar hours,
the hours into 60~mean solar minutes, and the minutes into
60~mean solar seconds. These are the hours, minutes, and
seconds in common use, and ordinary timepieces are made
to keep mean solar time as accurately as possible. It would
be very difficult, if not impossible, to construct a clock that
would keep true solar time with any high degree of precision.

\Article{109}{The Equation of Time.}---The difference between the
\index{Equation of time}%
\index{Time!equation of}%
true solar time and the mean solar time of a place is called
\textit{the equation of time}. It is taken with such an algebraic sign
that, when it is added to the mean solar time, the true solar
time is obtained.\footnote
  {This is the present practice of the American Ephemeris and Nautical
  Almanac; it was formerly the opposite.}
\index{American Ephemeris and Nautical Almanac}%

The date on which noon by mean solar time and true solar
time shall coincide is arbitrary, but it is so chosen that the
\begin{figure}[hbt]%[Illustration:]
\Input{206}{png}
\Caption[The equation of time.]{Fig}{64}
\end{figure}%
differences between the times in the two systems shall be
as small as possible. On the 24th of December the equation
of time is zero. It then becomes negative and increases
numerically until February~11, when it amounts to about
$-14$~minutes and $25$~seconds; it then increases and passes
through zero about April~15, after which it becomes positive
and reaches a value of $3$~minutes $48$~seconds on May~14;
it then decreases and passes through zero on June~14 and
becomes $-6$~minutes and $20$~seconds on July~26; it then
%% -----File: 207.png---Folio 177-------
increases and passes through zero on September~1 and
becomes $16$~minutes and $21$~seconds on November~2, after
which it continually decreases until December~24. The
results are given graphically in \Figref{64}. The dates may
vary a day or two from those given because of the leap year,
and the amounts by a few seconds because of the shifting of
the dates.

Some interesting results follow from the equation of time.
For example, on December~24 the equation of time is zero,
but the solar day is about $30$~seconds longer than the mean
solar day. Consequently, the next day the sun will be about
$30$~seconds slow; that is, noon by the mean solar clock has
shifted about $30$~seconds with respect to the sun. As the
sun has just passed the winter solstice, the period from sunrise
to sunset for the northern hemisphere of the earth is
slowly increasing, the exact amount depending upon the
latitude. For latitude $40°$~N. the gain in the forenoon resulting
from the earlier rising of the sun is less than the loss
from the shifting of the time of the noon. Consequently,
almanacs will show that the forenoons are getting shorter
at this time of the year, although the whole period between
sunrise and sunset is increasing. The difference in the
lengths of the forenoons and afternoons may accumulate
until it amounts to nearly half an hour.

\Article{110}{Standard Time.}---The mean solar time of a place
\index{Standard time}%
\index{Time!local}%
\index{Time!standard}%
is called its \textit{local time}. All places having the same longitude
have the same local time, but places having different longitudes
have different local times. The circumference of the
earth is nearly $25,000$ miles and $15°$~correspond\DPnote{[** "15°" is plural]} to a difference
of one hour in local time. Consequently, at the earth's
equator, $17$~miles in longitude give a difference of about one
minute in local time. In latitudes $40°$ to $45°$~north or south
$13$ to $12$~miles in longitude give a difference of one minute
in local time.

If every place along a railroad extending east and west
should keep its own local time, there would be endless confusion
%% -----File: 208.png---Folio 178-------
and great danger in running trains. In order to avoid
these difficulties, it has been agreed that all places whose
local times do not differ more than half an hour from that of
some convenient meridian shall use the local time of that
meridian. Thus, while the extreme difference in local time
of places using the local time of the same meridian may be
about an hour, neither of them differs more than about half
an hour from its standard time. In this manner a strip of
country about $750$~miles wide in latitudes $35°$ to $45°$ uses
the same time, and the next strip of the same width an hour
different, and so on. The local time of the standard meridian
of each strip is the \textit{standard time} of that strip.

At present standard time is in use in nearly every civilized
part of the earth. The United States and British America
are of such great extent in longitude that it is necessary to
use four hours of standard time. The eastern portion uses
what is called Eastern Time. It is the local time of the
meridian 5~hours west of Greenwich. This meridian runs
through Philadelphia, and in this city local time and standard
time are identical. At places east of this meridian it is later
by local time than by standard time, the difference being
one minute for $12$ or $13$~miles. At places west of this meridian,
but in the Eastern Time division, it is earlier by local time
than by standard time. The next division to the westward
is called Central Time. It is the local time of the meridian
6~hours west of Greenwich, which passes through St.~Louis.
The next time division is called Mountain Time. It is the
local time of the meridian 7~hours west of Greenwich. This
meridian passes through Denver. The last time division
is called Pacific Time. It is the local time of the meridian
8~hours west of Greenwich. This meridian passes about $100$~miles
east of San Francisco.

If the exact divisions were used, the boundaries between
one time division and the next would be $7°.5$~east and west of
the standard meridian. As a matter of fact, the boundaries
are quite irregular, depending upon the convenience of
%% -----File: 209.png---Folio 179-------
railroads, and they are frequently somewhat altered. The
change in time is nearly always made at the end of a railway
division; for, obviously, it would be unwise to have railroad
time change during the run of a given train crew. As
a result the actual boundaries of the several time divisions
are quite irregular and vary in many cases radically from the
\begin{figure}[hbt]%[Illustration:]
\Input{209}{png}
\Caption[Standard time divisions in the United States.]{Fig}{65}
\end{figure}%
ideal standard divisions. Moreover, many towns near the
borders of the time zones do not use standard time.

\Article{111}{The Distribution of Time.}---The accurate determination
\index{Distribution!of time}%
\index{Time!distribution of}%
of time and its distribution are of much importance.
There are several methods by which time may be
determined, but the one in common use is to observe the
transits of stars across the meridian and thus to obtain the
sidereal time. From the mathematical theory of the earth's
motion it is then possible to compute the mean solar time.
It might be supposed that it would be easier to find mean
solar time by observing the transit of the sun across the
meridian, but this is not true. In the first place, it is much
%% -----File: 210.png---Folio 180-------
more difficult to determine the exact time of the transit
of the sun's center than it is to determine the time of the
transit of a star; and, in the second place, the sun crosses the
meridian but once in $24$~hours, while many stars may be
observed. In the third place, observations of the sun give
true solar time instead of mean solar time, and the computation
necessary to reduce from one to the other is as difficult
as it is to change from sidereal time to mean solar time.

It remains to explain how time is distributed from the
places where the observations are made. In most countries
the time service is under the control of the government,
and the time signals are sent out from the national observatory.
For example, in the United States, the chief source
of time for railroads and commercial purposes is the Naval
Observatory, at Georgetown Heights, Washington, D.C\@.
There are three high-grade clocks keeping standard time
at this observatory. Their errors are found from observations
of the stars; and after applying corrections for these errors,
the mean of the three clocks is taken as giving the true
standard time for the successive $24$~hours. At $5$~minutes
before noon, Eastern Time, the Western Union Telegraph
Company and the Postal Telegraph Company suspend their
ordinary business and throw their lines into electrical connection
with the standard clock at the Naval Observatory.
\index{Naval Observatory}%
The connection is arranged so that the sounding key makes
a stroke every second during the $5$~minutes preceding noon
except the twenty-ninth second of each minute, the last $5$~seconds
 of the fourth minute, and the last $10$~seconds of the
fifth minute. This gives many opportunities of determining
the error of a clock. To simplify matters, clocks are
connected so as to be automatically regulated by these
signals, and there are at present more than $30,000$ of them
in use in this country. The time signals are sent out from
the Naval Observatory with an error usually less than $0.2$~of
 a second; but frequently this is considerably increased
when a system of relays must be used to reach great distances.
%% -----File: 211.png---Folio 181-------

These noon signals also operate time balls in $18$~ports in
the United States. This device for furnishing time, chiefly
to boat captains, consists of a large ball which is dropped at
noon, Eastern Time, from a considerable height at conspicuous
points, by means of electrical connection with the
Naval Observatory.
\index{Naval Observatory}%

Time for the extreme western part of the United States
is distributed from the Mare Island Navy Yard in California;
and besides, a number of college observatories have been
furnishing time to particular railroad systems. Naturally
most observatories regularly determine time for their own
use, though with the accurate distribution of time from
Washington the need for this work is disappearing except
in certain special problems of star positions.

\Article{112}{Civil and Astronomical Days.}---The civil day begins
\index{Day!astronomical}%
\index{Day!civil}%
at midnight, for then business is ordinarily suspended and
the date can be changed with least inconvenience. The
astronomical day of the same date begins at noon, $12$~hours
later; because, if the change were made at midnight, astronomers
might find it necessary to change the date in the
midst of a set of observations. It is true that many observations
of the sun and some other bodies are made in the daytime,
but of course most observational work is done at night.
The hours of the astronomical day are numbered up to $24$,
just as in the case of sidereal time.

\Article{113}{Place of Change of Date.}---If one should start at
\index{Date, place of change of}%
any point on the earth and go entirely around it westward,
the number of times the sun would cross his meridian would
be one less than it would have been if he had stayed at home.
Since it would be very inconvenient for him to use fractional
dates, he would count his day from midnight to midnight,
whatever his longitude, and correct the increasing difference
from the time of his starting point by arbitrarily changing
his date one day forward at some point in his journey. That
is, he would omit one date and day of the week from his
reckoning. On the other hand, if he were going around the
%% -----File: 212.png---Folio 182-------
earth eastward, he would give two days the same date and
day of the week. The change is usually made at the $180$th
meridian from Greenwich. This is a particularly fortunate
selection, for the $180$th meridian scarcely passes through any
land surface at all, and then only small islands. One can
easily see how troublesome matters would be if the change
were made at a meridian passing through a thickly populated
region, say the meridian of Greenwich. On one side
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4.5in]{212}{png}
\Caption[The change-of-date line.]{Fig}{66}
\end{figure}%
of it people would have a certain day and date, for example,
Monday, December~24, and on the other side of it a day
later, Tuesday, December~25.

The place of actual change of date does not strictly follow
the $180$th meridian from Greenwich, for travelers, going
eastward from Europe, lose half a day, while those going
westward from Europe and America arrive in the same
%% -----File: 213.png---Folio 183-------
longitude with a gain of half a day; hence their dates differ
by one day. The change-of-date line is shown in \Figref{66}.

\Article{114}{The Sidereal Year.}---The sidereal year is the time
\index{Sidereal!year}%
\index{Year!sidereal}%
required for the sun apparently to move from any position
with respect to the stars, as seen from the earth, around to
the same position again. Perhaps it is better to say that it
is the time required for the earth to make a complete revolution
around the sun, directions from the sun being determined
by the positions of the stars. Its length in mean
solar time is $365$~days, $6$~hours, $9$~minutes, $9.54$~seconds, or
just a little more than $365.25$~days.

\Article{115}{The Anomalistic Year.}---The anomalistic year is
\index{Year!anomalistic}%
the time required for the earth to move from the perihelion
of its orbit around to the perihelion again. If the perihelion
point were fixed, this period would equal the sidereal year.
But the attraction of the other planets causes the perihelion
point to move forward at such a rate that it completes a
revolution in about $108,000$ years; and the consequence is
that the anomalistic year is a little longer than the sidereal
year. It follows from the period of its revolution that the
perihelion point advances about $12''$ annually. Since the
earth moves, on the average, about a degree daily, it takes it
about $4$~minutes and $40$~seconds of time to move $12''$. The
actual length of the anomalistic year in mean solar time is
$365$~days, $6$~hours, $13$~minutes, $53.01$~seconds.

\Article{116}{The Tropical Year.}---The tropical year is the time
\index{Tropical year}%
\index{Year!tropical}%
required for the sun to move from a tropic around to the
same tropic again; or, better for practical determination,
from an equinox to the same equinox again. Since the
equinoxes regress about $50''.2$ annually, the tropical year is
about $20$~minutes shorter than the sidereal year. Its actual
length in mean solar time is $365$~days, $5$~hours, $48$~minutes,
$45.92$~seconds.

The seasons depend upon the sun's place with respect
to the equi\-nox\-es. Consequently, if the seasons are always
to occur at the same time according to the calendar, the
%% -----File: 214.png---Folio 184-------
tropical year must be used. This is, indeed, the year in
common use and, unless otherwise specified, the term \textit{year}
means the tropical year.

\Article{117}{The Calendar.}--In very ancient times the calendar
\index{Calendar}%
was based largely on the motions of the moon, whose phases
determined the times of religious ceremonies. The moon
does not make an integral number of revolutions in a year,
and hence it was occasionally necessary to interpolate a
month in order to keep the year in harmony with the seasons.

The week was another division of time used in antiquity.
The number of days in this period was undoubtedly based
upon the number of moving celestial bodies which were then
known. Thus, Sunday was the sun's day; Monday, the
moon's day; Tuesday, Mars' day; Wednesday, Mercury's
day; Thursday, Jupiter's day; Friday, Venus's day; and
Saturday, Saturn's day. The names of the days of the
week, when traced back to the tongues from which English
has been derived, show that these were their origins.

In the year 46~\BC\ the Roman calendar, which had
fallen into a state of great confusion, was reformed by
Julius Cæsar under the advice of an Alexandrian astronomer,
\index[xnames]{Caesar@{Cæsar}}%
Sosigenes. The new system, called the Julian Calendar,
\index[xnames]{Sosigenes}%
was entirely independent of the moon; in it there were $3$~years
of $365$~days each and then one year, the leap year, of
\index{Leap year}%
\index{Year!leap}%
$366$~days. This mode of reckoning, which makes the average
year consist of $365.25$~days, was put into effect at the
beginning of the year 45~\BC.

It is seen from the length of the tropical year, which was
given in \Artref{116}, that this system of calculation involves a
small error, averaging $11$~minutes and $14$~seconds yearly.
In the course of $128$~years the Julian Calendar gets one day
behind. To remedy this small error, in 1582, Pope Gregory~XIII
\index[xnames]{Gregory XIII, Pope}%
introduced a slight change. Ten days were omitted
from that year by making October~15 follow immediately
after October~4, and it was decreed that $3$~leap years out of
every $4$~centuries should henceforth be omitted. This again
%% -----File: 215.png---Folio 185-------
is not quite exact, for the Julian Calendar gets behind $3$~days
in $3 × 128 = 384$~years instead of $400$~years; yet
the error does not amount to a day until after more than $3300$~years
\index{Day!Julian}%
have elapsed.

To simplify the application, every year whose date
number is exactly divisible by~$4$ is a leap year, unless it is
exactly divisible by~$100$. Those years whose date numbers
are divisible by~$100$ are not leap years unless they are exactly
divisible by~$400$, when they are leap years. Of course, the
error which still remains could be further reduced by a rule
for the leap years when the date number is exactly divisible
by~$1000$, but there is no immediate need for it.

The calendar originated and introduced by Pope Gregory~XIII
\index[xnames]{Gregory XIII, Pope}%
in 1582, and known as the Gregorian Calendar, is now
in use in all civilized countries except Russia and Greece,
although it was not adopted in England until 1752. At that
time $11$~days had to be omitted from the year, causing considerable
disturbance, for many people imagined they were
in some way being cheated out of that much time. The
Julian Calendar is now $13$~days behind the Gregorian Calendar.
The Julian Calendar is called Old Style (O.S.), and
the Gregorian, New Style (N.S.).

In certain astronomical work, such as the discussion of the
observations of variable stars, where the difference in time of
the occurrence of phenomena is a point of much interest, the
Julian Day is used. The Julian Day is simply the number
of the day counting forward from January~1, 4713~\BC. This
particular date from which to count time was chosen because
that year was the first year in several subsidiary cycles,
which will not be considered here.

\Article{118}{Finding the Day of the Week on Any Date.}--An
ordinary year of $365$~days consists of $52$~weeks and one day,
and a leap year consists of $52$~weeks and $2$~days. Consequently,
in succeeding years the same date falls one day
later in the week except when a twenty-ninth of February
intervenes; and in this case it is two days later. These
%% -----File: 216.png---Folio 186-------
facts give the basis for determining the day of the week on
which any date falls, after it has been given in a particular
year.

Consider first the problem of finding the day of the week
on which January~1 falls. In the year~1900 January~1 fell
on Monday. To fix the ideas, consider the question for
1915. If every year had been an ordinary year, January~1
coming one day later in the week in each succeeding year,
it would have fallen, in 1915, $15$~days, or $2$~weeks and one
day, after Monday; that is, on Tuesday. But, in the
meantime there were $3$~leap years, namely, 1904, 1908,
and 1912, which put the date $3$~additional days forward in
the week, or on Friday. Similarly, it is seen in general
that the rule for finding the day of the week on which
January~1 falls in any year of the present century is to take
the number of the year in the century ($15$~in the example
just treated), add to it the number of leap years which have
passed (which is given by dividing the number of the year
by~$4$ and neglecting the remainder), divide the result by~$7$
to eliminate the number of weeks which have passed, and
finally, count forward from Monday the number of days
given by the remainder. For example, in 1935 the number
of the year is~$35$, the number of leap years is~$8$, the sum of
$35$~and~$8$ is~$43$, and $43$~divided by~$7$ equals~$6$ with the remainder
of~$1$. Hence, in 1935, January~1 will be one day later
than Monday; that is, it will fall on Tuesday.

In order to find the day of the week on which any date of
any year falls, find first the day of the week on which January~1
falls; then take the day of the year, which can be
obtained by adding the number of days in the year up to the
date in question, and divide this by~$7$; the remainder is the
number of days that must be added to that on which January~1
falls in order to obtain the day of the week. For
example, consider March~21, 1935. It has been found that
January~1 of this year falls on Tuesday. There are $79$~days
from January~1 to March~21 in ordinary years. If~$79$ is
%% -----File: 217.png---Folio 187-------
divided by~$7$, the quotient is~$11$ with the remainder of~$2$.
Consequently, March~21, 1935, falls $2$~days after Tuesday,
that is, on Thursday.


\Section{IX}{QUESTIONS}

1. Give three examples where intervals of time in which you
have had many and varied intellectual experiences now seem longer
than ordinary intervals of the same length. Have you had any contradictory
experiences?

2. If the sky were always covered with clouds, how should we
measure time?

3. What is your sidereal time to-day at $8$~\PM?

4. What would be the relations of solar time to sidereal time if
the earth rotated in the opposite direction?

5. What is the length of a sidereal day expressed in mean solar
time?

6. What is the standard time of a place whose longitude is $85°$~west
of Greenwich when its local time is $11$~\AM?

7. What is the local time of a place whose longitude is $112°$~west
of Greenwich when its standard time is $8$~\PM?

8. Suppose a person were able to travel around the earth in $2$~days;
what would be the lengths of his days and nights if he traveled
from east to west? From west to east?

9. If the sidereal year were in ordinary use, how long before
Christmas would occur when the sun is at the vernal equinox?

10. On what days of the week will your birthday fall for the next
$12$~years?

\normalsize

%% -----File: 218.png---Folio 188-------


\Chapter{VII}{The Moon}
\index{Moon}%

\Article{119}{The Moon's apparent Motion among the Stars.}---The
\index{Moon!apparent motion of}%
\index{Moon!orbit of}%
apparent motion of the moon can be determined by
observation without any particular reference to its actual
motion. In fact, the ancient Greeks observed the moon
with great care and learned most of the important peculiarities
of its apparent motion, but they did not know its
distance from the earth and had no accurate ideas of the
character of its orbit. The natural method of procedure is
first to find what the appearances are, and from them to
infer the actual facts.

The moon has a diurnal motion westward which is produced,
of course, by the eastward rotation of the earth.
Every one is familiar with the fact that it rises in the east,
goes across the sky westward, and sets in the west. Those
who have observed it except in the most casual way, have
noticed that it rises at various points on the eastern horizon,
crosses the meridian at various altitudes, and sets at various
points on the western horizon. They have also noticed that
the interval between its successive passages across the
meridian is somewhat more than $24$~hours.

Observations of the moon for two or three hours will show
that it is moving eastward among the stars. When its path
is carefully traced out during a whole revolution, it is found
that its apparent orbit is a great circle which is inclined to
the ecliptic at an angle of~$5°\,9'$. The point at which the
moon, in its motion eastward, crosses the ecliptic from south
to north is called the \textit{ascending node} of its orbit, and the
\index{Ascending node}%
\index{Nodes, ascending and descending}%
point where it crosses the ecliptic from north to south is
called the \textit{descending node} of its orbit. The attraction of the
%% -----File: 219.png---Folio 189-------
sun for the moon causes the nodes continually to regress on
the ecliptic; that is, in successive revolutions the moon
crosses the ecliptic farther and farther to the west. The
period of revolution of the line of nodes is $18.6$~years.

\Article{120}{The Moon's Synodical and Sidereal Periods.}--The
\index{Moon!periods of}%
\index{Period, of moon!sidereal}%
\index{Period, of moon!synodical}%
\index{Sidereal!period of moon}%
\index{Synodical period!of moon}%
synodical period of the moon is the time required for it to
move from any apparent position with respect to the sun
back to the same position again. The most accurate means
of determining this period is by comparing ancient and
modern eclipses of the sun; for, at the time of a solar eclipse,
the moon is exactly between the earth and the sun. The
advantages of this method are that, in the first place, at the
epochs used the exact positions of the moon with respect to
the sun are known; and, in the second place, in a long interval
during which the moon has made hundreds or even
thousands of revolutions around the earth, the errors in the
determinations of the exact times of the eclipses are relatively
unimportant because they are divided by the number
of revolutions the moon has performed. It has been found
in this way that the synodical period of the moon is $29$~days,
$12$~hours, $44$~minutes, and $2.8$~seconds; or $29.530588$~days,
with an uncertainty of less than one tenth of a second.

The sidereal period of the moon is the time required for
it to move from any apparent position with respect to the
stars back to the same position again. This period can be
found by direct observations; or, it can be computed from
the synodical period and the period of the earth's revolution
around the sun. Let~$S$ represent the moon's synodical
period, $M$~its sidereal period, and $E$~the period of the earth's
revolution around the sun, all expressed in the same units as,
for example, days. Then $1/M$~is the fraction of a revolution
that the moon moves eastward in one day, $1/E$~is the fraction
of a revolution that the sun moves eastward in one day,
and $1/M - 1/E$~is, therefore, the fraction of a revolution that
the moon gains on the sun in its eastward motion in one day.
Since the moon gains one complete revolution on the sun in
%% -----File: 220.png---Folio 190-------
$S$~days, $1/S$~is also the fraction of a revolution the moon
gains on the sun in one day. Hence it follows that
\[
\frac{1}{S} = \frac{1}{M} - \frac{1}{E},
\]
from which $M$~can be computed when $S$~and~$E$ are known.

It is easy to see that the synodical period is longer than
the sidereal. Suppose the sun, moon, and certain stars are
at a given instant in the same straight line as seen from the
earth. After a certain number of days the moon will have
made a sidereal revolution and the sun will have moved eastward
among the stars a certain number of degrees. Since
additional time is required for the moon to overtake it, the
synodical period is longer than the sidereal.

It has been found by direct observations, and also by the
equation above, that the moon's sidereal period is $27$~days,
$7$~hours, $43$~minutes, and $11.5$~seconds, or $27.32166$~days.
When the period of the moon is referred to, the sidereal
period is meant unless otherwise stated.

The periods which have been given are averages, for the
moon departs somewhat from its elliptical orbit, primarily
because of the attraction of the sun, and to a lesser extent
because of the oblateness of the earth and the attractions of
the planets. The variations from the average are sometimes
quite appreciable, for the perturbations, as they are called,
may cause the moon to depart from its undisturbed orbit
about~$1°.5$, and may cause its period of revolution to vary by
as much as $2$~hours.

\Article{121}{The Phases of the Moon.}---The moon shines entirely
by reflected sunlight, and consequently its appearance
as seen from the earth depends upon its position relative to
the sun. \Figureref{67} shows eight positions of the moon in its
orbit with the sun's rays coming from the right in lines which
are essentially parallel because of the great distance of the
sun. The left-hand side of the earth is the night side, and
similarly the left side of the moon is the dark side.
%% -----File: 221.png---Folio 191-------

The small circles whose centers are on the large circle
around the earth as a center show the illuminated and unilluminated
parts of the moon as they actually are; the
accompanying small circles just outside %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{221}{png}
\Caption[Explanation of the moon's phases.]{Fig}{67}
\index{Moon!phases of}%
\index{Phases!of moon}%
\end{wrapfigure}
of the large circle
show the moon as
it is seen from the
earth. For example,
when the moon is
at~$M_1$ between the
earth and sun, its
dark side is toward
the earth. In this
position it is said to
be \textit{in conjunction},
and the phase is \textit{new}.
At~$M_2$ half of the
illuminated part of the moon can be seen from the earth, and
it is in the \textit{first quarter}. In this position the moon is said
to be \textit{in quadrature}. Between the new moon and the first
\index{Quadrature}%
quarter the illuminated part of the moon as seen from the
earth is of crescent shape, and its convex side is turned
toward the sun.

When the moon is at~$M_3$ the illuminated side is toward the
earth. It is then \textit{in opposition}, and the phase is \textit{full}. If an
observer were at the sunset point on the earth, the sun
would be setting in the west and the full moon would be
rising in the east. At~$M_4$ the moon is again in quadrature,
and the phase is \textit{third quarter}.

To summarize: The moon is new when it has the same
right ascension as the sun; it is at the first quarter when
its right ascension is $6$~hours greater than that of the sun;
it is full when its right ascension is $12$~hours greater than
that of the sun; and it is at the third quarter when its right
ascension is $18$~hours greater than that of the sun.

\begin{wrapfigure}{\WLoc}{2.75in}%[Illustration:]
\Input[2.75in]{222a}{jpg}
\Caption[The moon partially illuminated
by light reflected from the
earth. \textit{Photographed by Barnard at
the Yerkes Observatory.}]{Fig}{68}
\index{Yerkes Observatory}%
\end{wrapfigure}
It is observed from the diagram that the earth would
have phases if seen from the moon. When the moon is
%% -----File: 222.png---Folio 192-------
new, as seen from the earth, the earth would be full as seen
from the moon. The phases of the earth corresponding to
every other position of
the moon can be inferred
from the diagram. The
phases of the moon and
earth are supplementary;
that is, the illuminated
portion of the moon as
seen from the earth plus
the illuminated portion of
the earth as seen from the
moon always equals $180°$.
When the moon is nearly
new, and, consequently,
the earth nearly full as
seen from the moon, the
dark side of the moon is
somewhat illuminated by sunlight reflected from the earth,
as is shown in \Figref{68}.

\Article{122}{The diurnal Circles of the Moon.}---Suppose first
\index{Moon!diurnal circles of}%
that the moon moves along the ecliptic and consider its
diurnal circles. Since they are parallel to the celestial
\begin{figure}[hbt]%[Illustration:]
\Input{222b}{png}
\Caption[The equator and ecliptic.]{Fig}{69}
\end{figure}%
equator (if the motion of the moon in declination between
rising and setting is neglected), it is sufficient, in view of the
discussion of the sun's diurnal circles (\Artref{58}), to give the
places where the moon crosses the meridian. Let~$VAV$,
\Figref{69}, represent the celestial equator spread out on a plane,
and~$VSAWV$ the ecliptic. Suppose, for example, that the
time of the year is March~21. Then the sun is at~$V$. If
%% -----File: 223.png---Folio 193-------
the moon is new, it is also at~$V$, because at this phase it has
the same right ascension as the sun. Since $V$ is on the celestial
equator, the moon crosses the meridian at an altitude
equal to $90°$~minus the latitude of the observer. In this
case it rises in the east and sets in the west. But if the moon
is at first quarter on March~21, it is at~$S$, because at this
phase it is $6$~hours east of the sun. It is then $23°.5$~north
of the equator, and, consequently, it crosses the meridian
$23°.5$~above the equator. In this case it rises north of east
and sets north of west. If the moon is full, it is at~$A$, and
if it is in the third quarter, it is at~$W$. In the former case it
is on the equator and in the latter $23°.5$~south of it.

Suppose the sun is at the summer solstice,~$S$. Then it
rises in the northeast, crosses the meridian $23°.5$~north of
the equator, and sets in the northwest. At the same time
the full moon is at~$W$, it rises in the southeast, crosses the
meridian $23°.5$~south of the equator, and sets in the southwest.
That is, when sunshine is most abundant, the light
from the full moon is the least. On the other hand, when
the sun is at the winter solstice~$W$, the full moon is at~$S$
and gives the most light. The other positions of the sun
and moon can be treated similarly.

Suppose the ascending node of the moon's orbit is at the
vernal equinox (\Figref{70}), and consider the altitude at which
\begin{figure}[hbt]%[Illustration:]
\Input{223}{png}
\Caption[Ascending node of the moon's orbit at the vernal equinox.]{Fig}{70}
\end{figure}%
the moon crosses the meridian when full at the time of the
winter solstice. The sun is at~$W$ and the full moon is in its
orbit $5°\,9'$~north of~$S$. If the latitude of the observer is~$40°$,
the moon then crosses his meridian at an altitude of $50° +
23°.5 + 5° = 78°.5$. That is, under these circumstances the
%% -----File: 224.png---Folio 194-------
full moon crosses the meridian higher in the winter time
than it would if its orbit were coincident with the ecliptic.
On the other hand, in the summer time, when the sun is at~$S$
and the full moon is at~$W$, the moon crosses the equator
farther south than it would if it were on the ecliptic. Under
these circumstances there is more moonlight in the winter
and less in the summer than there would be if the moon
were always on the ecliptic.

Now suppose the descending node is at $V$ and the ascending
node is at~$A$, \Figref{71}. For this position of its orbit the
\begin{figure}[hbt]%[Illustration:]
\Input{224}{png}
\Caption[Ascending node of the moon's orbit at the autumnal equinox.]{Fig}{71}
\end{figure}%
moon crosses the meridian lower in the winter than it would
if it moved along the ecliptic. The opposite is true when
the sun is at~$S$ in the summer. Of course, the ascending
node of the moon's orbit might be at any other point on
the ecliptic.

It is clear from this discussion that when the sun is on
the part of the ecliptic south of the equator, the full moon
is near the part of the ecliptic which is north of the equator,
and \textit{vice versa}. Therefore, when there is least sunlight there
is most moonlight, and there is the greatest amount of moonlight
when the moon's ascending node is at the vernal
equinox. When it is continuous night at a pole of the earth,
the gloom is partly dispelled by the moon which is above the
horizon that half of the month in which it passes from its
first to its third quarter.

\Article{123}{The Distance of the Moon.}---One method of determining
\index{Distance!of moon}%
\index{Moon!distance of}%
the distance of the moon is by observing the difference
in its directions as seen from two points on the earth's
surface, as $O_1$~and~$O_2$ in \Figref{72}. Suppose, for simplicity,
that $O_1$~and~$O_2$ are on the same meridian, and that the moon
%% -----File: 225.png---Folio 195-------
is in the plane of that meridian. The observer at~$O_1$ finds
that the moon is the angular distance~$Z_1O_1M$ south of his
zenith; and the observer at~$O_2$ finds that it is the angular
distance~$Z_2O_2M$ north of his zenith. Since the two observers
know their latitudes, they know the angle~$O_1EO_2$, and
consequently, the angles $EO_1O_2$~and~$EO_2O_1$. By subtracting
$Z_1O_1M$~plus~$EO_1O_2$ and $Z_2O_2M$~plus~$EO_2O_1$ from~$180°$,
the angles $MO_1O_2$~and~$MO_2O_1$ are obtained. Since the size
of the earth is known, the distance~$O_1O_2$ can be found. Then,
in the triangle~$O_1MO_2$ two angles and the included side are
known, and all the other parts of the triangle can be computed
\begin{figure}[hbt]%[Illustration:]
\Input{225}{png}
\Caption[Measuring the distance to the moon.]{Fig}{72}
\end{figure}%
by trigonometry. Suppose $O_1M$ has been found;
then, in the triangle~$EO_1M$ two sides and the included angle
are known, and the distance~$EM$ can be computed. In
general, the relations and observations will not be so simple
as those assumed here, but in no case are serious mathematical
or observational difficulties encountered. It is to
be noted that the result obtained is not guesswork, but
that it is based on measurements, and that it is in reality
given by measurements in the same sense that a distance
on the surface of the earth may be obtained by measurement.
The percentage of error in the determination of the
moon's distance is actually much less than that in most of
the ordinary distances on the surface of the earth.
%% -----File: 226.png---Folio 196-------

The mean distance from the center of the earth to the
center of the moon has been found to be $238,862$ miles, and
the circumference of its orbit is therefore $1,500,818$ miles.
On dividing the circumference by the moon's sidereal period
expressed in hours, it is found that its orbital velocity averages
\index{Moon!velocity of}%
\index{Velocity!of moon}%
$2288.8$~miles per~hour, or about $3357$~feet per~second.

A body at the surface of the earth falls about $16$~feet the
first second; at the distance of the moon, which is approximately
$60$~times the radius of the earth, it would, therefore,
fall $16 ÷ 60^2 = 0.0044$~feet, because the earth's attraction
varies inversely as the square of the distance from its center.
Therefore, in going $3357$~feet, or nearly two thirds of a mile,
the moon deviates from a straight-line path only about $\frac{1}{20}$~of
an~inch.

\Article{124}{The Dimensions of the Moon.}---The mean apparent
\index{Moon!dimensions of}%
diameter of the moon is $31'~5''.2$. Since its distance is
known, its actual diameter can be computed. It is found
that the distance straight through the moon is $2160$~miles,
or a little greater than one fourth the diameter of the earth.
Since the surfaces of spheres are to each other as the squares
of their diameters, it is found that the surface area of the
earth is $13.4$~times that of the moon; and since the volumes
of spheres are to each other as the cubes of their diameters,
it is found that the volume of the earth is $49.3$~times that
of the moon.

It has been stated that the mean apparent diameter of
the moon is $31'~5''.2$. The apparent diameter of the moon
varies both because its distance from the center of the earth
varies, and also because when the moon is on the observer's
meridian, he is nearly $4000$~miles nearer to it than when
it is on his horizon. In the observations of other celestial
objects the small distance of $4000$~miles makes no appreciable
difference in their appearance; but, since the distance
from the earth to the moon is, in round numbers, only
$240,000$ miles, the radius of the earth is $\frac{1}{60}$~of the whole
amount.
%% -----File: 227.png---Folio 197-------

In spite of the fact that the moon is nearer the observer
when it is on his meridian than when it is on his horizon,
every one has noticed that it appears largest when near
the horizon and smallest when near the meridian. The
reason that the moon appears to us to be larger when it is
near the horizon is that then intervening objects give us the
impression that it is very distant, and this influences our
unconscious estimate of its size.

\Article{125}{The Moon's Orbit with Respect to the Earth.}---The
\index{Moon!orbit of}%
moon's distance from the earth varies from about
$225,746$ miles to $251,978$ miles, causing a corresponding
variation in its apparent diameter. Its orbit is an ellipse,
having an eccentricity of~$0.0549$, except for slight deviations
due to the attractions of the sun, planets, and the equatorial
bulge of the earth. The moon moves around the earth,
which is at one of the foci of its elliptical orbit, in such a
manner that the line joining it to the earth sweeps over
equal areas in equal intervals of time. This statement requires
a slight correction because of the perturbations produced
by the attractions of the sun and planets. The
point in the moon's orbit which is nearest the earth is called
its \textit{perigee}, and the farthest point is called its \textit{apogee}.
\index{Apogee}%
\index{Moon!apogee of}%
\index{Moon!perigee of}%
\index{Perigee of moon's orbit}%

\Article{126}{The Moon's Orbit with Respect to the Sun.}---The
distance from the earth to the sun is about $400$~times that
from the earth to the moon. Consequently, the oscillations
of the moon back and forth across the earth's orbit as the
two bodies pursue their motion around the sun are so small
that they can hardly be represented to scale in a diagram.
As a consequence of the relative nearness of the moon and
its comparatively long period, its orbit is always concave
toward the sun. If the orbit of the moon were at any time
convex toward the sun, it would be when it is moving from
a position between the earth and sun to opposition, that
is, from $A$ to~$B$, \Figref{73}. It takes $14$~days for the moon
to move from the former position to the latter, and during
this time its distance from the sun increases by about $480,000$
%% -----File: 228.png---Folio 198-------
miles; but, in the meantime, the earth moves forward
about $14°$ in its orbit from $P$ to~$Q$, and it, therefore, is drawn
by the sun away from the straight line~$PT$ in which it was
originally moving by a distance of about $3,000,000$ miles.
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{228}{png}
\Caption[The orbit of the moon is concave to the sun.]{Fig}{73}
\end{figure}%
That is, in the $14$~days the moon actually moves in toward
the sun away from the original line of the earth's motion
$3,000,000 - 480,000 = 2,520,000$ miles, and its orbit, which
is represented by the broken line, is, therefore, concave toward
the sun at every point.

As a matter of fact, it is the center of gravity of the earth
and moon which describes what is called the earth's elliptical
orbit around the sun, and the earth and moon both
describe ellipses around this point as it moves on in its elliptical
path around the sun. Since the earth's mass is very
large compared to that of the moon, as will be seen in \Artref{127},
the center of the earth is always very near the center
of gravity of the two bodies.

\Article{127}{The Mass of the Moon.}---Although the moon is
\index{Mass!of moon}%
\index{Moon!mass of}%
comparatively near the earth, its mass cannot be obtained so
easily as that of many other objects farther away.

One of the best methods of finding the mass of the moon
depends upon the fact that the center of gravity of the
earth and moon describes an elliptical orbit around the sun
in accordance with the law of areas. Sometimes the earth
is ahead of the center of gravity, and at other times behind
it. When the earth is ahead of the center of gravity the
sun will be seen behind the position it would apparently
occupy if it were not for the moon. On the other hand,
%% -----File: 229.png---Folio 199-------
when the earth is behind the center of gravity, the sun will
be displaced correspondingly ahead of the position it would
otherwise apparently occupy. That is, the sun's apparent
motion eastward among the stars is not strictly in accordance
with the law of areas, for it sometimes is a little ahead
of, and at others a little behind, the position it would have
except for the moon. From very delicate observations it
has been found that the sun is displaced in this way about~$6''.4$.
Since the distance of the sun is known, the amount
of displacement of the earth in miles necessary to produce
this apparent displacement of the sun can be computed.
It has been found in this way that the distance of the center
of gravity of the earth and moon from the center of the earth
is $2886$~miles.

Now consider the problem of finding the ratio of the mass
of the earth to that of the moon. In \Figref{74} let $E$ represent
the earth, %[Illustration: Break]
\begin{wrapfigure}[10]{\WLoc}{3in}
\Input[3in]{229}{png}
\Caption[Center of gravity of the earth
and moon.]{Fig}{74}
\index{Center of gravity of earth and moon}%
\end{wrapfigure}
$C$~the center
of gravity of the earth
and moon, and $M$ the
moon. Let the distance~$EC$
be represented by~$x$,
and the distance~$EM$,
which is $238,862$ miles,
by~$r$. Since the mass of the earth multiplied by the distance
of its center from the center of gravity of the earth and moon
equals the mass of the moon multiplied by its distance from
the center of gravity of the earth and moon, it follows that
\[
x × E = (r - x)\, M.
\]
Since $x = 2886$ miles and $r = 238,862$ miles, it is found
that
\[
E = 81.8\, M.
\]
In round numbers the mass of the earth is $80$~times that of
the moon.

Since the orbit of the moon is inclined~$5°\,9'$ to the plane
of the ecliptic, the earth is sometimes above and sometimes
%% -----File: 230.png---Folio 200-------
below this plane. This causes an apparent displacement of
the sun from the ecliptic in the opposite direction. From
the amount of the apparent displacement of the sun in
latitude, as determined by observations, and from the inclination
of the moon's orbit and the distance of the sun,
it is possible to compute, just as from the sun's apparent
displacement in longitude, the mass of the moon relative to
that of the earth.

\Article{128}{The Rotation of the Moon.}---The moon always
\index{Moon!rotation of}%
\index{Rotation!of moon}%
presents the same side toward the earth, and therefore, as
seen from some point other than the earth or moon, it rotates
on its axis once in a sidereal month. For, in \Figref{67},
when the moon is at~$M_1$ a certain part is on the left toward
the earth, but when it has moved to~$M_3$ the same side is on
the right still toward the earth. Its direction of rotation
is the same as that of its revolution, or from west to east.
The plane of its equator is inclined about $1°\,32'$ to the plane
of the ecliptic, and the two planes always intersect in the
line of nodes of the moon's orbit.

It follows from what has been stated that the moon's
sidereal day is the same as its sidereal month, or $27.32166$
mean solar days. Its solar day is of the same length as its
synodical month, or $29.530588$ mean solar days, because its
synodical month is defined by its position with respect to
the earth and sun. Other things being equal, the temperature
changes from day to night on the moon would be much
greater than on the earth because its period of rotation is so
much longer; but the seasonal changes would be very slight
because of the small inclination of the plane of its equator
to the plane of its orbit.

It is a most remarkable fact that the moon rotates at
precisely such a rate that it always keeps the same face
toward the earth. It is infinitely improbable that it was
started exactly in this way; and, if it were not so started,
there must have been forces at work which have brought
about this peculiar relationship. It has been suggested that
%% -----File: 231.png---Folio 201-------
the explanation lies in the tidal reaction between the earth
and moon. Since the moon raises tides on the earth, it is
obvious that the earth also raises tides on the moon unless
it is absolutely rigid. Since the mass of the earth is more
than $80$~times that of the moon, the tides generated by the
earth on the moon, other things being equal, would be much
greater than those generated by the moon on the earth. If
a body is rotating faster than it revolves, and in the same
direction, one of the effects of the tides is to slow up its
rotation and to tend to bring the periods of rotation and
revolution to an equality. It has been generally believed
that the tides raised by the earth on the moon during millions
of years, part of which time it may have been in a
plastic state, have brought about the condition which now
exists. There are, however, serious difficulties with this
explanation (\Artref{265}), and it seems probable that the earth
and moon are connected by forces not yet understood.

\Article{129}{The Librations of the Moon.}---The statement that
\index{Librations of moon}%
\index{Moon!librations of}%
the moon always has the same side toward the earth is not
true in the strictest sense. It would be true if the planes
of its orbit and of its equator were the same, and if it moved
at a perfectly uniform angular velocity in its orbit.

The inclination of the moon's orbit to the ecliptic averages
about $5°\,9'$, and the inclination of the moon's equator to
the ecliptic is about $1°\,32'$. The three planes are so related
that the inclination of the moon's equator to the plane of
its orbit is $5°\,9' + 1°\,32' = 6°\,41'$. The sun shines alternately
over the two poles of the earth because of the inclination
of the plane of the equator to the plane of the ecliptic.
In a similar manner, if the earth were a luminous body it
would shine $6°\,41'$ over the moon's poles. Instead of shining
on them (except by reflected light), the tilting of the
moon's axis of rotation enables us to see $6°\,41'$ over the poles.
This is the \textit{libration} in latitude.

The moon rotates at a uniform rate,---at least the departures
from a uniform rate are absolutely insensible. It
%% -----File: 232.png---Folio 202-------
would take inconceivably great forces to make perceptible
short changes in its rate of rotation. On the other hand,
the moon revolves around the earth at a non-uniform rate,
for it moves in such a way that the law of areas is fulfilled.
Consider the moon starting from the perigee. It takes
about $6.5$~days, or considerably less than one quarter of its
period, for the moon to revolve through~$90°$; and, therefore,
the angle of rotation is considerably less than~$90°$. The
result is that the part of the moon on the side toward the
perigee, that is, the western edge, is brought partially into
view. On the opposite side of the orbit, the eastern edge of
the moon is brought partially into view. This is the libration
in longitude.

In addition to this, the moon is not viewed from the earth's
center. When it is on the horizon, the line from the observer
to the moon makes an angle of nearly $1°$ (the parallax
of the moon) with that from the earth's center to the moon.
This enables the observer to see nearly $1°$ farther around its
side than he could if it were on his meridian.

The result of the moon's librations is that there is only
$41$~per~cent of its surface which is never seen, while $41$~per~cent
is always in sight, and $18$~per~cent of it is sometimes
visible and sometimes invisible.

\Article{130}{The Density and Surface Gravity of the Moon.}---The
\index{Density!of moon}%
\index{Moon!density of}%
\index{Moon!surface gravity of}%
\index{Surface gravity!of moon}%
volume of the earth is about $50$~times that of the moon
and its mass is $81.8$~times that of the moon. Therefore the
density of the moon is somewhat less than that of the earth.
It is found from the relative volumes and masses of the earth
and moon that the density of the moon on the water standard
is about~$3.4$.

If the radius of the moon were the same as that of the
earth, gravity at its surface would be less than $\frac{1}{80}$ that at
the surface of the earth; but the small radius of the moon
tends to increase the attraction at its surface. If its mass
were the same as that of the earth, its surface gravity would
be nearly $16$~times that of the earth. On taking the two
%% -----File: 233.png---Folio 203-------
factors together, it is found that the surface gravity of the
moon is about~$\frac{1}{6}$ that of the earth. That is, a body on
the earth weighs by spring balances about $6$~times as much
as it would weigh on the moon.

If a body were thrown up from the surface of the moon
with a given velocity, it would ascend $6$~times as high as it
would if thrown up from the surface of the earth with the
same velocity. Perhaps this is the reason why the forces
to which both the earth and moon have been subjected have
produced relatively higher elevations on the moon than on
the earth. Also it would be possible for mountains of a
given material to be $6$~times as high on the moon as on the
earth before the rock of which they are composed would be
crushed at the bottom.

\Article{131}{The Question of the Moon's Atmosphere.}---The
\index{Atmosphere!of Moon}%
\index{Moon!atmosphere of}%
moon has no atmosphere, or at the most, an excessively rare
one. Its absence is proved by the fact that, at the time of
an eclipse of the sun, the moon's limb is perfectly dark and
sharp, with no apparent distortion of the sun due to refraction.
Similarly, when a star is occulted by the moon, it
disappears suddenly and not somewhat gradually as it
would if its light were being more and more extinguished
by an atmosphere.

Besides this, if the moon had an atmosphere, its refraction
would keep a star visible for a little time after it had been
occulted, just as the earth's atmosphere keeps the sun
visible about $2$~minutes after it has actually set. In a similar
way, the star would become visible a short time before
the moon had passed out of line with it. The whole effect
would be to make the time of occultation shorter than it
would be if there were no atmosphere.

If the moon had an atmosphere of any considerable
extent, there would be the effects of erosion on its surface;
but so far as can be determined, there is no evidence of such
action. Its surface consists of a barren waste, and it is,
perhaps, much cracked up because of the extremes of heat
%% -----File: 234.png---Folio 204-------
and cold to which it is subject. But there is nothing resembling
soil except, possibly, volcanic ashes. There can be
no water on the moon; for, if there were, it would be at least
partly evaporated, especially in the long day, and form an
atmosphere.

One cannot refrain from asking why the moon has no
atmosphere. It may be that it never had any. But the
evidence of great surface disturbances makes it not altogether
improbable that vast quantities of vapors have been emitted
from its interior. If this is true, they seem to have disappeared.
There are two ways in which their disappearance
can be explained. One is that they have united chemically
with other elements on the moon. As a possible example of
such action it may be mentioned that there are vast quantities
of oxygen in the rocks of the earth's crust, which may,
perhaps, have been largely derived from the atmosphere.
The second explanation is that, according to the kinetic
theory of gases, the moon may have lost its atmosphere by
the escape of molecule after molecule from its gravitative
control. This might be a relatively rapid process in the case
of a body having the low velocity of escape of $1.5$~miles per
second (\Artref{33}), especially if its days were so long that its
surface became highly heated.

It seems probable, therefore, that the moon could not
retain an atmosphere if it had one, and that whatever gases
it may ever have acquired from volcanoes or other sources
were speedily lost.

\Article{132}{The Light and Heat received by the Earth from the
Moon.}---The average distances of the earth and the moon
\index{Heat!from moon}%
\index{Light!from moon}%
\index{Moon!heat received from}%
from the sun are about the same; and, consequently, the
earth and the moon receive about equal amounts of light
and heat per unit area. The amount of light and heat that
the earth receives from the moon depends upon how much
the moon receives from the sun, what fraction it reflects,
its distance from the earth, and its phase. It is easy to see
that, if all the light the moon receives were reflected, the
%% -----File: 235.png---Folio 205-------
amount which strikes the earth could be computed for any
phase as, for example, when the moon is full. It is found by
taking into account all the factors involved that, if the moon
were a perfect mirror, it would give the earth, when it is
full, about $\frac{1}{100,000}$ as much light as the earth receives from
the sun. As a matter of fact, the moon is by no means a
perfect reflector, and the amount of light it sends to the
earth is very much less than this quantity.

It is not easy to compare moonlight with sunlight by direct
measurements, and the results obtained by different observers
are somewhat divergent. The measurements of Zöllner,
\index[xnames]{Zollner@{Zöllner}}%
which are commonly accepted, show that sunlight is
$618,000$ times greater than the light received from the full
moon. Sir John Herschel's observations gave the notably
\index[xnames]{Herschel, John}%
smaller ratio of $465,000$. At other phases the moon gives
not only correspondingly less light, but less than would be
expected on the basis of the part of the moon illuminated.
For example, at first quarter the illuminated area is half
that at full moon, but the amount of light received is less
than one eighth that at full moon. This phenomenon is
doubtless due to the roughness of the moon's surface. Moreover,
the amount of light received from the moon near first
quarter is somewhat greater than that received at the corresponding
phase at third quarter, the difference being due
to the dark spots on the eastern limb of the moon. On
taking into consideration the whole month, the average
amount of light and heat which the moon furnishes the earth
cannot exceed $\frac{1}{2,500,000}$ of that received from the sun. In
other terms, the earth receives as much light and heat from
the sun in $13$~seconds as it receives from the moon in the
course of a whole year.

\Article{133}{The Temperature of the Moon.}---The temperature
\index{Moon!temperature of}%
\index{Temperature!of moon}%
of the moon depends upon the amount of heat it receives,
the amount it reflects, and its rate of radiation. About $7$~per~cent
of the heat which falls on the moon is directly reflected,
and this has no effect upon its temperature. The
%% -----File: 236.png---Folio 206-------
remaining $93$~per~cent is absorbed and raises the temperature
of its surface. The rate of radiation of the moon's
surface materials for a given temperature is not known because
of the uncertainties of their composition and physical
condition. Nevertheless, it can be determined, at least
roughly, at the time of a total eclipse of the moon.

Consider the moon when it is nearly full and just before it
is eclipsed by passing into the earth's shadow, as at~$N$,
\Figref{81}. The side toward the earth is subject to the perpendicular
rays of the sun and has a higher temperature
than any other part of its surface. It is easy to measure
with some approximation the amount of heat received from
the moon, but it is not easy to determine what part of it is
reflected and what part is radiated. Now suppose the moon
passes on into the earth's shadow so that the direct rays of
the sun are cut off. Then all the heat received from the
moon is that radiated from a surface recently exposed to the
sun's rays. This can be measured; and, from the amount
received and the rate at which it decreases as the eclipse
continues, it is possible to determine approximately the
rate at which the moon loses heat by radiation, and from
this the temperature to which it has been raised. The observations
show that the amount of heat received from the
moon diminishes very rapidly after the moon passes into
\index[xnames]{Very}%
the earth's shadow. This means that its radiation is very
rapid and that probably its temperature does not rise very
high. It doubtless is safe to state that at its maximum it
is between the freezing and the boiling points. The recent
work of Very leads to the conclusion that the surface is
heated at its highest to a temperature of $200°$~Fahrenheit.

It is now possible to get a more or less satisfactory idea
of the temperature conditions of the moon. It must be
remembered, in the first place, that its day is $28.5$~times as
long as that of the earth. In the second place, it has no
atmospheric envelope to keep out the heat in the daytime
and to retain it at night. Consequently, when the sun rises
%% -----File: 237.png---Folio 207-------
for a point on the moon, its rays continue to beat down
upon the surface, which is entirely unprotected by clouds or
air, for more than $14$ of our days. During this time the
temperature rises above the freezing point and it may even
go up to the boiling point. When the sun sets, the darkness
of midnight immediately follows because there is no atmosphere
to produce twilight, and the heat rapidly escapes
into space. In the course of an hour or two the temperature
of the surface probably falls below the freezing point, and
in the course of a day or two it may descend to $100°$~below
zero. It will either remain there or descend still lower until
the sun rises again $14$~days after it has set.

The climatic conditions on the moon illustrate in the most
striking manner the effects of the earth's atmosphere and
the consequences of the earth's short period of rotation.

\Article{134}{General surface Conditions on the Moon.}---On the
whole, the surface of the moon is extremely rough, showing
no effects of weathering by air or water. It is broken by
several mountain chains, by numerous isolated mountain
peaks, and by more than $30,000$ observed craters. There
are several large, comparatively smooth and level areas,
which were called \textit{maria} (seas) by Galileo and other early
\index[xnames]{Galileo}%
observers, and the names are still retained though modern
instruments show that they not only contain no water but
are often rather rough. The smooth places are the areas
which are relatively dark as seen with the unaided eye or
through a small telescope. For example, the dark patch
near the bottom of \Figref{75} and a little to the left of the
center with a rather sharply defined lower edge is known as
\textit{Mare Serenitatis} (The Serene Sea). The light line running
out from the right of it and just under the big crater Copernicus
is the Apennine range of mountains. The most conspicuous
features which are visible with an ordinary inverting
telescope are shown on the map, \Figref{76}.

\Article{135}{The Mountains on the Moon.}---There are ten
\index{Lunar!mountains}%
\index{Moon!mountains of}%
ranges of mountains on the part of the moon which is visible
%% -----File: 238.png---Folio 208-------
from the earth. The mountains are often extremely slender
and lofty, in some cases attaining an altitude of more than
$20,000$ feet above the plains on which they stand. If the
\begin{figure}[hbt]%[Illustration:]
\Input{238}{jpg}
\Caption[The moon at $9\frac{3}{4}$~days. \textit{Photographed at the Yerkes Observatory.}]{Fig}{75}
\index{Yerkes Observatory}%
\end{figure}%
mountains on the earth were relatively as large, they would
be more than $15$~miles high. The height of the lunar mountains
is undoubtedly due, at least in part, to the low surface
%% -----File: 239.png---Folio 209-------
gravity on the moon, and to the fact that there has been
no erosion by air and water.

The height of a lunar mountain is determined from the
length of its shadow when the sun's rays strike it obliquely.
\begin{figure}[hbt]%[Illustration:]
\Input{239}{png}
\Caption[Outline map of the moon.]{Fig}{76}
\index{Moon!map of}%
\end{figure}%
For example, in \Figref{77} the crater Theophilus is a little
below the center, and in its interior are three lofty mountains
whose sharp, spirelike shadows stretch off to the left.
Since the size of the moon and the scale of the photograph
are both known, the lengths of the shadows can easily be
%% -----File: 240.png---Folio 210-------
\begin{figure}[hbt]%[Illustration:]
\Input{240}{jpg}
\Caption[The crater Theophilus and surrounding region (Ritchey).]{Fig}{77}
\index{Yerkes Observatory}%
\index[xnames]{Ritchey}%
\end{figure}%
determined. There is also no difficulty in finding the height
of the sun in the sky as seen from this position on the moon
when the picture was taken. Consequently, it is possible
from these data to compute the height of the mountains.
%% -----File: 241.png---Folio 211-------
In the particular case of Theophilus, the mountains in its
interior are more than $16,000$ feet above its floor. On the
earth the heights of mountains are counted from the sea
level, which, in most cases, is far away. For example, Pike's
Peak is about $14,000$ feet above the level of the ocean, which
is more than $1000$~miles away, but only about half that
height above the plateau on which it rests. The shadows
of the lunar mountains are black and sharp because the
moon has no atmosphere, and they are therefore well suited
for use in measuring the heights of objects on its surface.

\Article{136}{Lunar Craters.}---The most remarkable and the most
\index{Craters of moon}%
\index{Lunar!craters}%
\index{Moon!craters of}%
conspicuous objects of the lunar topography are the craters,
of which more than $30,000$ have been mapped. There have
been successive stages in their formation, for new ones in
many places have broken through and encroached upon the
old, as shown in \Figref{78}. Sometimes the newer ones are
precisely on the rims of the older, and sometimes they are
entirely in their interiors. The newer craters have deeper
floors and steeper and higher rims than the older, and one
of the most interesting things about them is that very often
they have near their centers lofty and spirelike peaks.

The term crater at once carries the impression to the mind
that these objects on the moon are analogous to the volcanic
craters on the earth. There is at least an immense
difference in their dimensions. Many lunar craters are from
$50$ to~$60$ miles in diameter, and, in a number of cases, their
diameters exceed $100$~miles. Ptolemy is $115$~miles across,
while Theophilus is $64$~miles in diameter and $19,000$ feet deep.
The lofty peak in the great crater Copernicus towers $11,000$
feet above the plains from which it rises. Some of these
craters are on such an enormous scale that their rims would
not be visible from their centers because of the curvature of
the surface of the moon.

The explanation of the craters is by no means easy, and
universal agreement has not been reached. If they are of
%% -----File: 242.png---Folio 212-------
\begin{figure}[hbtp]
\centering\Input{242}{jpg} %[Illustration: Fig. 78]
\Caption[The great crater Clavius with smaller craters on its rim
and in its interior. \textit{Photographed by Ritchey with the $40$-inch telescope
of the Yerkes Observatory.}]{Fig}{78}
\index{Yerkes Observatory}%
\end{figure}%
%% -----File: 243.png---Folio 213-------
volcanic origin, the activity which was present on the moon
enormously surpassed anything now known on the earth.
In view of the fact that there are no lava flows, and that in
most cases the material around a crater would not fill it,
the volcanic theory of their origin seems very improbable
and has been abandoned. Another suggestion is that the
craters have been formed by the bursting out of great masses
of gas which gathered under the surface of the moon and
became heated and subject to great tension because of its
contraction. According to this theory, the escaping gas
threw out large masses of the material which covered it and
thus made the rims of the craters. But it is hard to account
for the mountains which are so often seen in the interiors
of craters.

Gilbert suggested that the lunar craters may have been
\index[xnames]{Gilbert}%
formed by the impacts of huge meteorites, in some cases many
miles across. It is certain that such bodies, weighing hundreds
of pounds and even tons, now fall upon the earth
occasionally. It is supposed that millions of years ago the
collisions of these wandering masses with the earth and
moon were much more frequent than they are at the
present time. When they strike the earth, their energy is
largely taken up by the cushion of the earth's atmosphere;
when they strike the moon, they plunge in upon its surface
with a speed from $50$ to~$100$ times that of a cannon ball.
It does not seem improbable that masses many miles across
and weighing millions of tons might produce splashes in the
surface of the moon, even though it be solid rock, analogous
to the craters which are now observed. The heat
generated by the impacts would be sufficient to liquefy the
materials immediately under the place where the meteorites
struck, and might even cause very great explosions. The
mountains in the centers might be due to a sort of reaction
from the original splash, or from the heat produced
by the collision. At any rate, numerous experiments with
projectiles on a variety of substances have shown that pits
%% -----File: 244.png---Folio 214-------
closely resembling the lunar craters are very often obtained.
This view as to the cause of the craters is in harmony with
the theory that the earth and moon grew up by the accretion
of widely scattered material around nuclei which were
originally of much smaller dimensions (\Artref{250}).

An obvious objection to the theory that the craters on
the moon were produced by meteorites is that the earth has
no similar formations. Since the earth and moon are closely
associated in their revolution around the sun, it is clear that
the earth would have been bombarded at least as violently
as the moon. The answer to this objection is that, for millions
of years, the rains and snows and atmosphere have disintegrated
the craters and mountains on the earth, and their
powdered remains have been carried away into the valleys.
Whatever irregularities of this character the earth's surface
may have had in its early stages, all traces of them disappeared
millions of years ago. On the other hand, since air
and water are altogether absent from the moon, this nearest
celestial body has preserved for us the records of the forces
to which it, and probably also the earth, were subject in the
early stages of their development.

Probably the most serious objection to the impact theory
of the craters on the moon is that they nearly all appear to
have been made by bodies falling straight toward the moon's
center. It is obvious that a sphere circulating in space
would in a majority of cases be struck glancing blows by
wandering meteorites. The attraction of the moon would
of course tend to draw them toward its center, but their
velocities are so great that this factor cannot seriously
have modified their motions. The only escape from this
objection, so far as suggested, is that the heat generated by
the impacts may have been sufficient to liquefy the material
in the neighborhood of the places where the meteorites struck,
and thus to destroy all evidences of the directions of the blows.

\Article{137}{Rays and Rills.}---Some of the large craters, particularly
\index{Moon!rays and rills of}%
\index{Rays and rills}%
Tycho and Copernicus, have long light streaks,
%% -----File: 245.png---Folio 215-------
called \textit{rays}, radiating from them like spokes from the axle of
a wheel. They are not interfered with by hill or valley,
and they often extend a distance of several hundred miles.
They cast no shadows, which proves that they are at the
same level as the adjacent surface, and they are most conspicuous
\begin{figure}[hbt]%[Illustration:]
\Input{245}{jpg}
\Caption[The full moon. \textit{Photographed at the Yerkes Observatory \(Wallace\).}]{Fig}{79}
\index{Yerkes Observatory}%
\index[xnames]{Wallace, R. J.}%
\end{figure}%
at the time of full moon. They are easily seen in
\Figref{79}. It has been supposed by some that they are lava
streams and by others that they were great cracks in the
surface, formed at the time when the craters were produced,
which have since filled up with lighter colored material
from below.
%% -----File: 246.png---Folio 216-------

The rills are cracks in the moon's surface, a mile or so
wide, a quarter of a mile deep, and sometimes as much as
$150$~miles in length. They are very numerous, more than
1000 having been so far mapped. The only things at all
like them on the earth are such chasms as the Grand Canyon
of the Colorado and the cut below Niagara Falls. But
these gorges are the work of erosion, which has probably
been entirely absent from the surface of the moon. At any
rate, it is incredible that the rills have been produced by
erosion. The most plausible theory is that they are cracks
which have been caused by violent volcanic action, or by
the rapid cooling and shrinking of the moon.

The rays and rills are very puzzling lunar features which
seem to be fundamentally unlike anything in terrestrial
topography. Even our nearest neighbor thus differs very
radically from the earth.

\Article{138}{The Question of Changes on the Moon.}---There
\index{Moon!surface changes of}%
have been no observed changes in the larger features of the
lunar topography, although, from time to time, minor alterations
have been suspected. The most probable change of
any natural physical feature is in the small crater Linné, in
Mare Serenitatis. It was mapped about a century ago,
but in 1866 was said by Schmidt to be entirely invisible.
It is now visible as on the original maps. It is generally
believed that the differences in appearance at various times
have been due to slightly different conditions of illumination.

Since the moon's orbit is constantly shifting because of
the attraction of the sun, and since the month does not contain
an integral number of days, it follows that an observer
never gets at two different times exactly the same view of
the moon. W.~H. Pickering has noticed changes in some
\index[xnames]{Pickering, W. H.}%
small craters, depending upon the phase of the moon, which
he interprets as possibly being due to some kind of vegetation
which flourishes in the valleys where he supposes heavier
gases, such as carbon dioxide, might collect. Some of his
observations have been verified by other astronomers, but
%% -----File: 247.png---Folio 217-------
his rather bold speculations as to their meaning have not
been accepted.

It is altogether probable that the moon long ago arrived
at the stage where surface changes practically ceased. The
only known influences which could now disturb its surface
are the feeble tidal strains to which it is subject, and the
extremes of temperature between night and day. While it
would be too much to say that slight disintegration of the
surface rocks may not still be taking place, yet it is certain
that, on the whole, the moon is a body whose evolution is
essentially finished. The seasonal changes are unimportant,
but there is alternately for two weeks the blinding glare of
the sunlight, never tempered by passing clouds or even an
atmosphere, and the blackness and frigidity of the long lunar
night. Month succeeds month, age after age, with no important
variations in these phenomena.

\Article{139}{The Effects of the Moon on the Earth.}---The moon
\index{Moon!effects of on earth}%
reflects a relatively small amount of sunlight and heat to
the earth, and in conjunction with the sun it produces the
tides. These are the only influences of the moon on the
earth that can be observed by the ordinary person. It has
a number of very minor effects, such as causing minute
variations in the magnetic needle, the precession of the equinoxes,
and slight changes in the motion of the earth; but
they are all so small that they can be detected only by refined
scientific methods.

There are a great many ideas popularly entertained, such
as that it is more liable to rain at the time of a change of
the moon, or that crops grow best when planted in certain
phases, which have no scientific foundation whatever. It
follows from the fact that more light and heat are received
from the sun in $13$~seconds than from the moon in a whole
year, that its heating effects on the earth cannot be important.
The passing of a fleecy cloud, or the haze of Indian
summer, cuts off more heat from the sun than the moon
sends to the earth in a year. Consequently, it is entirely
%% -----File: 248.png---Folio 218-------
unreasonable to suppose that the moon has any important
climatic effects on the earth. Besides this, recorded observations
of temperature, the amount of rain, and the velocity
of the wind, in many places, for more than $100$~years, fail
to show with certainty any relation between the weather and
phases of the moon.

The phenomena of storms themselves show the essential
independence of the weather and the phases of the moon.
Storm centers move across the country in a northeasterly
direction at the rate of $400$ to~$500$ miles per day, and sometimes
they can be followed entirely around the earth. Consequently,
if a storm should pass one place at a certain phase
of the moon, it would pass another a few thousand miles
eastward at quite a different phase. The theory that a
storm occurred at a certain phase of the moon would then
be verified for one longitude and would fail of verification
at all the others.

\Article{140}{Eclipses of the Moon.}---The moon is eclipsed whenever
\index{Eclipses!of moon}%
\index{Moon!eclipses of}%
it passes into the earth's shadow so that it does not
\begin{figure}[hbt]%[Illustration:]
\Input{248}{png}
\Caption[The condition for eclipses of the moon and sun.]{Fig}{80}
\end{figure}%
receive the direct light of the sun. In \Figref{80}, $E$~represents
the earth and $PQR$ the earth's shadow, which comes to a
point at a distance of $870,000$ miles from the earth's center.
The only light received from the sun within this cone is
that small amount which is refracted into it by the earth's
atmosphere in the zone~$QR$. In the regions $TQP$ and~$SRP$
the sun is partially eclipsed, the light being cut off more and
more as the shadow cone is approached. The shadow cone~$PQR$
is called the \textit{umbra}, and the parts $TQP$ and~$SRP$, the
\index{Umbra!of earth's shadow}%
\textit{penumbra}.
\index{Penumbra!of earth's shadow}%
%% -----File: 249.png---Folio 219-------

When the moon is about to be eclipsed, it passes from full
illumination by the sun gradually into the penumbra, where
at first only a small part of the sun is obscured, and it then
proceeds steadily across the shadow of increasing density
until it arrives at~$A$, where the sun's light is entirely cut off.
The distance across the earth's shadow is so great that the
moon is totally eclipsed for nearly $2$~hours while it is passing
through the umbra, and the time from the first contact
with the umbra until the last is about $3$~hours and $45$~minutes.

It appears from \Figref{80} that the moon would be eclipsed
every time it is in opposition to the sun, but this figure is
drawn to show the relations as one looks perpendicularly
on the plane of the ecliptic, neglecting the inclination of the
moon's orbit. \Figureref{81} shows another section in which
\begin{figure}[hbt]%[Illustration:]
\Input{249}{png}
\Caption[Condition in which eclipses of the moon and sun fail.]{Fig}{81}
\end{figure}%
the plane of the moon's orbit, represented by~$MN$, is perpendicular
to the page. It is obvious from this that, when
the moon is in the neighborhood of~$N$, it will pass south of
the earth's shadow instead of through it. The proportions
in the figure are by no means true to scale, but a detailed
discussion of the numbers involved shows that usually the
moon will pass through opposition to the sun without encountering
the earth's shadow. But when the earth is $90°$
in its orbit from the position shown in the figure, that is,
when the earth as seen from the sun is at a node of the moon's
orbit, the plane of the moon's orbit will pass through the
sun, and consequently the moon will be eclipsed. At least,
the moon will be eclipsed if it is full when the earth is at or
near the node. The earth is at a node of the moon's orbit
at two times in the year separated by an interval of six
%% -----File: 250.png---Folio 220-------
months. Consequently, there may be two eclipses of the
moon a year; but because the moon may not be full when
the earth is at one of these positions, one or both of the
eclipses may be missed.

Since the sun apparently travels along the ecliptic in the
sky, the earth's shadow is on the ecliptic $180°$ from the sun.
The places where the moon crosses the ecliptic are the
nodes of its orbit, and, consequently, there can be an eclipse
of the moon only when it is near one of its nodes. Since
the nodes continually regress as a consequence of the sun's
attraction for the moon, the eclipses occur earlier year after
year, completing a cycle in $18.6$~years.

One scientific use of eclipses of the moon is that when they
occur, the heat radiated by the moon after it has just been
exposed to the perpendicular rays of the sun gives an opportunity,
as was explained in \Artref{133}, of determining its
temperature. Also, at the time of a lunar eclipse, the stars
in the neighborhood of the moon can easily be observed, and
it is a simple matter to determine the exact instant at which
the moon passes in front of a star and cuts off its light.
Since the positions of the stars are well known, such an
observation locates the moon with great exactness at the
time the observation is made. It is imaginable that the
\index{Moon!satellites of}%
\index{Satellites!of moon}%
moon may be attended by a small satellite. If the moon is
not eclipsed, its own light or that of the sun will make it
impossible to see a very minute body in its neighborhood;
but at the time of an eclipse, a satellite may be exposed to
the rays of the sun while the neighboring sky will not be
lighted up by the moon. Only at such a time would there
be any hope of discovering a small body revolving around
the moon. A search for such an attendant has been made,
but has so far proved fruitless.

\Article{141}{Eclipses of the Sun.}---The sun is eclipsed when
\index{Eclipses!of sun}%
\index{Eclipses!uses of}%
\index{Sun!eclipses of}%
the moon is so situated as to cut off the sun's light from at
least a portion of the earth. The apparent diameter of the
moon is only a little greater than that of the sun, and, consequently,
%% -----File: 251.png---Folio 221-------
eclipses of the sun last for a very short time.
This statement is equivalent to saying that the shadow cone
of the moon comes to a point near the surface of the earth,
as is shown in \Figref{80}. It is also obvious from this diagram
that the sun is eclipsed as seen from only a small part of the
earth. As the moon moves around the earth in its orbit
and the earth rotates on its axis, the shadow cone of the
moon describes a streak across the earth which may be
somewhat curved.

It follows from the fact that the path of the moon's shadow
across the earth is very narrow, as shown in \Figref{82}, that a
\begin{figure}[hbt]%[Illustration:]
\Input{251}{jpg}
\Caption[Path of the total eclipse of the sun, August 29--30, 1905.]{Fig}{82}
\end{figure}%
total eclipse of the sun will be observed very infrequently
at any given place. On this account, as well as because it
is a startling phenomenon for the sun to become dark in the
daytime, eclipses have always been very noteworthy occurrences.
Repeatedly in ancient times, in which the chronology
was very uncertain, writers referred to eclipses in connection
with certain historical events, and astronomers,
calculating back across the centuries, have been able to
%% -----File: 252.png---Folio 222-------
identify the eclipses and thus fix the dates for historians in
the present system of counting time. The infrequency of
eclipses at any particular place is evident from \Figref{83},
which gives the paths of all the total eclipses of the sun
from 1894--1973. In this long period the greater part of
the world is not touched by them at all.

So far the discussion has referred only to total eclipses of
the sun; but in the regions on the earth's surface which are
\begin{figure}[hbt]%[Illustration:]
\Input{252}{jpg}
\Caption[Paths of total eclipses of the sun. (From Todd's Total Eclipses.)]{Fig}{83}
\index{Total eclipses}%
\end{figure}%
near the path of totality, or in the penumbra of the moon's
shadow, which is entirely analogous to that of the earth,
there are partial eclipses of the sun. The region covered by
the penumbra is many times that where an eclipse is total;
and, consequently, partial eclipses of the sun are not very
infrequent phenomena.

There is not an eclipse of the sun every time the moon is
in conjunction with the sun because of the inclination of its
orbit. For example, when it is near~$M$, \Figref{81}, its shadow
%% -----File: 253.png---Folio 223-------
passes north of the earth. In fact, eclipses of the sun occur
only when the sun is near one of the moon's nodes, just as
eclipses of the moon occur only when the earth's shadow is
near one of the moon's nodes. Consequently, eclipses occur
twice a year at intervals separated by $6$~synodical months.
Since the moon's nodes regress, making a revolution in $18.6$~years,
eclipses occur, on the average, about $20$~days earlier
each year than on the preceding year.

The distance~$UV$, \Figref{80}, within which an eclipse of the
sun can occur is greater than~$AB$, within which an eclipse
of the moon can occur. Therefore it is not necessary that
the sun shall be as near the moon's node in order that an
eclipse of the sun may result as it is in order that there may
be an eclipse of the moon. When the relations are worked
out fully, it is found that there will be at least one solar
eclipse each time the sun passes the moon's node, and that
there may be two of them. Consequently, in a year, there
may be two, three, or four eclipses of the sun. If there are
only two eclipses, the moon's shadow is likely to strike
somewhere near the center of the earth and give a total
eclipse. On the other hand, if there are two eclipses while
the sun is passing a single node of the moon's orbit, they
must occur, one when the sun is some distance from the node
on one side, and the other when it is some distance from the
node on the other side. In this case the moon's shadow,
or at least its penumbra, strikes first near one pole of the
earth and then near the other. These eclipses are generally
only partial.

\Article{142}{Phenomena of \DPtypo{total}{Total} Solar Eclipses.}---A total eclipse
\index{Eclipses!phenomena of}%
of the sun is a startling phenomenon. It always occurs precisely
at new moon, and consequently the moon is invisible
until it begins to obscure the sun. The first indication of a
solar eclipse is a black slit or section cut out of the western
edge of the sun by the moon which is passing in front of it
from west to east. For some time the sunlight is not
diminished enough to be noticeable. Steadily the moon
%% -----File: 254.png---Folio 224-------
moves over the sun's disk; and, as the instant of totality
draws near, the light rapidly fails, animals become restless,
and everything takes on a weird appearance. Suddenly a
shadow rushes across the surface of the earth at the rate of
more than $1300$~miles an hour, the sun is covered, the stars
flash out, around the apparent edge of the moon are rose-colored
prominences (\Artref{236}) of vaporous material forced
up from the sun's surface to a height of perhaps $200,000$
miles, and all around the sun, extending out as far as half
its diameter, are the streamers of pearly light which constitute
the sun's corona (\Artref{238}). After about $7$~minutes,
at the very most, the western edge of the sun is uncovered,
daylight suddenly reappears, and the phenomena of a partial
eclipse take place in the reverse order.

Total eclipses of the sun afford the most favorable conditions
\index{Eclipses!uses of}%
for searching for small planets within the orbit of Mercury,
and it is only during them that the sun's corona can be
observed.


\Section{X}{QUESTIONS}

1. Verify by observations the motion of the moon eastward
among the stars, and its change in declination during a month.

2. For an observer on the moon describe, (\textit{a})~the apparent
motions of the stars; (\textit{b})~the motion of the sun with respect to the
stars; (\textit{c})~the diurnal motion of the sun; (\textit{d})~the motion of the earth
with respect to the stars; (\textit{e})~the motion of the earth with respect to
the sun; (\textit{f})~the diurnal motion of the earth; (\textit{g})~the librations of the
earth.

3. Describe the phases the moon would have throughout the
year if the plane of its orbit were perpendicular to the plane of the
ecliptic.

4. What would be the moon's synodical period if it revolved
around the earth from east to west in the same sidereal period?

5. Show by a diagram that, if the moon always presents the same
face toward the earth, it rotates on its axis and its period of rotation
equals the sidereal month.

6. Is it possible that the moon has an atmosphere and water on
the side remote from the earth?

7. Suppose you could go to the moon and live there a month.
%% -----File: 255.png---Folio 225-------
Give details regarding what you would observe and the experiences
you would have.

8. What are the objections to the theory that lunar craters are
of volcanic origin? That they were produced by meteorites?

9. How do you interpret rays and rills under the hypothesis that
lunar craters were produced by meteorites?

10. If the earth's reflecting power is $4$~times that of the moon,
how does earthshine on the moon compare with moonshine on the
earth?

\normalsize

%% -----File: 256.png---Folio 226-------


\Chapter{VIII}{The Solar System}

\Section{I}{The Law of Gravitation}

\Article{143}{The Members of the Solar System.}---The members
\index{Planets}%
of the solar system are the sun, the planets and their satellites,
the planetoids, the comets, and the meteors. It may
possibly be that some of the comets and meteors, coming in
toward the sun from great distances and passing on again,
are only temporary members of the system. The sun is
the one preëminent body. Its volume is nearly a thousand
times that of all the other bodies combined, its mass is so
great that it controls all their motions, and its rays illuminate
and warm then. It is impossible to treat of the planets
without taking into account their relations to the sun, but
the constitution and evolution of the sun are quite independent
of the planets.

The eight known planets are, in the order of their distance
from the sun, Mercury, Venus, Earth, Mars, Jupiter, Saturn,
Uranus, and Neptune. The first six are conspicuous objects
to the unaided eye when they are favorably located, and they
have been known from prehistoric times; Uranus and
Neptune were discovered in 1781 and~1846, respectively.
The planetoids (often called the small planets and sometimes
the asteroids) are small planets which, with a few exceptions,
revolve around the sun between the orbits of Mars and
Jupiter. The comets are bizarre objects whose orbits are
very elongated and lie in every position with respect to the
orbits of the planets. Probably at least a part of the meteors
are the remains of disintegrated comets; they are visible
only when they strike into the earth's atmosphere.
%% -----File: 257.png---Folio 227-------

\Article{144}{The Relative Dimensions of the Planetary Orbits.}---The
distance from the earth to the sun is called the \textit{astronomical
unit}. The distances from the planets to the sun can
\index{Astronomical unit}%
be determined in terms of the astronomical unit without
knowing its value in miles.

Consider first the planets whose orbits are interior to that
of the earth. They are called the \textit{inferior planets}. In
\index{Planets!inferior}%
\index{Planets!superior}%
\Figref{84} let $S$~represent the sun, $V$ the planet Venus, and
$E$ the earth. The
angle~$SEV$ is called
the \textit{elongation} of the
\index{Elongations of planets}%
planet, and may vary
from zero up to a
maximum which depends
upon the size of
the orbit of~$V$. When
the elongation is greatest,
the angle at~$V$ is a
right angle. Suppose
the elongation of~$V$ is
determined by observation
day after day
until it reaches its
maximum. Then, since
the elongation is measured and the angle at~$V$ %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3in}
\Input[3in]{257}{png}
\Caption[Finding the distance of an
inferior planet.]{Fig}{84}
\end{wrapfigure}
is~$90°$, the
shape of the triangle is determined, and $SV$ can be computed
by trigonometry in terms of~$SE$.

Now consider the planets whose orbits are outside that
of the earth. They are called the \textit{superior planets}. Suppose
the periods of revolution of the earth and Mars, for
example, have been determined from long series of observations.
This can be done without knowing anything about
their actual or relative distances. For, in the first place,
the earth's period can be obtained from observations of the
apparent position of the sun with respect to the stars; and
then the period of Mars can be found from the time required
%% -----File: 258.png---Folio 228-------
for it to move from a certain position with respect
to the sun back to the same position again. For example,
when a planet is exactly~$180°$ from the sun in the sky, as
seen from the earth, it is said to be in \textit{opposition}. The period
\index{Opposition!definition of}%
from opposition to opposition is called the \textit{synodical period}
(compare \Artref{120}). Let the sidereal period of the earth
be represented by~$E$, the sidereal period of the planet by~$P$,
and its synodical period by~$S$. Then, analogous to the case
of the moon in \Artref{120}, $P$~is defined by
\[
  \frac{1}{P} = \frac{1}{E} - \frac{1}{S}.
\]

Now return to the problem of finding the distance of a
superior planet in terms of the astronomical unit. In
\Figref{85}, let $S$~represent the
sun, and $E_1$ and~$M_1$ the
positions of the earth and
Mars when Mars is in opposition.
Let $E_2$ and~$M_2$ represent
the positions of the
earth and Mars when the
angle at~$E_2$ is, for example,
a right angle. Mars is then
said to be in \textit{quadrature}, and
\index{Quadrature}%
the time when it has this
position can be determined
by observation. The angles
$M_1SE_2$ and~$M_1SM_2$ can be
determined from the periods of the earth and Mars and the
interval %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.875in}
\Input[2.875in]{258}{png}
\Caption[Finding the distance of a
superior planet.]{Fig}{85}
\end{wrapfigure}
of time required for the earth and Mars to move
from $E_1$ and~$M_1$ respectively to $E_2$ and~$M_2$. The difference
of these two angles is~$M_2SE_2$, from which, together with
the right angle at~$E_2$, the distance~$SM_2$ in terms of~$SE_2$ can
be computed by trigonometry.

A little complication in the processes which have been
described arises from the fact that the orbit of the earth is
%% -----File: 259.png---Folio 229-------
not a circle. But the manner in which the distance of the
earth from the sun varies can easily be determined from
observations of the apparent diameter of the sun, for the
apparent diameter of an object varies inversely as its distance.
After the variations in the earth's distance have been
found, the results can all be reduced without difficulty to a
single unit. The unit adopted is half the length of the
earth's orbit, and is called its \textit{mean distance}, though it is a
\index{Mean distance, definition of}%
little less than the average distance to the sun.

\Article{145}{Kepler's Laws of Planetary Motion.}---The last
\index{Kepler's laws}%
great observer before the invention of the telescope was the
Danish astronomer Tycho
\index[xnames]{Tycho Brahe}%
Brahe (1546--1601). He was
an energetic and most painstaking
worker. He not only
catalogued many stars, but
he also observed comets,
proving they are beyond the
earth's atmosphere, and obtained
an almost continuous
record for many years of the
positions and motions of the
sun, moon, and planets.

Tycho Brahe's successor
was his pupil Kepler (1571--1630),
who spent more than
$20$~years in attempting to find
from the observations of his master the manner in which the
planets actually move. The results of an enormous amount
of calculation on his part are contained in the following three
laws of planetary motions:

I\@. \textit{Every planet moves so that the line joining it to the sun
sweeps over equal areas in equal intervals of time, whatever
their length.} This is known as the law of areas.
\index{Areas, law of}%
\index{Law!of areas}%

%[Illustration: Break]
\begin{wrapfigure}[18]{\WLoc}{2.25in}
\Input[2.25in]{259}{jpg}
\Caption[Johann Kepler.]{Fig}{86}
\index[xnames]{Kepler}%
\end{wrapfigure}

II\@. \textit{The orbit of every planet is an ellipse with the sun at
one of its foci.}
%% -----File: 260.png---Folio 230-------

III\@. \textit{The squares of the periods of any two planets are
proportional to the cubes of their mean distances from the
sun.}

All the complexities of the apparent motions of the planets
are explained by Kepler's three simple laws when taken in
\index[xnames]{Kepler}%
connection with the periods of the planets and the positions
of their orbits.

\Article{146}{The Law of Gravitation.}---Newton based his greatest
\index{Gravitation!discovery of}%
\index{Gravitation!law of}%
\index{Law!of gravitation}%
\index[xnames]{Newton}%
discovery, the law of gravitation, on Kepler's laws. From
each one of them he drew an important conclusion.

Newton proved by a suitable mathematical discussion,
based on his laws of motion, that it follows from Kepler's
first law that \textit{every planet is acted on by a force which is directed
toward the sun}. This was the first time that the sun
and planets were shown to be connected dynamically. Before
Newton's time it was generally supposed that there
was some force acting on the planets in the direction of their
motion which kept them going in their orbits.

The first law of Kepler led to the conclusion that the planets
are acted on by forces directed toward the sun, but gave no
information whatever regarding the manner in which the
forces depend upon the position of the planet. The second
law furnishes a basis for the answer to this question, and
from it Newton proved that the \textit{force acting on each planet
varies inversely as the square of its distance from the sun}.

The law of the inverse squares is encountered in many
phenomena besides gravitation. For example, it holds for
magnetic and electric forces, the intensity of light and of
sound, and the magnitudes of water and earthquake waves.
The reason it holds for the radiation of light is easily understood.
The area of the spherical surface which the rays
cross in proceeding from a point is proportional to the
square of its radius. Since the intensity of illumination is
inversely proportional to the illuminated area, it is inversely
as the square of the distance. If gravitation in some way
depended on lines of force extending out from matter radially,
%% -----File: 261.png---Folio 231-------
it would vary inversely as the square of the distance, but
nothing is positively known as to its nature.

Another interesting question remains, and that is whether
the gravitation of a body is strictly proportional to its
inertia, regardless of its constitution and condition, or
whether it depends upon its composition, temperature, and
other characteristics. All other known forces, such as magnetism,
depend upon other things than mass, and it might
be expected the same would be true of gravitation. But it
follows from Kepler's third law that the sun's attraction for
\index[xnames]{Kepler}%
the several planets is independent of their different constitutions,
motions, and physical conditions. Since the same
law holds for the $800$~planetoids as well, in which there is
opportunity for great diversities, it is concluded that gravitation
depends upon nothing whatever except the masses and
the distances of the attracting bodies.

Suppose the attraction between unit masses at unit distance
is taken as unity, and consider the attraction of a
body composed of many units for another of many units.
To fix the ideas, suppose one body has $5$~units of mass and
the other $4$~units; the problem is to find the number of
units of force between them at distance unity. Each of the
$5$~units exerts a unit of force on each of the $4$~units. That
is, each of the $5$~units exerts all together $4$~units of force on
the second body. Therefore, the entire first body exerts
$5 × 4 = 20$ units of force on the second body; or, the
whole force is proportional to the products of the masses.

On uniting the results obtained from Kepler's three laws
and assuming that they hold always and everywhere, the
universal law of gravitation is obtained:

\textit{Every particle of matter in the universe attracts every other
particle with a force which is proportional to the product of
their masses, and which varies inversely as the square of the
distance between them.}

\Article{147}{The Importance of the Law of Gravitation.}---The
\index{Gravitation!importance of law of}%
importance of a physical law depends upon the number of
%% -----File: 262.png---Folio 232-------
phenomena it coördinates and upon the power it gives the
scientist of making predictions. Consider the law of gravitation
\begin{figure}[hbt]%[Illustration:]
\Input{262}{jpg}
\Caption[Isaac Newton.]{Fig}{87}
\index[xnames]{Newton}%
\end{figure}%
in these respects. In his great work, \textit{Philosophiæ
Naturalis Principia Mathematica} (The Mathematical Principles
%[** http://www.gutenberg.org/etext/28233 :D]
\index{Principia}%
of Natural Philosophy), commonly called simply the
\textit{Principia}, Newton showed how every known phenomenon
of the motions, shapes, and tides of the solar system could be
explained by the law of gravitation. That is, the elliptical
%% -----File: 263.png---Folio 233-------
paths of the planets and the moon, the slow changes in their
orbits produced by their slight mutual attractions, the oblateness
of rotating bodies, the precession of the equinoxes, and
the countless small irregularities in planetary and satellite
motions that can be detected by powerful telescopes, are
all harmonious under the law of gravitation, and what once
seemed to be a hopeless tangle has been found to be an
orderly system. All the discoveries in this direction for more
than $200$~years have confirmed the exactness of the law
of gravitation until it is now by far the most certainly
established physical law.

Not only is the law of gravitation operative in the great
phenomena where its effects are easy to detect, but also in
everything in which the motion of matter is involved. It is
found on reflection that all phenomena depend either directly
or indirectly upon the motion of matter, for even changes
of the mental state of an individual are accompanied by
corresponding changes in the structure of his brain. When
a person moves, his changed relation to the remainder of
the universe causes a corresponding change in the gravitational
stress by which he is connected with it; indeed, when
he thinks, the alterations in his brain at once cause alterations
in the gravitational forces between it and matter
even in the remotest parts of space. These effects are certainly
real, though there is no known means of detecting
them.

The law of gravitation became in the hands of the successors
of Newton one of the most valuable means of discovery.
\index[xnames]{Newton}%
Time after time such great mathematicians as
Laplace and Lagrange, using it as a basis, predicted things
\index[xnames]{Lagrange}%
\index[xnames]{Laplace}%
which had not then been observed, but which invariably
were found later to be true. But scientific men are not
contented with simply making predictions and finding that
they come true. On the basis of their established laws they
seek to foresee what will happen in the almost indefinite
future, even beyond the time when the human race shall
%% -----File: 264.png---Folio 234-------
have become extinct, and, similarly, what the conditions were
back before the time when life on the earth began.

The law of gravitation was undoubtedly Newton's greatest
discovery, and the importance of it and his other scientific
work is indicated by the statements of competent judges.
The brilliant German scholar, Leibnitz (1646--1716), a contemporary
\index[xnames]{Leibnitz}%
of Newton and his greatest rival, said, ``Taking
mathematics from the beginning of the world to the time
when Newton lived, what he had done was much the better
half.'' The French mathematician, Lagrange (1736--1813),
\index[xnames]{Lagrange}%
one of the greatest masters of celestial mechanics, wrote,
``Newton was the greatest genius that ever existed, and the
most fortunate, for we cannot find more than once a system
of the world to establish.'' The English writer on the
history of science, Whewell, said, ``It [the law of gravitation]
\index[xnames]{Whewell}%
is indisputably and incomparably the greatest scientific
discovery ever made, whether we look at the advance which
it involved, the extent of the truth disclosed, or the fundamental
and satisfactory nature of this truth.'' Compare
these splendid and deserved eulogies with Newton's own
estimate of his efforts to find the truth: ``I do not know
what I may appear to the world; but to myself I seem to
have been only like a boy playing on the seashore, and
diverting myself in now and then finding a smoother pebble
or a prettier shell than ordinary, while the great ocean of
truth lay all undiscovered before me.'' There is every
reason to believe that this is the sincere and unaffected expression
of a great mind which realized the magnitude of
the unknown as compared to the known.

In Westminster Abbey, in London, Newton lies buried
among the noblest and the greatest English dead, and over
his tomb on a tablet they have justly engraved, ``Mortals,
congratulate yourselves that so great a man has lived for
the honor of the human race.''

\Article{148}{The Conic Sections.}---After having found that, if
\index{Conic sections}%
the orbit of a body is an ellipse with the center of force at a
%% -----File: 265.png---Folio 235-------
focus, then the force to which it is subject varies inversely as
the square of its distance, %[Illustration: Break]
\begin{wrapfigure}[27]{\WLoc}{1.5in}
\Input[1.5in]{265}{png}
\Caption[The conic sections.]{Fig}{88}
\end{wrapfigure}
Newton took up the converse
problem. Under the assumption that the attractive force
varies inversely as the square of the distance, he proved
that the orbit must be what is called a
\textit{conic section}, an example of which is the
ellipse.

The conic sections are highly interesting
curves first studied by the ancient Greeks.
They derive their name from the fact that
they can be obtained by cutting a circular
cone with planes. In \Figref{88} is shown a
double circular cone whose vertex is at~$V$.
A plane section perpendicular to the axis
of the cone gives a circle~$C$. An oblique
section gives an ellipse~$E$; however, the
plane must cut both sides of the cone.
When the plane is parallel to one side, or
element, of the cone, a parabola~$P$ is obtained.
\index{Parabola}%
When the plane cuts the two
branches of the double cone, the two
branches of an hyperbola~$HH$ are obtained.
\index{Hyperbola}%
There are in addition to these
figures certain limiting cases. One is that
in which the intersecting plane passes
only through the vertex~$V$ giving a
simple point; another is that in which the intersecting plane
touches only one element of the cone, giving a single straight
line; and the last is that in which the intersecting plane
passes through the vertex~$\DPtypo{B}{V}$ and cuts both branches of the
cone, giving two intersecting straight lines.

The character of the conic described depends entirely
upon the central force and the way in which the body is
started. For example, suppose a body is started from~$O$,
\Figref{89}, in the direction~$OT$, perpendicular to~$OS$. If
the initial velocity of the body is zero, it will fall straight to~$S$.
%% -----File: 266.png---Folio 236-------
If the initial velocity is not too great, it will describe the
ellipse~$E$, and $O$ will be the aphelion point. If the initial
velocity is just great enough so that the centrifugal acceleration
balances the attraction, the orbit will be the circle~$C$.
If the initial velocity is a little greater than that in the circle,
the body will describe the ellipse~$E'$, and $O$ will be the perihelion
point. If the initial velocity is exactly $\sqrt{2}$~times
\begin{wrapfigure}{\WLoc}{2.75in}%[Illustration:]
\Input[2.75in]{266}{png}
\Caption[Different conics depending
on the initial velocity.]{Fig}{89}
\end{wrapfigure}
that for the circular orbit,
the body will move in the
parabola~$P$. If the initial
velocity is still greater, the
orbit will be an hyperbola~$H$.
And finally, if the initial velocity
is infinite, the path will
be the straight line whose
direction is~$OT$. If the initial
direction of motion is
not perpendicular to~$OS$, the
results are analogous, except
that there is then no initial
velocity which will give a
circular orbit.

It is seen from this discussion
that it is as natural for a
body to move in one conic
section as in another. Some of the satellites move in orbits
which are very nearly circular; the planets move in ellipses
with varying degrees of elongation; many comets move in
orbits which are sensible parabolas; and there may possibly
be comets which move in hyperbolas.

\Article{149}{The Question of other Laws of Force.}---Many
\index{Laws!of force}%
other laws of force than that of the inverse squares are
conceivable. For example, the intensity of a force might
vary inversely as the third power of the distance. The character
of the curve described by a body moving subject to any
such force can be determined by mathematical processes.
%% -----File: 267.png---Folio 237-------
It is found that, if the force varied according to any other
power of the distance than the inverse square, except directly
as the first power, then (save in special initial conditions) the
orbits would be curves leading either into the center of force
or out to infinity. Such a law would of course be fatal to
the permanence of the planetary system.

If the force varied directly as the distance, the orbits
would all be exactly ellipses, in spite of the mutual attractions
of the planets, the sun would be at the center of all the
orbits, and all the periods would be the same. This would
imply an enormous speed for the remote bodies.

\Article{150}{Perturbations.}---If the planets were subject to no
\index{Perturbations}%
forces except the attraction of the sun, their orbits would be
strictly ellipses. But, according to the law of gravitation,
every planet attracts every other planet. Their mutual
attractions are small compared to that of the sun because
of their relatively small masses, but they cause sensible,
though small, deviations from strict elliptical motion, which
are called \textit{perturbations}.

The mutual perturbations of the planets are sometimes
regarded as blemishes on what would be otherwise a perfect
system. Such a point of view is quite unjustified. Each
body is subject to certain forces, and its motion is the result
of its initial position and velocity and these forces. If the
masses of the planets were not so small compared to that of
the sun, their orbits would not even resemble ellipses.

The problems of the mutual perturbations of the planets
and those of the perturbations of the moon are exceedingly
difficult, and have taxed to the utmost the powers of mathematicians.
In order to obtain some idea of their nature consider
the case of only two planets, $P_1$ and~$P_2$. The forces
that $P_1$ and~$P_2$ would exert upon each other if they both
moved in their unperturbed elliptical orbits can be computed
without excessive difficulty, and the results of these forces
can be determined. But the resulting departures from elliptical
motion cause corresponding alterations in the forces,
%% -----File: 268.png---Folio 238-------
which produce new perturbations. These new perturbations
in turn change the forces again. The forces give rise to new
perturbations, and the perturbations to new perturbing
forces, and so on in an unending sequence. In the solar
system where the masses of the planets are small compared
to that of the sun, the perturbations of the series decrease
very rapidly in importance. If the masses of the planets
were large compared to the sun so that Kepler's laws would
not have been even approximately true, it is doubtful if
even the genius of Newton could have extracted from the
\index[xnames]{Newton}%
intricate tangle of phenomena the master principle of the
celestial motions, the law of gravitation.

Although the perturbations may be small, the question
arises whe\-ther they may not be extremely important in the
long run. The subject was treated by Lagrange and Laplace
\index[xnames]{Lagrange}%
\index[xnames]{Laplace}%
toward the end of the eighteenth century. They
proved that the mean distances, the eccentricities, and the
inclinations of the planetary orbits oscillate through relatively
narrow ranges, at least for a long time. If these results
were not true, the stability of the system would be imperiled,
\index{Stability!of solar system}%
for with extreme variation of especially the first two
of these quantities the characteristics of the planetary orbits
would be entirely changed. On the other hand, the perihelion
points and the places where the planes of the orbits
of the planets intersect a fixed plane not only have small
oscillations, but they involve terms which continually change
in one direction. Examples of perturbations of precisely
this sort already encountered are the precession of the equinoxes
(\Artref{47}) and the revolution of the moon's line of
nodes (\Artref{119}).

\Article{151}{The Discovery of Neptune.}---Not only can the perturbations
\index{Discovery of Uranus and Neptune}%
\index{Neptune!discovery of}%
be computed when the positions, initial motions,
and the masses %[Illustration: Break, move up]
\begin{wrapfigure}[23]{\WLoc}{2.75in}
\Input[2.75in]{269}{jpg}
\Caption[William Herschel.]{Fig}{90}
\end{wrapfigure}
of the planets are given, but the converse
problem can be treated with some success. That is, if the
perturbations are furnished by the observations, the nature
of the forces which produce them can be inferred. The most
%% -----File: 269.png---Folio 239-------
celebrated example of this converse problem led to the discovery
of the planet Neptune.

In 1781 William Herschel discovered the planet Uranus
\index{Uranus!discovery of}%
\index[xnames]{Herschel, William}%
while carrying out his project of examining every object in
the heavens within reach of his telescope. After it had
been observed for some time its orbit was computed. In
order to predict its position exactly it was necessary to
compute the perturbations
due to all known
bodies. This was done
by Bouvard on the basis
\index[xnames]{Bouvard}%
of the mathematical
theory of Laplace. But
\index[xnames]{Laplace}%
by 1820 there were unmistakable
discordances
between theory and observation;
by 1830, they
were still more serious;
by 1840, they had become
intolerable. This does
not mean that prediction
assigned the planet to
one part of the sky and
observation found it in a
far different one; for, in
1840, its departure from
its calculated position
amounted to only two thirds the apparent distance between
the two components of Epsilon Lyræ (\Artref{88}), a quantity
\index{Epsilon Lyrae@{Epsilon Lyræ}}%
invisible to the unaided eye. It seems incredible that so
slight a discordance between theory and observation after $60$~years
of accumulation could have led to any valuable results.

By 1820 it began to be suggested that the discrepancies
in the motion of Uranus might be due to the attraction of
a more remote unknown %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.25in}
\Input[2.25in]{270a}{jpg}
\Caption[John Couch Adams.]{Fig}{91}
\end{wrapfigure}
planet. The problem was to find
the unknown planet. Such excessive mathematical difficulties
%% -----File: 270.png---Folio 240-------
were involved that it seemed insoluble. In fact, Sir
George Airy, Astronomer Royal of England, expressed himself
\index[xnames]{Airy}%
later than 1840 as not believing
the problem could be
solved. However, a young
Englishman, Adams, and a
\index[xnames]{Adams, J. C.}%
young Frenchman, Leverrier,
\index[xnames]{Leverrier}%
with all the enthusiasm of
youth, quite independently took
up the problem about 1845.
Adams finished his work first
and communicated his results
both to Challis, at Cambridge,
\index[xnames]{Challis}%
and to Airy, at Greenwich.
To say the least, they took
no very active interest in the
matter and allowed the search
for the supposed body to be
postponed. Adams continued
his work and made five separate
and very laborious computations.
In the meantime Leverrier
completed his work and
sent the results to a young
German astronomer, Galle.
\index[xnames]{Galle}%
Impatiently Galle waited for
the night and the stars. On
the first evening after receiving
Leverrier's letter, September~23,
1846, he looked for
the unknown body, and found
it within half a degree of
the position assigned to it
by Leverrier, which agreed
substantially with that indicated by Adams.

Neptune is nearly three thousand millions of miles from the
%% -----File: 271.png---Folio 241-------
earth, beyond the reach of all our senses except that of sight,
and it can be %[Illustration: Break, move down]
\begin{wrapfigure}{\WLoc}{2.25in}
\Input[2.25in]{270b}{jpg}
\Caption[Joseph Leverrier.]{Fig}{92}
\end{wrapfigure}
seen only with telescopic aid; its distance is
so great that more than four hours are required for its light
to come to us, yet it is bound to the remainder of the system
by the invisible bonds of gravitation. But its attraction
slightly influenced the motions of Uranus, and from
these slight disturbances its existence and position were
inferred. Notwithstanding the fact that both Adams and
\index[xnames]{Adams, J. C.}%
Leverrier made assumptions respecting the distance of the
\index[xnames]{Leverrier}%
unknown body which were somewhat in error, their work
stands as a monument to the reasoning powers of the human
mind, and to the perfection of the theory of the motions of
the heavenly bodies.

\Article{152}{The Problem of Three Bodies.}---While the problem
of two mutually attracting bodies presents no serious
mathematical troubles, because the motion is always in some
kind of a conic section, that of three bodies is one of the
most formidable difficulty. It is often supposed that it has
not been, and perhaps that it cannot be, solved. Such an
idea is incorrect, as will now be explained.

The theory of the perturbations of the planets is really a
problem of three, or rather of eight, bodies, and has been
completely solved for an interval of time not too great. That
is, while the orbits of the bodies cannot be described for an
indefinite interval of time because they are not closed curves
but wind about in a very complicated fashion, nevertheless
it is possible to compute their positions with any desired
degree of precision for any time not too remote. Therefore,
in a perfectly real and just sense the problem has been
solved.

There are particular solutions of the problem of three
bodies in which the motion can be described for any period
of time, however long. The first of these were discovered
by Lagrange, who found two special cases. In one of them
\index[xnames]{Lagrange}%
the bodies move so as to remain always in a straight line,
and in the other so as to be always at the vertices of an equilateral
%% -----File: 272.png---Folio 242-------
triangle. In both cases the orbits are conic sections.
In 1878 an American astronomer, Hill, in connection with
\index[xnames]{Hill}%
his work on the motion of the moon, discovered some less
simple but immensely more important special cases. Since
1890 Poincaré, universally regarded as the greatest mathematician
\index[xnames]{Poincare@{Poincaré}}%
of recent times, has proved the existence of an
infinite number of these special cases called periodic solutions.
In all of them the problem is exactly solved. Still more
recently Sundman, of Helsingfors, Finland, has in an important
\index[xnames]{Sundman}%
mathematical sense solved the general case. However,
in spite of all the results that have been achieved, the
problem still presents to the mathematician unsolved questions
of almost infinite variety.

\Article{153}{The Cause of the Tides.}---So far in the present
\index{Tides!cause of}%
discussion only the effect of one body on the motion of
another, taken as a whole, has been considered. There
remains to be considered the distortion of one body by
the attraction of another. These deformations give rise to
the tides.

Before proceeding to a direct discussion of the tidal problem
it is necessary to state an important principle, namely,
\textit{if two bodies are subject to equal parallel accelerations, their
relative positions are not changed}. The truth of this proposition
follows from the laws of motion, but it is better understood
from an illustration. Suppose two bodies of the
same or different dimensions are dropped from the top of a
high tower. They have initially a certain relation to each
other and they are subject to equal parallel accelerations,
namely, those produced by the earth's attraction. In their
descent they fall faster and faster; but, neglecting the effects
of the resistance of the air, they preserve the same relations
to each other.

Let $E$, \Figref{93}, represent the earth, and $O$ and~$O'$ two
points on its surface. Consider the tendency of the moon~$M$
to displace~$O$ on the surface of the earth. The moon attracts
the center of the earth~$E$ in the direction~$EM$. Let
%% -----File: 273.png---Folio 243-------
its acceleration be represented by~$EP$. In the same units
$OA$~represents the acceleration of~$M$ on~$O$ in direction and
amount. The line~$OA$ is greater than $EP$ because the
moon is nearer to~$O$ than it is to~$E$. Now resolve $OA$ into
two components, one of which, $OB$, shall be equal and parallel
to~$EP$. The other component is~$OC$. Since $OB$ and~$EP$
are equal and parallel, it follows from the principle stated
\begin{figure}[hbt]%[Illustration:]
\Input{273a}{png}
\Caption[Resolution of the tide-raising forces.]{Fig}{93}
\index{Tide-raising!forces}%
\end{figure}%
at the beginning of this article that they do not change the
relative positions of $E$ and~$O$. Therefore $OC$, the outstanding
component, represents the tide-raising acceleration both
in direction and amount.

The results for $O'$ are analogous, and the tide-raising
force~$O'C'$ is directed away from the moon because $O'A'$ is
shorter than~$EP$. \Figureref{94} shows the tide-raising accelerations
\begin{figure}[hbt]%[Illustration:]
\Input{273b}{png}
\Caption[The tide-raising forces.]{Fig}{94}
\end{figure}%
around the whole circumference of the earth.
This method of deriving the tide-raising forces is the elementary
geometrical counterpart of the rigorous mathematical
%% -----File: 274.png---Folio 244-------
treatment,\footnote
  {An analytical discussion proves that the tide-raising force is proportional
  to the product of the mass of the disturbing body and the radius of
  the disturbed body, and inversely proportional to the cube of the distance
  between the disturbing and disturbed bodies.}
and it can be relied on to give correctly
all that there is in this part of the subject.

A more detailed discussion than can be entered into here
shows that the tide-raising forces are about $5$~per~cent
greater on the side of the earth which is toward the moon
than on the side away from the moon. The forces outward
from the surface of the earth in the line of the moon are
about twice as great as those which are directed inward $90°$
from this line. The tidal forces due to the sun are a little
less than half as great as those due to the moon; no other
bodies have sensible tidal effects on the earth.

\Article{154}{The Masses of Celestial Bodies.}---The masses of
\index{Masses!determination of}%
celestial bodies are determined from their attractions for
other bodies. Suppose a satellite revolves around a planet
in an orbit of measured dimensions in an observed period.
From these data it is possible to compute the acceleration of
the planet for the satellite because the attraction balances
the centrifugal acceleration. It is possible to determine
what the earth's attraction would be at the same distance,
and, consequently, the relation of its mass to that of the
other planet. There has been much difficulty in finding
the masses of Mercury and Venus because they have no
known satellites. Their masses have been determined with
considerable reliability from their perturbations of each
other and of the earth, and from their perturbations of certain
comets that have passed near them.

A useful formula for the sum of the masses of any two
bodies $m_1$ and~$m_2$ which attract each other according to the
law of gravitation, for example, the two components of a
double star, is
\[
m_1 + m_2 = \frac{a^3}{P^2},
\]
where $a$ is the distance between the bodies expressed in
%% -----File: 275.png---Folio 245-------
terms of the earth's distance from the sun as unity, and
where $P$ is the period expressed in years. The sum of the
masses is expressed in terms of the sun's mass as unity.

\Article{155}{The Surface Gravity of Celestial Bodies.}---The
\index{Gravity!surface}%
\index{Sun!surface gravity of}%
\index{Surface gravity!determination of}%
\index{Surface gravity!of sun}%
surface gravity of a celestial body is an important factor in
the determination of its surface conditions, and is fundamental
in the question of its retaining an atmosphere. The
surface gravity of a spherical body depends only upon its
mass and dimensions.

Let $m$ represent the mass of the earth, $g$ its surface gravity,
and $r$ its radius. Then by the law of gravitation
\[
g = k^2 \frac{m}{r^2},
\]
where $k^2$ is a constant depending on the units employed.
Let $M$,~$G$, and~$R$ represent in the same units the corresponding
quantities for another body. Then
\[
G = k^2 \frac{M}{R^2}.
\]
On dividing the second equation by the first, it is found that
\[
\frac{G}{g} = \frac{M}{m} \left(\frac{r}{R}\right)^2,
\]
from which the surface gravity~$G$ can be found in terms of
that of the earth when the mass and radius of~$M$ are given.

It is sometimes convenient to have the expression for the
ratio of the gravities of two bodies in terms of their densities
and dimensions. Let $d$ and~$D$ represent the densities of
the earth and the other body respectively. Then, since
$m = \frac{4}{3} \pi d r^3$ and $M = \frac{4}{3} \pi D R^3$, it is found that
\[
\frac{G}{g} = \frac{D}{d} \frac{R}{r}.
\]
That is, the surface gravities of celestial bodies are proportional
to the products of their densities and radii. A
small density may be more than counterbalanced by a large
radius, as, for example, in the case of the sun, whose density
is only one fourth that of the earth but whose surface gravity
is about $27.6$~times that of the earth.
%% -----File: 276.png---Folio 246-------


\Section{XI}{QUESTIONS}

1. If the sidereal period of a planet were half that of the earth,
what would be its period from greatest eastern elongation to its next
succeeding greatest eastern elongation?

2. If the sidereal period of a planet were twice that of the earth,
what would be its period from opposition to its next succeeding
opposition?

3. What would be the period of a planet if its mean distance from
the sun were twice that of the earth?

4. What would be the mean distance of a planet if its period were
twice that of the earth?

5. The motion of the moon around the earth satisfies (nearly)
Kepler's first two laws. What are the respective conclusions which
follow from them?

6. The force of gravitation varies directly as the product of the
masses. Show that the acceleration of one body with respect to
another, both being free to move, is proportional to the sum of
their masses. \textit{Hint.} Use both the second and third laws of motion.

7. In Lagrange's two special solutions of the problem of three
bodies the law of areas is satisfied for each body separately with
respect to the center of gravity of the three. What conclusion
follows from this fact? How does the force toward the center of
gravity vary?

\normalsize


\Section{II}{The Orbits, Dimensions, and Masses of The
Planets}

\Article{156}{Finding the actual Scale of the Solar System.}---It
was seen in \Artref{144} that the relative dimensions of the
solar system can be determined without knowing any actual
distance. It follows from this that if the distance between
any two bodies can be found, all the other distances can be
computed.

The problem of finding the actual scale of the solar system
is of great importance, because the determination of the
dimensions of all its members depends upon its solution,
and the distance from the earth to the sun is involved in
measuring the distances to the stars. Not until after the
year 1700 had it been solved with any considerable degree
of approximation, but the distance from the earth to the sun
%% -----File: 277.png---Folio 247-------
is now known with an error probably not exceeding one
part in a thousand.

The direct method of measuring the distance to the sun,
\index{Distance!of sun}%
\index{Sun!distance of}%
analogous to that used in case of the moon (\Artref{123}), is of
no value because the apparent displacement to be measured
is very small, the sun is a body with no permanent surface
markings, and its heat seriously disturbs the instruments.
But, as has been seen (\Artref{144}), the distance from the earth
to any other member of the system is equally useful, and in
some cases the measurement of the distances to the other
bodies is feasible.

Gill, at the Cape of Good Hope, measured the distance
\index[xnames]{Gill}%
of Mars with considerable success, but its disk and red
color introduced difficulties. These difficulties do not arise
in the case of the smaller planetoids, which appear as starlike
points of light, but their great distances decrease the
accuracy of the results by reducing the magnitude of the
quantity to be measured. However, in 1898, Witt, of Berlin,
\index[xnames]{Witt}%
discovered a planetoid whose orbit lies largely within the
orbit of Mars and which approaches closer to the earth than
any other celestial body save the moon. Its nearness, its
minuteness, and its absence of marked color all unite to
make it the most advantageous known body for getting the
scale of the solar system by the direct method. Hinks, of
\index[xnames]{Hinks}%
Cambridge, England, made measurements and reductions of
photographs secured at many observatories, and found that
the parallax of the sun, or the angle subtended by the earth's
\index{Parallax!of sun}%
\index{Sun!parallax of}%
radius at the mean distance of the sun, is $8''.8$, corresponding
to a distance of $92,897,000$ miles from the earth to the sun.

The distance of the earth from the sun can also be found
from the aberration of light. The amount of the aberration
depends upon the velocity of light and the speed with which
the observer moves across the line of its rays. The velocity
of light has been found with great accuracy from experiments
on the surface of the earth. The amount of the aberration
has been determined by observations of the stars. From
%% -----File: 278.png---Folio 248-------
the two sets of data the velocity of the observer can be
computed. Since the length of the year is known, the length
of the earth's orbit can be obtained. Then it is an easy matter,
making use of the shape of the orbit, to compute the mean
distance from the earth to the sun. The results obtained in
this way agree with those furnished by the direct method.

Another and closely related method depends upon the
determination of the earth's motion in the line of sight
(\Artref{226}) by means of the spectroscope. Spectroscopic
technique has been so highly perfected that when stars best
suited for the purpose are used the results obtained give the
earth's speed with a high degree of accuracy. Its velocity
and period furnish the distance to the sun, as in the method
depending upon the aberration, and the results are about as
accurate as those furnished by any other method.

There are several other methods for finding the distance
to the sun which have been employed with more or less success.
One of them depends upon transits of Venus across
the sun's disk. Another involves the attraction of the sun
for the moon. But none of them is so accurate as those
which have been described.

\Article{157}{The Elements of the Orbits of the Planets.}---The
\index{Elements!of orbit}%
\index{Orbits!of planets, elements of}%
position of a planet at any time depends upon the size, shape,
and position of its orbit, together with the time when it was
at some particular position, as the perihelion point. These
quantities are called the \textit{elements of an orbit}, and when they
are given it is possible to compute the position of the planet
at any time.

The size of an orbit is determined by the length of its major
axis. It is an interesting and important fact that the period
of revolution of a planet depends only upon the major axis
of its orbit, and not upon its eccentricity or any other element.
The shape of an orbit is defined by its eccentricity.
The position of a planet's orbit is determined by its orientation
in its plane and the relation of its plane to some standard
plane of reference. The longitude of the perihelion point
%% -----File: 279.png---Folio 249-------
defines the orientation of an orbit in its plane. The plane
of reference in common use is the plane of the ecliptic. The
position of the plane of the orbit is defined by the location
of the line of its intersection with the plane of the ecliptic
and the angle between the two planes. The distance from
the vernal equinox eastward to the point where the orbit of
the body %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.25in}
\Input[2.25in]{279}{png}
\Caption[Elements of the orbit of a planet.]{Fig}{95}
\index{Elements!of orbit}%
\end{wrapfigure}
crosses the ecliptic
from south to north is called
the longitude of the ascending
node, and the angle between
the plane of the ecliptic
and the plane of the orbit
is called the inclination.

In \Figref{95}, $VNQ$ represents
the plane of the ecliptic and
$SNP$ the plane of the orbit.
The vernal equinox is at~$V$,
the angle $VSN$ is the longitude
of the ascending node,
\index{Ascending node}%
the angle $VSN + NSP$ is
the longitude of the perihelion, and the angle $QNP$ is the
inclination of the orbit.

The elements of the orbits of the planets, which change
very slowly, are given for January~1, 1916, in \Tableref{IV}.

\begin{table}[htb]
\begin{center}
\Caption{Table}{IV}
\index{Distance!of planets}%
\index{Eccentricity!of earth's orbit}%
\index{Eccentricity!of planetary orbits}%
\index{Elements!table of}%
\index{Inclination of earth's orbit!of planetary orbits}%
\index{Orbits!of planets, elements of}%
\index{Perihelion point!longitude of}%
\index{Period of planets}%
\index{Planetary orbits!dimensions of}%
\index{Planetary orbits!eccentricities of}%
\index{Planetary orbits!planes of}%
\index{Planets!distances of}%
\index{Planets!periods of}%
\index{Sidereal!period of planets}%
\makebox[0pt][c]{%
\TFontsize
\setlength{\tabcolsep}{4pt}%
\settowidth{\TmpLen}{\textsc{Period}}
\begin{tabular}{|l|*{3}{r|}*{4}{rr|}}
%[Elements of the orbits of the planets]
\hline
\textsc{Planet} &
  \TEntry{1.2\TmpLen}{\medskip\THead Dis-\\ tance, \\ Mil-\\ lions of \\ Miles\medskip} &
  \TEntry{\TmpLen}{\THead Period \\ in Years} &
  \TEntry{\TmpLen}{\THead Eccen\-tricity} &
  \TCEntry{2}{c|}{\textsc{Inclin-}}{\THead Inclin\-a\-tion \\ to Eclip\-tic} &
  \TCEntry{2}{c|}{\textsc{Period}}{\THead Longi\-tude of \\ Node} &
  \TCEntry{2}{c|}{\textsc{Perihe-}}{\THead Longi\-tude of \\ Perihe\-lion} &
  \TCEntry{2}{c|}{\textsc{Period}}{\THead Longi\-tude on \\ Jan.~1, \\ 1916} \\
\hline
\Strut%
Mercury &   $36.0$ &   $0.241$ & $0.20562$ & $\phantom{00}7$\rlap{$°$}& \rlap{$0'$}$\phantom{0}$&
 $\phantom{0}47$\rlap{$°$}&\rlap{$20'$}$\phantom{20}$&  $76$\rlap{$°$}& \rlap{$9'$}$\phantom{9}$&
  $334$\rlap{$°$}& \rlap{$2'$}$\phantom{2}$\\
Venus   &   $67.2$ &   $0.615$ & $0.00681$ &             $3$        &       $24$ &
            $75$        &       $55$               & $130$        &       $23$ &
  $345$        &       $50$ \\
Earth   &   $92.9$ &   $1.000$ & $0.01674$ &             $0$        &       $00$ &
 \multicolumn{2}{c|}{\rule{0.8\TmpLen}{0.5pt}} & $101$        &       $30$ &
   $99$        &       $49$ \\
Mars    &  $141.5$ &   $1.881$ & $0.09332$ &             $1$        &       $51$ &
            $48$        &       $55$               & $334$        &       $31$ &
  $116$        &       $25$ \\
Jupiter &  $483.3$ &  $11.862$ & $0.04836$ &             $1$        &       $18$ &
            $99$        &       $36$               &  $12$        &       $58$ &
    $3$        &       $51$ \\
Saturn  &  $886.0$ &  $29.458$ & $0.05583$ &             $2$        &       $30$ &
           $112$        &       $55$               &  $91$        &       $24$ &
  $102$        &       $20$ \\
Uranus  & $1781.9$ &  $84.015$ & $0.04709$ &             $0$        &       $46$ &
            $73$        &       $34$               & $169$        &       $18$ &
  $312$        &        $9$ \\
Neptune & $2791.6$ & $164.788$ & $0.00854$ &             $1$        &       $47$ &
           $130$        &       $51$               &  $43$        &       $54$ &
  $120$        &       $12$\rule[-2ex]{0pt}{0pt} \\
\hline
\end{tabular}}
\end{center}
\end{table}

%% -----File: 280.png---Folio 250-------
To the elements of the orbits of the planets must be
added the direction of their motion in order to be altogether
complete. The result is very simple, for they all revolve in
the same direction, namely, eastward.

The most interesting and important element of the planetary
orbits is the mean distance. The distance of Neptune
from the sun is $30$~times that of the earth and nearly $80$~times
that of Mercury. Since the amount of light and
heat received per unit area by a planet varies inversely as
\index{Heat!received by planets}%
\index{Planets!heat received by}%
the square of its distance from the sun, it follows that if the
units are chosen so that the amount received by the earth
is unity, then the respective amounts received by the several
planets are: Mercury,~$6.66$; Venus,~$1.91$; Earth,~$1.00$;
Mars,~$0.43$; Jupiter,~$0.037$; Saturn,~$0.011$; Uranus,~$0.0027$;
Neptune,~$0.0011$. It is seen that the earth receives more
than $900$~times as much light and heat per unit area as Neptune,
and that in the case of Mercury and Neptune the
ratio is more than~$6000$. Obviously, other things being
equal, the climatic conditions on planets differing so greatly
in distance from the sun would be enormous.

As seen from Neptune the sun presents a smaller disk
than Venus does to us when nearest to the earth. It is
sometimes supposed that Neptune is far away in the night
of space where the sun looks simply like a bright star. This
is far from the truth, for, since the sunlight received by the
earth is $600,000$ times full moonlight, and Neptune gets
$\frac{1}{900}$ as much light as the earth, it follows that the illumination
of Neptune by the sun is nearly $700$~times that of the
earth by the brightest full moon. Another erroneous idea
frequently held is that Neptune is so far away from the sun
that it gets a considerable fraction of its light from other
suns. The nearest known star is more than $9000$ times as
distant from Neptune as Neptune is from the sun, and, consequently,
Neptune receives more than $80,000,000$ times as
much light and heat as it would if the sun were at the distance
of the nearest star.
%% -----File: 281.png---Folio 251-------

It is almost impossible to get a correct mental picture of
the enormous dimensions of the solar system, and there are
often misconceptions in regard to the relative dimensions of
the orbits of the various planets. To assist in grasping these
distances; suppose one has traveled sufficiently to have
obtained some comprehension of the great size of the earth.
Then he is in a position to attempt to appreciate the distance
to the moon, which is so far that in spite of the fact it is more
than $2000$ miles in diameter, it is apparently covered by a
one-cent piece held at the distance of $6.5$~feet. In terms
of the earth's dimensions, its distance is about $10$~times the
circumference of the earth. It is so remote that about $14$~days
would be required for sound to come from it to the
earth if there were an atmosphere the whole distance to transmit
it at the rate of a mile in $5$~seconds.

Now consider the distance to the sun; it is $400$~times that
to the moon. If the earth and sun were put $4$~inches apart
on such a diagram as could be printed in this book, on the
same scale the distance from the earth to the moon would be
$\tfrac{1}{100}$ of an inch. If sound could come from the sun to the
earth with the speed at which it travels in air, $15$~years would
be required for it to cross the $92,900,000$ of miles between
the earth and sun. Some one, having found out at what
rate sensations travel along the nerve fibers from the hand
to the brain, proved by calculation that if a small boy with
a sufficiently long arm should reach out to the sun and burn
his hand off, the sensation would not arrive at his brain so
that he would be aware of his loss unless he lived to be more
than $100$~years of age.

The relative dimensions of the orbits of the planets can be
best understood from diagrams. Unfortunately, it is not
possible to represent them to scale all on the same diagram.
\Figureref{96} shows the orbits of the first four planets, together
with that of Eros, which occupies a unique position, and which
has been used in getting the scale of the system. \Figureref{97}
shows the orbits of the planets from Mars to Neptune on a
%% -----File: 282.png---Folio 252-------
scale which is about $\tfrac{1}{20}$ that of the preceding figure. The
most noteworthy fact is the relative nearness of the four
\begin{figure}[hbt]%[Illustration:]
\Input{282}{png}
\Caption[Orbits of the four inner planets.]{Fig}{96}
\end{figure}%
inner planets and the enormous distances that separate the
outer ones.

\Article{158}{The Dimensions, Masses, and Rotation Periods of
the Planets.}---The planets Mercury and Venus have no
known satellites and their masses are subject to some uncertainties.
The rotation periods of Mercury and Venus
are very much in doubt because of their unfavorable positions
for observation, while the distances of Uranus and
Neptune are so great that so far it has been impossible to
%% -----File: 283.png---Folio 253-------
see clearly any markings on their surfaces. There is some
uncertainty in the diameters of the planets on account of
\begin{figure}[hbt]%[Illustration:]
\Input{283}{png}
\Caption[Orbit of the outer planets.]{Fig}{97}
\end{figure}%
what is called irradiation, which makes a luminous object
appear larger than it actually is.

The data given in \Tableref{V} are based partly on Barnard's
many measures at the Lick Observatory, and partly on
those adopted for the American Ephemeris and Nautical
Almanac.
\index{American Ephemeris and Nautical Almanac}%
%% -----File: 284.png---Folio 254-------
\begin{table}[hbt]
\begin{center}
\Caption{Table}{V}
\index{Masses!of planets}%
\index{Moon!density of}%
\index{Moon!mass of}%
\index{Planets!density of}%
\index{Planets!dimensions of}%
\index{Planets!masses of}%
\index{Planets!surface gravity of}%
\index{Sun!density of}%
\index{Sun!mass of}%
\index{Sun!surface gravity of}%
\index{Surface gravity!of planets}%
\index{Surface gravity!of sun}%
\TFontsize%
%\caption[Data on sun, moon, and planets]{} %
\makebox[0pt][c]{%
\setlength{\tabcolsep}{2.5pt}%
\begin{tabular}{|l|r<{\ }|>{\ }r<{\quad\ }|c|c<{\ }|>{\ }l|c|}
\hline
\TCEntry{1}{|c|}{Neptune}{\THead Body}
  & \TCEntry{1}{c|}{\THF Diameter}{\THead Mean \\ Diameter}
  & \TCEntry{1}{c|}{\THF \scriptsize(Earth $= 1$)}{\THead Mass \\ \scriptsize(Earth $= 1$)}
\index{Mass!of sun}%
  & \TCEntry{1}{c|}{\THF \scriptsize(Water $= 1$)}{\THead Density \\ \scriptsize(Water $ = 1$)}
\index{Density!of moon}%
\index{Density!of sun}%
  & \TCEntry{1}{c|}{\THF Surface}{\THead Surface \\ Gravity \\ ($g = 1$)}
\index{Gravity!of planets}%
  & \TCEntry{1}{c|}{\THF Period of}{\THead Period of \\ Rotation}
  & \TCEntry{1}{c|}{\THF to Orbit}{\medskip\THead Inclina- \\ tion of \\ Equator \\ to Orbit\medskip}
\\
\hline
\rule{0pt}{3.5ex}%
Sun     & $864,392$  &  $329,390$  & $1.40$           & \llap{$2$}$7.64$           & $25$~d.\ $8$~h.   &  $7°\,15'$ \\
Moon    &   $2,160$  &  $0.0122$   & $3.34$           &  $0.16$           & $27$~d.\ $7.7$~h. &  $6°\,41'$ \\
Mercury &   $3,009$  &  $0.045$\rlap{(?)} & $4.48$\rlap{(?)} &  $0.31$\rlap{(?)} & \QMark            & \QMark    \\
Venus   &   $7,701$  &  $0.807$\rlap{(?)} & $4.85$\rlap{(?)} &  $0.85$           & \QMark            & \QMark    \\
Earth   &   $7,918$  &  $1.0000$   & $5.53$           &  $1.00$           & $23$~h.\ $56$~m.  & $23°\,27'$ \\
Mars    &   $4,339$  &  $0.1065$   & $3.58$           &  $0.36$           & $24$~h.\ $37$~m.  & $23°\,59'$ \\
Jupiter &  $88,392$  &  $314.50$   & $1.25$           &  $2.52$           &$\Z9$~h.\ $55$~m.  &  $3°$     \\
Saturn  &  $74,163$  & $\Z94.07$   & $0.63$           &  $1.07$           & $10$~h.\ $14$~m.  & $27°$     \\
Uranus  &  $30,193$  & $\Z14.40$   & $1.44$           &  $0.99$           & \QMark            & \QMark    \\
Neptune &  $34,823$  & $\Z16.72$   & $1.09$           &  $0.86$           & \QMark            & \QMark    \\
\hline
\end{tabular}}
\end{center}
\end{table}

Some interesting facts are revealed by this table. The
first four planets are very small compared to the outer four,
and since their volumes are as the cubes of their diameters,
\begin{figure}[hbt]%[Illustration:]
\Input{284}{png}
\Caption[Relative dimensions of sun and planets.]{Fig}{98}
\index{Dimensions!of sun, moon, and planets}%
\end{figure}%
the latter average more than a thousand times greater in
volume than the former. The inner planets are much denser
than the outer ones and, so far as known, rotate on their
axes more slowly.

\Figureref{98} shows an arc of the sun's circumference and the
%% -----File: 285.png---Folio 255-------
eight planets to the same scale. It is apparent from this
diagram how insignificant the earth is in comparison with
the larger planets, and how small they are all together in
comparison with the sun.

\Article{159}{The Times for Observing the Planets.}---Mercury
and Venus are most conveniently situated for observation
when they are near their greatest elongations, for then they
are not dimmed by the more brilliant rays of the sun. When
they are east of the sun they can be seen in the evening, and
when they are west of the sun they are observable only in
the morning. Ordinarily the evening is more convenient
for making observations than the morning, and therefore
the results will be given only for this time.

Those planets which are farther from the sun than the
earth can be observed best when they are in opposition, or
$180°$ from the sun, for then they are nearest the earth and
their illuminated sides are toward the earth. When a planet
is in opposition it crosses the meridian at midnight, and it
can be observed late in the evening in the eastern or southeastern
sky.

The problem arises of determining at what times Mercury
and Venus are at greatest eastern elongation, and at
what times the other planets are in opposition. If the time
at which a planet has its greatest eastern elongation is once
given, the dates of all succeeding eastern elongations can
be obtained by adding to the original one multiples of its
synodical period. If $S$ represents the synodical period of an
inferior planet, $P$ its sidereal period, and $E$ the earth's period,
the synodical period is given by (Arts.\ \hyperref[art:120]{120},~\hyperref[art:144]{144})
\[
\frac{1}{S} = \frac{1}{P} - \frac{1}{E};
\]
and in the case of a superior planet the corresponding formula
for the synodical period is
\[
\frac{1}{S} = \frac{1}{E} - \frac{1}{P}.
\]
On the basis of the sidereal periods given in \Tableref{IV}, these
%% -----File: 286.png---Folio 256-------
formulas, and data from the American Ephemeris and Nautical
Almanac, the following table has been constructed:\footnote
  {In this table the tropical year is used and $30$~days are taken as constituting
  a month.}
\begin{table}[hbt]
\begin{center}
\Caption{Table}{VI}
\index{Elongations of planets!dates of}%
\index{Opposition!of planets, dates of}%
\index{Planets!dates of elongation of}%
\index{Planets!dates of opposition of}%
\index{Planets!synodical periods of}%
\index{Synodical period!of planets}%
\makebox[0pt][c]{%
\setlength{\tabcolsep}{3pt}%
%\caption[Dates of eastern elongation and oppostion]{} %
\settowidth{\TmpLen}{\textsc{Elongation}}%
\begin{tabular}{|l|>{\quad}l@{ }r<{\quad}|r@{ }r@{ }r@{${}={}$}l@{}|}
\hline
\TEntry{\TmpLen}{\TFontsize\THead Planet}
  &\TCEntry{2}{c|}{\TFontsize\THF tion or Opposition}{\medskip\TFontsize\THead Eastern Elonga-\\ tion or Opposition\medskip}
  &\TCEntry{4}{c|}{$0\text{ yr. } 3\text{ mo. }99.9\text{ d.} = 9.99999\text{ yr.}$}{\TFontsize\THead Synodical Period} \\
  \hline
\Strut
Mercury\MyDotFill   &Sept.& 9, 1916  &0 yr. &3 mo. &24.2~d. &0.31726~yr\DPtypo{}{.} \\
Venus\MyDotFill     &April&23, 1916  &1 yr. &7 mo. &5.7~d.  &1.59882~yr. \\
Mars\MyDotFill      &Feb. & 9, 1916  &2 yr. &1 mo. &18.7~d. &2.13523~yr. \\
Jupiter\MyDotFill   &Oct. &23, 1916  &1 yr. &1 mo. &3.1~d.  &1.09206~yr. \\
Saturn\MyDotFill    &Jan. & 4, 1916  &1 yr. &0 mo. &12.6~d. &1.03514~yr. \\
Uranus\MyDotFill    &Aug. &10, 1916  &1 yr. &0 mo. &4.3~d.  &1.01205~yr. \\
Neptune\MyDotFill   &Jan. &22, 1916  &1 yr. &0 mo. &2.2~d.  &1.00611~yr.\rule[-2ex]{0pt}{0pt} \\
\hline
\end{tabular}}
\end{center}
\end{table}

The superior planets are most brilliant when they are
in opposition; the inferior planets are brightest some time
after their greatest eastern elongation because they are
then relatively approaching the earth and their decrease in
distance more than offsets their diminishing phase. For
example, in 1916 Venus was at its greatest eastern elongation
April~23, but kept getting brighter until May~27.

Mercury is so much nearer the sun than the earth that
its greatest elongation averages only $23°$, though it varies
from $18°$ to $28°$ because of the eccentricity of the orbit of
the planet. Consequently, it can be observed only for a
very short time after the sun is far enough below the horizon
for the brightest stars to be visible. Mercury at its brightest
is somewhat brighter than a first-magnitude star. There is
no difficulty in observing any of the other planets except
Uranus and Neptune, Uranus being near the limits of visibility
without optical aid, and Neptune being quite beyond
them. Venus is brilliantly white and at its brightest quite
surpasses every other celestial object except the sun and
moon. Mars is of the first magnitude and decidedly red.
%% -----File: 287.png---Folio 257-------
Jupiter is white and next to Venus in brilliance. Saturn is
of the first magnitude and slightly yellowish.

\Article{160}{The Planetoids.}---On examination it is found that
the distance of each planet from the sun is roughly twice
that of the preceding, with the exception of Jupiter, whose
distance is about $3.5$ times that of Mars. In 1772 Titius
\index[xnames]{Titius}%
derived a series of numbers by a simple law which gave the
distances of the planets (Uranus and Neptune were not
known then) with considerable accuracy, except that there
was a number for the vacant space between Mars and
Jupiter. The law is that if $4$ is added to each of the numbers
$0$, $3$, $6$, $12$, $24$, $48$, the sums thus obtained are nearly
proportional to the distances of the planets from the sun.
This law, commonly called Bode's law, because the writings
\index{Bode's law}%
\index[xnames]{Bode}%
of Bode made it widely known, rests on no scientific basis
and entirely breaks down for Neptune, but it played an
important rôle in two discoveries. One of these was that
both Adams and Leverrier assigned distances to the planet
\index[xnames]{Adams, J. C.}%
\index[xnames]{Leverrier}%
Neptune on the basis of this law, and computed the other
elements of its orbit from its perturbations of Uranus (\Artref{151}).
 The other discovery to which Bode's law contributed
was that of the planetoids.

Toward the end of the eighteenth century the idea became
\index{Ceres, discovery of}%
wide\-spread among astronomers that there was probably an
undiscovered planet between Mars and Jupiter whose distance
would agree with the fifth number of the Bode series.
In 1800 a number of German astronomers laid plans to search
for it, but before their work was actually begun Piazzi, at
\index[xnames]{Piazzi}%
Palermo, on January~1, 1801, the first day of the nineteenth
century, made the discovery when he noticed an object (apparently
a star) where none had previously been seen. Piazzi
called the new planet, which was of small dimensions, Ceres.

After the discovery of Ceres had been made, but before
the news of it had reached Germany by the slow processes
of communication of those days, the philosopher Hegel
\index[xnames]{Hegel}%
published a paper in which he claimed to have proved by
%% -----File: 288.png---Folio 258-------
the most certain and conclusive philosophical reasoning that
there were no new planets, and he ridiculed his astronomical
colleagues for their folly in searching for them.

Piazzi observed Ceres for a short time and then he was
\index[xnames]{Piazzi}%
taken ill. By the time he had recovered, the earth had
moved forward in its orbit to a position from which the
planetoid could no longer be seen. In a little less than a
year the earth was again in a favorable position for observations
of Ceres, but the problem of picking it up out of the
countless stars that fill the sky, and from which it could not
be distinguished except by its motions, was almost as difficult
as that of making the original discovery. The difficulty
was entirely overcome by Gauss, then a young man of~24,
\index[xnames]{Gauss}%
but later one of the greatest mathematicians of his time, for,
under the stimulus of this special problem, he devised a
practical method of determining the elements of the orbit
of a planet from only three observations. After the elements
of the orbit of a body are known, its position can be
computed at any time. Gauss determined the elements of
the orbit of Ceres, and his calculation of its position led to its
rediscovery on the last day of the year.

On March~28, 1802, Olbers discovered a second planetoid,
\index[xnames]{Olbers}%
which he named Pallas; on September~2, 1804, Harding
\index{Pallas, discovery of}%
\index[xnames]{Harding}%
found Juno; and on March~29, 1807, Olbers picked up a
\index{Juno, discovery of}%
fourth, Vesta. No other was found until 1845, when Hencke
\index{Vesta, discovery of}%
\index[xnames]{Hencke}%
discovered Astræa, after a long search of $15$~years. In 1847
three more were discovered, and every year since that time
at least one has been discovered.

In 1891 a new epoch was started by Wolf, of Heidelberg,
\index[xnames]{Wolf, Max}%
who discovered a planetoid by photography. The method
is simply to expose a plate two or three hours with the
telescope following the stars. The star images are points,
but the planetoids leave short trails, or streaks, \Figref{99},
because they are moving among the stars. There are now
all together more than $800$ known planetoids.

After the first two planetoids had been discovered it was
%% -----File: 289.png---Folio 259-------
\index{Orbits!of planetoids}%
\index{Origin!of planetoids}%
\index{Planetoids!orbits of}%
\index{Planetoids!origin of}%
supposed that they might be simply the fragments of an
original large planet which had been torn to pieces by an
explosion. If such were the case, the different parts in their
orbits around the sun would all pass through the position
occupied by the planet at the time of the explosion; therefore,
for some time the search for new planetoids was largely
confined to the regions about the points where the orbits
of Ceres and Pallas intersect. But this theory of their
\begin{figure}[hbt]%[Illustration:]
\Input{289}{jpg}
\Caption[Photograph of stars showing a planetoid (Egeria) trail in the
center of the plate. \textit{Photographed by Parkhurst at the Yerkes Observatory.}]{Fig}{99}
\index{Yerkes Observatory}%
\index[xnames]{Parkhurst}%
\end{figure}%
origin has been completely abandoned. The orbits of Eros
and two other planetoids are interior to the orbit of Mars,
while many are within $75,000,000$ miles of this planet; on
the other hand, many others are nearly $300,000,000$ miles
farther out, and the aphelia of four are even beyond the orbit
of Jupiter. Their orbits vary in shape from almost perfect
circles to elongated ellipses whose eccentricities are $0.35$ to~$0.40$.
The average eccentricity of their orbits is about~$0.14$,
or approximately twice that of the orbits of the planets.
Their inclinations to the ecliptic range all the way from zero
to~$35°$, with an average of about~$9°$.
%% -----File: 290.png---Folio 260-------

The orbits of the planetoids are distributed by no means
uniformly over the belt which they occupy. Kirkwood long
\index[xnames]{Kirkwood}%
ago called attention to the fact that the planetoids are infrequent,
or entirely lacking, at the distances at which their
periods would be $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$,~$\dots$ of Jupiter's period. The
numerous discoveries since the application of photography
have still further emphasized the existence of these remarkable
gaps. It is supposed that the perturbations by Jupiter
during indefinite ages have cleared these regions of the
bodies that may once have been circulating in them, but the
question has not received rigorous mathematical treatment.

The diameters of Ceres, Pallas, Vesta, and Juno were
\index{Planetoids!diameters of}%
measured by Barnard with the $36$-inch telescope of the Lick
\index{Lick Observatory}%
\index[xnames]{Barnard}%
Observatory, and he found that they are respectively $485$,
$304$, $243$, and $118$~miles. There are probably a few more
whose diameters exceed $100$~miles, but the great majority
are undoubtedly much smaller. Probably the diameters of
the faintest of those which have been photographed do not
exceed $5$~miles.

By 1898 the known planetoids were so numerous and their
orbits caused so much trouble, because of their large perturbations
by Jupiter, that astronomers were on the point
of neglecting them, when Witt, of Berlin, found one within
\index[xnames]{Witt}%
the orbit of Mars, which he named Eros. At once great
\index{Eros}%
interest was aroused. On examining photographs which
had been taken at the Harvard College Observatory in
\index{Harvard College Observatory}%
1893, 1894, and 1896, the image of Eros was found several
times, and from these positions a very accurate orbit was
computed by Chandler. The mean distance of Eros from
\index[xnames]{Chandler}%
the sun is $135,500,000$ miles, but its distance varies considerably
because its orbit has the high eccentricity of~$0.22$;
its inclination to the ecliptic is about~$11°$. At its nearest,
Eros is about $13,500,000$ miles from the earth, and then conditions
are particularly favorable for getting the scale of the
solar system (\Artref{156}); and at its aphelion it is $24,000,000$
miles beyond the orbit of Mars (\Figref{96}).
%% -----File: 291.png---Folio 261-------

Not only is Eros remarkable because of the position of
\index{Variability!of Eros}%
its orbit, but in February and March of 1901 it varied in
brightness both extensively and rapidly. The period was
$2$~hr.\ $38$~min., and at minimum its light was less than one
third that at maximum. By May the variability ceased.
Several suggestions were made for explaining this remarkable
phenomenon, such as that the planetoid is very different in
reflecting power on different parts, or that it is really composed
of two bodies very near together, revolving so that the
plane of their orbit at certain times passes through the
earth, but the cause of this remarkable variation in brightness
is as yet uncertain.

\Article{161}{The Question of Undiscovered Planets.}---The
\index{Planets!possible undiscovered}%
great planets Uranus and Neptune have been discovered in
modern times, and the question arises if there may not be
others at present unknown. Obviously any unknown planets
must be either very small, or very near the sun, or beyond
the orbit of Neptune, for otherwise they already would have
been seen.

The perihelion of the orbit of Mercury moves somewhat
\index{Planets!intra-Mercurian}%
faster than it would if this planet were acted on only by
known forces. One explanation offered for this peculiarity
of its motion is that it may be disturbed by the attraction of
a planet whose orbit lies between it and the sun. A planet
in this position would be observed only with difficulty because
its elongation from the sun would always be small.
Half a century ago there was considerable belief in the
existence of an intra-Mercurian planet, and several times
it was supposed one had been observed. But photographs
have been taken of the region around the sun at all recent
total eclipses, and in no case has any object within the orbit
of Mercury been found. It is reasonably certain that there
is no object in this region more than $20$~miles in diameter.

The question of the existence of trans-Neptunian planets
\index{Planets!trans-Neptunian}%
is even more interesting and much more difficult to answer.
There is no reason to suppose that Neptune is the most remote
%% -----File: 292.png---Folio 262-------
planet, and the gravitative control of the sun extends
enormously beyond it. There are two lines of evidence,
besides direct observations, that bear on the question. If
there is a planet of considerable mass beyond the orbit of
Neptune, it will in time make its presence felt by its perturbations
of Neptune. Since Neptune was discovered it has
made less than half a revolution, and the fact that its observed
motion so far agrees with theory is not conclusive evidence
against the existence of a planet beyond. In fact, there are
some very slight residual errors in the theory of the motion
of Uranus, and from them Todd inferred that there is
\index[xnames]{Todd}%
probably a planet revolving at the distance of about $50$~astronomical
units in a period of about $350$~years. The conclusion
is uncertain, though it may be correct. A much
more elaborate investigation has been made by Lowell, who
\index[xnames]{Lowell}%
finds that the slight discrepancies in the motion of Uranus
are notably reduced by the assumption of the existence of
a planet at the distance of $44$~astronomical units (period $290$~years)
whose mass is greater than that of the earth and less
than that of Neptune.

It will be seen (\Artref{196}) that planets sometimes capture
comets and reduce their orbits so that their aphelia are
near the orbits of their captors. Jupiter has a large family
of comets, and Saturn and Uranus have smaller ones. As
far back as 1880, Forbes, of Edinburgh, inferred from a
\index[xnames]{Forbes}%
study of the orbits of those comets whose aphelia are beyond
the orbit of Neptune that there are two remote members of
the solar family revolving at the distances of $100$ and~$300$
astronomical units in the immense periods of $1000$ and~$5000$
years. W.~H. Pickering has made an extensive statistical
\index[xnames]{Pickering, W. H.}%
study of the orbits of comets and infers the probable existence
of three or four trans-Neptunian planets. The data
are so uncertain that the correctness of the conclusion is
much in doubt.

\Article{162}{The Zodiacal Light and the Gegenschein.}---The
\index{Gegenschein}%
\index{Light!zodiacal}%
\index{Zodiacal light}%
zodiacal light is a soft, hazy wedge of light stretching up
%% -----File: 293.png---Folio 263-------
from the horizon along the ecliptic just as twilight is ending
or as dawn is beginning. Its base is $20°$~or~$30°$ wide and it
generally can be followed $90°$~from the sun, and sometimes
it can be seen as a narrow, very faint band $3°$~or~$4°$ wide entirely
around the sky. It is very difficult to decide precisely
what its limits are, for it shades very gradually from an
illumination perhaps a little brighter than the Milky Way
into the dark sky.

The best time to observe the zodiacal light is when the
ecliptic is nearly perpendicular to the horizon, for then it is
less interfered with by the dense lower air. In the spring
the sun is very near the vernal equinox. At this time of
the year the ecliptic comes up after sunset from the western
horizon north of the equator, and makes a large angle with
the horizon. Consequently, the spring months are most
favorable for observing the zodiacal light in the evening, and
for analogous reasons the autumn months are most favorable
for observing it in the morning. It cannot be seen in strong
moonlight.

The \textit{gegenschein}, or counterglow, is a very faint patch of
light in the sky on the ecliptic exactly opposite to the sun.
It is oval in shape, from~$10°$ to~$20°$ in length along the ecliptic,
and about half as wide. It was first discovered by Brorsen
\index[xnames]{Brorsen}%
in 1854, and later it was found independently by Backhouse
\index[xnames]{Backhouse}%
and Barnard. It is so excessively faint that it has been
\index[xnames]{Barnard}%
observed by only a few people.

The cause of the gegenschein is not certainly known.
It has been suggested that it is a sort of swelling in the
zodiacal band which appears to be a continuation of the
zodiacal light. This explanation calls for an explanation of
the zodiacal light, which, of course, might well be independently
asked for. The zodiacal light is almost certainly due
to the reflection of light from a great number of small particles
circulating around the sun in the plane of the earth's
orbit, and extending a little beyond the orbit of the earth.
An observer at~$O$, \Figref{100}, would see a considerable number
%% -----File: 294.png---Folio 264-------
of these illuminated particles above his horizon~$H$; %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.25in}
\Input[2.25in]{294}{png}
\Caption[Explanation of
the zodiacal light.]{Fig}{100}
\end{wrapfigure}
and
with the conditions as represented in the diagram, the zodiacal
band would extend faintly
beyond the zenith and across the
sky.

It is not clear from this theory
of the zodiacal light why there
should be a condensation exactly
opposite the sun. But at a point
$930,000$ miles from the earth,
which is beyond the apex of its
shadow, there is a region where,
in consequence of the combined
forces of the earth and sun,
wandering particles tend to circulate in a sort of dynamic
whirlpool. It has been suggested that the circulating particles
which produce the zodiacal light are caught in this
whirl and are virtually condensed enough to produce the
observed phenomenon of the gegenschein.


\Section{XII}{QUESTIONS}

1. Which of the methods of measuring the distance from the earth
to the sun depend upon our knowledge of the size of the earth, and
which are independent of it?

2. Make a single drawing showing the orbits of all the planets
to the same scale. On this scale, what are the diameters of the earth
and of the moon's orbit?

3. If the sun is represented by a globe $1$~foot in diameter, what
would be the diameters and distances of the planets on the same
scale?

4. How long would it take to travel a distance equal to that from
the sun to the earth at the rate of $60$~miles per~hour? How much
would it cost at $2$~cents per~mile?

5. The magnitude of the sun as seen from the earth is~$-26.7$.
What is its magnitude as seen from Neptune? As seen from Neptune,
how many times brighter is the sun than Sirius?

6. If Jupiter were twice as far from the sun, how much fainter
would it be when it is in opposition?

%% -----File: 295.png---Folio 265-------

7. How great are the variations in the distances of the planets
from the sun which are due to the eccentricities of their orbits?

8. Suppose the earth and Neptune were in a line between the
sun and the nearest star; how much brighter would the star appear
from Neptune than from the earth?

9. In what respects are all the planets similar? In what respects
are the four inner planets similar and different from the four outer
planets? In what respects are the four outer planets similar and
different from the four inner planets?

10. Find the velocities with which the planets move, assuming
their orbits are circles.

11. Find the next dates at which Mercury and Venus will have
their greatest eastern elongations, and at which Mars, Jupiter, and
Saturn will be in opposition.

12. If possible, observe the zodiacal light and describe its location
and characteristics.

\normalsize

%% -----File: 296.png---Folio 266-------


\Chapter{IX}{The Planets}

\Section{I}{Mercury and Venus}
\index{Mercury}%

\Article{163}{The Phases of Mercury and Venus.}---The inferior
planets Mercury and Venus are alike in several respects and
may conveniently be treated together. They both have
phases somewhat analogous to those of the moon. When
they are in inferior conjunction, that is, %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3.25in}
\Input[3.25in]{296}{png}
\Caption[Phases of an inferior planet.]{Fig}{101}
\index{Mercury!phases of}%
\index{Phases!of Mercury and Venus}%
\index{Venus!phases of}%
\end{wrapfigure}
at~$A$, \Figref{101}, their
dark side is toward
the earth and their
phase is new. Since
the orbits of these
planets are inclined
somewhat to the plane
of the ecliptic, they
do not in general pass
across the sun's disk.
If they do not make
a transit, they present an extremely thin crescent when they
have the same longitude as the sun. As they move out
from~$A$ toward~$B$ their crescents increase, and their disks,
as seen from the earth, are half illuminated when they have
their greatest elongation at~$B$. During their motion from
inferior conjunction at~$A$ to their greatest elongation at~$B$,
and on to their superior conjunction at~$C$, their distances
from the earth constantly increase, and this increase of
distance to a considerable extent offsets the advantage
arising from the fact that a larger part of their illuminated
areas are visible. In order that an inferior planet may be
seen, not only must its illuminated side be at least partly
%% -----File: 297.png---Folio 267-------
toward the earth, but it must not be too nearly in a line with
the sun. For example, a planet at~$C$, \Figref{101}, has its illuminated
side toward the earth, but it is invisible because
it is almost exactly in the same direction as the sun.

The variations in the apparent dimensions of Venus are
greater than those of Mercury because, when Venus is nearest
the earth, it is much nearer than the closest approach of
Mercury, and when it is farthest from the earth, it is much
farther than the most remote point in Mercury's orbit.
At the time of inferior conjunction the distance of Venus is
$25,700,000$ miles, while that of Mercury is $56,900,000$ miles;
and at superior conjunction their respective distances are
$160,100,000$ and $128,900,000$ miles. These numbers are
modified somewhat by the eccentricities of the orbits of
these three bodies, and especially by the large eccentricity
of the orbit of Mercury.

Mercury and Venus transit across the sun's disk only
\index{Mercury!transits of}%
\index{Transits of Mercury and Venus}%
\index{Venus!transits of}%
when they pass through inferior conjunction with the sun
near one of the nodes of their orbits. The sun is near the
nodes of Mercury's orbit in May and November, and consequently
this planet transits the sun only if it is in inferior
conjunction at one of these times. Since there is no simple
relation between the period of Mercury and that of the
earth, the transits of Mercury do not occur very frequently.
A transit of Mercury is followed by another at the same node
of its orbit after an interval of $7$, $13$, or $46$ years, according
to circumstances, for these periods are respectively very
nearly $22$, $41$, and $145$ synodical revolutions of the planet.
Moreover, there may be transits also when Mercury is near
the other node of its orbit. The next transits of Mercury will
occur on May~7, 1924, and on November~8, 1927. Mercury
is so small that its transits can be observed only with a
telescope.

The transits of Venus, which occur in June and December,
are even more infrequent than those of Mercury. The
transits of Venus occur in cycles whose intervals are, starting
%% -----File: 298.png---Folio 268-------
with a June transit, $8$,~$105.5$, $8$,~and $121.5$~years. The last two
transits of Venus occurred on December~8, 1874, and on
December~6, 1882. The next two will occur on June~8,
2004, and on June~5, 2012.

The chief scientific uses of the transits of Mercury and
Venus are that they give a means of determining the positions
of these planets, they make it possible to investigate
their atmospheres, and they were formerly used indirectly
for determining the scale of the solar system (\Artref{156}).

\Article{164}{The Albedoes and Atmospheres of Mercury and
Venus.}---The albedo of a body is the ratio of the light which
\index{Atmosphere!of Mercury and Venus}%
\index{Mercury!albedo of}%
\index{Mercury!atmosphere of}%
\index{Venus!atmosphere of}%
it reflects to that which it receives. The amount of light
reflected depends to a considerable extent upon whether or
not the body is surrounded by a cloud-filled atmosphere. A
body which has no atmosphere and a rough and broken
surface, like the moon, has a low albedo, while one covered
with an atmosphere, especially if it is filled with partially
condensed water vapor, has a higher reflecting power. Every
one is familiar with the fact that the thunderheads which
often appear in the summer sky shine as white as snow
when illuminated fully by the sun's rays. It was found by
Abbott that their albedo is about~$0.65$. If an observer could
\index[xnames]{Abbott}%
see the earth from the outside, its brightest parts would
undoubtedly be those portions of its surface which are
covered either by clouds or by snow.

The albedo of Mercury, according to the careful work of
Müller, of Potsdam, is about~$0.07$, while that of Venus is~$0.60$.
\index[xnames]{Muller@{Müller}}%
This is presumptive evidence that the atmosphere
of Mercury is either very thin or entirely absent, and that
that of Venus is abundant.

It follows from the kinetic theory of gases (\Artref{32}) and
the low surface gravity of Mercury (\Artref{158}, \Tableref{V}) that
Mercury probably does not have sufficient gravitative control
to retain a very extensive atmospheric envelope. This
inference is confirmed by the fact that, when Mercury
transits the sun, no bright ring is seen around it such as would
%% -----File: 299.png---Folio 269-------
be observed if it were surrounded by any considerable atmosphere.
Moreover, Müller found that the amount of light
\index[xnames]{Muller@{Müller}}%
received from Mercury at its various phases proves that it
is reflected from a solid, uneven surface. Therefore there is
abundant justification for the conclusion that Mercury has
an extremely tenuous atmosphere, or perhaps none at all.

The evidence regarding the atmosphere of Venus is just
the opposite of that encountered in the case of Mercury.
Its considerable mass and surface gravity, approximating
those of the earth, naturally lead to the conclusion that it
can retain an atmosphere comparable to our own. But the
conclusions do not rest alone upon such general arguments;
for, when Venus transits the sun, its disk is seen to be surrounded
by an illuminated atmospheric ring. Besides this,
when it is not in transit, but near inferior conjunction, the
illuminated ring of its atmosphere is sometimes seen to
extend considerably beyond the horns of the crescent. Also,
the brilliancy of Venus decreases somewhat from the center
toward the margin of its disk where the absorption of light
would naturally be the greatest. Spectroscopic observations,
\index{Spectroscope}%
which are as yet somewhat doubtful, point to the
conclusion that the atmosphere of Venus contains water
vapor. Taking all the evidence together, we are justified in
the conclusion that Venus has an atmospheric envelope
corresponding in extent, and possibly in composition, to that
of the earth.

\Article{165}{The Surface Markings and Rotation of Mercury.}---The
\index{Mercury!markings of}%
\index{Mercury!rotation of}%
\index{Rotation!of Mercury}%
first astronomer to observe systematically and continuously
the surface markings of the sun, moon, and planets
was Schröter (1745--1816). He was an astronomer of rare
\index[xnames]{Schroeter@{Schröter}}%
enthusiasm and great patience, but seems sometimes to
have been led by his lively imagination to erroneous
conclusions.

Schröter concluded from observations of Mercury made
in 1800, that the period of rotation of this planet is $24$~hours
and $4$~minutes. This result was quite generally accepted
%% -----File: 300.png---Folio 270-------
until after Schiaparelli took up his systematic observations
\index[xnames]{Schiaparelli}%
of the planets, at Milan, about 1880. Schiaparelli found
that Mercury could be much better seen in the daytime,
when it was near the meridian, than in the evening, because
the illumination of the sky was found to be a much less
serious obstacle than the absorption and irregularities of
refraction which were encountered when Mercury was near
the horizon. His experience in this matter has been confirmed
by later astronomers.

Schiaparelli came to the conclusion, from elusive and
vague markings on the planet, that its axis is essentially
perpendicular to the plane of its orbit, and that its periods
of rotation and revolution are the same. These results are
agreed to by Lowell, who has carefully observed the planet
\index[xnames]{Lowell}%
with an excellent $24$-inch telescope at Flagstaff, Ariz.
Although the observations are very difficult, we are perhaps
entitled to conclude that the same face of Mercury is always
toward the sun.

\Article{166}{The Seasons of Mercury.}---If the period of rotation
\index{Mercury!seasons of}%
\index{Seasons!of Mercury}%
of Mercury is the same as that of its revolution, its seasons
are due entirely to its varying distance from the sun
and the varying rates at which it moves in its orbit in harmony
with the law of areas. The eccentricity of the orbit
of Mercury is so great that at perihelion its distance from the
sun is only two thirds of that at aphelion. Since the amount
of light and heat the planet receives varies inversely as
the square of its distance from the sun, it follows that the
illumination of Mercury at aphelion is only four ninths of
that at perihelion. It is obvious that this factor alone would
make an important seasonal change.

Whatever the period of rotation of Mercury may be, its
rate of rotation must be essentially uniform. Since it
moves in its orbit so as to fulfill the law of areas, its motion
of revolution is sometimes faster and sometimes slower than
the average. The result of this is that not exactly the same
side of Mercury is always toward the sun, \emph{even if its periods
%% -----File: 301.png---Folio 271-------
of revolution and rotation are the same}. The mathematical
discussion shows that, at its greatest, it is $23°.7$~ahead of its
mean position in its orbit, and consequently, at such a time,
the sun shines around the surface of Mercury $23°.7$~beyond
the point its rays would reach if its orbit were strictly a
circle. Similarly, the planet at times gets $23°.7$~behind its
mean position. That is, Mercury has a libration (\Artref{129})
\index{Libration of Mercury}%
\index{Mercury!librations of}%
of~$23°.7$. If Mercury's period of rotation equals its period
of revolution, there are, therefore, $132°.6$ of longitude on
the planet on which the sun always shines, an equal amount
on which it never shines, and two zones $47°.4$~wide in which
there is alternating day and night with a period equal to
the period of the planet's revolution around the sun.

If the periods of rotation and revolution of Mercury are
the same, the side toward the sun is perpetually subject to
its burning rays, which are approximately ten times as
intense as they are at the distance of the earth, and, moreover,
they are never cut off by clouds or reduced by an
appreciable atmosphere. The only possible conclusion is
that the temperature of this portion of the planet's surface
is very high. On the side on which the sun never shines the
temperature must be extremely low, for there is no atmosphere
to carry heat to it from the warm side or to hold in
that which may be conducted to the surface from the interior
of the planet. The intermediate zones are subject to alternations
of heat and cold with a period equal to the period
of revolution of the planet, and every temperature between
the two extremes is found in some zone.

\Article{167}{The Surface Markings and Rotation of Venus.}---The
\index{Rotation!of Venus}%
\index{Venus!markings of}%
\index{Venus!rotation of}%
history of the observations of Venus and the conclusions
regarding its rotation are almost the same as in the
case of Mercury. As early as 1740 J.~J. Cassini inferred from
\index[xnames]{Cassini, J.}%
the observations of his predecessors that Venus rotates on its
axis in $23$~hours and $20$~minutes. About 1790 Schröter concluded
\index[xnames]{Schroeter@{Schröter}}%
that its rotation period is about $23$~hours and $21$~minutes,
and that the inclination of the plane of its equator
%% -----File: 302.png---Folio 272-------
to that of its orbit is~$53°$. These results were generally
accepted until 1880, when Schiaparelli announced that Venus,
\index[xnames]{Schiaparelli}%
like Mercury, always has the same face toward the sun.

The observations of Schiaparelli were verified by himself
in 1895, and they have been more or less definitely confirmed
by Perrotin, Tacchini, Mascari, Cerulli, Lowell, and others.
\index[xnames]{Cerulli}%
\index[xnames]{Lowell}%
\index[xnames]{Mascari}%
\index[xnames]{Perrotin}%
\index[xnames]{Tacchini}%
However, it must be remarked that the atmosphere interferes
with seeing the surface of Venus and that the observations
are very doubtful. Moreover, recent direct observations
by a number of experienced astronomers point to a period of
rotation of about $23$~or $24$~hours.

The spectroscope can also be applied under favorable
conditions to determine the rate at which a body rotates.
In 1900 Bélopolsky concluded from observations of this sort
\index[xnames]{Belopolsky@{Bélopolsky}}%
that the period of rotation of Venus is short. More accurate
observations by Slipher, at the Lowell Observatory, show no
\index{Lowell Observatory}%
\index[xnames]{Slipher, V. M.}%
evidence of a short period of rotation. The preponderance
of evidence seems to be in favor of the long period of rotation,
but the conclusion is at present very uncertain.

\Article{168}{The Seasons of Venus.}---The character of the seasons
\index{Seasons!of Venus}%
\index{Venus!seasons of}%
of Venus depends very much upon whether the planet's
period of rotation is approximately $24$~hours or is equal
to its period of revolution. If the planet rotates in the
shorter period and if its equator is somewhat inclined to
the plane of its orbit, the seasons must be quite similar to
those of the earth, though the temperature is probably somewhat
higher because the planet is nearer to the sun. On the
other hand, if the same face of Venus is always toward the
sun, the conditions must be more like those on Mercury,
though the range of temperatures cannot be so extreme
because its atmosphere reduces the temperature on the side
toward the sun and raises it on the opposite side by carrying
heat from the warmer side to the cooler.

Suppose the periods of rotation and revolution of Venus are
equal. Since the orbit of Venus is very nearly circular, it is
subject to only a small libration and only a very narrow zone
%% -----File: 303.png---Folio 273-------
around it has alternately day and night. The position of
the sun in its sky is nearly fixed and the climate at every
place on its surface is remarkably uniform. There must be a
system of atmospheric currents of a regularity not known on
the earth, and it has been suggested that all the water on
the planet was long ago carried to the dark side in clouds and
precipitated there as snow. This conclusion is not necessarily
true, for it seems likely that the air would ascend on
the heated side and lose its moisture by precipitation before
the high currents which would go to the dark side had proceeded
far on their way.

Considered as a whole, Venus is more like the earth than
any other planet; and, so far as can be determined, it is in
a condition in which life can flourish. In fact, if any other
planet than the earth is inhabited, that one is probably
Venus. It must be added, however, that there is no direct
evidence whatever to support the supposition that there
is life upon its surface.


\Section{II}{Mars}

\Article{169}{The Satellites of Mars.}---In August, 1877, Asaph
\index{Mars!satellites of}%
\index{Satellites!of Mars}%
Hall, at Washington, discovered two very small satellites
\index[xnames]{Hall}%
revolving eastward around Mars, sensibly in the plane of its
equator. They are so minute and so near the bright planet
that they can be seen only with a large telescope, and usually
it is advantageous, when observing them, to obscure Mars
by a small screen placed in the focal plane. These satellites
are called Phobos and Deimos. The only way of determining
\index{Deimos}%
\index{Phobos}%
their dimensions is from the amount of light they reflect
to the earth. Though Phobos is considerably brighter
than Deimos, its diameter probably does not exceed $10$~miles.

Not only are the satellites of Mars very small, but in other
respects they present only a rough analogy to the moon
revolving around the earth. The distance of Phobos from
the center of Mars is only $5850$ miles, while that of Deimos
is $14,650$ miles. That is, Phobos is only $3680$ miles from the
%% -----File: 304.png---Folio 274-------
surface of the planet. The curvature of the planet's surface
is such that Phobos could not be seen by an observer from
latitudes greater than $68°\,15'$ north or south of the planet's
equator. The relative dimensions of Mars and the orbits
of its satellites are shown in \Figref{102}.

As was seen in \Artref{154}, the period of a satellite depends
upon the mass of the planet around which it revolves and
upon its distance %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3in}
\Input[3in]{304}{png}
\Caption[Mars and the orbits of its
satellites.]{Fig}{102}
\end{wrapfigure}
from
the planet's center. Notwithstanding
the small
mass of Mars, its satellites
are so close that
their periods of revolution
are very short, the
period of Phobos being
$7$~hrs.\ $39$~m.\ and that
of Deimos being $30$~hrs.\
$18$~m. Since Mars rotates
on its axis in $24$~hrs.\
and $37$~m., Phobos makes
more than $3$~revolutions
while Mars is making
one rotation. It therefore
rises in the west, passes eastward across the sky, and
sets in the east. Here is an example in which the direction
of apparent motion and actual motion are the same. The
period of Phobos from meridian to meridian is $11$~hrs.\ and
$7$~m. On the other hand, Deimos rises in the east and sets
in the west with a period from meridian to meridian of
$131$~hrs.\ and $14$~m.

\Article{170}{The Rotation of Mars.}---In 1666 Hooke, an English
\index{Mars!rotation of}%
\index{Rotation!of Mars}%
\index[xnames]{Hooke}%
observer, and Cassini, at Paris, saw dark streaks on the
\index[xnames]{Cassini, G. D.}%
ruddy disk of Mars, and these features of the planet's surface
are so definite and permanent that even to-day astronomers
can recognize the objects which these men observed
and drew. Some of them are shown in \Figref{103}, which is a
%% -----File: 305.png---Folio 275-------
series of 9~photographs, taken one after the other at short
intervals, by Barnard, at the Yerkes Observatory. By
comparing observations at one time with those made at a
later date the period of rotation of the planet can be found.
In fact, considerable rotation is observable in the short
interval covered by the photographs in \Figref{103}. Hooke
\index[xnames]{Hooke}%
\begin{figure}[hbt]%[Illustration:]
\Input{305}{jpg}
\Caption[Mars. \textit{Photographed by Barnard with the $40$-inch telescope of
the Yerkes Observatory, Sept.~28, 1909.}]{Fig}{103}
\index{Yerkes Observatory}%
\end{figure}%
and Cassini soon discovered that Mars turns on its axis in
\index[xnames]{Cassini, G. D.}%
a period of a little more than $24$~hrs. By comparing their
observations with those of the present day it is found that
its period of rotation is $24$~hrs.\ $37$~m.\ $22.7$~secs. The
high order of accuracy of this result is a consequence of the
fact that the importance of the errors of the observations
diminishes as the time over which they extend increases.

The inclination of the plane of the equator of Mars to
the plane of its orbit is between $23°$ and~$24°$. The inclination
cannot be determined as accurately as the period of
%% -----File: 306.png---Folio 276-------
rotation because the only advantage of a long series of
observations consists in their number. But, in spite of its
uncertainty, the obliquity of the ecliptic of Mars to its
equator is certainly approximately equal to that of the
earth, and, consequently, the seasonal changes are qualitatively
much like those of the earth. One important
difference is that the period of Mars is about $23$~months,
and, therefore, while its day is only a little longer than that
of the earth, its year is nearly twice as long. It is not meant
to imply by these statements that the climate of Mars is
similar to that of the earth. Its distance from the sun is
so much greater that the amount of light and heat it receives
per unit area is only about $0.43$ of that which the earth
receives.

\Article{171}{The Albedo and Atmosphere of Mars.}---According
\index{Atmosphere!of Mars}%
\index{Mars!atmosphere of}%
to the observations of Müller, the albedo of Mars is~$0.15$,
\index[xnames]{Muller@{Müller}}%
which indicates probably a thin atmosphere on the planet.

The surface gravity of Mars is only $0.36$ that of the earth,
and, consequently, it would be expected on the basis of the
kinetic theory of gases that it might retain some atmosphere,
though not a very extensive one. Direct observations of
the planet confirm this conclusion. In the first place, its
surface can nearly always be seen without appreciable interference
from atmospheric phenomena. If the earth were
seen from a distant planet, such as Venus, not only would
the clouds now and then entirely shut off its surface from
view, but the reflection and absorption of light in regions
where there were no clouds would probably make it impossible
to see distinctly anything on its surface.

The fact that Mars has a rare atmosphere is also proved
by the suddenness with which it cuts off the light from
stars when it passes between them and the earth. Those
planets which have extensive atmospheres, such as Jupiter,
extinguish the light from the stars more gradually. If the
atmosphere of Mars, relatively to its mass, were of the same
density as that of the earth, it would be rarer at the surface
%% -----File: 307.png---Folio 277-------
of the planet than our atmosphere is at the top of the loftiest
mountains.

A number of lines of evidence have been given for the
conclusion that the atmosphere of Mars is not extensive.
The question occasionally arises whether it has any atmosphere
at all. The answer to this must be in the affirmative,
because faint clouds, possibly of dust or mist, have often
been observed on its surface. They are very common along
the borders of the bright caps which cover its poles. Another
related phenomenon which is very remarkable and not
easy to explain is that, sometimes for considerable periods,
the planet's whole disk %[Illustration: Break]
\begin{wrapfigure}[13]{\WLoc}{3.5in}
\Input[3.5in]{307}{jpg}
\Caption[Barnard's drawings of Mars.]{Fig}{104}
\index{Lick Observatory}% [** TN: Presumed reference]
\end{wrapfigure}
is dim and obscure as though covered
by a thin mist.
While the cause
of this obscuration
is not
known, it is supposed
that it is
a phenomenon
of the atmosphere
of the
planet. Besides
this, Mars undergoes seasonal changes, not only in the polar
caps, which will be considered in the next article, but also
even in conspicuous markings of other types. \Figureref{104}
gives three drawings of the same side of Mars by Barnard,
on September~23, October~22, and October~28, 1894, showing
notable temporary changes in its appearance.

\Article{172}{The Polar Caps and the Temperature of Mars.}---The
\index{Mars!polar caps of}%
\index{Mars!seasons of}%
\index{Mars!temperature of}%
\index{Polar caps of Mars}%
\index{Seasons!of Mars}%
\index{Temperature!of Mars}%
surface of Mars on the whole is dull brick-red in color,
but its polar regions during their winter seasons are covered
with snow-white mantles. One of these so-called polar
caps sometimes develops in the course of two or three days
over an area reaching down from the pole $25°$~to~$35°$; it
remains undiminished in brilliancy during the long winter
of the planet; and, as the spring advances, it gradually
%% -----File: 308.png---Folio 278-------
diminishes in size, contracting first around the edges; it
then breaks up more or less, and it sometimes entirely disappears
in the late summer.

After the southern polar cap has shrunk to the dimensions
given by Barnard's observation of August~13, 1894, \Figref{105},
an elongated white
patch is found to be left
behind the retreating white
sheet. The same thing was
observed in the same place
at the corresponding Martian
season in 1892, and also
at later oppositions. This
means that the phenomenon
is not an accident, but
that it depends upon the
nature of the surface of
Mars. Barnard has suggested
that there may be
an elevated region in the
place on which the spot is
observed where the snow
or frost remains until after
it has entirely disappeared
in the valleys. At any rate,
this phenomenon strongly
points to the conclusion
that there are considerable
irregularities in the surface
of Mars, though on the
whole it is probably much
smoother than the earth.
This is an important point
which must be borne in
mind in interpreting other %[Illustration: Break, move down]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{308}{jpg}
\Caption[Disappearance of polar cap
of Mars (Barnard).]{Fig}{105}
\index{Lick Observatory}% [** TN: Presumed reference]
\index{Mars!polar caps of}%
\index{Polar caps of Mars}%
\index[xnames]{Barnard}%
\end{wrapfigure}
things observed upon the surface
of the planet.
%% -----File: 309.png---Folio 279-------

The polar cap around the south pole of Mars has been
more thoroughly studied than the one at the north pole
because the south pole is turned toward the earth when Mars
is in opposition near the perihelion point of its orbit. The
eccentricity of the orbit of this planet is so great that its
distance from the orbit of the earth when it is at its perihelion
(which is near the aphelion of the earth's orbit) is more than
$23,000,000$ miles less than when it is at its aphelion. However,
in the course of immense time the mutual perturbations
of the planets will so change the orbit of Mars that its northern
polar region will be more favorably situated for observations
from the earth than its southern.

If the polar caps of Mars are due to snow, there must be
water vapor in its atmosphere. The spectroscope is an
\index{Spectroscope}%
instrument which under suitable conditions enables the
astronomer to determine the constitution of the atmosphere
of a celestial body from which he receives light. Mars is
not well adapted to the purpose because, in the first place,
the light received from it is only reflected sunlight which
may have traversed more or less of its shallow and tenuous
atmosphere; and, in the second place, the atmosphere of
the earth itself contains usually a large amount of water
vapor. It is not easy to make sure that the absorption
spectral lines (\Artref{225}) may not be due altogether to the
water vapor in the earth's atmosphere.

The early spectroscopic investigations of Huggins and
\index{Mars!water on}%
\index[xnames]{Huggins}%
Vogel pointed toward the existence of water on Mars; the
\index[xnames]{Vogel}%
later ones by Keeler and Campbell, with much more powerful
\index[xnames]{Campbell}%
\index[xnames]{Keeler}%
instruments and under better atmospheric conditions, gave
the opposite result; but the \DPtypo{spectograms}{spectrograms}
obtained by Slipher
\index[xnames]{Slipher, V. M.}%
at the Lowell Observatory, under exceptionally favorable
\index{Lowell Observatory}%
instrumental and climatic conditions, again indicate water
on Mars. In view of the difficulties of the problem, a negative
result could scarcely be regarded as being conclusive
evidence of the entire absence of water on Mars, while
evidence of a small amount of water vapor in its atmosphere
%% -----File: 310.png---Folio 280-------
is not unreasonable and is quite in harmony with the phenomena
of its polar caps.

The distance of Mars from the sun is so great that it
receives only about $0.43$~as much light and heat per unit
area as is received by the earth. The question then arises
how its polar caps can nearly, or entirely, disappear, while
the poles of the earth are perpetually buried in ice and
snow. The responses to this question have been various,
many of them ignoring the fundamental physical principles
on which a correct answer must be based.

In the first place, consider the problem of determining
what the average temperature of Mars would be if its atmosphere
and surface structure were exactly like those of the
earth. That is, let us find what the temperature of the earth
would be if its distance from the sun were equal to that of
Mars. The amount of heat a planet radiates into space on
the average equals that which it receives, for otherwise its
temperature would continually increase or diminish. Therefore,
the amount of heat Mars radiates per unit area is on
the average $0.43$~of that radiated per unit area by the earth.
Now the amount of heat a body radiates depends on the
character of its surface and on its temperature. In this
calculation the surfaces of Mars and the earth are assumed
to be alike. According to Stefan's law, which has been verified
\index{Stefan's law}%
\index[xnames]{Stefan}%
both theoretically and experimentally, the radiation of
a black body varies as the fourth power of its absolute
temperature. Or, the absolute temperatures of two black
bodies are as the fourth roots of their rates of radiation.

Now apply this proportion to the case of Mars and the
earth. On the Fahrenheit scale the mean annual surface
temperature of the whole earth is about~$60°$, or $28°$~above
freezing. The absolute zero on the Fahrenheit scale is
$491°$~below freezing. Therefore, the mean temperature of
the earth on the Fahrenheit scale counted from the absolute
zero is about $491° + 28° = 519°$. Let $x$ represent the
absolute temperature of Mars; then, under the assumption
%% -----File: 311.png---Folio 281-------
that its surface is like that of the earth, the proportion becomes
\[
x: 519 = \sqrt[4]{0.43}: \sqrt[4]{1},
\]
from which it is found that $x = 420°$. That is, under these
hypotheses, the mean surface temperature of Mars comes
out $491° - 420° = 71°$ below freezing, or $71° - 32° = 39°$
below zero Fahrenheit.

The results just obtained can lay no claim to any particular
degree of accuracy because of the uncertain hypotheses
on which they rest. But it does not seem that the hypothesis
that the surfaces of Mars and the earth radiate similarly
can be enough in error to change the results by very many
degrees. If the atmosphere of Mars were of the same constitution
as that of the earth, but simply more tenuous, its
actual temperature would be lower than that found by the
computation. On the other hand, if the atmosphere of
Mars contained an abundance of gases which strongly
absorb and retain heat, such as water vapor and carbon
dioxide, its mean temperature might be considerably above~$-39°$.
But, taking all these possibilities into consideration,
it seems reasonably certain that the mean temperature of
Mars is considerably below zero Fahrenheit. The question
then arises how its polar caps can almost, or entirely,
disappear each summer.

The fundamental principles on which precipitation and
evaporation depend can be understood by considering these
phenomena in ordinary meteorology. If there is a large
quantity of water vapor in the air and the temperature
falls, there is precipitation before the freezing point is reached,
and the result is rain. On the other hand, if the amount of
water vapor in the air is small, there is no precipitation
until after the temperature has descended below the freezing
point of water. In this case when precipitation occurs it
is in the form of snow or hoar frost.

The reverse process is similar. Suppose the temperature
of snow is slowly being increased. If there is only a very
%% -----File: 312.png---Folio 282-------
little water vapor in the air surrounding it, the snow evaporates
into water vapor without first melting. On the other
hand, if the atmosphere contains an abundance of water
vapor, the snow does not evaporate until after its temperature
has risen above the freezing point. But at the freezing
point the snow turns into water.

The gist of the whole matter is this: If the water vapor
in the atmosphere is abundant, precipitation and evaporation
take place above the freezing point; and if it is scarce,
precipitation and evaporation take place below the freezing
point. The temperature at which these processes begin
depends only on the density of water vapor present and not
at all upon the constitution and density of the remainder of
the atmosphere. For example, snow evaporates (or sublimes)
at $-35°$~Fahrenheit when the density of the water
vapor surrounding it is such that its pressure is less than
$0.00018$ of ordinary atmospheric pressure; or, if this is the
water-vapor pressure and the temperature falls below~$-35°$,
snow is precipitated. Similarly, water evaporates at~$80°$
Fahrenheit if the pressure of the water above it is less than
$0.034$~of atmospheric pressure; or, with this pressure of
water vapor, precipitation occurs if the temperature falls
below~$80°$. This explains why the earth's atmosphere on
the whole is much dryer in the winter than it is in the summer.

The application to Mars is simple. Suppose its polar
caps are actually due to snow or hoar frost, as they appear
to be. The fact that they nearly or entirely disappear in
the long summers means only that the atmosphere is dry
enough for evaporation to take place at the temperature
which prevails on the planet. If the temperature of Mars
were known, the amount of water vapor in its atmosphere
could be computed from the phenomena of the polar caps; and
conversely, if the amount of water vapor in the atmosphere
of Mars were known, its temperature could be computed.

Some direct considerations on the possible temperature
of Mars have been given, and reference has been made to
%% -----File: 313.png---Folio 283-------
the possibility of determining the water content of its
atmosphere by means of the spectroscope. There is an
additional line of evidence which bears on the question in
hand. The surface of the planet is largely of a brick-red
color, and is interpreted as being in a desert condition. While
there are dark regions which have been supposed possibly to
be marshes, there are certainly no large bodies of water on
its surface comparable to the oceans and seas upon the
earth. These things confirm the conclusion that water is
not abundant on Mars and that its mean temperature may
be below zero; but, in the equatorial regions in the long summers,
the temperature may rise for a considerable time even
above the freezing point.

\Article{173}{The Canals of Mars.}---In 1877, Schiaparelli, an
\index{Canals of Mars}%
\index{Mars!canals of}%
\index[xnames]{Schiaparelli}%
Italian observer of Milan, made the first of a series of important
discoveries respecting the surface markings of Mars.
Although he had only a modest telescope of $8.75$~inches' aperture,
he found that the brick-red regions, which had been
supposed to be continental areas, were crossed and recrossed
by many straight, dark, greenish streaks which always
ended in the darker regions known as ``seas.'' These streaks
were of great length, extending in uniform width from a few
hundred miles up to three or four thousand miles. While
they appeared to be very narrow, they must have been at
least $20$~miles across. Schiaparelli called them ``canali''
(channels), which was translated as ``canals,'' a designation
unfortunately too suggestive, for they have no analogy to
\begin{figure}[hbt]%[Illustration: Moved up]
\centering\Input[4.5in]{314}{jpg}
\Caption[Lowell's map of Mars.]{Fig}{106}
\end{figure}%
anything on the earth. When a number of them intersect,
there is generally a dark spot at the point of intersection
which is called a ``lake.'' Sometimes a number of them
intersect at a single point; and, according to Lowell, the
\index[xnames]{Lowell}%
junctions of canals are always surrounded by lakes, while
lakes are found at no other places.

In the winter of 1881--82 Mars was again in opposition,
though not so near the earth as in 1877. Schiaparelli not
only verified his earlier observations, but he also found the
%% -----File: 314.png---Folio 284-------
remarkable fact that a number of the canals had doubled;
that is, that, in a number of cases, two canals ran parallel
to each other at a distance of from~$200$ to $400$~miles, as shown
on Lowell's map in \Figref{106}, which is a photograph of a
\index[xnames]{Lowell}%
globe on which he had drawn all the markings he had
observed. The doubling was found to depend upon the seasons
and to develop with great rapidity when the sun was
at the Martian equinox.

The history of the observations of the markings of Mars
since the time of Schiaparelli is filled with the most remarkable
\index[xnames]{Schiaparelli}%
contradictions. %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{315}{jpg}
\Caption[Drawings of Mars in 1894 by
Barnard at the Lick Observatory.]{Fig}{107}
\end{wrapfigure}
The observations of the keen-eyed
Italian have been confirmed by a number of other astronomers,
among whom may be mentioned Perrotin and Thollon,
\index[xnames]{Perrotin}%
\index[xnames]{Thollon}%
of Nice, Williams, of England, W.~H. Pickering, of Harvard,
\index[xnames]{Pickering, W. H.}%
\index[xnames]{Williams}%
and especially Lowell, who has a large $24$-inch telescope
%% -----File: 315.png---Folio 285-------
favorably located at Flagstaff, Arizona. On the other hand,
\index{Lowell Observatory}%
many of the foremost observers working with the very largest
telescopes, such as Antoniadi, with the $32.75$-inch Meudon
\index[xnames]{Antoniadi}%
refractor, the Lick observers, with the great $36$-inch
telescope, Barnard, with the $40$-inch Yerkes telescope, and
\index{Yerkes Observatory}%
\index[xnames]{Barnard}%
Hale, with the $60$-inch reflector of the Solar Observatory at
\index{Solar!Observatory}%
\index[xnames]{Hale}%
Mt.~Wilson, California, have been entirely unable to see the
\index{Mount Wilson Solar Observatory}%
canals. This does not mean that they have not seen markings
on Mars, for they
\index{Mars!explanation of canals of}%
have observed many of
them; but they do not
find the narrow, straight
lines observed by Schiaparelli,
\index[xnames]{Schiaparelli}%
Lowell, and
\index[xnames]{Lowell}%
others. In \Figref{107} four
views of Mars are shown
as seen by Barnard with
the great telescope of the
Lick Observatory, and
\index{Lick Observatory}%
\Figref{108} is a photograph
made with the $60$-inch
reflecting telescope of the
Mt.~Wilson Solar Observatory.
In the midst
of these conflicting results
it is difficult to draw any certain conclusion; but it must
be remembered in considering such a subject that reliable
positive evidence ought to outweigh a large amount of negative
evidence.

\Article{174}{Explanations of the Canals of Mars.}---The explanations
of the canals of Mars have been extremely varied.
Many astronomers believe they are illusions of some sort.
They think the eye in some way integrates the numerous
faint markings which certainly exist on Mars into straight
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4.5in]{316}{jpg}
\Caption[Photograph of Mars (the $60$-inch reflector of the Mt.~Wilson
Solar Observatory).]{Fig}{108}
\end{figure}%
lines and geometrical figures. The experiments of Maunder
\index[xnames]{Maunder}%
and Evans and the more recent ones of Newcomb of having
\index[xnames]{Evans}%
\index[xnames]{Newcomb}%
%% -----File: 316.png---Folio 286-------
a number of persons make drawings of what they could see
on a disk covered with irregular marks and held slightly
beyond the limits of distinct vision, strikingly confirm this
conclusion. Antoniadi states in the most unequivocal terms
\index[xnames]{Antoniadi}%
that the observations of Mars at the opposition of 1909 give
to the theory of the objective existence of canals on Mars
an unanswerable confutation. Other astronomers hold that
such a network of markings on a planet whose surface is
certainly somewhat uneven is inherently improbable, and
should not be accepted without the most conclusive evidence.

At the other extreme stands Lowell, who maintains that
\index[xnames]{Lowell}%
%% -----File: 317.png---Folio 287-------
not only are the canals real but that they prove the existence
on the planet of highly intelligent beings. He argues for
the reality of the canals on the ground that they always
appear at well-defined positions on the planet and that they
change in a systematic way with the seasons. He argues that
they are artificial because they always run along the arcs of
great circles, because several of them sometimes cross at a
point with the utmost precision, and because in many cases
two of them run perfectly parallel for more than a thousand
miles. Obviously this remarkable regularity could not be
the result of such processes as the erosion of rivers or the
cracking of the surface.

W.~H. Pickering first suggested that the canals may be
\index[xnames]{Pickering, W. H.}%
due to vegetation, and Lowell's theory is an elaboration of
\index[xnames]{Lowell}%
this idea. Lowell believes the streaks, known as canals,
are strips of vegetation $20$~or more miles wide, which grow
on a region irrigated by lateral ditches from a large central
canal. This explains their seasonal character. Moreover,
he finds the streaks first developing near the dark (marshy?)
regions and extending gradually out from them even across
the equator of the planet to regions having the opposite season.
The explanation given for this phenomenon is that
when the snow of the polar caps melts, the resulting water
first collects in the marshes and is led thence out into the
waterways which extend through the centers of the canals.
The observations of Lowell show that, according to his
explanation, water must flow along the canals at the rate
of $2.1$~miles per hour. He infers from the elaborate system
of irrigated regions that Mars is inhabited by creatures
possessing a high order of intelligence.

Although Lowell's theory seems highly improbable and may
be altogether wrong, life may nevertheless exist upon Mars.
But if there is life on this planet, the creatures which inhabit
it must be very different physically from those on the earth,
because it would be necessary for them to be adapted to an
entirely different environment. On Mars the surface gravity
%% -----File: 318.png---Folio 288-------
is less than on the earth, the light and heat received from
the sun are less and the temperature is probably far lower,
the atmosphere is much less abundant, and it may be quite
different in constitution, and the seasonal changes are
nearly twice as long. The plants and animals which inhabit
the earth are more or less perfectly adapted to the
conditions existing on its surface, and the conditions have
not been made to fit them, as was once generally believed.
Similarly, life on other planets must be adapted to the
environment in which it is placed or it would shortly perish.

Further, if Mars or any other world is inhabited, there is
no reason to suppose that its highest intelligence has reached
the precise stage attained by the human race. The most
intelligent creatures on another planet may be in the condition
corresponding to that in which our ancestors were when
they lived in caves and ate uncooked food; or, millions of
years ago they may have passed through the stage of strife
and deadly competition in which the human race is to-day.

It is a curious fact that those who know but little about
astronomy are nearly always very much interested in the
question whether other worlds are inhabited, while as a rule
astronomers who devote their whole lives to the subject
scarcely give the question of the habitability of other planets
a thought. Astronomers are doubtless influenced by the
knowledge that such speculations can scarcely lead to certainty,
and they are deeply impressed by the fundamental
laws which they find operating in the universe. Nevertheless,
there seems to be no good reason why we should not now
and then consider the question of the existence of life, not
only on the other planets of the solar system, but also on the
millions of planets that possibly circulate around other suns.
Such speculations help to enlarge our mental horizon and
to give us a better perspective in contemplating the origin
and destiny of the human race, but we should never forget
that they are speculations.
%% -----File: 319.png---Folio 289-------


\Section{III}{Jupiter}

\Article{175}{Jupiter's Satellite System.}---The first objects discovered
\index{Jupiter!satellite system of}%
\index{Satellites!of Jupiter}%
by Galileo when he pointed his little telescope to
\index[xnames]{Galileo}%
the sky in 1610 were the four brightest moons of Jupiter.
They are barely beyond the limits of visibility without optical
aid and, indeed, could be seen with the unaided eye if they
were not obscured by the dazzling rays of Jupiter. No other
satellite of Jupiter was discovered until 1892, when Barnard,
\index[xnames]{Barnard}%
then at the Lick Observatory, caught a glimpse of a fifth
\index{Lick Observatory}%
one very close to the planet. It is so small and so buried
in the rays of the neighboring brilliant planet that it can
be seen only by experienced observers with the aid of the
most powerful telescopes in the world.

{\stretchyspace%
Early in 1905 Perrine found by photography that Jupiter
\index[xnames]{Perrine}%
has still two more satellites which are more remote from the
planet than those previously known. Their distances from
Jupiter are both about $7,000,000$ miles and their periods of
revolution are about $0.75$~of a} year. The eccentricities of
their orbits are considerable and their paths actually loop
through one another. The mutual inclination of their
orbits is~$28°$ and they do not pass nearer than $2,000,000$
miles of each other.

The seven satellites so far enumerated revolve around
Jupiter from west to east, but two more have been discovered
whose motion is in the opposite direction. The
eighth was found by Melotte, at Greenwich, England, in
\index[xnames]{Melotte}%
January,~1908. It revolves around Jupiter at a mean
distance of approximately $14,000,000$ miles in a period
of about $740$~days. Its orbit is inclined to Jupiter's equator
by about~$28°$. The ninth was discovered by S.~B.
Nicholson, in July,~1914, at the Lick Observatory. Its
\index[xnames]{Nicholson}%
mean distance from Jupiter is about $15,400,000$ miles and its
period is nearly $3$~years. These remote satellites are very
small and faint, the ninth being of the nineteenth magnitude,
and the eighth about one magnitude brighter.
%% -----File: 320.png---Folio 290-------

%[** TN: Moved up one paragraph]
\begin{table}[hbt]
\begin{center}
\Caption{Table}{VII}
%\caption[Jupiter's satellite system]{}%
\setlength{\tabcolsep}{3pt}%
\makebox[0pt][c]{%
\begin{tabular}{|r@{~}l|r@{~}l|c|r|}
\hline
\TCEntry{2}{|c|}{VIII (Unnamed).}{\TFontsize\THead Satellite}
  & \TCEntry{2}{c|}{\TFontsize Center of Jupiter}{%
    \medskip\TFontsize\THead Distance from \\ Center of
    Jupiter\medskip%
  }
  & \settowidth{\TmpLen}{about $277$~days}%
    \TEntry{\TmpLen}{\TFontsize\THead Period \\ of Revolution}
  & \settowidth{\TmpLen}{about $100$~mi.}%
    \TEntry{\TmpLen}{\TFontsize\THead Diameter}
\\
\hline
\Strut%
   V & (Unnamed) &    $112,500$ &     mi. & $\Z0$d.\ $11$h.\ $57$m. & about $100$~mi.           \\
   I & (Io)      &    $261,000$ &     mi. & $\Z1$d.\ $18$h.\ $28$m. & $2452$~mi.                \\
  II & (Europa)  &    $415,000$ &     mi. & $\Z3$d.\ $13$h.\ $14$m. & $2045$~mi.                \\
 III & (Ganymede)&    $664,000$ &     mi. & $\Z7$d.\ $\Z3$h.\ $43$m.& $3558$~mi.                \\
  IV & (Callisto)&  $1,167,000$ &     mi. & $16$d.\ $16$h.\ $32$m.  & $3345$~mi.                \\
  VI & (Unnamed) &  $7,300,000$ &     mi. & about $266$~days & \multicolumn{1}{c |}{small}      \\
 VII & (Unnamed) &  $7,500,000$ &     mi. & about $277$~days & \multicolumn{1}{c |}{small}      \\
VIII & (Unnamed) & $14,000,000$ & $±$ mi. & about $740$~days & \multicolumn{1}{c |}{very small} \\
  IX & (Unnamed) & $15,400,000$ & $±$ mi. & nearly $3$~years & \multicolumn{1}{c |}{very small} \\
\hline
\end{tabular}}
\end{center}
\end{table}

The first four satellites discovered are numbered I,~II,
III,~IV in the order of their distance from Jupiter. The
fifth, although it is very close to Jupiter, was given the
number~V. The orbits of these five satellites, shown in
\Figref{109}, are nearly circular and lie in the plane of Jupiter's
equator. The four larger satellites are of considerable
dimensions and their diameters have been determined by
Barnard, the results being given in the \hyperref[Table:VII]{following table}.\DPnote{** TN: Change ":" to "."}
\index[xnames]{Barnard}%

\Article{176}{Markings on Jupiter's Satellites.}---The great distance
of Jupiter makes it difficult to detect any but large
and distinctly colored %[Illustration: Break, moved up]
\begin{wrapfigure}{\WLoc}{3.125in}
\Input[3.125in]{321a}{png}
\Caption[Orbits of first four satellites of
Jupiter.]{Fig}{109}
\end{wrapfigure}
markings on its satellites. In 1890
Barnard found satellite~I to be elongated parallel to the
equator of Jupiter when transiting its darker portions and
elongated, or double, in the opposite direction when passing
over its brighter parts. He interpreted this as meaning that
the poles of the satellite are dark and that the equatorial
belt is light colored. The accompanying drawing, \Figref{110},
showing the satellite transiting a light region above and a
dark one below, exhibits the observed appearance at the
left and the probable actual condition at the right. When
held at some distance from the eye, the two appear the
same.
%% -----File: 321.png---Folio 291-------

Some observers have thought that satellites III and~IV are
somewhat elliptical in shape, but Barnard has observed
\index[xnames]{Barnard}%
them repeatedly with
the great Lick and
Yerkes telescopes and
\index{Lick Observatory}%
\index{Yerkes Observatory}%
has been quite unable
to detect in them any
departures from strict
sphericity. Various
markings have been
at times observed on
the satellites, and
Douglas inferred from
\index[xnames]{Douglas}%
his observations of
satellite~III that its
period of rotation is
about $7$~hours. At
present these are matters
of speculation.

\Article{177}{Discovery of the Finite Velocity of Light.}---A very
\index{Light!velocity of}%
\index{Velocity!of light}%
important discovery was made in connection with observations
of Jupiter's satellites. The periods of revolution of the
four largest satellites naturally
were determined when Jupiter
was in opposition, and therefore
\begin{wrapfigure}[19]{\WLoc}{2.25in}%[Illustration:]
\Input[2.25in]{321b}{jpg}
\Caption[Barnard's drawings of Jupiter's satellite~I.]{Fig}{110}
\end{wrapfigure}
nearest the earth. Since the
satellites are in the plane of
Jupiter's equator, which is only
slightly inclined to the ecliptic,
they are eclipsed when they
pass behind Jupiter. From their
periods of revolution the times
at which they will be eclipsed
can be predicted.

Suppose the periods of revolution
of the satellites and the
%% -----File: 322.png---Folio 292-------
times at which they are eclipsed are determined when the
earth is in the vicinity of~$E_1$, \Figref{111}. Six months later,
when the earth has arrived at~$E_2$, its distance from Jupiter
is greater by approximately the diameter of the earth's
orbit, and then the eclipses of the satellites are found to
be behind their predicted times by the time required for
light to travel across the earth's orbit. From such observations,
in 1675, the Danish astronomer Römer inferred that
\index[xnames]{Roemer@{Römer}}%
it takes light $600$~seconds to travel a distance equal to that
from the sun to the earth. Later observations have shown
that the correct time is $498.58$~seconds. When the distance
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4.5in]{322}{png}
\Caption[Discovery of velocity of light from eclipses of Jupiter's satellites.]{Fig}{111}
\end{figure}%
from the earth to the sun has been determined by independent
means, the velocity of light can be found from this
interval, which is called the light equation.

At the present time the velocity of light can be determined
much more accurately by physical experiments on
the surface of the earth than it can from observations of
Jupiter's satellites. The work of Fizeau, Michelson, and
\index[xnames]{Fizeau}%
\index[xnames]{Michelson}%
Newcomb shows that it is very approximately $186,324$ miles
\index[xnames]{Newcomb}%
per second. From this velocity and the light equation of
$498.58$~seconds, the distance to the sun can be computed.

\Article{178}{The Rotation of Jupiter.}---The surface of Jupiter
\index{Jupiter!rotation of}%
\index{Rotation!of Jupiter}%
is covered with a great number of semi-permanent markings
from which its rotation can be determined. The period
%% -----File: 323.png---Folio 293-------
of rotation for spots near the equator has been found to be
about $9$~hrs.\ and $50$~m., and for those in higher latitudes about
$9$~hrs.\ and $57$~m., with an average of $9$~hrs.\ and $54$~m.; that
is, between the equatorial zone and high latitudes there is a
difference in the period of about $\frac{1}{85}$ of the whole period.
In $85$~rotations the equator gains a rotation on the higher
latitudes. Moreover, as Barnard has found, the rates of
\index[xnames]{Barnard}%
rotation in corresponding northern and southern latitudes
are quite different in several zones.

The circumference of Jupiter is nearly $300,000$ miles, and
it follows from this and its rate of rotation that the motion
at its equator is about $30,000$ miles per hour. Consequently,
if two spots whose periods of rotation differ by $7$~minutes
were both near the equator, they would pass each other
with the relative speed of $30,000 ÷ 85 = 353$ miles per
hour. Though spots whose periods differ by $7$~minutes are
probably in no case in approximately the same latitude, yet
they must have large relative motions. Compare these
results with the speed of from $70$ to $100$~miles per hour with
which tornadoes sweep along the surface of the earth.

The fact that the equatorial belt of Jupiter rotates in a
shorter period than its higher latitudes is a most remarkable
phenomenon. If it were an isolated case, one would naturally
suppose that the peculiarity was due to irregularities
of motion inherited from the time of its origin. Such currents
in a body in a fluid condition would be destroyed by
friction only very slowly; but the same phenomenon is
also found in the case of Saturn and the sun. It can hardly
be supposed that the three are mere coincidences. If they
are not, the implication is that these peculiarities of
rotation have been produced by similar causes. It has
been suggested, as will be explained in Arts.\ \hyperref[art:253]{253},~\hyperref[art:254]{254}, that
the cause may be the impacts of circulating meteors or other
material.

\Article{179}{Surface Markings of Jupiter.}---The characteristic
\index{Jupiter!belts of}%
\index{Jupiter!markings on}%
markings of Jupiter are a series of conspicuous dark and
%% -----File: 324.png---Folio 294-------
bright belts which stretch around the planet parallel to its
equator as shown in Figs.\ \Fref{112},~\Fref{113}, and~\Fref{114}. The central
equatorial belt is usually very light and about $10,000$ miles
wide; on each side is a belt of reddish-brown color generally
of about the same width. Several other alternately light
and dark belts can be made out in higher latitudes, though
not as distinctly as the equatorial belts, partly, at least,
because they are observed obliquely. The belts vary considerably
in width from year to year as the drawings,
\Figref{114}, by Hough
\index[xnames]{Hough, G. W.}%
show. On the whole,
the %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3.25in}
\Input[3.25in]{324}{jpg}
\Caption[Jupiter, Sept.~7, 1913 (Barnard).]{Fig}{112}
\end{wrapfigure}
southern dark belt
of Jupiter is wider
and more conspicuous
than the northern
one.

A good telescope
under favorable atmospheric
conditions
reveals in the belts
many details which
continually change as
though what we see
is cloudlike in structure.
In fact, it follows
from the low mean density of the planet and the almost
certain central condensation that its exterior parts, to a
depth of many thousands of miles, must have a very low
density; and it is improbable that anything which is visible
from the earth approaches the solid state.

Dark spots often appear on Jupiter, especially in the northern
\index{Jupiter!great red spot on}%
hemisphere, which gradually turn red and finally vanish.
The most remarkable and permanent spot so far known
appeared in 1878 just beneath the southern red belt. When
first discovered it was a pinkish oval about $7000$ miles across
in the direction perpendicular to the equator, and about
%% -----File: 325.png---Folio 295-------
$30,000$~miles long parallel to the equator. In a year it had
changed to a bright red color and was by far the most conspicuous
object on the planet. It has since then been known
as ``the great red spot,'' but it has undergone many changes,
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4.75in]{325}{jpg}
\Caption[Photographs of Jupiter (E.~C. Slipher, Lowell Observatory).]{Fig}{113}
\index{Lowell Observatory}%
\index[xnames]{Slipher, E. C.}%
\end{figure}%
both in color and brightness. At the present time it has
become rather inconspicuous, and the material of which it is
composed seems to be sinking back beneath the vapors which
surround the planet.

A very remarkable thing in connection with the red spot
was that its period of rotation increased $7$~seconds the first
eight years following its discovery, but it has remained essentially
constant since that time. Possibly the increase in
period of rotation of the red spot, which was somewhat
longer than that of the surrounding material which continually
flowed by it, was due to its being elevated so that its
distance from the axis of rotation of the planet was increased.
Under these conditions the rate of rotation would be reduced
%% -----File: 326.png---Folio 296-------
in harmony with the principle of the conservation of moment
of momentum (\Artref{45}). At any rate, changes in rotation
are always accompanied by considerable changes in color
and visibility of the parts affected.

\Article{180}{The Physical Condition and Seasonal Changes of
Jupiter.}---In considering the physical condition of Jupiter
\index{Atmosphere!of Jupiter}%
\index{Jupiter!atmosphere of}%
\index{Jupiter!physical condition of}%
\index{Jupiter!seasons of}%
\index{Seasons!of Jupiter}%
it should be remembered that it has the low average density
of~$1.25$ on the water
\begin{wrapfigure}[27]{\WLoc}{3in}%[Illustration:]
\Input[3in]{326}{jpg}
\Caption[Drawings of Jupiter showing
variations in widths of dark belts
(Hough).]{Fig}{114}
\index[xnames]{Hough, G. W.}%
\end{wrapfigure}
standard, that its surface
markings are not permanent,
and that there are
violent relative motions
of its visible parts. All
these things indicate that
Jupiter is largely gaseous
near its surface.

The surface gravity of
Jupiter is $2.52$~times that
of the earth, and this
produces great pressures
in its atmosphere at
moderate depths. These
pressures are sustained
by the expansive tendencies
of the interior gases
which may be composed
of light elements, or
which may have high
temperatures. It has
sometimes been supposed that the surface of Jupiter is very
hot and that it is self-luminous, but such cannot be the case,
for the shadows cast on the planet by the satellites are
perfectly black, and when a satellite passes into the shadow
of Jupiter it becomes absolutely invisible.

In conclusion, we shall probably not be far from the
truth if we infer that Jupiter is still in an early stage of its
%% -----File: 327.png---Folio 297-------
evolution, rather than far advanced like the terrestrial
planets, that it contains enormous volumes of gases which
are in rapid circulation both along and perpendicular to its
surface, and that possibly the energy of its internal fires
gives rise to violent motions.

The eccentricity of Jupiter's orbit is very small and the
plane of its equator is inclined only~$3°\,5'$ to the plane of its
orbit. The factors which produce seasonal changes are,
therefore, unimportant in the case of this planet. Its distance
from the sun is so great that it receives per unit area
only $\frac{1}{27}$~as much light and heat as is received by the earth;
and, consequently, its surface must be cold unless it is
warmed by internal heat.


\Section{IV}{Saturn}

{\stretchyspace%
\Article{181}{Saturn's Satellite System.}---Saturn, like Jupiter,
\index{Satellites!of Saturn}%
\index{Saturn!satellite system of}%
has $9$~satellites.} The largest one was discovered by Huyghens
\index[xnames]{Huyghens}%
in 1655, then four more were found by J.~D. Cassini between
\index[xnames]{Cassini, G. D.}%
1671 and 1684, two by William Herschel in 1789, one by
\index[xnames]{Herschel, William}%
G.~P. Bond and Lassell in 1848; and the ninth by W.~H.
\index[xnames]{Bond}%
\index[xnames]{Lassell}%
Pickering in 1899. Pickering suspected the existence of
\index[xnames]{Pickering, W. H.}%
a tenth in 1905, but the supposed discovery has not been
confirmed.

Saturn is so remote that the dimensions of its satellites are
only roughly known from their apparent brightness. All
their masses are unknown except that of Titan, which, from
its perturbation of its neighboring satellite Hyperion, was
found by Hill to be $\frac{1}{4714}$~that of Saturn. %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3in}
\Input[3in]{329}{png}
\Caption[Orbit of Saturn's satellites.]{Fig}{115}
\end{wrapfigure}
The $7$~satellites
\index[xnames]{Hill}%
which are nearest to Saturn revolve sensibly in the plane
of its equator, while the orbit of the eighth, Japetus, is
inclined about~$10°$, and that of the ninth about~$20°$.

When the eighth satellite, Japetus, is on the western side
\index{Variability!of Japetus}%
of Saturn it always appears considerably brighter than when
it is on the eastern side. This difference in brightness is
undoubtedly due to the fact that this satellite, like the moon,
always has the same side toward the planet around which
%% -----File: 328.png---Folio 298-------
it revolves, and that its two sides reflect light very unequally.
Similar, but less marked, phenomena have been observed by
Lowell and E.~C. Slipher in connection with the first two
\index[xnames]{Lowell}%
\index[xnames]{Slipher, E. C.}%
satellites, and the explanation is the same as in the case of
Japetus.

\Tableref{VIII} gives the list of Saturn's satellites, together
with their mean distances from its center, their periods, and
their approximate diameters. It will be observed that an
enormous gap separates the first eight from the ninth.

\Figureref{115} gives to scale the orbits of Saturn's satellites,
with the exception of the ninth, which is too remote to be
shown. The eight satellites revolve around Saturn from
west to east, the direction in which it rotates, but the ninth,
like the eighth and ninth satellites of Jupiter, revolves in
the retrograde direction. This satellite was the first object
discovered in the solar system having retrograde motion,
and it aroused great interest. These retrograde revolutions
have a fundamental bearing on the question of the origin
of the satellite systems.

\begin{table}[htb]
\begin{center}
\Caption{Table}{VIII}
%\caption[List of Saturn's satellites]{} %
\makebox[0pt][c]{%
\setlength{\tabcolsep}{4pt}%
\begin{tabular}{|r@{ }l|r<{\ }|*{3}{r@{}l@{ }}|c|}
\hline
\TCEntry{2}{|c|}{VIII (Enceladus)}{\TFontsize\THead Satellite} &
\TCEntry{1}{c|}{$9,999,999$~mi.}{\medskip\TFontsize\THead Distance from center of Saturn\medskip} &
\TCEntry{6}{c|}{$99999999999$}{\TFontsize\THead Period of Revolution} &
\TCEntry{1}{c|}{about $9999$ mi}{\TFontsize\THead Diameter} \\
\hline
\Strut%
I    & (Mimas)     & $\phantom{1,}117,000$ mi. &   $0$ & d. & $22$ & h. & $37$ & m. & about\; $\phantom{0}600$ mi. \\
II   & (Enceladus) & $\phantom{1,}157,000$ mi. &   $1$ &    &  $8$ &    & $53$ &    & about\; $\phantom{0}800$ mi. \\
III  & (Tethys)    & $\phantom{1,}186,000$ mi. &   $1$ &    & $21$ &    & $18$ &    & about\;           $1200$ mi. \\
IV   & (Dione)     & $\phantom{1,}238,000$ mi. &   $2$ &    & $17$ &    & $41$ &    & about\;           $1100$ mi. \\
V    & (Rhea)      & $\phantom{1,}332,000$ mi. &   $4$ &    & $12$ &    & $25$ &    & about\;           $1500$ mi. \\
VI   & (Titan)     & $\phantom{1,}771,000$ mi. &  $15$ &    & $22$ &    & $41$ &    & about\;           $3000$ mi. \\
VII  & (Hyperion)  & $\phantom{1,}934,000$ mi. &  $21$ &    &  $6$ &    & $39$ &    & about\; $\phantom{0}500$ mi. \\
VIII & (Japetus)   &           $2,225,000$ mi. &  $79$ &    &  $7$ &    & $54$ &    & about\;           $2000$ mi. \\
IX   & (Ph\oe{}be) &           $7,996,000$ mi. & $546$ &    & $12$ &    & $0$  &    & about\; $\phantom{0}200$ mi.%
\rule[-2ex]{0pt}{0pt} \\
\hline
\end{tabular}}
\end{center}
\end{table}

The question may be asked why the remote satellites of
both Jupiter and Saturn revolve in the retrograde direction.
This question cannot be answered with certainty at the
%% -----File: 329.png---Folio 299-------
present time. But it is certain that the farther a satellite
is from a planet, the less securely is it held under the gravitative
control of its primary; and there is a distance beyond
which a satellite cannot permanently revolve because it
would abandon
the planet in
obedience to the
greater attraction
of the sun. A
mathematical discussion
of the
problem shows
that, at a given
distance from a
planet, motion in
the retrograde direction
is much
more stable than
in the forward
direction; and
consequently, out
near the region
where instability begins, it would be expected that only
retrograde satellites would be found. The orbit of the ninth
satellite of Saturn is in the region of stability even for direct
\index{Stability!of satellites}%
motion; but Jupiter's eighth and ninth satellites would
both have unstable orbits if they revolved in the forward
direction at the same distances from Jupiter.

\Article{182}{Saturn's Ring System.}---Saturn is distinguished from
\index{Rings of Saturn}%
\index{Saturn!ring system of}%
all the other planets by three wide, thin rings which extend
around it in the plane of its equator. They were first seen
by Galileo in 1610, but their true character was not known
\index[xnames]{Galileo}%
until the observations of Huyghens in 1655. The dimensions
\index[xnames]{Huyghens}%
of Saturn and its ring system according to the extensive
measurements of Barnard are given in \Tableref{IX}.
\index[xnames]{Barnard}%
%% -----File: 330.png---Folio 300-------

\begin{table}[htb]
\begin{center}
\Caption{Table}{IX}
%\caption[Saturn's ring system]{}
\begin{tabular}{|p{3.75in}@{}l|}%[** TN: Hard-coded width]
\hline
\Strut%
Equatorial radius of Saturn\MyDotFill                   & $38,235$ miles \\
Center of Saturn to inner edge of crape ring\MyDotFill  & $44,100$ miles \\
Center of Saturn to inner edge of bright ring\MyDotFill & $55,000$ miles \\
Center of Saturn to outer edge of bright ring\MyDotFill & $73,000$ miles \\
Center of Saturn to inner edge of outer ring\MyDotFill  & $75,240$ miles \\
Center of Saturn to outer edge of outer ring\MyDotFill  & $86,300$ miles \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}[hbt]%[Illustration:]
\centering\Input[4.75in]{330}{jpg}
\Caption[Saturn with rings tilted at greatest angle (drawing by Barnard).]{Fig}{116}
\index[xnames]{Barnard}%
\index{Yerkes Observatory}%
\end{figure}%
The distance from the surface of Saturn to the inner edge
of the thin, faint ring, known as ``the crape ring,'' is nearly
$6000$ miles. The width of the crape ring is about $11,000$
miles. Outside of the crape ring is the main bright ring,
whose width is about $18,000$ miles. Its brightness increases
from its junction with the crape ring outward nearly to its
outer margin. At its brightest place it is as luminous as the
planet itself. Beyond the main bright ring there is a dark
gap about $2200$ miles across. It is known as ``Cassini's
\index[xnames]{Cassini, G. D.}%
Division'' because it was first observed by Cassini. Outside
of this dark space is the outer bright ring with a width of
%% -----File: 331.png---Folio 301-------
about $11,000$ miles. The distance across the whole ring
system from one side to the other is about $172,600$~miles.

The rings of Saturn are inclined about $27°$ to the plane of
the planet's orbit and about $28°$ to the plane of the ecliptic.
Consequently, they are observed from the earth at a great
variety of angles. When their inclination is high, Saturn
and its ring system present through a good telescope one of
the finest sights in the heavens, as is evident from Figs.\
\begin{figure}[ht]%[Illustration:]
\Input{331}{jpg}
\Caption[Saturn. \textit{Photographed Nov.~19, 1911, with the $60$-inch telescope
of the Mount Wilson Solar Observatory.}]{Fig}{117}
\end{figure}%
\Fref{116}~and~\Fref{117}. When their plane passes through the earth,
they appear to be a very thin line and even entirely disappear
from view for a few hours, as Barnard found when
\index{Yerkes Observatory}%
\index[xnames]{Barnard}%
observing them with the great $40$-inch telescope in 1907.
It follows that the rings must be very thin, their thickness
probably not exceeding $50$~miles. When the rings were nearly
edgewise to the earth, Barnard could see them faintly; but
the places which are entirely vacant when they are highly
inclined to the earth, were found to be brighter than the places
where the rings are really brilliant (\Figref{118}). Barnard
%% -----File: 332.png---Folio 302-------
made the suggestion that this appearance is due to the fact
that light shining from the sun through the open regions is
reflected back from the interior edges of the denser parts of
the rings.

\Article{183}{The Constitution of Saturn's Rings.}---The bright
\index{Rings of Saturn!constitution of}%
rings of Saturn have the same appearance of solidity and
continuity as the planet itself. It was generally believed
until about a century ago that they were solid or fluid. Yet
since 1715, when J.~Cassini first mentioned the possibility,
\index[xnames]{Cassini, J.}%
it has frequently been suggested that the rings may be simply
\begin{figure}[hb]%[Illustration:]
\Input{332}{jpg}
\Caption[Rings of Saturn, December~12, 1907 (drawing by Barnard).]{Fig}{118}
\index{Yerkes Observatory}%
\index[xnames]{Barnard}%
\end{figure}%
swarms of meteors, or exceedingly minute satellites, revolving
around the planet in the plane of its equator. Such
small bodies would exert only negligible gravitational influences
upon one another, and their orbits would be sensibly
independent of one another except for collisions.

The meteoric theory of the constitution of Saturn's rings
was first rendered probable by Laplace, who showed that
\index[xnames]{Laplace}%
a symmetrical, solid ring would be dynamically unstable.
That is, solid rings would be something like spans of enormous
bridges, whose ends do not rest upon the planet but upon
other portions of the rings. They would have to be composed
of inconceivably strong material to withstand the
%% -----File: 333.png---Folio 303-------
strains due to their motion and the gravitational forces to
which they would be subjected. In 1857, Clerk-Maxwell
\index[xnames]{Clerk-Maxwell}%
proved from dynamical considerations that the rings could
be neither solid nor fluid, and that they were, therefore, composed
of small independent particles. Now, if they are
meteoric, those parts which are nearest the planet must
move fastest, just as those planets which are nearest the sun
move fastest; while, if they are solid, the opposite must be
the case. In 1895, Keeler showed by line-of-sight observations
\index[xnames]{Keeler}%
with the spectroscope (\Artref{226}) that the inner parts
\index{Spectroscope}%
not only move fastest, but that all parts move precisely
as they would move if they were made up of totally disconnected
particles, the innermost particles of the crape ring
performing their revolution in about $5$~hours, while the outermost
particles of the outer bright ring require $137$~hours to
complete a revolution. Moreover, Barnard found that they
\index[xnames]{Barnard}%
do not cast perfectly black shadows, for he saw Japetus
faintly illuminated by the rays of the sun which filtered
through the ring. Hence it may be considered as firmly
established that the rings of Saturn are swarms of meteors.

Rings are strange substitutes for satellites, but a probable
\index{Roche's limit}%
explanation of their existence in place of satellites is at
hand. A planet exerts tidal strains upon satellites in its
vicinity, and these tendencies to rupture increase very
rapidly as the distance of the satellite decreases. In 1848,
Roche proved that these tidal forces would break up a fluid
\index[xnames]{Roche}%
satellite of the same density as the planet around which it
revolved if its distance were less than $2.44$\,\ldots\ radii of
the planet. The limit would be less for denser satellites,
and a little less for solid satellites, but not much less if they
were of large dimensions. It is seen from the numbers
in \Tableref{IX}, or from \Figref{116}, that the rings are within this
limit. It is not supposed that they are the pulverized
remains of satellites that ever did actually exist, but rather
that the material of which they are composed is subject
to such forces that the mutual gravitation of the separate
%% -----File: 334.png---Folio 304-------
particles can never draw them together into a single body.
If they should unite into a satellite, it would probably be
small, for they are not massive enough to have produced
by their attraction any disturbance of the motions of the
satellites which can so far be observed.

One more interesting thing remains to be mentioned. If
a meteor were to revolve in the vacant space between the
rings known as Cassini's division, its period would be nearly
commensurable with the periods of four of the satellites,
and would be one half that of Mimas. Kirkwood called
\index[xnames]{Kirkwood}%
attention to this relation, which is entirely analogous to that
found in the case of the planetoids (\Artref{160}). Encke and
\index[xnames]{Encke}%
other astronomers have suspected that there is a narrow
division between the crape ring and the inner edge of the
bright ring, where the period of a revolving meteor would
be one third that of Mimas. More recently Lowell has been
\index[xnames]{Lowell}%
convinced by his observations at Flagstaff of the existence
of several other very narrow divisions at places where the
periods of revolving particles would be simply commensurable
with the periods of Mimas or Enceladus. But in order
to secure perfect commensurability he was led to the conclusion
that Saturn is composed of layers of different densities,
and that the inner ones are more oblate, and, therefore,
rotate faster, than the outer ones.

\Article{184}{On the Permanency of Saturn's Rings.}---The question
\index{Rings of Saturn!permanency of}%
at once arises whether the meteoric constitution of the
rings, in which there is abundant opportunity for collisions,
is a permanent one. The fact that the rings exist and are
separated from the planet by a number of thousands of
miles, while beyond them there are 9~satellites, indicates
that they are not transitory in character. The only circumstance
that distinguishes them dynamically from the
satellites is the possibility of their collisions. If a collision
occurred, at least some heat would be generated at the
expense of their energy of motion. When the revolutionary
energy of a body is decreased, its orbit diminishes in size.
%% -----File: 335.png---Folio 305-------
Therefore, when two of the small bodies of which Saturn's
ring is composed collide, the orbit of at least one of them
must be diminished in size. These collisions with the accompanying
degradation of energy are probably taking place at
a very slow rate. If so, the rings of Saturn are slowly shrinking
down on the planet. It may be that the crape ring is
the result of particles whose orbits have been reduced from
the larger dimensions of the bright ring by collisions with
other particles.

\Article{185}{The Surface Markings and the Rotation of Saturn.}---The
\index{Rotation!of Saturn}%
\index{Saturn!rotation of}%
\index{Saturn!surface markings on}%
surface markings of Saturn are much like those of
Jupiter, though, of course, they are not seen so well because
of the great distance of this planet. There are a bright
equatorial belt and a number of darker and broader belts in
the higher latitudes, though they are less conspicuous than
the belts on Jupiter.

It has been rather difficult for observers to find spots on
Saturn conspicuous and lasting enough to enable them to
determine the period of its rotation. From observations
made in 1794 Herschel concluded that its period of rotation
\index[xnames]{Herschel, William}%
is $10$~hrs.\ and $16$~m.; Hall's observation of a bright equatorial
\index[xnames]{Hall}%
spot in 1876 gave for this spot a period of $10$~hrs.\ and $14$~m.
This was generally adopted as the period of Saturn's rotation,
particularly after it had been verified by a number of other
observers. But, in 1903, Barnard discovered some bright
\index[xnames]{Barnard}%
spots in northern latitudes, and his observations of them,
together with those of several other astronomers, showed
that these spots were passing around Saturn in $10$~hrs.\ and
$38$~m. This difference in period means that there is a relative
drift between the material of Saturn's equatorial belt and
that of its higher latitudes of $800$ or $900$ miles per hour.

In sharp contrast to the planet Jupiter, the plane of the
equator of Saturn is inclined to the plane of its orbit by an
angle of~$27°$. This is a still higher inclination than those
found in the case of the earth and Mars, and would hardly
be expected in so large a planet as Saturn after finding that
%% -----File: 336.png---Folio 306-------
the axis of Jupiter is almost exactly perpendicular to the
plane of its orbit.

\Article{186}{The Physical Condition and Seasonal Changes of
Saturn.}---The density of Saturn is about $0.63$ on the water
\index{Atmosphere!of Saturn}%
\index{Saturn!physical condition of}%
\index{Saturn!seasons of}%
\index{Seasons!of Saturn}%
standard. Consequently, it must be largely in a gaseous
condition. Probably no considerable portion of it is purely
gaseous, for it seems more likely, in view of the fact that it
is opaque, that the gases of which it is composed are filled
with minute liquid particles, just as our own atmosphere
becomes charged with globules of water, forming clouds.

The remarkable relative motions of the different parts of
the surface of Saturn show that it is at least in a fluid state
and that it is a place of the wildest turmoil. Doubtless it is
a world whose evolution has not yet sufficiently advanced to
give it any permanent markings, much less to fit it as a place
in any way suitable for the abode of even the lowest forms of
life.

The high inclination of the plane of Saturn's equator to
that of its orbit gives it marked seasonal changes. Moreover,
its orbit is rather more eccentric than the orbits of
the other large planets. But it is so far from the sun that
it receives only $\frac{1}{90}$ as much light and heat per unit area as
the earth receives; and it follows that its surface is very cold
unless it has an atmosphere of remarkable properties, or unless
a large amount of heat is conveyed to it from a hot interior.

A consequence of the rapid rate of rotation and low density
of Saturn is that it is very oblate. The difference between
its equatorial and polar diameters is nearly $6700$ miles,
or about $10$~per~cent of its whole diameter. Its oblateness
is so great that it is conspicuous even through a telescope of
$6$~inches' aperture.


\Section{V}{Uranus and Neptune}

\Article{187}{The Satellite Systems of Uranus and Neptune.}---Uranus
has four known satellites, two of which were discovered
\index{Neptune!satellite of}%
\index{Satellites!of Neptune}%
\index{Satellites!of Uranus}%
\index{Uranus!satellites of}%
by William Herschel, in 1787, and the other two
\index[xnames]{Herschel, William}%
%% -----File: 337.png---Folio 307-------
by Lassell, in 1851. Their distances are respectively $120,000$,
\index[xnames]{Lassell}%
$167,000$, $273,000$ and $365,000$ miles, and their periods of
revolution are respectively $2.5$, $4.1$, $8.7$, and $13.5$~days.
Their diameters probably range between $500$ and~$1000$
miles. They all move sensibly in the same plane, but this
plane is inclined about $98°$ to the plane of the planet's
orbit; that is, if the plane of the orbits of the satellites is
thought of as having been turned up from that of the planet's
orbit, the rotation has been continued $8°$ beyond perpendicularity,
and the satellites revolve in the retrograde direction.

Neptune has one known satellite which was discovered
by Lassell, in 1846. It revolves at a distance of $221,500$
miles in a period of $5$~days $21$~hours. Its diameter is probably
about $2000$ miles. The plane of its orbit is inclined about
$145°$ to that of the planet's orbit; that is, the inclination
between the two planes is about $35°$ and the satellite revolves
in the retrograde direction.

\Article{188}{Atmospheres and Albedoes of Uranus and Neptune.}---Very
\index{Atmosphere!of Uranus and Neptune}%
\index{Neptune!atmosphere of}%
\index{Uranus!atmosphere of}%
little is known directly respecting the atmospheres
of Uranus and Neptune. Their low mean densities imply
that their exterior parts are largely in the gaseous state.
As confirmatory of this conclusion, the spectroscope shows
\index{Spectroscope}%
that the light which we receive from them must have passed
through an extensive absorbing medium in addition to the
sun's atmosphere and that of the earth, through which the
light from all planets passes. The absorbing effects of the
element hydrogen and water vapor are shown in the spectra
of both planets, but, according to the recent results of Slipher,
\index[xnames]{Slipher, V. M.}%
more strongly in the case of Neptune than in that of Uranus.
A number of the other absorption bands are due to unknown
substances.

The albedo of Uranus is~$0.63$, and that of Neptune,~$0.73$.

\Article{189}{The Periods of Rotation of Uranus and Neptune.}---Surface
\index{Neptune!rotation of}%
\index{Rotation!of Neptune}%
\index{Rotation!of Uranus}%
\index{Uranus!rotation of}%
markings have been seen on Uranus by Buffham,
\index[xnames]{Buffham}%
Young, the Andre brothers, Perrotin, Holden, Keeler, and
\index[xnames]{Holden}%
\index[xnames]{Keeler}%
\index[xnames]{Perrotin}%
\index[xnames]{Young, C. A.}%
other observers, but they have been so indefinite and fleeting
%% -----File: 338.png---Folio 308-------
that it has not been possible to draw any certain conclusions
from them. Nevertheless, so far as they go, they indicate
that the period of rotation of Uranus is $10$ or $12$~hours, and
\index{Uranus!physical condition of}%
that the plane of its equator is inclined something like $10°$ to
$30°$ to the plane of the orbits of the satellites. In 1894,
Barnard detected a slight flattening of the disk, with the
\index[xnames]{Barnard}%
equatorial diameter inclined $28°$ to the plane of the orbits
of the satellites. Finally, in 1912, V.~M. Slipher, at the
\index[xnames]{Slipher, V. M.}%
Lowell Observatory, found by spectroscopic means that
\index{Lowell Observatory}%
Uranus rotates in the direction of revolution of its satellites
in a period of $10$~hrs.\ $50$~m. This result is entitled to
considerable confidence.

No certain markings have been seen on Neptune, and,
\index{Neptune!physical condition of}%
consequently, its rate of rotation has not been found by
direct means. But by indirect processes both the position
of the plane of its equator and its rate of rotation have
been found, at least approximately. The dimensions and
mass of Neptune are known with considerable accuracy.
Now, if the rate of rotation were known, the equatorial
bulging could be computed. Suppose the plane of the orbit
of the satellite were inclined to that of the planet's equator.
Then the equatorial bulge would perturb the motion of the
satellite; in particular, it would cause a revolution of its
nodes, and the rate could be computed.

The problem of determining the rate of rotation of Neptune
is about the converse of that which has just been
described. The nodes of the orbit of its satellite revolve,
and the manner of their motion shows the existence of a
certain equatorial bulge inclined about $20°$ to the plane of
the satellite's orbit. The bulging, or ellipticity, of the
planet is~$\frac{1}{85}$, indicating, according to the work of Tisserand
\index[xnames]{Tisserand}%
and Newcomb, a rather slow rotation as compared to the
\index[xnames]{Newcomb}%
rates of rotation of Jupiter and Saturn.

\Article{190}{Physical Condition of Uranus and Neptune.}---We
can infer the physical conditions of Uranus and Neptune only
from that of other planets which are more favorably situated
%% -----File: 339.png---Folio 309-------
for observation. They are probably in much the same state
as Jupiter and Saturn, though, possibly, somewhat further
advanced in their evolution because of their smaller dimensions.
One thing to be noticed is that they receive relatively
little light and heat from the sun. The amounts per
unit area are about $\frac{1}{368}$ and $\frac{1}{904}$ that received by the earth.
If their capacity for absorbing and radiating heat were the
same as that of the earth, their temperatures (\Artref{172}) would
be respectively about $-340°$ and $-364°$ Fahrenheit. Nevertheless,
it must not be imagined that even Neptune would
receive only feeble illumination from the sun. Although
the sun, as seen from that vast distance, would subtend a
smaller angle than Venus does to us when nearest the earth,
the noonday illumination would be equal to $700$~times our
brightest moonlight.


\Section{XIII}{QUESTIONS}

1. Find by the method of \Artref{172} what the mean temperatures
of the earth would be at the distances of Mercury and Venus.

2. If the earth always presented the same face toward the sun,
and if there were no distribution of heat by the atmosphere, what
would be the mean temperature of its illuminated side? What
would be the result if the earth were at the distance of Venus from
the sun?

3. If the mean temperature of the equatorial zone of the earth
is~$85°$, and if it receives, per unit area, $2.5$~times as much heat as the
polar regions, what is the mean temperature of the polar regions,
neglecting the transfer of heat by the atmosphere?

4. What would be the mean temperature of the equatorial
zone of the earth at the mean distance of Mars?

5. Suppose the mean temperature of the Thibetan plateau at a
height of $15,000$ feet above sea level is~$40°$; what would it be if
the earth were at the distance of Mars from the sun?

6. Suppose the atmosphere which a planet can hold is proportional
to its surface gravity; how does the atmosphere of Mars
compare with that of the earth at an altitude of $15,000$ feet above
sea level?

7. Waiving the temperature difficulties in the hypothesis regarding
the habitability of Mars, what reasonable explanation can
%% -----File: 340.png---Folio 310-------
you give for the fact that the canals are always along the arcs of
great circles?

8. Try the experiment of Maunder and Evans.

9. What would be the total area of $400$~canals having an average
width of $20$~miles and an average length of $300$~miles? Suppose
to irrigate this area for a season a foot of water is required; how
much would this water weigh on the earth? On Mars? Suppose
a fall of four feet per mile is required to get a flow in the canals at
the necessary rate; suppose it is necessary to pump the water out
of the ``marshes'' to a higher level to get the fall; suppose the
pumps work $10$~hours a day for $300$~days; how many horsepower
of work must they deliver?

\normalsize

%% -----File: 341.png---Folio 311-------


\Chapter{X}{Comets and Meteors}

\Section{I}{Comets}

\Article{191}{General Appearance of Comets.}---The planets are
\index{Comets!appearance of}%
characterized by the invariability of their form, the simplicity
of their motions, and their general similarity to one
another. In strong contrast to these relatively stable bodies
are the comets, whose bizarre appearance, complex motions,
and temporary visibility have led astronomers to devote to
them a great amount of attention. Until the last two centuries
they were objects of superstitious terror which were
supposed to portend calamities. At least so far as their
motions are concerned, they are now known to be as lawful
as the other members of the solar system.

The typical comet is composed of a head, or \textit{coma}, a
brighter nucleus within the head which is often starlike in
appearance, and a tail streaming out in the direction opposite
to the sun. The apparent size of the head may be anywhere
from almost starlike smallness to the angular dimensions
of the sun. The nucleus is usually very small and
bright, but the tail often extends many degrees from the
head before it gradually fades out into the darkness of the
sky. The head is the most distinctive part of the comet, for
it is always present and looks much like a circular nebula.
Either the nucleus or tail, or both, may be absent, especially
if the comet is a small one. Comets vary in brightness from
those which are so faint that they are barely visible through
large telescopes to those which are so bright that they may
be observed in full daylight, even when almost in the direction
of the sun. In spite of their being sometimes very
%% -----File: 342.png---Folio 312-------
\begin{figure}[hbtp]%[Illustration:]
\centering\Input{342}{jpg}
\Caption[Brooks' Comet, Oct.~19, 1911. \textit{Photographed by Barnard at the
 Yerkes Observatory.}]{Fig}{119}
\index{Yerkes Observatory}%
\index[xnames]{Barnard}%
\index[xnames]{Brooks}%
\end{figure}%
%% -----File: 343.png---Folio 313-------
bright, they are so nearly transparent that faint stars are
visible through them without the slightest appreciable
diminution of their light.

There are records of about $400$~comets having been seen
\index{Comets!naming of}%
before the invention of the telescope, in 1609, and more
than the same number have been observed since that date.
Astronomers now keep a close watch of the sky, and only
very faint ones can escape their notice. From $3$ to~$10$ are
found yearly. They are lettered for each year $a$,~$b$, $c$,\,\ldots\ in
the order of their discovery, and are numbered I,~II, III,\,\ldots\ in
the order that they pass their perihelia. Besides
this, they are generally named after their discoverers.

\Article{192}{The Orbits of Comets.}---In ancient times it was
\index{Comets!orbits of}%
\index{Orbits!of comets}%
supposed that comets were malevolent visitors prowling
through the earth's atmosphere, bent on mischief. Kepler
\index[xnames]{Kepler}%
supposed they moved in straight lines, but Doerfel showed
\index[xnames]{Doerfel}%
that the comet of 1681 moved in a parabola around the sun
as a focus. In 1686 Newton invented a graphical method
\index[xnames]{Newton}%
of computing comets' orbits from three or more observations
of their apparent positions. Better methods have been
devised by Lambert, Laplace, Gauss, and later astronomers,
\index[xnames]{Gauss}%
\index[xnames]{Lambert}%
\index[xnames]{Laplace}%
and now there is usually no difficulty in determining the
elements of an orbit from three complete observations which
are separated by a few days.

The orbits of about $400$~comets have been computed,
and as nearly as can be determined from the imperfect observations
on which the computations of many of them are
based, the orbits of about $300$~of them are essentially parabolic.
In fact, they are so generally parabolic, or, at least,
extremely elongated, that it has been customary in the preliminary
computations to assume they are parabolas. Of
the remaining cometary orbits, nearly~$100$ have been shown
to be distinctly elliptical in shape.

A conic section is an ellipse if its eccentricity is less than
\index{Conic sections}%
unity, a parabola if its eccentricity equals unity, and an
hyperbola if its eccentricity exceeds unity. Since a body
%% -----File: 344.png---Folio 314-------
moving subject to gravitation may describe any one of these
three classes of orbits, and since the eccentricity of a parabola
is the limiting case between that of an ellipse and that
of an hyperbola, it is infinitely improbable that the orbit of
any comet is exactly parabolic.

It is important to determine whether the eccentricities
of the orbits of comets are slightly less than unity or slightly
greater than unity. In the former case comets are permanent
members of the solar system; in the latter, they are
only temporary visitors. The difficulty in answering the
question is not theoretical, but practical. In the first place,
comets are more or less fuzzy bodies and it is difficult to
locate the exact positions of their centers of gravity. In
the second place, they are observed during only a very small
part of their whole periods while they are in the neighborhood
of the earth's orbit. Generally they are not seen much
beyond the orbit of Mars and very rarely at the distance of
Jupiter. For such a small arc the motion is sensibly the
same in a very elongated ellipse, in a parabola, and in an
hyperbola whose eccentricity is near unity, as is evident
from \Figref{120}.

More than $80$~comets move in orbits whose major axes
are so short that they will certainly return to the sun. The
remainder move in exceedingly elongated orbits, and the
character of their motion is less certain. But it is significant
that the recent computations of Strömgren show that
\index[xnames]{Stromgren@{Strömgren}}%
in all cases in which comets have been sufficiently observed
to give accurate results respecting their orbits, they were
moving in ellipses when they entered the solar system. At
the present time there is no known case of a comet which
was well observed for a long time whose orbit was hyperbolic,
and astronomers are becoming united in the opinion
that they are permanent members of the solar system.

The orbits of all the planets are nearly in the same plane;
on the other hand, the planes of the orbits of the comets lie
in every possible direction and exhibit no tendency to parallelism.
%% -----File: 345.png---Folio 315-------
The perihelia of the orbits of comets are distributed
all around the sun, but show a slight tendency to cluster in
the direction in which the sun is moving among the stars, a
fact which probably has some connection with the sun's
motion.

Some comets have perihelion points only a few hundred
thousand miles from the surface of the sun, and when nearest
\begin{figure}[hbt]%[Illustration:]
\Input{345}{png}
\Caption[Similarity of elongated ellipses, parabolas, and hyperbolas in
the vicinity of the orbit of the earth.]{Fig}{120}
\end{figure}%
the sun they actually pass through its corona (\Artref{238}).
About $25$~comets pass within the orbit of Mercury; nearly
three fourths of those which have been observed come
within the orbit of the earth; very few so far seen are permanently
without the orbit of Mars, and all known comets
%% -----File: 346.png---Folio 316-------
come within the orbit of Jupiter. This does not mean that
there are no comets with great perihelion distances, or even
that those with perihelion distances greater than the distance
from the earth to the sun are not very numerous. Comets
are relatively inconspicuous objects until they come considerably
within the orbit of Mars. Sometimes their brightness
increases a hundred thousandfold while they move from
the orbit of Mars to that of Mercury. Consequently, even
if comets whose perihelia are beyond the orbit of Mars were
very numerous, not many of them would be observed.

\Article{193}{The Dimensions of Comets.}---After the orbits of
\index{Comets!dimensions of}%
\index{Dimensions!of comets}%
comets have been computed so that their distances from the
earth are known, their actual dimensions can be determined
from their apparent dimensions. It has been found that
the head of a comet may have any diameter from $10,000$
miles up to more than $1,000,000$ miles. The most remarkable
thing about the head of a comet is that it nearly always
contracts as the comet approaches the sun, and expands
again when the comet recedes. The variation in volume is
very great, the ratio of the largest to the smallest sometimes
being as great as $100,000$ to~$1$. John Herschel suggested
\index[xnames]{Herschel, John}%
that the contraction may be only apparent, the outer layers
of the comet becoming transparent as it approaches the sun.
This suggestion contradicts the appearances and seems to be
extremely improbable.

The nucleus of a comet may be so small as to be scarcely
visible, say $100$~miles in diameter, or it may be as large as
the earth. For example, William Herschel observed the
\index[xnames]{Herschel, William}%
great comet of 1811 when its head was more than $500,000$
\index{Comet!of 1811}%
miles in diameter, while its nucleus measured only $428$~miles
across. The nuclei vary in size during the motion of comets,
but the change is quite irregular and no law of variation has
been discovered.

The tails of comets are inconceivably large. Their diameters
are counted by thousands and tens of thousands of
miles where they leave the heads of comets, and by tens of
%% -----File: 347.png---Folio 317-------
thousands or hundreds of thousands of miles in their more
remote parts. They vary in length from a few million
miles, or even less, up to more than a hundred million of
miles. In volume, the tails of comets are thousands of
times greater than the sun and all the planets together.
The strangest thing about them is that they point almost
directly away from the sun whichever way the comet may
be going. That is, when the comet is approaching the sun,
\begin{wrapfigure}{\WLoc}{1.75in}%[Illustration:]
\Input[1.75in]{347}{png}
\Caption[The tails of
comets are always directed
away from the
sun.]{Fig}{121}
\end{wrapfigure}
the tails trail behind like the smoke
from a locomotive; when the comet
is receding, they project ahead like
the rays from the head light on a
misty night. When a comet is far
from the sun, its tail is small, or may
be entirely absent; as it approaches
the sun, the tail develops in dimensions
and splendor, and then diminishes
again on its recession from the sun.

\Article{194}{The Masses of Comets.}---Comets
\index{Comets!masses of}%
give visible evidence of remarkable
tenuity, but their volumes
are so great that, if their densities
were one ten-thousandth of that of
air at the surface of the earth, their masses in many cases
would be comparable to the masses of the planets.

The masses of comets are determined from their attractions
for other bodies (Arts.\ \hyperref[art:19]{19},~\hyperref[art:154]{154}). Or, rather, their lack
of appreciable mass is shown by the fact that they do not
produce observable disturbing effects in the motions of
bodies near which they pass. Many comets have had their
orbits entirely changed by planets without producing any
sensible effects in return. Since, according to the third law
of motion, action between two bodies is equal and opposite,
it follows that the masses of comets are very small, probably
not exceeding one millionth that of the earth.

One of the most striking examples of the feeble gravitational
%% -----File: 348.png---Folio 318-------
power of comets was furnished by the one discovered
by Brooks in 1889. It had passed through Jupiter's satellite
\index{Brooks' comet}%
system in 1886 without interfering sensibly with the motions
of these bodies, although its own orbit was so transformed
that its period was reduced from $27$~years to $7$~years.

\Article{195}{Families of Comets.}---Notwithstanding the great
diversities in the orbits of comets, there are a few groups
whose members seem to have some intimate relation to one
another, or to the planets. There are two types of these
groups, and they are known as \textit{comet families}.
\index{Comets!families of}%

Families of the first type are made up of comets which
pursue nearly identical paths. The most celebrated family
of this type is composed of the great comets of 1668, 1843,
\index{Comet!of 1668}%
\index{Comet!of 1843}%
\index{Comet!of 1880 and 1882}%
1880, and 1882. A much smaller one seen in 1887 probably
should be added to this list. Their orbits were not only
nearly identical, but the comets themselves were very similar
in every respect. They came to the sun from the direction
of Sirius---that is, from the direction away from which
the sun is moving with respect to the stars---and escaped
the notice of observers in the northern hemisphere until
they were near perihelion. They passed half way around
the sun in a few hours at a distance of less than $200,000$
miles from its surface, moving at the enormous velocity of
more than $350$~miles per second. Their tails extended out
in dazzling splendor $100,000,000$ miles from their heads.

One might think that the various members of a comet
family are but the successive appearances of the same comet;
but such is not the case, for the observations show that
though their orbits may be ellipses, their periods are at
least $600$ or $800$~years. This means that they recede to
something like five times the distance of Neptune from the
sun. The most plausible theory seems to be that they are
the separate parts of a great comet which at an earlier visit
to the sun was broken up by tidal disturbances.

Families of the second type are made up of comets whose
orbits have their aphelion points and the ascending and
%% -----File: 349.png---Folio 319-------
descending nodes of their orbits near the orbits of the planets.
About $30$~comets have their aphelia near Jupiter's orbit,
and are known as Jupiter's family of comets, \Figref{122}.
Their orbits are, of course, all elliptic, and their periods are
from $3$ to $8$~years. They move around the sun in the same
direction that the planets revolve. Half of them have been
\begin{figure}[hbt]%[Illustration:]
\Input{349}{png}
\Caption[Jupiter's family of comets (Popular Astronomy).]{Fig}{122}
\end{figure}%
seen at two or more perihelion passages. These comets are
all inconspicuous objects and entirely invisible to us except
when they are near the earth.

Saturn has a family of $2$~comets, Uranus a family of~$3$,
and Neptune a family of $6$~members. The terrestrial planets
do not possess comet families. There are, according to the
statistical study of W.~H. Pickering, two or three groups of
\index[xnames]{Pickering, W. H.}%
%% -----File: 350.png---Folio 320-------
comets whose aphelia are several times the distance of Neptune
from the sun, suggesting, possibly, the existence of
planets at these respective distances.

\Article{196}{The Capture of Comets.}---A very great majority
\index{Comets!capture of}%
of comets move in sensibly parabolic orbits whose positions
have no special relations to the positions of the orbits of the
planets. But the orbits of nearly all those comets which
are elliptical and not exceedingly elongated lie near the
plane of the planetary orbits and have their aphelia near
the orbits of the planets. These facts suggest that the
orbits of comets moving in these ellipses have been changed
from parabolas or very elongated ellipses by the disturbing
action of the planet near whose orbit their aphelion points
lie. This question of the transformation of orbits of comets
was first discussed by Laplace, who found that if a comet
\index[xnames]{Laplace}%
which is approaching the sun on a parabolic or elongated
elliptical orbit passes closely in front of a planet, its motion
will be retarded so that it will subsequently move in a
shortened elliptical orbit, at least until it is disturbed
again.

Suppose a comet approaches the sun in a sensibly parabolic
orbit and passes closely in front of a planet so that its
orbit is reduced to an ellipse. It is then said to have been
\textit{captured}. It will in the course of time pass near the planet
again, when its orbit may be still further reduced; or, its
orbit may be elongated and it may possibly be driven from
the solar system on a parabola or an hyperbola.

It is a generally accepted theory that the members of the
comet families of the various planets have been captured
by the method described. Jupiter has a larger family of
comets than any other planet because of its greater mass
and also because, if a comet were captured originally by any
planet beyond the orbit of Jupiter, it would yet be possible
for Jupiter to reduce its orbit still further. On the other
hand, when Jupiter has captured a comet and made it a
member of its own family, it is far within the orbit of the
%% -----File: 351.png---Folio 321-------
remoter planets and is no longer subject to capture by them.
The planets beyond the orbit of Jupiter have a few comets
each, and the clustering of the aphelia of comets at still
more remote distances has suggested the existence of planets
as yet undiscovered (\Artref{161}). The terrestrial planets have
no comet families partly because their masses are small compared
to that of the sun, and partly because comets cross
their orbits at very great speed.

The masses of the planets are not great enough to reduce
a parabolic comet to membership in their own families at
one disturbance. The matter is illustrated by Brooks' comet,
\index{Brooks' comet}%
\index[xnames]{Brooks}%
1889-V, whose period, according to the computations of
Chandler, was reduced by Jupiter, in 1886, from $27$~years
\index[xnames]{Chandler}%
to $7$~years. Lexell's comet, of 1770, furnishes an example
\index{Lexell's comet}%
\index[xnames]{Lexell}%
of a disturbance of the opposite character. In 1770 it was
moving in an elliptical orbit with a period of $5.5$~years; but
in 1779 it approached near to Jupiter, its orbit was enlarged,
and it has never been seen again.

When a planet captures a comet, the former reduces the
dimensions of the orbit of the latter, but the latter still revolves
around the sun. The question arises whether a planet
might not capture a comet in a more fundamental sense;
that is, reduce its orbit so that it would become a satellite of
the planet. It has been repeatedly suggested that the
planets may have captured their satellites in this manner.
The answer to this suggestion is that a planet cannot capture
a comet and make it into a satellite simply by its own gravitation
and that of the sun. The only possibility is that the
comet should encounter resistance in a very special manner,
and even then the problem presents serious difficulties. No
small resistance would be sufficient because the motion of a
comet around the sun in a parabolic orbit is much greater
than it would be in a satellite orbit; and, in order that resistance
should reduce the velocity by the required amount, it
would be necessary for the comet to encounter so much
material that its mass would grow several fold.
%% -----File: 352.png---Folio 322-------

\Article{197}{On the Origin of Comets.}---The similarities of the
\index{Comets!origin of}%
\index{Origin!of comets}%
motions of the various planets point to the conclusion that
they had a common origin, and the agreement of the direction
of the rotation of the sun with their direction of revolution
indicates that they have been associated with the sun
throughout their whole evolution. This line of reasoning
does not lead to the inference that the comets belong to the
planetary family. They may have had quite a different
origin; at any rate, most of them recede from the sun to
regions several times as remote as the planet Neptune.

It was formerly supposed that comets are merely small,
wandering masses which pass from star to star, visiting our
sun but once. The intervals of time required for such excursions
are enormously greater than has generally been supposed.
For example, the great comet of 1882 came almost
exactly from the direction of Sirius and returned again in
\index{Sirius}%
the same direction. Suppose the comet moved under the
attraction of Sirius until it had passed over half of the distance
from Sirius to the sun, and that it then moved sensibly
under the attraction of the sun. Although Sirius is
one of the nearest known stars in all the sky, it is found that
it would take $70,000,000$ years to describe this part of its
orbit. About twice this period of time would be required
for it to come from Sirius to the sun, and eight times this
immense interval for a comet to come from a star four times
as far away. These figures do not disprove the theory that
comets wander from star to star, but they show that if this
hypothesis is true, then comets spend most of their time in
traveling and but little in visiting.

If the comets moved from star to star, their orbits with
respect to the sun would never be elliptical until after they
had been captured; they would, indeed, nearly always be
strongly hyperbolic because the stars are moving with respect
to one another with velocities which correspond to hyperbolic
speed for comets at such great distances. The fact
that no comet out of the hundreds whose orbits have been
%% -----File: 353.png---Folio 323-------
computed has moved in a sensibly hyperbolic orbit points
strongly to the conclusion that comets have been permanent
members of the solar system. They are possibly the remains
of the far outlying masses of a nebula from which the solar
system may have been developed. With increasing proof
that they are actually permanent members of the solar system,
their importance in connection with the question of its
origin and evolution continually increases.

\Article{198}{Theories of Comets' Tails.}---The fact that the tails
\index{Comets' tails, theories of}%
\index{Tails of comets, theories of}%
of comets usually project almost directly away from the
sun indicates that they are in
some way acted upon by a
repelling force emanating from
the %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.25in}
\Input[2.25in]{353}{png}
\Caption[The repulsion theory
of the origin of comets' tails.]{Fig}{123}
\end{wrapfigure}
sun. The intensity of this
repulsion has been computed in
a number of cases by Barnard
and others from the accelerations
which masses have undergone
which were receding from
the heads of comets along their
tails. These accelerations have been determined by comparing
photographs of the comets taken at different times
separated by short intervals.

It was suggested by Olbers as early as 1812 that the repulsive
\index{Electrical repulsion}%
\index[xnames]{Olbers}%
force which apparently produces the tails of comets may
be electrical in character. This theory has been taken up
and systematically developed by Bredichin, of Moscow.
\index[xnames]{Bredichin}%
According to it, the sun and comet nuclei both repel the
material of which the tails of comets are composed. Those
particles which leave the nuclei in the direction away from
the sun continue on in straight lines; those which leave in
other directions are gradually bent back by the force from
the sun and form the outer parts of the tails, as shown in
\Figref{123}. The resulting tails, especially if they are very
long, are slightly curved because the motion of the comet
is somewhat athwart the line along which the repelled particles
%% -----File: 354.png---Folio 324-------
move, that is, the line from the sun through the nucleus
(see \Figref{121}).

Electrical repulsion acts on the surfaces of particles, while
gravitation depends on their masses. Therefore, while large
masses are attracted by the sun more than they are electrically
repelled, the opposite may be true for small particles,
and the electrical repulsion is relatively stronger the smaller
they are. Consequently, the tails which are produced out
of small particles will be more nearly straight than those
which are composed of larger particles. Bredichin advanced
\index[xnames]{Bredichin}%
the theory that the long, straight tails are due to hydrogen
gas, the ordinary slightly curved tails to hydrocarbon gases,
and the short, stubby, and much curved tails to vapors of
metals. Spectroscopic observations have to a considerable
extent confirmed these conclusions. Some comets have tails
of more than one type, as for example Delavan's comet
\index{Delavan's comet}%
\index[xnames]{Delavan}%
(\Figref{124}).

If the electrical repulsion theory is adopted, the question
at once arises why the sun and the materials of which the
tails of comets are composed are similarly electrified. A
plausible answer to this question can be given. At least
the hydrogen in the sun's atmosphere seems to be negatively
electrified. Suppose a comet approaches the sun from a
remote part of space without an electrical charge. Laboratory
experiments show that the ultra-violet rays from the
sun, striking on the nucleus of the comet, will probably drive
off negatively charged particles which will be repelled by
the negative charge of the sun, and they will thus form a
tail for the comet. The repulsion will depend upon the
size of the particles and the electrical potential of the sun.
After the negatively electrified particles have been driven
off, the nucleus will be positively charged and, consequently,
will be electrically attracted by the sun. But since the particles
driven off will be only an exceedingly small part of the
whole comet, this attraction will not be great enough sensibly
to alter the comet's motion.
%% -----File: 355.png---Folio 325-------
\begin{figure}[hbtp]%[Illustration]
\centering\Input{355}{jpg}
\Caption[Delavan's comet, Sept.~28, 1914, showing a long, straight tail
and one having considerable curvature (Barnard).]{Fig}{124}
\index[xnames]{Barnard}%
\index{Delavan's comet}%
\end{figure}
%% -----File: 356.png---Folio 326-------

Another theory which merits careful attention is that the
particles which constitute comets' tails are driven off by the
pressure of the sun's light. According to Clerk-Maxwell's
\index{Light!pressure of}%
\index[xnames]{Clerk-Maxwell}%
electromagnetic theory, light exerts a pressure upon bodies
upon which it falls which is proportional to the light energy
in a unit of space. For bodies of considerable magnitude
the pressure is relatively very small, though it has been
detected by Nichols and Hull; but for minute bodies, say a
\index[xnames]{Hull}%
\index[xnames]{Nichols}%
ten-thousandth of an inch in diameter, the light pressure
may greatly exceed the sun's attraction. For still smaller
bodies the light pressure becomes relatively larger until their
diameters are approximately equal to a wave length of light,
say, one fifty-thousandth of an inch. Then, as Schwarzschild
\index[xnames]{Schwarzschild}%
has shown, the light pressure decreases relatively to the
force of gravitation. Consequently, if the particles are very
small the attraction will more than equal the repulsion.

But it has been shown more recently by Lebedew that
\index[xnames]{Lebedew}%
there is light pressure upon gases, in which the diameters
of the molecules are always a very small fraction of a
wave length of light, and that the pressure is proportional to
the amount of energy which the gas absorbs. Consequently,
it is not necessary to assume that the particles of which the
tails of comets are composed are larger than molecules.

It is generally supposed by astronomers that both electrical
repulsion and light pressure are factors in the production
of comets' tails. Nevertheless, there are outstanding
phenomena which these theories do not explain. In the
first place, there is no adequate explanation of the luminosity
of comets' tails. As comets approach the sun, their tails
increase in brightness much more rapidly than they should
if they were shining only by reflected light. The luminosity
of such exceedingly tenuous bodies whose density is doubtless
far less than that in the best vacuum tubes of the present
time can scarcely be explained as a temperature effect.
And still more embarrassing to these theories are the facts
that comets' tails do not always point directly away from
%% -----File: 357.png---Folio 327-------
the sun, and that sometimes they change their direction by
a number of degrees in a very short time. For example,
Barnard took photographs of Brooks's comet, 1893-IV,
\index[xnames]{Barnard}%
\index[xnames]{Brooks}%
on November~2 and November~3. In this interval the comet
moved forward in its orbit about~$1°$; and, consequently,
according to these theories, the direction of its tail should
have changed about~$1°$. But there was an actual change of
direction of the tail of~$16°$ which has not been explained.
There are also sudden and great changes in the character
and luminosity of comets' tails which no theory explains.
Sometimes secondary tails are developed with great rapidity,
making an angle of as much as $45°$ with the line joining the
comet with the sun. Obviously much remains to be learned
in connection with the tails of comets.

\Article{199}{The Disintegration of Comets.}---The particles that
\index{Comets!disintegration of}%
\index{Disintegration!of comets}%
leave the head of a comet to form its tail never unite with
it again. In this way, at each reappearance of a comet, that
part of the material which goes to form its tail is dispersed
into space; and, as the quantity remaining becomes reduced,
the comet becomes less and less conspicuous. Possibly this
is one of the reasons why Halley's comet in 1910 was not
\index{Halley's comet}%
\index[xnames]{Halley}%
such a remarkable object as it seems to have been in some
of its earlier apparitions.

There is another way in which comets disintegrate. Since
their masses are very small, the mutual attractions of their
parts are not sufficient to hold them together if they are
subject to strong disturbing forces. When they pass near
the sun, they are elongated by enormous tides. In fact, if
they pass within Roche's limit (\Artref{183}), the tidal forces exceed
\index{Roche's limit}%
\index[xnames]{Roche}%
their self gravitation unless they are as dense as the sun.
Comets have such exceedingly low density that the limits of
tidal disintegration for them must be very great. Consequently,
when a comet passes near the sun, the tidal forces
to which it is subject tend to tear it into fragments, which,
of course, may be assembled again by their mutual gravitation
after they have receded far from the sun. But on
%% -----File: 358.png---Folio 328-------
their way out they may pass near a planet which will exert
analogous forces, and may so disorganize them that they
will never again be united into a single body.

The theory which has just been outlined is clear. Now
what have been the observed facts? Biela's comet was
\index{Biela's comet}%
\index[xnames]{Biela}%
broken into two parts by some unknown forces, and the two
components subsequently traveled in independent paths.
The great comet of 1882 was seen to have a number of outlying
fragments when it was in the vicinity of the sun, and
many other comets have exhibited analogous phenomena.

Another source of disturbance to which comets are subject
is the scattered meteoric material which may more or
less fill the space among the planets. The phenomenon of
the zodiacal light gives an almost certain proof of its extensive
\index{Light!zodiacal}%
\index{Zodiacal light}%
existence. Such scattered particles would have little
effect on a dense body like a planet, but might cause serious
disturbances in a tenuous comet. In fact, there are many
instances in which comets and comets' tails seem to have
been subjected to unknown exterior forces. They are now
and then more or less broken up, and occasionally the tails
of comets have been apparently cut off and brushed aside.

Many comets which have been observed at two or three
perihelion passages have been found to be fainter at each
successive return than they were at the preceding, and some
have eventually entirely disappeared. It seems to be a safe
conclusion that comets are slowly disintegrated under the
disturbing forces of the sun and planets and the resisting
meteoric material which they may encounter. As confirmatory
of this view, it may be noted that the members of Jupiter's
family have small tails or none at all; that this comet
family does not contain as many members as might be expected;
and that a number of comets have totally disappeared,
presumably by disintegration.

\Article{200}{Historical Comets.}---In this article some of those
comets will be briefly described which have exhibited phenomena
of unusual interest. The enumeration of their
%% -----File: 359.png---Folio 329-------
peculiarities will illustrate the general statements which have
preceded, and will give additional information respecting
these remarkable objects.

\textit{The Comet of} 1680.---The comet of 1680 was the first one
\index{Comet!of 1680}%
whose orbit was computed on the basis of the law of gravitation.
Newton made the calculations and found that its
\index[xnames]{Newton}%
period of revolution was about $600$~years. It is one of the
family of comets mentioned in \Artref{195}. At its perihelion
it passed through the sun's corona at a distance of only
$140,000$ miles from its surface. It flew along this part of its
orbit at the rate of $370$
miles per second, and
its tail, $100,000,000$ miles
long, changed its direction
to correspond with
the motion of the comet
in its orbit.

\textit{The Great Comet of}
1811.---The great comet
\index{Comet!of 1811}%
of 1811 was visible from
March~26, 1811, until
August~17, 1812, and was
carefully observed by William Herschel. He discovered from
\index[xnames]{Herschel, William}%
the changes in its brightness, that it shone partly by its own
light; for its brilliance increased as it approached the sun
more rapidly than it would have done if it had been shining
entirely by reflected light. At one time its tail was
$100,000,000$ miles long and $15,000,000$ miles in diameter.
The phenomena connected with it suggested to Olbers the
electrical repulsion theory of comets' tails.

\textit{Encke's Comet} (1819).---Encke's comet was the first
member of Jupiter's family to be discovered, and it has a
shorter period ($3.3$~years) %[Illustration: Break, moved down]
\begin{wrapfigure}[16]{\WLoc}{3in}
\Input[3in]{359}{jpg}
\Caption[Encke's comet (Barnard).]{Fig}{125}
\index{Encke's comet}%
\index[xnames]{Encke}%
\end{wrapfigure}
than any other known comet.
At its brightest it was an inconspicuous telescopic object
(\Figref{125}), but it is noted for the fact that its period was
shortened, presumably by encountering some resistance,
%% -----File: 360.png---Folio 330-------
about $2.5$~hours at each revolution until 1868; since that
time the change in the period of revolution has been only
one half as great. The change in volume of Encke's comet
\index{Encke's comet}%
\index[xnames]{Encke}%
at times was extraordinary. On October~28, 1828, it was
$135,000,000$ miles from the sun and had a diameter of $312,000$
miles; on December~24, its distance was $50,000,000$ miles
from the sun, and its diameter was only $14,000$ miles; while
at its perihelion passage, on December~17, 1838, at a distance
of $32,000,000$ miles, its diameter was only $3000$ miles.
That is, at one time its volume was more than a million
times greater than it was at another.

\textit{Biela's Comet} (1826).---Biela's comet is also a small
\index{Biela's comet}%
\index[xnames]{Biela}%
member of Jupiter's family and has a period of about $6.6$~years.
At its appearance in 1846, it presented no unusual
phenomena until about the 20th~of December, when it was
considerably elongated. By the first of January it had separated
into two distinct parts which traveled along in parallel
orbits at a distance of about $160,000$ miles from each other.
At this time the two parts were undergoing considerable
changes in brightness, usually alternately, and sometimes
they were connected by a faint stream of light. At their
appearance in 1852 the two components were $1,500,000$ miles
apart, and they have never been seen again, although searched
for very carefully. De~Vico's comet, of 1844, and Brorsen's
\index{Brorsen's comet}%
\index[xnames]{Brorsen}%
\index[xnames]{Devico@{De Vico}}%
comet, of 1846, are also comets which have disappeared,
the former having been observed but once, and the latter
but four times after its discovery.

\textit{Donati's Comet} (1858).---Donati's comet was one of the
\index{Donati's comet}%
\index[xnames]{Donati}%
greatest comets of the nineteenth century. It was visible
with the unaided eye for $112$~days, and through a telescope
for more than $9$~months. Its tail, which was more than
$54,000,000$ miles long, at one time subtended an angle of more
than $30°$ as seen from the earth. It moved in the retrograde
direction in an orbit with a period of more than $2000$
years, and at its aphelion its distance from the sun was
more than $5.3$~times that of Neptune.
%% -----File: 361.png---Folio 331-------

\textit{Tebbutt's Comet} (1861).---Tebbutt's comet was of great
\index{Tebbutt's comet}%
\index[xnames]{Tebbutt}%
dimensions, but is noteworthy chiefly because the earth
passed through its tail. As could have been anticipated
from the excessive tenuity of comets' tails, the earth experienced
no sensible effects from the encounter. The earth
must have passed through the tails of comets many times in
geological history, and there is no evidence whatever that it
has ever been disturbed by them. In fact, if a comet should
strike the earth, head on, it is probable that the result would
not be disastrous to the earth.

\textit{The Great Comets of} 1880 \textit{and} 1882.---The comets of 1880
\index{Comet!of 1880 and 1882}%
and 1882 were two splendid members of the most remarkable
known family of comets which travel in the same orbit.
Both of these comets, as well as the earlier members of the
same family, are noteworthy for their vast dimensions, their
great brilliancy, and their close approach to the sun. The
comet of 1882 was observed both before and after perihelion
passage. Although it swept through several hundred
thousand miles of the sun's corona, its orbit was not sensibly
altered. Yet it gave evidence of having been subject to
violent disrupting forces. After perihelion passage it was
observed to have as many as 5~nuclei, while Barnard and
\index[xnames]{Barnard}%
other observers saw in the immediate vicinity as many as
6~or 8~small comet-like masses, apparently broken from the
large body, traveling in orbits parallel to it.

\textit{Morehouse's Comet} (1908).---On September~1, 1908,
Morehouse, at the Yerkes Observatory, discovered the third
comet of the year. It was found on photographic plates
taken for other purposes, and is one of the few examples in
which comets have been discovered by photography. This
comet was never bright, but was one of the most remarkable
comets ever observed in the extent and variety of its activities.
It was well situated for observation, and Barnard
obtained $239$~photographs of it on $47$~different nights. The
material which went into the tail of the comet was often
evolved with the most startling rapidity. For example, on
%% -----File: 362.png---Folio 332-------
the 30th~of September, in the early part of the night, the
comet presented an almost normal appearance. Before the
night was over, the tail had become cyclonic in form and
was attached to the head, which then was small and starlike,
by a very slender, curved, tapering neck. On the
succeeding night the material that then constituted the tail
was entirely detached from the head. On October~15, there
was another large outbreak of material which was shown
by successive photographs to be swiftly receding from the
comet (\Figref{126}).

Not only was Morehouse's comet noteworthy for the extraordinary
activities exhibited by its tail, but it changed in
brightness in a very remarkable manner. It was generally
considerably below the limits of visibility with the unaided
eye, but now and then it would flash up, without apparent
reason, for a day or so until it could be seen very faintly
without a telescope. While a number of larger comets have
been observed in recent years, no other has given evidence
of such remarkable changes in the forces that produce comets'
tails, and no other has exhibited such mysterious variations
in brightness.

\Article{201}{Halley's Comet.}---Halley's comet is the most celebrated
\index{Halley's comet}%
\index[xnames]{Halley}%
one in all the history of these objects. It is named
after Halley, not because he discovered it, but because he
computed its orbit from observations made in 1682 by the
methods which had been developed by his friend Newton.
\index[xnames]{Newton}%
Halley found that the orbit of this comet was almost identical
with the orbits of the comets of 1607 and~1531. He
came to the conclusion that these various comets were only
different appearances of the same one which was revolving
around the sun in a period of about $75$~years. The records
of comets in 1456, 1301, 1145, and~1066 confirmed this view
because these dates differ from 1682 by nearly integral multiples
of $75$~or $76$~years. From his computations Halley predicted
that the comet would appear again and pass its perihelion
point on March~13, 1759.
%% -----File: 363.png---Folio 333-------
\begin{figure}[hbtp]%[Illustration]
\centering\Input{363}{jpg}
\Caption[Morehouse's comet, Oct.~15, 1908. \textit{Photographed by Barnard
at the Yerkes Observatory.}]{Fig}{126}
\index{Yerkes Observatory}%
\index[xnames]{Barnard}%
\end{figure}
%% -----File: 364.png---Folio 334-------

Many of Halley's contemporaries were very skeptical
\index[xnames]{Halley}%
regarding this prediction. The law of gravitation had only
recently been discovered and the certainty with which it
had been established was not yet fully comprehended.
Halley was accused by skeptics of seeking notoriety by
making a prophecy and cleverly putting forward the date
of its fulfillment so far that he would be dead before his
failure became known. However, before the $75$~years had
passed away, the law of gravitation had become so firmly
established, and the mathematical processes employed in
astronomical work had become so well understood, that
astronomers, at least, had implicit faith in the correctness
\begin{figure}[hbt]%[Illustration:]
\Input{364}{png}
\Caption[The orbit of Halley's comet.]{Fig}{127}
\index{Halley's comet}%
\end{figure}%
of Halley's prediction, although since its last appearance the
comet had been invisible for the lifetime of a man and had
gone out $3,000,000,000$ miles from the sun to beyond the
orbit of Neptune. There was great popular interest in the
comet as the date for its return approached. It actually
passed its perihelion within one month of the time predicted
by Halley. The slight error in the prediction was due to
the imperfect observations of its positions in 1682, and to
the perturbations by planets which were then unknown.
This was the first verification of such a prediction; and the
definiteness and completeness with which it was fulfilled
had been entirely unapproached in the case of all the
prophecies which the world had known up to that time.

Halley's comet passed the sun again in 1835. At this
%% -----File: 365.png---Folio 335-------
time it was so accurately observed that its subsequent orbit
could be computed with a high degree of precision. If it
had made its next revolution in the same period as the one
\begin{figure}[hbt]%[Illustration:]
\Input{365}{jpg}
\Caption[Halley's comet, May~29, 1910. \textit{Photographed by Barnard at
the Yerkes Observatory.}]{Fig}{128}
\index{Halley's comet}%
\index[xnames]{Barnard}%
\index[xnames]{Halley}%
\end{figure}%
ending in 1835, it would have passed its perihelion in July,
1912. Instead of this, it passed its perihelion on April~19,
1910. The perturbations of the remote planets reduced its
%% -----File: 366.png---Folio 336-------
period by more than two years. The most accurate computations
of its orbit and predictions of the time of its return
were made by Cowell and Cromellin, of Greenwich,
\index[xnames]{Cowell}%
\index[xnames]{Cromellin}%
who missed the time of perihelion passage by only $2.7$~days.
Their computations were so accurate that even this small
discrepancy could not be the result of accumulated errors,
and they believe that the comet has been subject to some
\begin{figure}[hbt]%[Illustration:]
\Input{366}{png}
\Caption[The relations of the sun, earth, and Halley's comet in 1910.]{Fig}{129}
\index{Halley's comet}%
\index[xnames]{Halley}%
\end{figure}%
unknown forces. Its next return will be about 1985, and
\Figref{127} shows the position in its orbit for various epochs
during this interval. In order to get the precise time of its
return, it will be necessary to take into account the perturbations
of the planets.

While Halley's comet is a very large one (\Figref{128}), its
latest appearance was somewhat disappointing, especially to
the general public, who had been led to expect that it would
%% -----File: 367.png---Folio 337-------
rival the sun in brightness. One of the reasons for the disappointment
was that the earth was not very near the comet
when it was at its perihelion where it was brightest and had
the longest tail. The relations of the earth, comet, and
sun in this part of its orbit are shown in \Figref{129}, drawn by
Barnard. On May~5, the length of the comet's tail was
\index[xnames]{Barnard}%
$37,000,000$ miles. On May~18 the comet passed between
the earth and the sun and was entirely invisible when projected
on the sun's disk. This shows that even its nucleus
was extremely tenuous and transparent. At this time the
earth passed through at least the outlying part of its tail.
Neither at this time nor at any other did the comet have
any sensible influence upon the earth. On the whole, it was
altogether devoid of interesting features.


\Section{II}{Meteors}

\Article{202}{Meteors, or Shooting Stars.}---An attentive watch
\index{Meteors}%
\index{Shooting stars}%
of the sky on almost any clear, moonless night will show one
or more so-called ``shooting stars.'' They are little flashes
of light which have the appearance of a star darting across
the sky and disappearing. Instead of being actual stars,
which are great bodies like our sun, they are, as a matter
of fact, tiny masses so small that a person could hold one
in his hand. Under certain circumstances of motion and
position, they dash into the earth's atmosphere at a speed
\index{Velocity!of meteors}%
of from $10$~to $40$~miles per second, and the heat generated
by the friction with the upper air vaporizes or burns them.
The products of the combustion and pulverization slowly
fall to the earth if they are solid, or are added to the atmosphere
if they are gaseous. Since it is misleading to call them
``shooting stars,'' they will always be called ``meteors''
hereafter.

The distances of meteors were first determined in 1798
by Brandes and Benzenberg, at Göttingen. They made
\index[xnames]{Benzenberg}%
\index[xnames]{Brandes}%
simultaneous observations of them from positions separated
by a few miles, and from the differences in their apparent
%% -----File: 368.png---Folio 338-------
directions they computed their altitudes above the surface
of the earth (\Artref{29}). Their observations and those
of many succeeding astronomers, among whom may be
mentioned Denning, of England, and Olivier, of Virginia,
\index[xnames]{Denning}%
\index[xnames]{Olivier}%
have shown that meteors rarely, if ever, become visible at
\index{Meteors!height of}%
altitudes as great as $100$~miles, and nearly all of them disappear
before they have descended to within $30$~miles of the
earth's surface.

The velocity with which a meteor enters the atmosphere
can be found by determining the point at which it becomes
visible, the point at which it disappears, and the interval of
time during which it is visible. The total amount of light
energy given out by a meteor can be determined from its
apparent brightness, its distance from the observer, and the
time during which it is radiant. The energy radiated by a
meteor has its source in the heat generated by the friction
of the meteor with the earth's atmosphere, and it cannot
exceed the kinetic energy of the meteor when it entered
the atmosphere. Suppose all the kinetic energy of a meteor
is transformed into light. This assumption is not strictly
true, but it will be approximately true for matter moving
with the high speed of a meteor. Then, since the energy
of motion of a body is one half its mass multiplied by the
square of its velocity, the mass of the meteor can be computed
because its light energy and velocity can be determined
directly from observations by the methods which
have just been described. By such means it has been found
that ordinarily the masses of meteors do not exceed a few
tenths of an ounce. However, the observational data are
difficult to determine and the subject has received relatively
less attention than it deserves. Consequently, no great
reliance should be placed on the precise numerical results.

\Article{203}{The Number of Meteors.}---If a person scans the
\index{Meteors!number of}%
sky an hour or so and finds that he can see only a few meteors,
he is tempted to draw the conclusion that the number of
them which strike the earth's atmosphere daily is not very
%% -----File: 369.png---Folio 339-------
large. He bases his conclusion mostly on the fact that half
of the celestial sphere is within his range of vision, but a
diagram representing the earth and its atmosphere to scale
will show him that he can see by no means half the meteors
which strike the earth's atmosphere. As a matter of fact,
he can see the atmosphere over only a few square miles of
the earth's surface.

From very many counts of the number of meteors which
can be seen from a single place during a given time, it has
been computed that between $10$ and $20$~millions of them
strike into the earth's atmosphere daily. There are probably
several times this number which are so small that they
escape observation. Often when astronomers are working
with telescopes they see faint meteors dart across the field
of vision which would be quite invisible with the unaided eye.

Meteors enter the earth's atmosphere from every direction.
The places where they strike the earth and the velocities
of their encounter depend both upon their own velocities
and also upon that of the earth around the sun. The
side of the earth which is ahead in its motion encounters
more meteors than the opposite, for it receives not only those
which it meets, but also those which it overtakes, while the
part of the earth which is behind receives only those which
overtake it. The meridian is on the forward side of the
earth in the morning and on the rearward side in the evening.
It is found by observation that more meteors are seen
in the morning than in the evening, and that the relative
velocities of impact are greater.

\Article{204}{Meteoric Showers.}---Occasionally unusual numbers
\index{Meteoric showers}%
of meteors are seen, and then it is said that there is a
meteoric shower. There have been a few instances in which
meteors were so numerous that they could not be counted,
but usually not more than one or two appear in a minute.

At the time of a meteoric shower the meteors are not
only more numerous than usual, but a majority of them
move so that when their apparent paths are projected backward,
%% -----File: 370.png---Folio 340-------
they pass through, or very near, a point in the sky.
This point is called the \textit{radiant point} of the shower, for the
\index{Radiant point of meteors}%
meteors all appear to radiate from it. A number of meteor
trails which clearly define a radiant point are shown in
\Figref{130}.

The most conspicuous meteoric showers occur on November~15
and November~24 yearly. The former have their
radiant in Leo, within the sickle, and are called the \textit{Leonids}.
\index{Leo}%
\index{Leonid meteors}%
\begin{figure}[hbt]%[Illustration:]
\Input{370}{png}
\Caption[Meteor trails defining a radiant point (Olivier).]{Fig}{130}
\index[xnames]{Olivier}%
\end{figure}%
From the position of this constellation (Arts.\ \hyperref[art:82]{82},~\hyperref[art:93]{93}), it
follows that they can be seen only in the early morning hours.
The latter have their radiant in Andromeda, and are called
the \textit{Andromids}. They can be seen only in the early part
\index{Andromid meteors}%
of the night. The Leonids and Andromids are not equally
numerous every year. Great showers of the Leonids occurred
in 1833 and~1866, and less remarkable ones, though
greater than the ordinary, from 1898 to~1901. The Andromids
appear in unusual numbers every thirteen years.

Besides these meteoric showers, according to Denning,
\index[xnames]{Denning}%
%% -----File: 371.png---Folio 341-------
nearly $3000$ other less conspicuous ones have been found.
The Perseids appear for a week or more near the middle of
\index{Perseid meteors}%
August, the Lyrids on or about April~20, the Orionids on or
about October~20, etc.
\index{Lyrid meteors}%
\index{Orionid meteors}%

\Article{205}{Explanation of the Radiant Point.}---In 1834 Olmsted
\index{Radiant point of meteors}%
showed that the apparent radiation of meteors from a
point is due to the fact that they move in parallel lines,
and that we see only the projection of their motion on the
celestial sphere. Thus, in \Figref{131}, the actual paths of the
meteors are~$AB$, but their apparent paths as seen by an
observer at~$O$ are~$AC$. When these lines are all continued
\begin{figure}[hbt]%[Illustration:]
\Input{371}{png}
\Caption[Explanation of the radiant point of meteors.]{Fig}{131}
\end{figure}%
backward, they meet in the point which is in the direction
from which the meteors come.

It follows that the meteors which give rise to the meteoric
showers are moving in vast swarms along orbits which intersect
the orbit of the earth. When the earth passes through
the point of intersection, it encounters the meteors and a
shower occurs. Thus, the orbit of the Leonids touches the
\index{Leonid meteors}%
orbit of the earth at the point which the earth occupies on
November~14. In this case the earth meets the meteors
(\Figref{132}), while the Andromids overtake the earth.
\index{Andromid meteors}%

\Article{206}{Connection between Comets and Meteors.}---The
fact that the volatile material of which comets' tails are
composed gradually becomes exhausted, after which the
comets themselves become invisible, and the fact that
meteoric showers are due to wandering swarms of small
%% -----File: 372.png---Folio 342-------
particles which revolve around the sun in elongated elliptical
orbits, suggest the hypothesis that comets and meteors
are related. The hypothesis is confirmed and virtually
proved by the identity of the orbits of certain meteoric
swarms and comets.

In 1866 Schiaparelli showed that the August meteors
\index{August meteors}%
\index[xnames]{Schiaparelli}%
move in the same orbit as Tuttle's comet of~1862. That is,
\index{Tuttle's comet}%
\index[xnames]{Tuttle}%
in addition to the comet, which is a member of Saturn's
family, there are many other small bodies (meteors) traveling
in the same orbit. In 1867 Leverrier found that the
\index[xnames]{Leverrier}%
Leonids move in the same orbit as Tempel's comet of 1866,
\index{Tempel's comet of 1866}%
\index[xnames]{Tempel}%
while Weiss showed that the meteors of April~20 and the
\index[xnames]{Weiss}%
\begin{figure}[hbt]%[Illustration:]
\Input{372}{png}
\Caption[Orbit of the Leonid meteors.]{Fig}{132}
\index{Leonid meteors}%
\end{figure}%
comet of 1861 move in the same orbit, and that the paths
of the Andromids and Biela's comet were likewise the same.
\index{Andromid meteors}%
\index{Biela's comet}%
\index[xnames]{Biela}%
It has recently been claimed that the Aquarid meteors of
\index{Aquarid meteors}%
early May have an orbit almost identical with that of
Halley's comet.
\index{Halley's comet}%
\index[xnames]{Halley}%

While it is not possible to be certain as to the origin of
comets, the history of their later evolution and final end is
tolerably clear. The elongated orbits in which they may
have originally moved are reduced when they are captured
by the planets. Their periods of revolution are subsequently
shorter, their volatile material wastes away in the form of
tails, and the remaining material is scattered along their
orbits by the dispersive forces to which they are subject.
%% -----File: 373.png---Folio 343-------
If these orbits cross the orbit of a planet, the remains of the
comets are gradually swept up by the larger body. If an orbit
of a comet does not originally cross the orbit of a planet,
the perturbations of the planets will, in general, in the course
of time, cause it to do so. The result will be that the planets
sweep up more and more of the remains of disintegrated
comets and undergo a gradual growth in this manner.

\Article{207}{Effects of Meteors on the Solar System.}---The
\index{Meteors!effects of on solar system}%
most obvious effect of the numerous meteors which swarm
in the solar system is a resistance both to the rotations and
the revolutions of all the bodies. As was stated in \Artref{45},
the effects of meteors upon the rotation of the earth are at
present exceedingly slight, and it is very probable that their
influences upon the rotations of the other members of the
system are also inappreciable. A retardation in the translatory
motion of a body causes its orbit to decrease in size.
Hence, so far as the meteors affect the planets in this way,
they cause them continually to approach the sun.

Another effect of meteors upon the members of the solar
system is to increase their masses by the accretion of matter
which may have come originally from far beyond the orbit
of Neptune. As the masses of the sun and planets increase,
their mutual attractions increase and the orbits of
the planets become smaller. Looking backward in time, we
are struck by the possibility that the accretion of meteoric
matter may have been more rapid in former times, and that
it may have been an important factor in the growth of the
planets from much smaller bodies.

\Article{208}{Meteorites.}---Sometimes bodies weighing from a
\index{Aerolites@{Aërolites}}%
\index{Meteorites}%
\index{Siderites}%
few pounds up to several hundred pounds, or even a few
tons, dash into the earth's atmosphere, glow brilliantly from
the heat generated by the friction, roar like a waterfall,
occasionally produce violent detonations, and end by falling
on the earth. Such bodies are called \textit{meteorites}, \textit{siderites}, or
\textit{aërolites}.

About two or three meteorites are seen to fall yearly; but,
%% -----File: 374.png---Folio 344-------
since a large part of the earth is covered with water or is
uninhabited for other reasons, it is probable that in all
at least~100 strike the earth annually. The outside of a
meteorite during its passage through the air is subject to
intense and sudden heating, and the rapid expansion of its
surface layers often breaks it into many fragments. The
surface is fused and on striking cools rapidly. The result is
that it has a black, glossy structure, usually with many
small pits where the less refractive material has been melted
\begin{figure}[ht]%[Illustration:]
\Input{374}{jpg}
\Caption[Stony meteorite which fell at Long Island, Kansas; weight,
700~pounds (Farrington).]{Fig}{133}
\index[xnames]{Farrington}%
\end{figure}%
out. Since meteors pass entirely through the atmosphere in
a few seconds, only their surfaces give evidence of the extremes
of heat and pressure to which they have been subjected
in their final flight.

Most meteors are composed of stone, though it is often
\index{Meteorites!composition of}%
mixed with some metallic iron. Even where pure iron is not
present, some of its compounds are usually found. About
three or four out of every hundred are nearly pure iron
with a little nickel. All together about $30$~elements which
occur elsewhere on the earth have been found in meteorites,
but no strange ones. Yet in some respects their structure
is quite different from that of terrestrial substances. They
%% -----File: 375.png---Folio 345-------
have peculiar crystals, they show but little oxidation and
no action of water, and they contain in their interstices relatively
large quantities of occluded gases, some of which are
\begin{figure}[hb]%[Illustration:]
\centering\Input[4in]{375a}{jpg}
\Caption[Iron meteorite from Cañon Diablo, Arizona; weight, $265$~pounds
(Farrington).]{Fig}{134}
\index[xnames]{Farrington}%
\end{figure}%
combustible. According to Farrington, some meteors give
evidence of fragmentation and recementation, others show
faulting (fracture and sliding of one surface on another) with
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{375b}{jpg}
\Caption[Durango, Mexico. Meteorite showing peculiar crystallization
characteristic of certain meteorites (Farrington).]{Fig}{135}
\end{figure}%
recementation, and others, veins where foreign material has
been slowly deposited.

\Article{209}{Theories respecting the Origin of Meteorites.}---If
\index{Meteorites!origin of}%
\index{Origin!of meteorites}%
it were known that meteorites are but meteors which are so
large that they reach the earth before they are completely
%% -----File: 376.png---Folio 346-------
oxidized and pulverized, we might justly conclude that they
are probably the remains of disintegrated comets. This
would enable us to learn certain things about comets which
cannot be settled yet. But no meteorite is known certainly
to have been a member of any meteoric swarm. However,
two meteorites have fallen during the time of meteoric
showers, one in France, at the time of the Lyrids in 1905,
and the other in Mexico, just before the Andromids in 1885.
\index{Andromid meteors}%
\index{Lyrid meteors}%

The structure of some meteorites is more like that of lava
from deep volcanoes than anything else found on the earth.
An old theory was that they have been ejected by volcanic
explosions from the moon, planets, or perhaps the sun.
This theory would account for some of their characteristics,
and would explain why they contain only familiar elements,
at least if the other bodies of the solar system contain only
those found on the earth; but it does not at all explain the
fragmentation, faulting, and veins, for forces great enough
to produce ejections would scarcely be found without heat
enough to produce at least fusion.

Chamberlin has maintained that meteorites may be the
\index[xnames]{Chamberlin}%
débris of bodies, perhaps of planetary dimensions, which
have been broken up by tidal strains when they have passed
some larger mass within Roche's limit. When suns pass by
\index{Roche's limit}%
\index[xnames]{Roche}%
other suns, it is probable that at rare intervals they pass so
near each other that their planets (if they have any) are
broken up. More rarely, the suns themselves may be disintegrated.
Indeed, this may be the origin of all cometary
and meteoric matter. Whether it is or not, there is here
a possibility of disintegration which must be taken into
account in any theory of cosmical evolution.

The present desiderata are more accurate determinations
of comets' orbits to find whether any of them are really
hyperbolic, more accurate determinations of the velocities of
meteors to find whether they ever come into our system on
parabolic or hyperbolic orbits, and finally the answer to the
question whether meteors and meteorites are really related.
%% -----File: 377.png---Folio 347-------

The suggestion that a meteorite may be a fragment of a
world which was disrupted before the origin of the earth
makes some demands on the imagination, but it seems no
more incredible to us than seemed the suggestion to our
predecessors a century ago that great mountains have been
utterly destroyed by the rains and snows and winds.


\Section{XIV}{QUESTIONS}

1. What observations would prove that comets are not in the
earth's atmosphere, as the ancients supposed they were?

2. Suppose two small masses are moving around the sun in the
same elongated orbit, but that one is somewhat ahead of the other.
How will their distance apart vary with their position in their orbit
(use the law of areas)? Does this suggest an explanation of the
variations in the dimensions of comets' heads?

3. The velocity of a comet moving in a parabolic orbit is inversely
as the square root of its distance from the sun. At the distance
of the earth a comet has a velocity of about $25$~miles per
second. What is the distance between the comets of 1843 and 1882
when they are $100,000$ astronomical units from the sun?

4. Suppose the particles of which a comet is composed have
almost exactly the same perihelion point but somewhat different
aphelion points. How would the dimensions of the comet vary
with its position in its orbit?

5. By means of Kepler's third law compute the period of a
comet whose aphelion point is at a distance of $140,000$ astronomical
units, which is about half the distance of the nearest known star.

6. What objections are there to the theory that originally all
comets had an aphelion distance equal to that of Neptune, and that
the orbits of some have been increased and others diminished by the
action of the planets?

7. On the repulsion theory should a comet's tail be equally long
when it is approaching the sun and when it is receding?

8. Draw the diagram mentioned in the first paragraph of \Artref{203}.

9. Count the number of meteors you can observe in an hour on
some clear, moonless night.

10. If possible, observe the Leonid or Andromid meteors.

11. Make a list of the fairly well-explained cometary phenomena,
and of those for which no satisfactory theory exists.

\normalsize

%% -----File: 378.png---Folio 348-------
\begin{figure}[hbtp]%[Illustration]
\centering\Input{378}{jpg}
\Caption[The tower telescope of the solar observatory of the Carnegie
Institution of Washington, Pasadena, California.]{Fig}{136}
\index{Mount Wilson Solar Observatory}%
\index{Solar!Observatory}%
\end{figure}
%% -----File: 379.png---Folio 349-------


\Chapter{XI}{The Sun}

\Section{I}{The Sun's Heat}

\Article{210}{The Problem of the Sun's Heat.}---The light and
\index{Light!from sun}%
\index{Sun!light and heat of}%
heat radiated by the sun are essential for the existence of life
on the earth, and consequently the question of the source
of the sun's energy, how long it has been supplied, and how
long it will last are of vital interest. Not only are these
questions of importance because the sun is the dominant
member of the solar system, governing the motions of the
planets and illuminating and heating them with its abundant
rays, but also because the sun is a star, and the only
one of the hundreds of millions in the sky which is so near
that its surface can be studied in detail.

Obviously the first thing to do in studying the heat of the
sun is to measure the amount received from it by the earth;
then, the amount which the sun radiates can be computed.
The amount of heat given out by the sun gives the basis for
determining its temperature. Then naturally follows the
question of the origin of the sun's heat. The answers to
these questions are of great importance in considering the
the evolution of the solar system and the stars.

\Article{211}{The Amount of radiant Energy received by the
Earth from the Sun.}---Light is a wave motion in the ether
\index{Light!wave lengths of}%
\index{Wave length of light}%
whose wave lengths vary from about $\frac{1}{65,000}$ of an inch, in
the violet, to about $\frac{1}{40,000}$ of an inch, in the red. Radiant
heat differs from light physically only in that its waves are
longer. The circumstance that human eyes are sensitive
to ether waves of certain lengths and not to those that are
longer or shorter is, of course, of no importance in discussing
%% -----File: 380.png---Folio 350-------
the physical question of the sun's heat. Consequently, in
\index{Heat!from sun}%
\index{Sun!heat received from}%
the problem of solar radiation rays of all wave lengths are
included, and together they constitute the radiant energy
emitted by the sun.

Physicists have devised various methods of measuring
the amount of energy received from a radiating source.
In applying them to the problem of determining the amount
of energy received from the sun the chief difficulty consists
in making correct allowance for the absorption of light and
\index{Absorption of light}%
heat by the earth's atmosphere. The best results have been
\index{Atmosphere!absorption of light by}%
obtained by making simultaneous measurements from near
sea level, from the summits of lofty mountains, and from
balloons. Langley measured the intensity of solar radiation
\index[xnames]{Langley}%
at the top of Mount Whitney, $14,887$ feet above the
sea, and at its base. He arrived at the conclusion that $40$~per~cent
of the rays striking the atmosphere perpendicularly,
when it is free from clouds, are absorbed before they reach
the surface of the earth; later investigations have reduced
this estimate to $35$~per~cent. The work initiated by Langley
has been continued most successfully by Abbott, Fowle, and
\index[xnames]{Abbott}%
\index[xnames]{Fowle}%
Aldrich, and they find that the rate at which radiant energy
\index[xnames]{Aldrich}%
of all wave lengths is received by the earth from the sun at
the outer surface of our atmosphere when the sun is at its
mean distance is, in terms of mechanical work, $1.51$~horse
power per square yard.

The earth intercepts a cylinder of rays from the sun whose
cross section is equal to a circle whose diameter equals the
diameter of the earth. The area of this circle is, therefore,
$\pi r^2$, where $r$~equals $3955 × 1760 = \DPtypo{6,960,000}{6,960,800}$
yards.\footnote
  {The mean radius of the earth is $3955$ miles and there are $1760$ yards in a mile.}
Hence
the rate at which solar energy is intercepted by the whole
earth is in round numbers $230,000,000,000,000$ horse power.

In the evolution of life upon the earth the sun has been as
important a factor as the earth itself. Consequently, geologists
and biologists have a deep interest in the sun, and particularly
%% -----File: 381.png---Folio 351-------
in the question whether or not its rate of radiation
is constant. It has long been supposed that probably the
sun is slowly cooling off and that the light and heat received
from it are gradually diminishing, but it was a distinct surprise
when Langley and Abbott found that its rate of radiation
\index[xnames]{Abbott}%
sometimes varies in a few days by as much as $10$~per~cent.
If a change of this amount in the rate of radiation
of the sun were to persist indefinitely, the mean temperature
of the earth would be changed about $13°$~Fahrenheit; but
a variation of $10$~per~cent for only a few days has no important
\index{Variation!of sun's radiation}%
effect on the climate. Abbott, Fowle, and Aldrich
\index[xnames]{Aldrich}%
\index[xnames]{Fowle}%
have continued the investigation of this question, and by
making observations simultaneously in Algiers, in Washington,
and in California, so as to eliminate the effects of local
and transitory atmospheric conditions, they have firmly
established the reality of small and rapid variations in the
sun's rate of radiation.

The question of variation in the amount of energy received
from the sun can also be considered in the light of geological
evidence. The fossils preserved in the rocks of all geological
ages prove that there has been an unbroken life chain upon
the earth for many tens of millions of years. This means
that during all this vast period of time the temperature of
the earth has been neither so high nor so low as to destroy
all life. Moreover, the record is clear that, in spite of glacial
epochs and intervening warmer eras, the temperature changes
have not been very great, and there is no evidence of a progressive
cooling of the sun.

\Article{212}{Sources of the Energy used by Man.}---One of the
\index{Energy!of wind}% [** TN: Moving up one page]
earliest extensive sources of energy for mechanical work used
by man was the wind. It has turned, and still turns, millions
of windmills for driving machinery or pumping water.
Until the last few decades it moved nearly all of the ocean-borne
commerce of the whole world, and it is still an important
factor in shipping. But that part of the energy of the
wind which is used is an insignificant fraction of all that
%% -----File: 382.png---Folio 352-------
exists. For example, if, in a breeze blowing at the rate of
$20$~miles an~hour, all the energy in the air crossing an area
$100$~feet square perpendicular to its direction of motion were
used, it would do about $560$~horse power of work.

What is the origin of the energy in the wind? The sun
warms the atmosphere over the equatorial regions of the
earth more than that over the higher latitudes, and the
resulting convection currents constitute the wind. Consequently,
all the energy in every wind that blows came originally
from the sun.

Another source of energy which has been of great practical
\index{Energy!of water}%
value is water power. The source of this energy is also the
sun, because the sun's heat evaporates the water and raises
it into the air a half mile or more, the winds carry part of
it out over the land, where it falls as rain or snow, and in
descending again to the ocean it may now and then plunge
over a precipice, where its energy can be utilized by men.
Amazing as are the figures for such great waterfalls as
Niagara, they give but a faint idea of the enormous work the
sun has done in raising water into the sky, and the equally
great amount of work the water does in falling back to the
earth. During a heavy rain an inch of water may fall.
An inch of water on a square mile weighs over $60,000$ tons.
In the eastern half of the United States, where the annual
rainfall is about $35$~inches, every year over $2,000,000$ tons of
water fall on each square mile from a height of half a mile or
more.

The great modern source of energy for mechanical work is
coal. The coal has formed from vegetable matter which
accumulated in peat beds ages and ages ago. Consequently,
the immediate source of its energy is the plants out of which
it has developed. But the plants obtained their energy from
the sun. In millions of tiny cells the sun's energy broke
up the carbon dioxide which they inhaled from the atmosphere;
then the oxygen was exhaled and the carbon was
stored up in their tissues. When a plant is burned, as much
%% -----File: 383.png---Folio 353-------
energy is developed and given up again as the sun put into
it when it grew.

Thus it is seen that all the great sources of energy can be
traced back to the sun; it is true of the minor ones also.
One naturally inquires whether these sources of energy are
perpetual. The winds will certainly continue to blow and
the rains to descend as long as the earth and sun exist in
their present conditions, but the coal and petroleum will
\index{Energy!of coal}%
eventually be exhausted. They will last several centuries
and perhaps a few thousand years. This period seems long
compared to the lifetime of an individual, or perhaps of a
nation, but it is only a minute fraction of the time during
which our successors will probably occupy the earth. It
follows that they will be compelled to depend upon sources
of energy at present but little utilized. Perhaps some great
benefactor of mankind will discover a means of putting to
direct use the enormous quantities of energy which the sun
is now sending to the earth. At present we are depending
on that infinitesimal residue of the energy which the earth
received in earlier geological times and which has been
stored up and preserved in petroleum and coal.

\Article{213}{The Amount of Energy radiated by the Sun.}---The
\index{Energy!radiated by sun}%
\index{Solar!energy}%
\index{Sun!radiation of}%
earth as seen from the sun subtends an angle of only
$17''.6$. That is, its apparent area is about $\frac{1}{15}$ the greatest
apparent area of Venus as seen from the earth. A glance
at Venus will show that this is an exceedingly small part of
the whole celestial sphere. Since the little earth at a distance
of $93,000,000$ of miles receives the enormous quantity
of heat given in \Artref{211}, it follows that the amount which
is radiated by the sun must be inconceivable. It can be
brought within the range of our understanding only by contemplating
some of the things it might do.

The energy radiated per square yard from the sun's surface
is equivalent to $70,000$ horse power. This amount of heat
energy would melt a layer of ice $2200$~feet thick every hour
all over the surface of the sun; and it would melt a globe of
%% -----File: 384.png---Folio 354-------
ice as large as the earth in $2$~hours and $40$~minutes. Less
than one two-billionth of the energy poured forth by the sun
is intercepted by the earth, and less than ten times this
amount by all the planets together; the remainder travels
on through the ether to the regions of the stars at the rate
of $186,000$ miles per second.
\index{Light!velocity of}%
\index{Velocity!of light}%

\Article{214}{The Temperature of the Sun.}---Stefan's law (\Artref{172})
\index{Stefan's law}%
\index{Sun!temperature of}%
\index{Temperature!of sun}%
\index[xnames]{Stefan}%
that a black body radiates as the fourth power of its
absolute temperature, gives a basis for determining the
temperature of a body whose rate of radiation is known.
While the sun is probably not an ideal radiator, such as is
contemplated in the statement of Stefan's law, and while
it radiates from layers at various depths below its surface,
with the upper layers absorbing part of the energy coming
from the lower, yet an approximate idea of the temperature
of its radiating layers can be obtained from its rate of radiation.
On using Stefan's law as a basis for computation, it
is found that the temperature of the radiating layers of the
sun is at least $10,000°$ Fahrenheit. Or, it would be more
accurate to say that an ideal radiating surface at this temperature
would have the same rate of radiation as the sun,
and since the sun is not a perfect radiator, its temperature
is probably still higher. This temperature is several thousand
degrees higher than has been obtained in the most
efficient electrical furnaces, and is far beyond that required
to melt or vaporize any known terrestrial substance; yet,
the temperature of the interior of the sun is undoubtedly
far higher.

Another method of determining the temperature of the
sun is from the proportion of energy of different wave lengths
which it radiates. A body of low temperature radiates
relatively a large amount of red light and a small amount of
blue light. As the temperature rises the relative proportion
of blue light increases. The uncertainties in the results obtained
by this method of determining the temperature of
the sun arise, in the first place, from the fact that, at the
%% -----File: 385.png---Folio 355-------
best, it is not very precise, and, in the second place, from the
fact that both the sun's and the earth's atmospheres absorb
very unequally radiant energy of various wave lengths.
After making the necessary allowances for the absorption,
the results obtained by this method confirm those found by
the other.

There have been a number of other methods of obtaining
the temperature of the sun from the time of Newton, but
\index[xnames]{Newton}%
most of them have rested on physical principles which are
unsound, and in some cases they have led to most extravagant
results.

\Article{215}{The Principle of the Conservation of Energy.}---Before
\index{Conservation of energy}%
\index{Energy!conservation of}%
taking up the question of the origin of the sun's heat,
it is advisable to consider the principle of the conservation
of energy. It is comparable in importance and generality
to the principle of the conservation (indestructibility) of
matter. It was once supposed that when inflammable material,
as wood, is burned, it is utterly annihilated. But it
has been known for about $150$~years that if the ashes, the
smoke, and the gases produced by the combustion were all
gathered up and weighed in a vacuum, their weight would
exactly equal that of the original wood together with the
oxygen which united with it in burning.

Similarly, it was supposed until after 1840 that energy
might be destroyed as well as transformed. For example,
it was supposed that the energy lost by friction ceased to
exist. But it had been noted that friction produced heat,
and heat was known to be equivalent to mechanical energy,
for it had been turned into work, for example, by means
of the steam engine. It does not seem now to have been
a large step to have conjectured that the heat produced by
the friction is exactly equivalent to the energy lost. But
many elaborate experiments were required (made mostly by
Mayer and Joule) to prove the correctness of this conjecture
\index[xnames]{Joule}%
\index[xnames]{Mayer}%
and to lead to the generalization, now universally accepted,
that \emph{the total amount of energy in the universe is always the
%% -----File: 386.png---Folio 356-------
same}. This is one of the most far-reaching principles of science,
and, like the law of gravitation, is involved in every
phenomenon in which there is motion of matter.

The energy of a body as used in the principle of the conservation
\index{Energy!kinetic}%
\index{Energy!potential}%
\index{Energy!radiant}%
\index{Kinetic energy}%
\index{Potential energy}%
of energy means both its energy of motion (kinetic
energy) and also its energy of position, or the power it may
have of doing work because of its position (potential energy).
It is the sum of the potential and kinetic energies of the
universe which is constant. Since energy may be in a radiant
form and in transit from one body to another, or from
a body out into endless space, the principle holds only when
the energy which is in the ether is also included.

\Article{216}{The Contraction Theory of the Sun's Heat.}---The
\index{Contraction of sun}%
\index{Sun's heat!contraction theory of}%
mutual attractions of the particles of which the sun is composed
tend to cause it to contract. A contraction of the sun
would be equivalent to a fall of all of its particles toward
its center. If they should fall the whole distance one at a
time, they would generate a certain amount of heat upon
their impacts. If they should fall simultaneously, first a
fraction of the distance and then another, the same total
amount of heat would be generated. It might be supposed
without computation that an enormous contraction would
be necessary in order to produce enough heat to change
appreciably the temperature of the sun.

The effect of the sun's contraction can be considered more
exactly in terms of energy. The sun in an expanded condition
would have more potential energy with respect to
the force of gravitation than if it were contracted, because
work would be done on it by gravitation in changing it from
the first state to the second. Therefore the kinetic energy,
or temperature, of the sun must rise on its contraction. It
is analogous to a falling body. The higher it is above the
surface of the earth, the greater its potential energy; the
farther it falls, the more potential energy it loses and the
more kinetic energy it acquires.

The problem is to determine whether the contraction of
%% -----File: 387.png---Folio 357-------
the sun might not supply it with heat to take the place of
that which it radiates so lavishly. With the insight of
genius, Helmholtz saw the nature of the question and foresaw
\index[xnames]{Helmholtz}%
its probable answer. In 1854, at a celebration in commemoration
of the philosopher Kant, he gave a solution of
\index[xnames]{Kant}%
the problem under the assumption that the sun contracts
in such a way as to remain always homogeneous. With
our present data regarding its rate of radiation, its volume,
and its mass, it is found by the methods of Helmholtz that,
under the assumption that it is homogeneous and remains
homogeneous during its shrinking, a contraction of its radius
of $120$~feet per year would produce as much heat as it radiates
annually. This contraction is so small that it could
not be detected from the distance of the earth with our most
powerful telescopes in less than $10,000$~years.

So far in this discussion it has been assumed that the sun
contracts, and the consequences of the contraction have been
deduced. It remains to consider the question whether under
the conditions which prevail it actually does contract. The
reason it does not at once shrink under the mutual gravitation
of its parts is that its high temperature gives it a great
tendency to expand. As it radiates energy into space its
temperature doubtless falls a little; the decrease in temperature
permits it to contract a little; the contraction produces
heat which momentarily restores the equilibrium;
and so on in an endless cycle. This conclusion is certainly
correct, as Ritter and Lane proved about 1870, provided
\index[xnames]{Lane}%
\index[xnames]{Ritter}%
the sun behaves as a monatomic gaseous body. Moreover,
Lane established the fact, known as Lane's paradox, that
\index{Lane's law!paradox}%
so long as a purely gaseous body cools and contracts, its
temperature rises, because, with decreasing volume and
greater concentration of matter, the gravitational forces
can withstand stronger expansive tendencies due to high
temperature. If, with increasing concentration, the laws of
gases fail because the deep interior becomes liquid or solid,
the temperature might no longer increase.
%% -----File: 388.png---Folio 358-------

The question of the variation in the rate of radiation of a
contracting sun with increasing age is an important one.
Lane showed that, so long as the sun obeys the law of gases,
\index{Lane's law}%
\index[xnames]{Lane}%
its temperature is inversely as its radius. By Stefan's law
\index{Stefan's law}%
\index[xnames]{Stefan}%
the rate of radiation is proportional to the fourth power of
the absolute temperature. Consequently the rate of radiation,
per unit area, of a contracting gaseous sphere is inversely
as the fourth power of its radius. But the whole
radiating surface is proportional to the square of the radius.
Therefore the rate of radiation of the entire surface of a
contracting gaseous sphere is inversely as the square of its
radius. That is, according to this theory, the earth received
continually more and more heat until the sun ceased to be
perfectly gaseous, if, indeed, it has yet reached that stage.
When the sun's radius was twice as great as it is at present
it gave the earth one fourth as much heat, and the theoretical
temperature of the earth (\Artref{172}) was about $200$~degrees
lower than at present.

\Article{217}{Other Theories of the Sun's Heat.}---A number of
\index{Sun's heat!combustion theory of}%
\index{Sun's heat!meteoric theory of}%
other hypotheses as to the source of the sun's energy have
been advanced, but they are all inadequate. They will be
enumerated here in order that the reader may not suppose
that they are important, and that astronomers have failed
to consider them.

The most obvious suggestion is that the sun started hot
and is simply cooling. If it had the very high specific heat
of water, at its present rate of radiation its mean temperature
would fall $2.57$~degrees annually. On referring to its present
temperature, it is seen that its radiation could not continue
more than a few thousand years, and that a few thousand
years ago its rate of radiation must have been several times
that at present. These results are absurd and show the
falsity of the suggestion.

It is natural to associate heat with something burning,
and one naturally inquires whether the heat of the sun
cannot be accounted for by the combustion of the material
%% -----File: 389.png---Folio 359-------
of which it is composed. In considering this hypothesis the
first thing to be noted is that the same material will burn
only once. It is found from the amount of heat produced
by coal that if the sun were entirely made up of the best
anthracite coal and oxygen in such proportion that when
the combustion was completed there would be no residue
of either, the heat generated would supply the present rate
of radiation less than $1500$~years. If none of the heat produced
by the combustion were radiated away, and if
the specific heat of the sun were unity, the temperature of
the sun would rise to only about one third of its present
value. Consequently this theory is even less satisfactory
than the preceding.

Shortly after the discovery of the law of the conservation
of energy the large amount of heat generated by the impact
of meteors was established. The heat generated by a
meteor striking into the earth's atmosphere at the average
rate of $25$~miles per second is about $100$~times as great as
would be produced by its combustion if it were oxygen and
anthracite coal. A meteor would fall into the sun from
the distance of the earth with a velocity of about $380$~miles
per second, and since the energy is proportional to the square
of the velocity, the heat generated would be about $23,000$~times
that produced by the combustion of an equal amount
of carbon and oxygen. Lord Kelvin supposed that possibly
\index[xnames]{Kelvin}%
enough meteors strike into the sun to replenish the energy
it loses by radiation.

A complete answer to the meteoric theory of the sun's
heat is that it requires an impossibly large total mass for
the meteors. They could not possibly exist in sufficient
numbers within the earth's orbit; and, if they came from
without, they would strike the earth in enormously greater
numbers than are observed. In fact, computation shows
that if the heat of the sun were due to meteors coming into
it from all directions and from beyond the earth's orbit, the
earth would receive $\frac{1}{236}$~as much heat directly from the
%% -----File: 390.png---Folio 360-------
meteors as it receives from the sun. This is millions of times
more heat than the earth receives from meteors, and, consequently,
the theory that the sun's heat is maintained by the
impact of meteors is untenable.

\Article{218}{The Past and the Future of the Sun on the Basis
of the Contraction Theory.}---The contraction theory of
\index{Sun!past and future of}%
the sun's heat is the only one of those considered which
even begins to satisfy the conditions a successful theory must
meet. If it is the only important source of the sun's heat,
it is possible to determine, at least roughly, how long the
sun can have been radiating at its present rate, and how
long it can continue to radiate in the future.

Computation shows that if the sun had contracted from
infinite expansion, the widest possible dispersion, the total
amount of heat generated would have been less than $20,000,000$~times
the amount now radiated annually. If it had contracted
only from the distance of the earth's orbit, the amount
of heat that would have been generated would have been
about one half of one per cent less. Therefore, according
to the contraction theory, the earth can have received heat
from the sun at its present rate only about $20,000,000$~years.
If the sun is strongly condensed at its center, this
time limit should be increased about $5,000,000$~years.

In the future, according to this theory, the sun will contract
more and more until it ceases to be gaseous. Probably
by the time its mean density equals~5 its temperature will
begin to fall. A contraction to this density will produce
enough heat to supply the present rate of radiation only
$10,000,000$~years. Then, \emph{if the sun's contraction is the only
important source of its energy}, its temperature will begin to
fall, its rate of radiation will diminish, the temperature of
the earth will gradually decline, and all life on the earth will
eventually become extinct. The sun, a dead and invisible mass,
will speed on through space with its retinue of lifeless planets.

\Article{219}{The Age of the Earth.}---After the development of
\index{Age of earth}%
\index{Earth!age of}%
the contraction theory of the sun's heat, physicists, among
%% -----File: 391.png---Folio 361-------
whom Lord Kelvin was especially prominent, informed the
\index[xnames]{Kelvin}%
{\stretchyspace%
geologists and biologists in rather arbitrary terms that the
earth was not more than $25,000,000$ years of age, and that}
all the great series of changes with which their sciences
had made them familiar must have taken place within this
time. But no one science or theory should be placed above
all others, and other lines of evidence as to the age of the
earth are entitled to a full hearing. If they should unmistakably
agree that the earth is much more than $25,000,000$ years
of age, the inevitable conclusion would be that the
contraction theory is not the whole truth. This is a matter
of the greatest importance, for not only is it at the foundation
of the interpretation of geological and biological evolution,
but it bears vitally on the question of the age of the
stars and on the past and the future of the sidereal universe.

One of the simplest methods employed by geologists for
determining the age of the earth is that of computing the
time necessary for the oceans to acquire their salinity. The
\index{Salinity of the oceans}%
rivers that flow into the oceans carry to them various kinds
of salts in solution; the water that is evaporated from them
leaves these minerals behind. Consequently the salinity of
the oceans continually increases. It is clear that it is
possible to compute the age of the oceans from the present
amount of salt in them and the rate at which it is being
carried into them. Of course, it is necessary to make some
assumptions regarding the rate at which salt was carried to
the sea in earlier geological ages. The last factor is somewhat
uncertain, but this method has led to the conclusion
that the interval which has elapsed since the oceans were
formed and salt began to be carried down into them is
more than $60,000,000$~years, and that it is probably from~$90,000,000$
to $140,000,000$~years.

Nearly all the rocks that are exposed on the surface of
the earth are stratified. This means that, on the whole,
they have been formed from silt carried by the wind and
water and deposited on the bottoms of lakes or oceans.
%% -----File: 392.png---Folio 362-------
These stratified deposits are in many places of enormous
thickness. When it is remembered that the present rocks are
usually not the result of the simple disintegration and deposition
of the original earth material, but that most of them
have been repeatedly broken up and redeposited, it is evident
that the time required for the great stratification which
is now observed is enormous. There is obviously much chance
for divergence of views regarding the rates at which these
processes have gone on, but nearly every calculation on this
basis has led to the conclusion that the time since the disintegration
and stratification of the earth's rocks began is
at least $100,000,000$~years, and most of them have reached
much larger figures. The disintegration and total destruction
of mountains and plateaus is a closely related process
and leads to the same results.

The rocks of the earliest geological formations contain
\index{Fossils, occurrence of}%
only a few fossils, and they are of primitive forms of life.
Later rocks contain the remains of higher forms of plants
and animals, until finally the vertebrates and the highest
types existing at the present time are found. Obviously
an enormous interval of time has been required for all this
great series of changes in life forms to have taken place, but
it is difficult to make a numerical estimate. Huxley gave the
\index[xnames]{Huxley}%
question much attention and thought a billion years would be
necessary for the evolution. The recent discovery of mutation
has shown that the process of evolution, at least in plants,
may be more rapid than he supposed; but, on the whole,
biologists feel that the contraction theory of the sun's heat
sets much too restricted limits for the age of the earth.

The most recent, and possibly the best, method of arriving
at the age of the earth has followed the discovery of radioactive
substances. Uranium degenerates by a slow breaking
\index{Uranium}%
up of its atoms in which radium, lead, and helium are evolved.
\index{Helium}%
\index{Radium}%
From the relative proportions of these products in certain
rocks it is possible to compute the time during which degeneration
has been going on in them. This method has led
%% -----File: 393.png---Folio 363-------
to a greater age for the earth than any other. Strutt, in England,
\index[xnames]{Strutt}%
Boltwood, of Yale, and many others have given this
\index[xnames]{Boltwood}%
method a large amount of study, and have obtained figures
reaching up into several hundreds of millions of years.
Boltwood, especially, has found that the geologically older
rocks show greater antiquity by this method of determining
their age, and he reaches the conclusion that some of them
are nearly $2,000,000,000$~years old.

It is difficult to reach a positive conclusion regarding the
age of the earth from this conflicting evidence. The geological
methods point to an age for the earth since erosion
began of at least $100,000,000$~years. Geologists do not see
how the facts in any of their lines of attacking the problem
can be brought into harmony with the theory that the sun
has been furnishing light and heat to the earth for only
$25,000,000$~years. This discrepancy between their figures
and those given by the contraction theory cannot be ignored,
and therefore we are forced to the conclusion that
the sun has other important sources of heat energy besides
its contraction. Aside from this, the fact that a contracting
gaseous mass radiates inversely as the square of its
radius gives a distribution of the radiation of solar energy
altogether at variance with geological evidence.

A possible source of energy for the sun which has not been
considered here as yet is that liberated in the degeneration
of radioactive elements. It is not certain that uranium and
\index{Radioactivity in sun}%
radium exist in the sun, but helium, which is one of the
\index{Helium}%
\index{Radium}%
products of the disintegration of these elements, exists there
\index{Disintegration!of matter}%
in abundance; in fact, it is called helium because it was
first discovered in the sun (Greek, \textit{helios}~=~sun), and gives
presumptive evidence of uranium and radium being there,
too. The disintegration of uranium and radium is accompanied
\index{Energy!from radium}%
by the evolution of an enormous quantity of heat,
the energy liberated by radium being about $260,000$~times
that produced by the combustion of an equal weight of coal
and oxygen. These results are startling, and at first it
%% -----File: 394.png---Folio 364-------
seems that if a small fraction of the sun were radium or
uranium, its radiation of energy would be almost indefinitely
prolonged.

If one part in~$800,000$ of the sun were radium, heat would
\index{Sun's heat!subatomic energy theory of}%\DPnote{** sub-atomic}
be produced from this source alone as fast as it is now being
radiated, but in less than $2000$~years half of the radium would
be gone and the production of heat would correspondingly
diminish. Or, to go backward in time, only $2000$~years
ago the amount of radium would have been twice as great
as at present, and the production of heat would have been
twice as rapid. Since this conclusion is not in harmony
with the facts, the hypothesis that the sun's heat is largely
due to the disintegration of radium is untenable.
\index{Disintegration!of matter}%

Now consider uranium, which degenerates $3,000,000$~times
more slowly than radium. In the case of this element
the slowness of the rate of degeneration presents a difficulty.
If the sun were entirely uranium, heat would not be produced
more than one third as fast as it is now being radiated.
But in the deep interior of the sun where the temperature
and pressure are inconceivably high, the release of
the subatomic energies may possibly be much more rapid
than under laboratory conditions, and the process may not
be confined to the elements which are radioactive at the surface
of the earth. There is no laboratory experience to support
this suggestion because within the range of experiment
the rates of the radioactive processes have been found to
be independent of temperature and other physical conditions.
But, if there is something in the suggestion, and
especially if under the conditions prevailing in the sun the
subatomic energies of all elements are released, the amount
of energy may be sufficient for hundreds and even thousands
of millions of years. But at once the question regarding the
origin of the subatomic energies arises, and, at present, there
is no answer to it.
%% -----File: 395.png---Folio 365-------


\Section{XV}{QUESTIONS}

1. How many horse power of energy per inhabitant is received
by the earth from the sun?

2. What is the average amount of energy per square yard received
by the whole earth from the sun?

3. Does the energy which is manifested in the tides come from
the sun? What becomes of the energy in the tides?

4. What becomes of that part of the sun's energy which is
absorbed by the earth's atmosphere?

5. If the earth's atmosphere absorbs $35$~per cent of the energy
which comes to it from the sun, how can the atmosphere cause the
temperature of the earth's surface to be higher than it would otherwise
be?

6. Show from the rate at which the earth receives energy from
the sun, the size of the sun, and the earth's distance from the sun,
that the sun radiates $70,000$~horse power of energy per square
yard.

7. Taking the earth's mean temperature as $60°$~F. and the rates
of radiation of the earth (see question~2) and of the sun, compute
the temperature of the sun on the basis of Stefan's law.

8. All scientists agree that the earth is more than $5,000,000$~years
old. On the hypothesis that the contraction of the sun is its
only source of heat, and that during the last $5,000,000$~years
it has radiated at its present rate, what were its radius and density
at the beginning of this period? On the basis of Lane's law, what
was its temperature? On the basis of Stefan's law, what was its
rate of radiation per unit area and as a whole? On the basis of the
method of \Artref{172}, what was the mean temperature of the earth?

\normalsize


\Section{II}{Spectrum Analysis}

\Article{220}{The Nature of Light.}---In order to comprehend
\index{Light!nature of}%
the principles of spectrum analysis it is necessary to understand
the nature of light. A profound study of the fundamental
properties of light was begun by Newton, but,
\index[xnames]{Newton}%
unfortunately, some of his basal conclusions were quite
erroneous. Thomas Young (1773--1829) laid the foundation
\index[xnames]{Young, Thomas}%
of the modern undulatory theory of light. That is, he
established the fact that light consists of waves in an all-pervading
medium known as the \textit{ether}, by showing that when
%% -----File: 396.png---Folio 366-------
two similar rays of light meet they destroy each other where
their phases are different, and add where their phases are
the same. These phenomena, which are analogous to those
exhibited by waves in water, would not be observed if
Newton's idea were correct that light consisted of minute
\index[xnames]{Newton}%
particles shot out from a radiating body.

Physical experiments prove that light waves in the ether
are at right angles to the line of their propagation, like the
up-and-down waves which travel along a steel beam when
it is struck with a hammer, or the torsional waves that are
transmitted along a solid elastic body when one of its ends
is suddenly twisted. In an ordinary beam of light the
vibrations are in every direction perpendicular to the line
of propagation. If the vibrations in one direction are destroyed
while those at right angles to it remain, the light
is said to be \textit{polarized}. Many substances have the property
\index{Light!polarized}%
of polarizing light which passes through them.

The distance from one wave to the next for red light is
\index{Light!wave lengths of}%
\index{Wave length of light}%
about $\frac{1}{40,000}$~of an inch, and for violet light about $\frac{1}{70,000}$~of an
inch. There are vibrations both of smaller and greater wave
lengths. The range beyond the violet\footnote
  {Excepting the so-called X-rays, which are much shorter.}
is not very great,
for, even though very short waves are emitted by a body,
they are absorbed and scattered by the earth's atmosphere
before reaching the observer; but there is no limit in the
other direction to the lengths of rays. Langley explored the
\index[xnames]{Langley}%
so-called heat rays of the sun with his bolometer far beyond
\index{Bolometer}%
those which are visible to the human eye. The waves
used in wireless telegraphy, which differ from light waves
only in their length, are often hundreds of yards long.

\Article{221}{On the Production of Light.}---A definite conception
\index{Light!production of}%
of the way in which matter emits radiant energy is
important for an understanding of the principles of spectrum
analysis, but, unfortunately, the fundamental properties
of matter are involved, and physicists are not yet in agreement
on the subject. However, the theory that radiant
%% -----File: 397.png---Folio 367-------
energy is due to accelerated electrons is in good standing and
\index{Electrons}%
gives a correct representation of the principal facts.

The molecules of which substances are composed are
themselves made up of atoms. The atoms were generally
supposed to be indivisible until the year 1895, when the
cathode and X-rays prepared the way for the recent discoveries
in radioactivity and subatomic units. In connection
with these discoveries it was found that the atoms
are made up of numerous still smaller particles, called \textit{electrons}
or \textit{corpuscles}. An atom, according to the hypothesis
\index{Corpuscles}%
of Rutherford, is composed of a small central nucleus, carrying
\index[xnames]{Rutherford}%
a positive charge of electricity, and one or more rings
\begin{figure}[hbt]%[Illustration:]
\Input{397}{png}
\Caption[Model of atom, non-radiating at left and radiating at right.]{Fig}{137}
\end{figure}%
of electrons carrying (or perhaps consisting of) negative
charges of electricity, which revolve around the positive
nucleus at great speed. Under ordinary circumstances the
electrons revolve in circular paths with uniform speed, all
those of a given ring traveling in the same circle. Under
these circumstances, represented in the left of \Figref{137},
the atom is not radiating.

When a body is highly heated the molecules and atoms
of which it is composed are in very rapid motion and jostle
against one another with great frequency. These impacts
disturb the motions of the electrons and cause them to
describe wavy paths in and out across the circles in which
they ordinarily move. This condition is shown at the
%% -----File: 398.png---Folio 368-------
right in \Figref{137}. These small vibrations, which are
periodic in character, produce light waves in the ether;
and light waves are also produced by the impacts themselves,
but they are not periodic.

The character of the motions of the corpuscles can be
understood by considering a bell. Suppose it is suspended
by a twisted cord which is rapidly untwisting. A ring of
particles around the bell corresponds to a ring of corpuscles
in an atom. If the bell is simply rotating, it gives out no
sound. Suppose it strikes something. The particles of
which it is composed vibrate rapidly in and out; this, combined
with its rotation, causes them to describe wavy paths
across their former circular orbits. These waves produce
the sound. Of course, it is not necessary that the bell should
be rotating in order to produce sound, and in this respect
the analogy is imperfect.

The frequency of the vibrations of a corpuscle in an atom
is astounding. The length of a light wave of yellow light
is in round numbers $\frac{1}{50,000}$~of an inch. In a second of time
enough waves are emitted to make a line of them $186,000$~miles
along. Therefore, the number of oscillations per
second of the corpuscles in an atom is in round numbers
$600,000,000,000,000$.

It has often been suggested that the atoms of all the
chemical elements are made out of exactly the same kind of
electrons. Certainly there is as yet no evidence to the contrary.
If the electrons are not composite structures themselves,
the idea is reasonable enough; but if they are made
up of still smaller units, the hypothesis seems improbable.

The dynamics of an atom, according to the corpuscular
theory, is of much interest. The positive nucleus attracts
the revolving negative corpuscles. They are kept from falling
in on the nucleus both by the centrifugal force due to
their rapid revolution, and also by their mutual repulsions
which result from their being similarly electrified. If the
number of corpuscles in a ring is small, the atom is stable.
%% -----File: 399.png---Folio 369-------
With an increasing number of corpuscles the stability of the
atom diminishes. Finally, the atom is stable only if the
corpuscles revolve in two or more rings. The regions of
instability which separate atoms having a certain number
of rings from those having other numbers possibly give a
clue to the celebrated periodic law of the chemical elements
discovered by Mendeléeff.
\index[xnames]{Mendeleeff@{Mendeléeff}}%

\Article{222}{Spectroscopes and the Spectrum.}---The energy
\index{Spectrum!analysis}%
\index{Spectroscope}%
which a body radiates is completely characterized by the
wave lengths which it includes and their respective intensities.
The spectroscope is an instrument which enables us
to analyze light into its parts of different wave lengths, and
to study each one separately.

There are three principal types of spectroscopes. In the
first and oldest type the light passes through one or more
prisms; in the second, perfected by Rowland and Michelson,
\index[xnames]{Michelson}%
\index[xnames]{Rowland}%
the light is reflected from a surface on which are ruled
many parallel equidistant lines; and in the third, invented
\index{Grating spectroscope}%
by Michelson, the light passes through a pile of equally
thick plane pieces of glass piled up like a stairway. The
first type is most advantageous when the source of light is
faint, like a small star, comet, or nebula. Its chief fault is
that the scale of the spectrum is not the same in all parts.
The second type is advantageous for bright sources of light
like the sun or the electric arc in the laboratory. It gives
the same scale for all parts of the spectrum, but uses only
a small part of the incident light. The third type, known
as the \textit{echelon}, gives high dispersion without great loss of
\index{Echelon spectroscope}%
light. Only the first type, which is most used in astronomy,
will be more fully described here.

The basis of the prism spectroscope is the refraction and
\index{Prism spectroscope}%
the dispersion of light when it passes through a prism. Let~$L$,
\Figref{138}, represent a beam of white light which passes
through the prism~$P$. As it enters at~$A$ from a rarer to a
denser medium, it is bent \emph{toward} the perpendicular to the
surface; and as it emerges at~$B$ from a denser to a rarer
%% -----File: 400.png---Folio 370-------
medium, it is bent \emph{from} the perpendicular to the surface.
This change in the direction of the beam of light is its
\textit{refraction}.
\index{Light!refraction of}%
\index{Refraction}%

Not only is the beam of light refracted, but it is also spread
out into its colors. As it enters the prism the violet light
is refracted the most and the red the least, and the same thing
is true when it emerges. %[Illustration: Break]
\begin{wrapfigure}[13]{\WLoc}{3in}
\Input[3in]{400}{png}
\Caption[Refraction and dispersion of light
by a prism.]{Fig}{138}
\end{wrapfigure}
Consequently, instead of a beam
of white light falling on the screen~$S$ there is found a band of
colors which, in order from the most refracted to the least
refracted, are violet,
indigo, blue, green,
yellow, orange, and
red. This separation
of light into its colors
is called \textit{dispersion}.
\index{Light!dispersion of}%

In the diagram only
the visible part of the
spectrum is indicated.
Beyond the red are
the infra-red, or heat, rays \textit{I-R}, and beyond the violet are
the ultra-violet rays \textit{U-V}. The colors are not separated by
sharp boundaries, but shade from one to another by insensible
gradations. The ultra-violet part of the spectrum is
several times as long as the visible part, and the infra-red
part is several times as long as the ultra-violet part.

While \Figref{138} shows exactly the way in which a spectrum
\index{Light!absorption of}%
might be formed, it would be too faint to be of any
value in practice. In order to obtain a bright spectrum the
apparatus is arranged as sketched in \Figref{139}, though in
practice several prisms, one after the other, are often employed.
The rays which pass through the screen at~$O$ are
made parallel by the lens~$L_1$. They strike the prism~$P$ in
parallel lines, and those of a given color continue through~$P$
and to the lens~$L_2$ in parallel lines (the dispersion is not indicated
in the diagram). The lens~$L_2$ brings the rays to a
focus at~$F$, and the eyepiece~$E$ sends all those of each color
%% -----File: 401.png---Folio 371-------
out in a small bundle of parallel lines (only one color is represented
in the diagram). The eye is placed just to the right
of~$E$, and all the parallel rays of each bundle are brought to
a focus at a point on the retina. In this way many rays of
each color are brought to a focus at the same place in the
observer's eye.

While strictly white light gives all colors, it is not necessary
that a luminous body should emit all kinds of light, or
that all colors emitted should be given out in equal intensity.
\begin{figure}[hbt]%[Illustration:]
\Input{401}{png}
\Caption[A spectroscope having only one prism.]{Fig}{139}
\end{figure}%
In fact, it is well known that if a body is simply warm but
not self-luminous, it gives out in sensible quantities only
infra-red rays. If it is extremely hot, it may radiate mostly
ultra-violet rays.

\Article{223}{The First Law of Spectrum Analysis.}---The first
\index{Laws!of spectrum analysis}%
\index{Spectrum!analysis, laws of}%
theoretical discussion of the principles of spectrum analysis
which reached approximately correct conclusions was made
by Ångström in 1853. The work of Bunsen, and especially
\index[xnames]{Angstrom@{Ångström}}%
\index[xnames]{Bunsen}%
of Kirchhoff in 1859, put the subject on essentially its present
\index[xnames]{Kirchhoff}%
basis. The laws of spectrum analysis as formulated here
are consequences of a general law due to Kirchhoff, and of
certain experimental facts. After they have been stated,
they will be seen to be simple consequences of the mode of
production of radiant energy.

The first law of spectrum analysis is: \emph{A radiating solid,
liquid, or gas under high pressure gives a continuous spectrum
whose position of maximum intensity depends upon the temperature
of the source; and conversely, if a spectrum is continuous,
%% -----File: 402.png---Folio 372-------
the source of light is a solid, liquid, or gas under high
pressure, and the position of radiation of maximum intensity
determines the temperature of the source.}

This law means, in the first place, that a radiating solid,
liquid, or gas under high pressure gives out light, or more
generally radiant energy, of all wave lengths; and, in the
second place, the wave length at which the radiation is most
intense depends upon the temperature of the source. It is
clear from the way in which light is produced that the first
part of the law should be true. When a body is in a solid
or liquid state, or when it is a gas under high pressure, the
molecules are so close together that they continually interfere
with one another. Under these circumstances the oscillations
of the corpuscles cannot take place in their natural
periods, but they are altered in all possible manners. This
results in vibrations of all periods, and therefore the spectra
are continuous.

The way in which the wave length of maximum radiation
depends upon the temperature is given by Wien's law\footnote
  {Experiments show that this law does not give good results for low
  temperatures, but the applications in astronomy are to high temperatures.}---
\index{Wien's law}%
\index[xnames]{Wien}%
\[
\lambda = \frac{ 0.2076 }{ T },
\]
where $\lambda$~is the wave length in inches and $T$~is the absolute
temperature on the Fahrenheit scale. For example, if the
temperature of the sun is~$10,000°$, its wave length of maximum
radiation is about $\frac{1}{50,000}$~of an inch.

\Article{224}{The Second Law of Spectrum Analysis.}---The
second law of spectrum analysis is: \emph{A radiating gas under
low pressure gives a spectrum which consists of bright lines
whose relations to one another and whose positions in the
spectrum\footnote
  {The positions of lines in a spectrum determine, of course, their relations
  to one another; but in practice the lines of an element are usually identified
  by their relations to one another, just as a constellation is recognized by the
  relative positions of its stars.}
depend upon the nature of the gas (and in some
%% -----File: 403.png---Folio 373-------
cases to some extent upon its temperature, density, electrical
and magnetic condition); and conversely, if a spectrum consists
of bright lines, then the source is a radiating gas (or
gases) under low pressure, and the composition of the gas (or
gases) can be determined from the relations of the lines to one
another and from their positions in the spectrum.}

When molecules are free from all restraints the oscillations
of their electrons take place in fixed periods which
depend upon the internal forces involved, just as free bells
of given structure vibrate in definite ways and give forth
sounds of definite pitch. Consequently, free radiating molecules
emit light of one or more definite wave lengths depending
on the structure of the molecules, and there are
\begin{figure}[hbt]%[Illustration:]
\Input{403}{png}
\Caption[A bright-line spectrum above and a reversed spectrum below.]{Fig}{140}
\end{figure}%
bright lines at corresponding places in the spectrum and no
light whatever at other places. A bright-line spectrum is
shown in the top part of \Figref{140}. Some elements give
only a few lines and others a great many. For example,
sodium has but two lines, both in the yellow, and iron more
than $2000$~lines. It is needless to say that all these facts
are established by laboratory experiments.

It may be objected that in a gas, even under low pressure,
the molecules are not free from outside interference, for they
collide with one another many millions of times per second.
But the intervals during which they are in collision are very
short compared with the intervals between collisions. Consequently,
while there will be some light of all wave lengths,
it will be inappreciable compared to that which is characteristic
of the radiating gas, and the spectrum will seem to consist
%% -----File: 404.png---Folio 374-------
of bright lines of various colors on a perfectly black
background.

\Article{225}{The Third Law of Spectrum Analysis.}---The third
law of spectrum analysis is: \emph{If light from a solid, liquid, or
gas under great pressure passes through a cooler gas (or gases),
then the result is a bright spectrum which is continuous except
where it is crossed by dark lines, and the dark lines have the
positions which would be occupied by bright lines if the
intervening cooler gas were the source of light, and conversely,
if a bright spectrum is continuous except where it is
crossed by dark lines, then the source of light is a solid, liquid,
or gas under great pressure, and the light has passed through
a cooler intervening gas (or gases) whose constitution can be
determined from the relations of the dark lines to one another
and from their positions in the spectrum.}

In a word, a cool gas absorbs the same kinds of rays it
would give out if it were incandescent, and no others. Similarly,
a musical instrument absorbs tones of the same pitch
as those which it can produce. For example, if the key for
middle~\textit{C} on a piano is held down and this tone is produced
near by, the piano will respond with the same tone; but if
\textit{D} is produced, the piano will give no response. This phenomenon
occurs in many branches of physics and is very
important. For example, it is at the basis of wireless telegraphy.
The receiving instrument and the sending instrument
are tuned together, and only in this way do the effects
of the feeble waves which reach to great distances become
sensible. The fact that the sending and receiving instruments
must be tuned the same explains how it is that many
different wireless instruments can be working at the same
time without sensible interference.

When the intervening cooler gas absorbs certain parts of
the energy which passes through it, it becomes heated and
its rate of radiation is increased. It might be supposed that
this new radiation would make up for the energy which has
been absorbed. That which has been absorbed and that
%% -----File: 405.png---Folio 375-------
which is radiated are, indeed, exactly equal, but the radiated
energy is sent out in every direction and not alone in
the direction of the original light passing through the gas.
That is, certain parts of the original energy are taken out
and scattered in every direction. Therefore, in a spectrum
crossed by dark lines the dark lines are not absolutely black,
but only black relatively to the remainder of the spectrum.
A spectrum of this sort is called an \textit{absorption}, or \textit{dark-line},
or \textit{reversed} spectrum. The reverse of the bright-line spectrum
given in the top of \Figref{140} is shown in the bottom
part of the figure.

\Article{226}{The Fourth Law of Spectrum Analysis.}---The fourth
\index{Absorption spectrum}%
\index{Doppler-Fizeau law}%
\index{Radial velocity}%
\index{Spectrum!absorption}%
law of spectrum analysis was first discovered by Doppler
\index[xnames]{Doppler}%
and was experimentally established by Fizeau. It is commonly
\index[xnames]{Fizeau}%
called the Doppler principle, or the Doppler-Fizeau
law. It is: \emph{If the source (radiating gas in the case of a spectrum
of bright lines, and an intervening cooler gas in case of
an absorption spectrum) and receiver are relatively approaching
toward, or receding from, each other, then the lines of the
spectrum are displaced respectively in the direction of the
violet or the red by an amount which is proportional to the
relative speed of approach or recession; and conversely, if the
lines of a spectrum are displaced toward the violet or the red,
the source and receiver are respectively approaching toward,
or receding from, each other, and the relative speed of approach
or recession can be determined from the amount of the displacement.}

The explanation of the shift of the lines of the spectrum
when there is relative motion of the source and the receiver
is very simple. If the source is stationary, it sends
out wave after wave separated by a given interval; if it is
moving toward the receiver, it follows up the waves which it
emits and the intervals between them are diminished. That
is, the wave lengths have become shorter, which is only another
way of stating that the corresponding spectral lines
have been shifted toward the violet. Of course, for motion
%% -----File: 406.png---Folio 376-------
in the opposite direction the spectral lines are shifted toward
the red.

If the receiver moves toward the source, he receives not
only the waves which would reach him if he were
stationary, but also those which he meets as a consequence
of his motion. The distances between the waves
are diminished and the spectral lines are shifted toward
the violet. Motion in the opposite direction produces the
opposite results.

The formula for the shift in the spectral lines is
\[
\Delta\lambda = \frac{v}{V} \lambda,
\]
where $\Delta\lambda$~is the amount of the shift, $\lambda$~is the wave length
of the line in question, $v$~the relative velocity of the source
and receiver, and $V$~the velocity of light. Suppose $v$~is 18.6
miles per second; then, since $V$~is $186,000$~miles per second
and the greatest wave length in the visible spectrum is
nearly twice that of the shortest, the displacement is about
$\frac{1}{10,000}$~of the distance between the ends of the visible spectrum.
It follows that for the velocities with which the
planets move the displacements of the spectral lines are
very small, and that refined means must be employed in
order to determine them accurately. The usual method is
to photograph the spectrum of the distant object and at
the same time to send through the spectroscope beside it
the light from some suitable laboratory source. The lines
of the latter will of course have their normal positions.
The displacements of the lines of the celestial object with
respect to them are measured with the aid of a microscope.

When the spectral lines of an object are well defined, displacement
results of astonishing precision can be obtained.
In the case of stars of certain types the relative velocities
toward or from the earth, called \textit{radial velocities}, can be determined
to within one tenth of a mile per second.
%% -----File: 407.png---Folio 377-------


\Section{XVI}{QUESTIONS}

1. What problems can be solved approximately for the sun and
stars by the first principle of spectrum analysis?

2. What would be the character of the spectrum of moonlight?

3. Comets have continuous bright spectra crossed by still brighter
lines; what interpretation is to be made of these facts, remembering
that comets shine partly by reflected light?

4. The spectra of Uranus and Neptune contain dark lines and
bands of great intensity at the positions of the less intense hydrogen
lines of the solar spectrum; what interpretation is to be placed on
these phenomena?

5. Can the motion of the earth with respect to the sun and moon
be determined by spectroscopic means? The motion of the earth
with respect to the planets?

6. If an observer were approaching a deep red star with the velocity
of light, what color would the star appear to have? If he were
receding with the velocity of light?
\index{Radial velocity}%

7. What effect would the rapid rotation of a star have on its spectral
lines?

8. Suppose an observer examines the spectra of the eastern and
western limbs of the sun; how would the spectral lines be related?
Could they be distinguished from lines due to absorption by the
earth's atmosphere?

\normalsize


\Section{III}{The Constitution of the Sun}

\Article{227}{Outline of the Sun's Constitution.}---The apparent
\index{Sun!constitution of}%
surface of the sun is called the \textit{photosphere} (light sphere).
It has the appearance of being rather sharply defined, \Figref{141},
and it is the boundary used to define the size of the sun,
but the sun is disturbed by such violent vertical motions
that it is probably very broken in outline. At the distance
of the sun from the earth an object $500$~miles across
subtends an angle of only one second of arc, and, therefore,
irregularities in the photosphere would not be visible unless
they amounted to several hundred miles. The part of the
sun interior to the photosphere is always invisible.

Above the photosphere lies a sheet of gas, probably from~$500$
to $1000$~miles thick, which is called the \textit{reversing layer}
because, as will be seen (\Artref{233}), it produces a reversed,
%% -----File: 408.png---Folio 378-------
or absorption, spectrum. It contains many terrestrial substances,
such as calcium and iron, in a vaporous state.

Outside of the reversing layer is another layer of gas,
\index{Reversing layer}%
from~$5000$ to $10,000$~miles deep, called the \textit{chromosphere}
\index{Chromosphere}%
\begin{figure}[hbt]%[Illustration:]
\Input{408}{jpg}
\Caption[The Sun. \textit{Photographed by Fox with the $40$-inch telescope of the
Yerkes Observatory.}]{Fig}{141}
\index{Yerkes Observatory}%
\index[xnames]{Fox}%
\end{figure}%
(color sphere). At the time of a total eclipse of the sun it
is seen as a brilliant scarlet fringe whose outer surface seems
%% -----File: 409.png---Folio 379-------
to be covered with leaping flames. There are often eruptions,
called \textit{prominences}, which break up into it and ascend to
\index{Prominences}%
great heights.

The outermost portion of the sun is the \textit{corona} (crown),
\index{Corona, of sun}%
a halo of pearly light which is so much fainter than the illumination
of the earth's atmosphere that it can be seen only
at the time of a total solar eclipse. It is irregular in form and
gradually fades out into the blackness of the sky at the distance
of from~$1,000,000$ to $3,000,000$~miles from the surface
of the sun.

\Figureref{142} shows an ideal section through the sun. The
upper surface of the invisible interior is the photosphere,
\begin{figure}[hbt]%[Illustration:]
\Input{409}{png}
\Caption[Ideal section of the sun.]{Fig}{142}
\index{Photosphere}%
\end{figure}%
$R$~is the reversing layer, $S$~is a spot, $K$~is the chromosphere,
$P$~is a prominence, and $C$~is the corona.

\Article{228}{The Photosphere.}---When the sun is examined
\index{Photosphere}%
through a good telescope it presents a finely mottled appearance
instead of the uniform luster which might be expected.
The brighter parts are intensely luminous nodules,
somewhat irregular in form, $500$~or $600$~miles across. These
``rice grains,'' as they are sometimes called, have been resolved
into smaller elements having a diameter of not over
$100$~miles; and although all these granules together do not
constitute over one fifth of the sun's surface, yet, according
%% -----File: 410.png---Folio 380-------
to Langley's estimates, they radiate about three fourths of
\index[xnames]{Langley}%
the light. A small portion of the sun's surface highly
magnified is shown in \Figref{143}.

The photosphere of the sun gives a continuous spectrum.
Therefore, according to the first law of spectrum analysis,
it is a solid, liquid, or gas under great pressure. Since the
photosphere is not transparent there is a strong inclination
to infer that it is liquid, or at least consists of clouds of
liquid particles (carbon, iron, calcium, etc.) floating in a
vapor of similar substances.
But the temperature of the
sun is so high that this
conclusion is not certain.

In considering the sun it
must be remembered that
its surface gravity is nearly
$28$~times that of the earth,
and that the pressure %[Illustration: Break]
\begin{wrapfigure}[18]{\WLoc}{2.5in}
\Input[2.5in]{410}{jpg}
\Caption[Small portion of the sun's
surface, highly magnified.]{Fig}{143}
\end{wrapfigure}
under
equal masses of atmosphere
is correspondingly greater.
Hence, it is not unreasonable
to suppose that the
pressure down under the
corona, chromosphere, and
reversing layer is great enough to produce a continuous
spectrum. The conclusion that the photosphere is almost
entirely, if not altogether, gaseous is supported by the fact
that the cooler, overlying reversing layer is gaseous and
contains some of the most refractory known substances.
The ``rice-grain'' structure of the photosphere is explained
by Abbott as being due to relative motions of layers at
\index[xnames]{Abbott}%
different levels analogous to those which produce a mackerel
sky in the earth's atmosphere. He supposes that the dark
places between the ``rice-grains'' correspond to those places
where clouds form in our own atmosphere, and that they
are regions where the temperature has fallen somewhat
%% -----File: 411.png---Folio 381-------
below that of the remainder of the photosphere. There are
other astronomers, however, who believe that the bright nodules
are the summits of ascending convection currents, which,
by expansion and cooling, are reduced to the state where the
most refractory substances partially condense and radiate
most brilliantly, while the darker spaces between are where
the cooler currents descend.

The photosphere is the region from which the sun loses
energy by radiation. This energy must be supplied from
the interior. There are three processes by which heat may
be transferred from one position to another, viz., by conduction,
by convection, and by radiation. Conduction is entirely
too slow to be quantitatively adequate for bringing
heat to the surface of the sun. Convection currents might
be violent enough and might reach deep enough to bring to
the surface the requisite amount of heat. In order to get
a quantitative idea of the requirements suppose that essentially
all of the sun's radiation is from a layer of the photosphere,
of average density one tenth, $500$~miles thick. Suppose
its specific heat is unity. At the rate at which the sun
radiates, the temperature of this layer would decrease one
degree Fahrenheit in $1.6$~hours if fresh energy were not supplied
from below. Hence the requirements do not seem to
be unreasonably severe.

In a body as nearly opaque as the sun seems to be, radiation
probably is of no importance in the escape of heat from the
deep interior to the surface layers.

\Article{229}{Sun Spots.}---The most conspicuous markings ever
seen on the sun are relatively dark spots which occasionally
appear in the photosphere and last from a few days up to
several months, with an average duration of a month or two.
The typical spot consists of a round, relatively black nucleus,
called the \textit{umbra}, and a surrounding less dark belt called the
\index{Sun spots!umbrae of@{umbræ of}}%
\index{Umbra!of sun spots}%
\textit{penumbra}, \Figref{144}. The penumbra is made up of converging
\index{Penumbra!of sun spots}%
\index{Sun spots!penumbra of}%
filaments, or ``willow leaves,'' of brighter material,
which look as though the intensely luminous photospheric
%% -----File: 412.png---Folio 382-------
columns were tipped over so as to make their sides visible.
The umbra and penumbra do not gradually merge into each
other, and likewise the penumbra and surrounding photosphere
have a fairly definite line of separation.
\begin{figure}[hb]%[Illustration: Moved up]
\Input{412}{jpg}
\Caption[Great sun spot of July~17, 1905. \textit{Photographed by Fox with the
$40$-inch telescope of the Yerkes Observatory.}]{Fig}{144}
\index[xnames]{Fox}%
\end{figure}

The umbra of a sun spot may be anywhere from~$500$ to
$50,000$~miles across; the diameter of the penumbra may be
as great as $200,000$~miles. When the spots are of these
dimensions they can be seen simply with the aid of a smoked
glass to reduce the glare of the sun. The Chinese claim to
have records of observations of sun spots made centuries
before their discovery by Galileo in 1610.
\index[xnames]{Galileo}%

The umbra of a sun spot is dark only in comparison with
the glowing photosphere which surrounds it, for a calcium
light projected on it appears black. In fact, it sometimes
shows many details of darker spots and brighter streaks which
most often appear shortly before it breaks up. In the
neighborhood of spots the brightness of the photosphere is
usually above the average, and there are nearly always in
their vicinity very bright elevated masses of calcium which
constitute the \textit{faculæ}. These faculæ are especially conspicuous
\index{Faculae@{Faculæ}}%
%% -----File: 413.png---Folio 383-------
when near the sun's apparent margin, or limb, as
it is called, for in these regions the photosphere is greatly
dimmed by the extensive absorbing material through which
its rays must pass, while on the other hand the faculæ project
out through the absorbing material and shine with but
slightly diminished luster.

\Article{230}{The Distribution and Periodicity of Sun Spots.}---Sun
\index{Distribution!of sun spots}%
\index{Periodicity of sun spots}%
\index{Sun spots!distribution and periodicity of}%
\index{Sun spots!periodicity of}%
spots are rarely seen except in two belts extending from
latitude~$6°$ to latitude~$35°$ on each side of the sun's equator.
Moreover, they are not always equally numerous. For
three or four years they appear with great frequency, then
they become less numerous and decline to a minimum for
three or four years, after which they are more numerous
again. The interval from maximum number to maximum
number averages about $11.11$~years, though the period varies
from about $7$~years to more than $16$~years. When a period
is short the maximum which follows it is very marked, as
though a large amount of sun-spot activity had been crowded
into a short interval; on the other hand, when a maximum
is delayed it is below normal in activity.

There is a connection between the positions of sun spots
and their numbers, first pointed out by Schwabe in 1852.
\index[xnames]{Schwabe}%
After a sun-spot maximum has passed, the spots appear
year after year for about five years, on the average, in successively
lower latitudes, and they are continually less
numerous. At the sixth year a few are still visible in about
latitudes~$±6°$, and a new cycle starts in latitudes about~$±35°$.
After this the spots in the low latitudes disappear, those in
the higher latitudes increase in numbers, but gradually descend
in latitude until the maximum activity is reached in
latitudes~$±16°$. The areas covered by spots in years of
maximum activity are from $15$~to $45$~times those covered in
years of minimum activity. The results from 1876 to 1902
are shown in \Figref{145}.

Since accurate records of the numbers and dimensions of
sun spots have been kept, the sun's southern hemisphere
%% -----File: 414.png---Folio 384-------
has been somewhat more active than the northern. For the
\begin{figure}[hbt]%[Illustration: Moved up]
\Input{414}{jpg}
\Caption[Distribution and magnitudes of sun spots for the period from
1876 to 1902 (Maunder).]{Fig}{145}
\index[xnames]{Maunder}%
\end{figure}%
period from 1874 to 1902, $57$~per cent of the total spot area
was in the southern hemisphere of the sun and only $43$~per
cent in the northern. That is, the activity in the southern
hemisphere was about one third greater than that in the
northern. Whether this difference is permanent and what it
means cannot at present be determined.

\Article{231}{The Motions of Sun Spots.}---Individual sun spots
may drift both in latitude and in longitude, and they often
have complicated and violent internal motions. As a rule,
those spots whose latitudes are less than~$20°$ drift slowly
toward the sun's equator, and those which are in higher
latitudes drift away from it. When two spots are near together
they are generally on the same parallel of latitude.
The spot which is ahead usually moves forward with respect
to the sun's surface, while the one which is behind lags continually
%% -----File: 415.png---Folio 385-------
farther in the rear. If a large spot divides into two
components, they generally recede from each other, sometimes
at the rate of $1000$ miles an hour.

Sun spots sometimes have spiral motions, but until
recently the phenomenon was thought to be hardly characteristic
because it was observed in only a small percentage
of cases. Hale's invention of the spectroheliograph (\Artref{237})
\index{Spectroheliograph}%
\index[xnames]{Hale}%
furnished a new and powerful means of studying solar
phenomena, and it has led in recent years to a discovery
of great interest and importance in this connection.

In 1908 Hale proved the existence of magnetic fields in
\index{Sun!magnetic field of}%
the high levels of sun spots. One may well wonder how such
a result could be established, since we receive only light and
heat from the sun. Naturally it must be done from the
characteristics of the radiant energy which the sun sends to
the earth. About 1896 Zeeman found that most spectral
\index[xnames]{Zeeman}%
lines are doubled, or at least widened, when observed along
the lines of force of a magnet, and that the two components
are circularly polarized in opposite directions. Hale examined
the widened spectral lines belonging to sun spots
and proved that they have the properties of spectral lines in
a magnetic field. Then he took up the question of the
origin of the magnetic fields. It was shown by Rowland in
\index[xnames]{Rowland}%
1876 that static electric charges in revolution produce electromagnetic
effects like those produced by electric currents.
Consequently Hale concluded that the magnetic fields in
sun spots are due to vortical motions of particles carrying
static electric charges, and the explanation is almost certainly
correct.

More recently the whole sun has been found to be involved
in a magnetic field whose poles agree approximately with
its poles of rotation; it may be analogous to that which
envelopes the earth. Schuster has suggested that the magnetic
\index[xnames]{Schuster}%
states of the earth and sun may be a consequence of
their rotations, and that all rotating bodies must be magnets.

Hale's discovery is a proof of cyclonic motion in the
%% -----File: 416.png---Folio 386-------
upper parts of sun spots. Unlike cyclones on the earth, the
\index{Sun spots!polarity of}%
direction of motion in a hemisphere is not always the same.
Hale found numerous examples where two spots seemed to
\index[xnames]{Hale}%
be connected, one having one polarity and the other the
opposite (\Figref{146}). It has been suggested they are the
two ends of a cylindrical whirl. This idea is confirmed, at
least to some extent, by the fact that, so far as observational
evidence goes at present, when two spots are near together,
they always have opposite polarity. Another remarkable
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4.5in]{416}{jpg}
\Caption[Sun spots having opposite polarity. \textit{Photographed at the
Mt.~Wilson Solar Observatory with the spectroheliograph} (Hale).]{Fig}{146}
\index{Solar!Observatory}%
\end{figure}%
fact is that if two neighboring spots are in the northern hemisphere
of the sun, the one which is ahead has a counter-clockwise
%[** TN: Only instance, broken across a line in original.]
vortical motion, while the motion in the other is
in the opposite direction. The conditions are the opposite
in the sun's southern hemisphere.

Evershed, in India, announced in 1909 that at the lowest
\index[xnames]{Evershed}%
visible levels there is radial motion outward from spots
parallel to the surface of the sun. More recently St.~John,
\index[xnames]{Stjohn@{St.\ John}}%
at the Mt.~Wilson Solar Observatory (\Figref{147}), has made
extensive studies of the motions in sun spots with the advantage
%% -----File: 417.png---Folio 387-------
of most powerful instruments, and he concludes
that at the lower levels there is motion radially outward
from spot centers, at levels about $2500$~miles higher there is
\index{Sun spots!motions of}%
no horizontal motion, and in the high levels of the chromosphere
($10,000$~to $15,000$~miles) the motion is inward toward
the centers of the spots. This suggests that spots are produced
by cooler gases from high levels rushing in toward a
center, descending some thousands of miles, and then spreading
out at lower levels, but the consideration of the quality
and quantity of the materials involved in the two movements,
\begin{figure}[hbt]%[Illustration:]
\Input{417}{jpg}
\Caption[The Mt.~Wilson Solar Observatory of the Carnegie Institution
of Washington. Pasadena, California.]{Fig}{147}
\index{Mount Wilson Solar Observatory}%
\index{Solar!Observatory}%
\end{figure}%
together with their kinetic energies, led St.~John to
\index[xnames]{Stjohn@{St.\ John}}%
the conclusion that the material flowing inward and downward
by no means equals that flowing outward at lower
levels from the axes of spots. He believes, rather, that a
spot is formed by currents ascending from the sun's interior
and spreading out just above the photosphere. The in-rushing
and descending chromospheric material is a secondary
result of the primary currents. The spots are dark because
the expanding gases of which they are composed are
cooler than those which constitute the photosphere.

Independent evidence of a conclusive character shows
that spots are cooler than the ordinary photosphere. There
%% -----File: 418.png---Folio 388-------
is evidence from the so-called enhanced spectral lines which
has been brought out by Hale, Adams, and Gale; the lines
\index[xnames]{Adams, W. S.}%
\index[xnames]{Gale}%
\index[xnames]{Hale}%
in the spectrum of spots are related to those in the spectrum
of the remainder of the sun just as the spectra with low temperatures
in the electric furnace are related to those with
high temperatures; and finally, the spectra of spots contain
flutings, or bands, which are believed to be due to absorption
by chemical compounds which would be broken up
into their constituent elements in the higher temperatures of
the photosphere.

\Article{232}{The Rotation of the Sun.}---The rotation of at least
\index{Rotation!of sun}%
\index{Sun!rotation of}%
that part of the sun in which the spots occur can be found
from their apparent transits across its disk. The first
systematic investigation of the sun's rotation was made by
Carrington and Spoerer about the middle of the nineteenth
century. They found that the sun rotates from west to east
about an axis inclined $7°$~to the perpendicular to the plane
of the ecliptic. The sun's axis points to a position whose
right ascension and declination are respectively $18$~h.\ $44$~m.\ and~$+46°$,
which is almost exactly midway between Vega
and Polaris. The period of the solar rotation depends upon
the latitude. Spots near the sun's equator complete a revolution
in about $25$~days; in latitude~$30°$, in about $26.5$~days;
in latitude~$45°$, in about $27.5$~days; in higher latitudes spots
are not seen.

Reference has already been made to the faculæ, or bright
\index{Faculae@{Faculæ}!periodicity of}%
clouds, which are especially abundant in the neighborhood
of sun spots. The positions of the faculæ are easily determined
on photographs of the sun, and from photographs
made at sufficiently short intervals the rotation of the sun
can be found. This method has given results in accord with
those obtained from observations of spots.

The remarkable developments of spectroscopic methods
which followed Hale's invention of the spectroheliograph
have furnished a third method of measuring the rotation of
the sun. By its use bright clouds of calcium vapor, called
%% -----File: 419.png---Folio 389-------
\textit{flocculi} by Hale, and both bright and dark flocculi of hydrogen
\index{Flocculi}%
\index[xnames]{Hale}%
have been photographed. The rotation of the sun has
been determined by Hale and Fox from photographs of
flocculi.

Finally, the rotation of the sun has been determined by
the Doppler-Fizeau effect. One limb of the sun at the equator
\index{Doppler-Fizeau law}%
\index[xnames]{Doppler}%
\index[xnames]{Fizeau}%
approaches the earth at the rate of $1.3$~miles per second,
while the other recedes at the same velocity. The spectroscopic
method is so highly developed that it not only gives
the rate of rotation of the sun approximately, but it shows
that the period is shorter at the equator than it is in higher
latitudes.

The results for the periods of rotation of the sun by the
various methods are given in the \hyperref[Table:X]{following table}, in which
the results are expressed in mean solar days.\DPnote{** TN: Change ":" to "."}

\begin{table}[hbtp]
\begin{center}
\Caption{Table}{X}% Periods of rotation of the sun
\TFontsize
\setlength{\tabcolsep}{4pt}%
\settowidth{\TmpLen}{\THF Sun Spots}
\begin{tabular}{|*{6}{c|}}
\hline
\TEntry{\TmpLen}{\THead Latitude}  &
\TEntry{\TmpLen}{\THead Sun Spots}  &
\TEntry{\TmpLen}{\THead Faculæ}  &
\TEntry{\TmpLen}{\THead Calcium Flocculi} &
\TEntry{\TmpLen}{\THead Hydrogen Flocculi} &
\TEntry{\TmpLen}{\medskip\THead Doppler-Fizeau Method\medskip} \\
\hline
\Strut
$\phantom{0}0\rlap{$°$}$~~to $\phantom{0}5$\rlap{$°$} &     $25.00$   &    $24.73$  &    $24.76$   &    $25.7$   &       $24.67$ \\
$\phantom{0}5$~~to $10$         &     $25.09$   &    $24.79$   &   $24.98$   &    $25.0$   &       $24.86$ \\
$10$~~to $15$         &     $25.26$   &    $25.12$   &   $25.17$   &    $24.7$   &       $25.12$ \\
$15$~~to $20$         &     $25.48$   &    $25.33$   &   $25.48$   &    $24.8$   &       $25.44$ \\
$20$~~to $25$         &     $25.75$   &    $25.37$   &   $25.73$   &    $24.5$   &       $25.81$ \\
$25$~~to $30$         &     $26.09$   &    $25.64$   &   $25.77$   &    $24.5$   &       $26.20$ \\
$30$~~to $35$         &     $26.47$   &    $26.47$   &   $26.18$   &    $24.2$   &       $26.67$\rule[-2ex]{0pt}{0pt} \\
\hline
\end{tabular}
\end{center}
\end{table}

By the Doppler-Fizeau method Adams found the periods
\index[xnames]{Adams, W. S.}%
of rotation of the sun in latitudes $45°$,~$60°$, and~$74°$, to be
respectively $28.1$,~$31.3$, and $32.2$~days.

The reason that the sun rotates in its peculiar manner is
not certainly known, though Elliott Smith has attempted
\index[xnames]{Smith}%
to show that the more rapid rotation of the equatorial zone
is an inevitable consequence of the contraction of a rotating
mass of gas. The question deserves further quantitative
examination.
%% -----File: 420.png---Folio 390-------

Under the hypothesis that the sun is a mixture of fluids
in equilibrium, Wilsing, Sampson, and Wilczynski have
\index[xnames]{Sampson}%
\index[xnames]{Wilczynski}%
\index[xnames]{Wilsing}%
reached the conclusion from hydrodynamical considerations
that cylindrical layers of it rotate with the same speed.
According to this view the outermost cylinder, which includes
only the equatorial zone, rotates fastest, and successive
cylinders toward the axis rotate more and more slowly.
It is supposed that this condition is inherited from some
primitive state and that friction has not yet reduced the
rotation to uniformity. Wilczynski showed that friction
between the different layers would not wear down the differences
of motion appreciably in many millions of years.
But he neglected the convection currents which must certainly
exist to great depths and which would quickly destroy
the supposed different rotations in different cylinders.
Notwithstanding these difficulties, no other theory at present
is more satisfactory than that the sun's peculiar rotation has
been inherited from more extreme conditions which prevailed
in the remote past.

\Article{233}{The Reversing Layer.}---Newton began the analysis
\index{Reversing layer}%
\index[xnames]{Newton}%
of light by passing it through a small circular opening. In
1802 Wollaston passed the light from the sun through a
\index[xnames]{Wollaston}%
narrow slit, instead of a pinhole, and found that the solar
spectrum was crossed by $7$~dark lines. In a few years the
subject was taken up by Fraunhofer, who soon found that
\index[xnames]{Fraunhofer}%
the spectrum was crossed by an immense number of dark
lines. In 1815 he mapped $324$~of them, and they have since
been known as ``Fraunhofer lines.'' A greatly improved
\index{Fraunhofer lines}%
map of these lines was made by Kirchhoff in 1862, and still
\index[xnames]{Kirchhoff}%
another by Ångström in 1868. In 1886 Langley mapped
\index[xnames]{Angstrom@{Ångström}}%
\index[xnames]{Langley}%
the solar spectrum with the aid of his bolometer far into the
infra-red region, and in 1886, 1889, and 1893 Rowland published
\index[xnames]{Rowland}%
extensive and very accurate maps from measurements
of the positions and characteristics obtained with his powerful
grating spectroscope. In 1895 Rowland published his
great ``Preliminary Table of Solar Spectrum Wave Lengths,''
%% -----File: 421.png---Folio 391-------
containing the results for about $14,000$~spectral lines. A
portion of the solar spectrum is shown in \Figref{148} with a
bright-line comparison spectrum above.

The spectrum of the sun is continuous except for the very
numerous dark lines which cross it. Therefore, in accordance
with the third law of spectrum analysis, there is between
the photosphere and the observer cooler gas, and its
constitution can be determined from the relations among the
dark lines and from their positions. The lines prove the
existence of sodium, iron, and other heavy metals in this
intervening gas, and since they cannot remain in the gaseous
state in our own atmosphere they must be in that of the sun.
\begin{figure}[hbt]%[Illustration:]
\Input{421}{jpg}
\Caption[Portion of solar spectrum below with a Titanium comparison
spectrum above.]{Fig}{148}
\end{figure}%
This absorbing material which overlies the photosphere
constitutes the \textit{reversing layer}.

If the reversing layer could be viewed not projected against
the brilliant photosphere, it would give a spectrum of bright
lines exactly at the places occupied by the dark lines under
the conditions as they normally exist. At the total eclipse
of the sun in 1870, Young placed the slit of his spectroscope
\index[xnames]{Young, C. A.}%
tangent to the limb of the sun. Just as the moon cut off the
last of the photosphere the spectrum suddenly flashed out
in bright lines where an instant before the dark ones had
been. Since 1895, during nearly every total eclipse of the
sun, this ``flash spectrum'' has been photographed, and
\index{Flash spectrum}%
\index{Spectrum!flash}%
there is no doubt that the positions of its lines are identical
with those of the corresponding dark Fraunhofer lines. From
the duration of their appearance as bright lines and the known
rate at which the moon apparently passes across the disk of
%% -----File: 422.png---Folio 392-------
the sun, it has been found that the reversing layer is $500$~or
$600$~miles deep.

As a rule the effect of pressure on an absorbing gas is to
cause the dark lines to shift slightly toward the red end of
the spectrum. Extensive studies by various astronomers of
the displacements of the Fraunhofer lines have led to the
conclusion that the pressure of the reversing layer, even at
its lower levels, does not exceed $5$~or $6$~times that of the
earth's atmosphere at sea level. This is a very remarkable
result in view of the great extent of the sun's atmosphere
and the fact that gravity at the surface of the sun is nearly
$28$~times as great as it is at the surface of the earth. Possibly
electrical repulsion from the sun and light pressure
partly offset the great surface gravity of the sun.

\Article{234}{Chemical Constitution of the Reversing Layer.}---Of
\index{Reversing layer!constitution of}%
\index{Sun!constitution of}%
the $14,000$ lines in Rowland's spectrum about one third
are due to the absorption by the earth's atmosphere, and
the remainder are produced by the sun's reversing layer
and chromosphere. By comparing the positions of the lines
of the sun's spectrum with those given by the various elements
in laboratory experiments, it is possible to infer the
chemical constitution of the material which produces the
absorption. In this manner $38$~elements are known certainly
to exist in the sun, but more than $6000$~of the lines
mapped by Rowland have not as yet been identified as
belonging to any element.

The presence of iron is established by more than $2000$~line
coincidences, carbon by more than~$200$, calcium by more
than~$75$, magnesium by~$20$, sodium by~$11$, copper by~$2$, and
lead by~$1$. It will be noticed that nearly all the elements
in the table which follows are metals, the exceptions being
hydrogen, helium, carbon, and oxygen. On the other hand,
a number of heavy metals, such as gold and mercury, are
missing. The \hyperref[Table:XI]{following table} gives the elements found in the
sun and their atomic weights.\DPnote{** TN: Change ":" to "."}
%% -----File: 423.png---Folio 393-------
\begin{table}[hbt]
\begin{center}
\Caption{Table}{XI}
% \caption[Atomic weights of the elements]{}
\index{Chemical constitution of sun}%
\index{Sun!constitution of}%
\index{Elements!in sun}%
\TFontsize
\settowidth{\TmpLen}{\textsc{Atomic Weight}}
\begin{tabular}{|l|c||l|c|}
\hline
\TEntry{\TmpLen}{\THead Element} &
\TEntry{\TmpLen}{\THead Atomic Weight} &
\TEntry{\TmpLen}{\THead Element} &
\TEntry{\TmpLen}{\medskip\THead Atomic Weight\medskip} \\
\hline
\Strut%
\DTE{Hydrogen} & $1$ & \DTE{Copper} & $64$ \\
\DTE{Helium} & $4$  & \DTE{Zinc} & $65$ \\
\DTE{Glucinum} & $9$ &  \DTE{Germanium} & $72$ \\
\DTE{Carbon} & $12$ &  \DTE{Strontium} & $88$ \\
\DTE{Oxygen} & $16$ &  \DTE{Yttrium} & $89$ \\
\DTE{Sodium} & $23$ &  \DTE{Zirconium} & $91$ \\
\DTE{Magnesium} & $24$ &  \DTE{Niobium} & $93$ \\
\DTE{Aluminum} & $27$ &  \DTE{Molybdenum} & $96$ \\
\DTE{Silicon} & $28$ &  \DTE{Rhodium} & $103$ \\
\DTE{Potassium} & $39$ &  \DTE{Palladium} & $107$ \\
\DTE{Calcium} & $40$ &  \DTE{Silver} & $108$ \\
\DTE{Scandium} & $44$ &  \DTE{Cadmium} & $112$ \\
\DTE{Titanium} & $48$ &  \DTE{Tin} & $119$ \\
\DTE{\DPtypo{Venadium}{Vanadium}} & $51$ &  \DTE{Barium} & $137$ \\
\DTE{Chromium} & $52$ &  \DTE{Lanthanum} & $139$ \\
\DTE{Manganese} & $55$ &  \DTE{Cerium} & $140$ \\
\DTE{Iron} & $56$ &  \DTE{Neodymium} & $144$ \\
\DTE{Nickel} & $59$ &  \DTE{Erbium} & $168$ \\
\DTE{Cobalt} & $59$ &  \DTE{Lead} & $207$\rule[-2ex]{0pt}{0pt} \\
\hline
\end{tabular}
\end{center}
\end{table}

While the presence of the spectral lines of an element proves
its existence, their absence does not show that it is not present.
In the first place, heavy elements, like gold, mercury,
and platinum, would probably sink far below the level of
the reversing layer, and consequently would give no lines in
the solar spectrum. Then, again, the characteristic spectra
of some of the elements, particularly non-metals, are suppressed
by the presence of some other elements, particularly
metals. Sometimes the spectrum of an element is entirely
obliterated by the presence of a small percentage of another
element. This may be the explanation of the fact that the
spectra of fluorine, chlorine, bromine, iodine, sulphur, selenium,
tellurium, nitrogen, phosphorus, arsenic, antimony,
and boron are not found in the sun, although most of these
elements occur abundantly in the earth. Some elements
have spectra that change radically with alterations in their
%% -----File: 424.png---Folio 394-------
conditions of temperature, pressure, and electrical excitation.
One of these elements is oxygen, which was long sought for
in the sun before it was certainly found. Of course, the proof
of its existence was complicated by the fact that it occurs
in abundance in the earth's atmosphere. Finally, as Lockyer
\index[xnames]{Lockyer}%
suggested, some of the so-called elements may be in
reality compounds which are broken up under the extreme
conditions of temperature prevailing in the sun, and their
characteristic spectra may be in this manner destroyed.

The reversing layer is undoubtedly constantly receiving
material from below and above, and therefore it is safe to
conclude that its composition is not qualitatively different
from that of the remainder of the sun. It is interesting
that nearly $40$~terrestrial elements are found, for it points
strongly to the conclusion that the sun and the earth have
had a common origin.

The distribution of the elements in distance above the
sun's photosphere was determined by Mitchell from excellent
photographs of the flash spectrum which he secured in
the eclipse of 1905, and by St.~John from considerations of
\index[xnames]{Stjohn@{St.\ John}}%
the Doppler-Fizeau effect. On the whole the lighter elements
\index{Doppler-Fizeau law}%
\index[xnames]{Doppler}%
\index[xnames]{Fizeau}%
extend to high altitudes while the heavier elements
are confined to the lower levels. A peculiar exception is
that calcium, whose atomic weight is~$40$, extends in abundance
up into the chromosphere $10,000$~miles, even as high
as hydrogen. Iron and the heavier metals are found only
down in the reversing layer.

\Article{235}{The Chromosphere.}---As has been stated, the chromosphere
\index{Chromosphere}%
is a gaseous envelope around the sun above the
reversing layer whose depth is from~$5000$ to $10,000$~miles. It
gets its scarlet color from the incandescent hydrogen and
calcium of which it is largely composed.

The spectrum of the chromosphere consists of many lines,
some of which are permanent while others come and go.
The permanent lines are due mostly to hydrogen, helium, and
calcium; the intermittent lines are due to many elements
%% -----File: 425.png---Folio 395-------
which seem to have been temporarily thrown up into it
through the reversing layer.

The existence of the element helium was first inferred from
\index{Helium}%
the presence of a bright yellow line in the solar spectrum near
the two yellow lines of sodium. It is universally prevalent
in the chromosphere, giving a bright line when the sun is
eclipsed, or at any time when the slit of the spectroscope is
put on the chromosphere parallel to the sun's limb. For
some unknown reason helium does not give a dark-line absorption
spectrum when the light from the photosphere
passes through it. This seems to be a direct contradiction
to the third law of spectrum analysis, which holds true in
all other known cases. But helium is a very remarkable
element in several other respects. Next to hydrogen, it
has the lowest atomic weight, it is very inactive, and enters
into no known chemical combinations with other elements,
it has the lowest known refractive index, it is an excellent
conductor of electricity, its rate of diffusion is $15$~times its
theoretical value, its solubility in water is nearly zero, and it
is liquefied only with the utmost difficulty. It has already
been explained that helium is one of the products of the disintegration
of uranium, radium, and other radioactive substances.
It was not discovered on the earth until 1895, when
Ramsay, on examining the spectrum of the mineral clevite,
\index[xnames]{Ramsay}%
found the yellow spectral line of helium.

\Article{236}{Prominences.}---Vast eruptions, called \textit{prominences},
\index{Prominences}%
shoot up from the sun's photosphere through its chromosphere
to heights ranging from $20,000$~miles up to $300,000$~miles,
or even to greater elevations in extreme cases. One
$80,000$~miles in height is shown in \Figref{149}. They usually
occur in the neighborhood of sun spots and are never seen
near the sun's poles. They leap up in jets and flames, often
changing their appearance greatly in the course of $10$~or $15$~minutes,
as is shown in \Figref{150}. Their velocity of ascent
is frequently $100$~miles per second and sometimes exceeds
$200$~miles per second.
%% -----File: 426.png---Folio 396-------

If eruptive prominences should leave the photosphere with
a velocity of more than $380$~miles per second, and if they
should suffer no resistance from the reversing layer and
chromosphere, they would escape entirely from the sun and
pass out beyond the planets to the distances of the stars.
It is very difficult to account for their great velocities. No
satisfactory theory has been developed for explaining how
such violent explosive forces are long held in restraint and
\begin{figure}[hbt]%[Illustration:]
\Input{426}{jpg}
\Caption[Solar prominence, August~21, 1909, reaching to a height of
$80,000$~miles. \textit{Photographed at the Mt.~Wilson Solar Observatory.}]{Fig}{149}
\index{Mount Wilson Solar Observatory}%
\index{Solar!Observatory}%
\end{figure}%
then suddenly released. Perhaps under the extreme conditions
of temperature and pressure prevailing in the interior
of the sun, all elements, like radium under terrestrial conditions,
explode because of their subatomic energies. Julius
\index[xnames]{Julius}%
has maintained that the prominences may be mirage-like
appearances due to unusual refraction, and that they are not
actual eruptions from the sun as they seem to be. But their
velocities are determined both from their motion perpendicular
to the line of sight when they are seen on the sun's
limb, and also from spectral line displacements in accordance
%% -----File: 427.png---Folio 397-------
with the Doppler-Fizeau principle, and it seems very
\index{Doppler-Fizeau law}%
\index[xnames]{Doppler}%
\index[xnames]{Fizeau}%
improbable that they are not real.

The spectra of eruptive prominences show many lines,
especially in the lower levels. In them the bright lines of
sodium, magnesium, iron, and titanium are conspicuous,
while those of calcium, chromium, and manganese are
\begin{figure}[hbt]%[Illustration:]
\Input{427}{jpg}
\Caption[Changes in a solar prominence in an interval of ten minutes.
\textit{Photographed by Slocum at the Yerkes Observatory.}]{Fig}{150}
\index{Yerkes Observatory}%
\index[xnames]{Slocum}%
\end{figure}%
generally found. In the higher levels calcium is the predominating
element, a remarkable fact in view of its atomic
weight of~$40$.

Prominences were formerly observed only when the sun
was totally eclipsed, for at other times the illumination of
the sky made them altogether invisible. But since the development
of the spectroscope they can be observed at any
time. If the light from the limb of the sun is passed through
%% -----File: 428.png---Folio 398-------
the spectroscope, the continuous illumination of the earth's
atmosphere is spread out and correspondingly enfeebled;
on the other hand, the light from the prominences consists
of single colors and is not diminished in intensity by passing
through the spectroscope. Consequently, if the dispersion
is sufficient, the atmospheric illumination is reduced until
the prominences become visible.

Not all the prominences are eruptive. Besides those
which burst out suddenly, rising to great heights and soon
disappearing or subsiding again, there are others, called
\textit{quiescent} prominences, which spread out, like the tops of
banyan trees, with here and there a stem reaching below.
They often develop far above the surface of the sun, without
apparent connections with it, and seem to be due to material
which for some mysterious reason suddenly becomes visible.
They rest quietly at great altitudes, somewhat like terrestrial
clouds, often for many days, notwithstanding the sun's
gravity. They are made up of hydrogen, helium, and
calcium.

\Article{237}{The Spectroheliograph.}---The photosphere radiates
\index{Spectroheliograph}%
a continuous spectrum, while above it is the reversing
layer which produces the dark absorption lines. Some of the
lines, as the $K$-line due to calcium, are broad because of the
great extent of the absorbing layer. Now, calcium is abundant
in the prominences, and, moreover, it shines with an
intensity greater than that of the reversing layer. The result
is that the reversing layer makes a broad, dark line, say
the $K$-line, and above it is more luminous calcium in a rarer
state which produces a narrow bright line in the midst of
the dark one. The line is said to be ``doubly reversed.''

The spectroheliograph is an instrument invented and perfected
by Hale in 1891 for the purpose of photographing
\index[xnames]{Hale}%
the sun with the light from a single element. The ideas on
which it depends were almost simultaneously developed and
applied by Deslandres. In this instrument, or rather combination
\index[xnames]{Deslandres}%
of instruments, the sunlight is passed through a
%% -----File: 429.png---Folio 399-------
spectroscope and is spread out into a spectrum. The $K$-line,
which is most frequently used, is doubly reversed in
the regions of faculæ and prominences. All the spectrum
is cut off by an opaque screen except the bright part of the
$K$-line which passes through a second narrow slit. That is,
the only light which passes through both slits is the calcium
light from that portion of the sun's image which falls on the
first slit of the spectroscope. In \Figref{151}, $S$~is the image
of the sun at the focal plane of the telescope, $A$~is the slit
of the spectroscope (the prisms are not shown), $T$~is the
spectrum which falls on the screen~$B$, $R$~is a slit in the screen~$B$
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{429}{png}
\Caption[The spectroheliograph.]{Fig}{151}
\end{figure}%
which is adjusted so that it admits the bright center of
the doubly reversed $K$-line, and $P$~is a photographic plate on
which the $K$-line falls. The apparatus is so made that the
slit~$A$ may be moved across the image of the sun~$S$, and the
slit~$R$ simultaneously moved so that the $K$-line falls on
successively different parts of the photographic plate~$P$. In
this manner a photograph of the hot calcium vapors which
lie above the reversing layer may be obtained; such a photograph
is shown in \Figref{152}. Some other spectral lines have
also been used in this way. For example, two photographs
of a spot with the so-called $H$-line are shown in \Figref{153}.

The width of a spectral line depends upon the density of
the gas which is emitting the light. Suppose a thick layer
of calcium gas which is rare at the top and denser at the bottom
gives a bright $K$-line. The central part will be due to
%% -----File: 430.png---Folio 400-------
light coming from all depths, particularly from the higher
layers where absorption is unimportant. On the other
hand, the marginal parts of the line will be due to light
\begin{figure}[hbt]%[Illustration:]
\Input{430}{jpg}
\Caption[Spectroheliogram of the sun taken with the doubly reversed
calcium line. \textit{Photographed by Hale and Ellerman at Yerkes Observatory.}]{Fig}{152}
\index{Yerkes Observatory}%
\index[xnames]{Ellerman}%
\index[xnames]{Hale}%
\end{figure}%
coming from the lower levels where the gas is denser. Following
out these principles, and using a very narrow slit,
Hale first obtained photographs of different levels of the
solar atmosphere.
%% -----File: 431.png---Folio 401-------

\Article{238}{The Corona.}---During total eclipses the sun is
\index{Corona, of sun}%
seen to be surrounded by a halo of pearly light, called the
\textit{corona}, extending out $200,000$~or $300,000$~miles, while some
of the streamers reach out at least $5,000,000$~miles. So far
it has not been possible to find any observational evidence
of the corona except at the times of total eclipses of the sun.
One of the reasons that eclipses are of great scientific interest
is that they afford an opportunity of studying this
remarkable solar appendage. The brief duration of total
eclipses and their infrequency have made progress in the
researches on the corona rather slow. The corona is not
\begin{figure}[hbt]%[Illustration:]
\Input{431}{jpg}
\Caption[Spectroheliograms of a sun spot with the doubly reversed H-line
of calcium. \textit{Hale and Ellerman, Solar Observatory, Aug.~7 and~9, 1915.}]{Fig}{153}
\index{Mount Wilson Solar Observatory}%
\index{Solar!Observatory}%
\index[xnames]{Ellerman}%
\index[xnames]{Hale}%
\end{figure}%
arranged in concentric layers like an atmosphere, but is
made up of complicated systems of streamers (\Figref{154}), in
general stretching out radially from the sun, but often simply
and doubly curved, and somewhat resembling auroræ.
Many observers have declared that its finely detailed structure
resembles the Orion nebula.

The coronal streamers often, perhaps generally, have
their bases in the regions of active prominences, but exceptions
have been noted. That they are in some way connected
with activity on the sun is shown by the fact that the
form of the corona changes in a cycle of about eleven years,
the same as that of sun-spot activity. At sun-spot maxima
the coronal streamers radiate from all latitudes nearly
%% -----File: 432.png---Folio 402-------
\begin{sidewaysfigure}%[Illustration]
\centering\Input[0.95\linewidth]{432}{jpg}
\Caption[Photograph of the corona at the total eclipse of the sun, May~28, 1900 (Barnard and Ritchey).]{Fig}{154}
\index[xnames]{Barnard}%
\index[xnames]{Ritchey}%
\end{sidewaysfigure}
%% -----File: 433.png---Folio 403-------
equally. As the maxima pass, the coronal streamers gradually
withdraw from the poles of the sun and extend out to
greater distances in the sun-spot zones. At the sun-spot
minima, the corona consists of short rays in the polar regions,
curved away from the solar axis, and long streamers extending
out in the equatorial plane.

The spectroscope shows that the corona emits three kinds
of light. First, there is a small quantity which is known
to be reflected sunlight, for it gives, though faintly, the
Fraunhofer absorption lines, and it is polarized. Second,
\index[xnames]{Fraunhofer}%
there is white light whose source, according to the first law
of spectrum analysis, must be incandescent solid or liquid
particles. Lastly, there is a bright-line spectrum whose
source, according to the second law of spectrum analysis,
must be an incandescent gas. The most conspicuous line
is in the green and is emitted by an element, called \textit{coronium},
\index{Coronium}%
which is not yet known on the earth. There seems to be
at least one other substance present, but no known elements.

According to present ideas, the corona consists of dust
particles, liquid globules, and small masses of gas which
are widely scattered. From the amount of light and heat
radiated, and from the temperature which masses so near the
sun must have, Arrhenius computed that there is one dust
\index[xnames]{Arrhenius}%
particle, on the average, in every $14$~cubic yards of the corona.
The excessive rarity of the corona is shown by the fact that
comets have plunged through hundreds of thousands of miles
of it without being sensibly retarded. The dust particles
and liquid globules give the reflected light; the liquid, the
continuous spectrum; and the gases, the bright-line spectrum.
The form of the corona shows that its condition of equilibrium
is not at all similar to that of an atmosphere like the one surrounding
the earth. Its increase of density toward the sun
is inexplicably slow, though doubtless light pressure and
electrical forces are opposed to gravity. Its radial structure
and periodical variation in general form are without satisfactory
explanation.
%% -----File: 434.png---Folio 404-------

\Article{239}{The Eleven-Year Cycle.}---It has been explained that
\index{Eleven-year cycle}%
\index{Magnetic storms, periodicity of}%
\index{Sun's eleven-year cycle}%
sun spots vary in frequency and distribution on the sun's
surface in a period averaging a little more than $11$~years.
There are a number of other phenomena which undergo
changes in the same period.

The faculæ are most numerous in the sun-spot zones,
\index{Faculae@{Faculæ}!periodicity of}%
although they occur all over the sun. Both their number
and the positions of the zones where they are most numerous
vary periodically with the sun-spot period. This is quite
to be expected, for the sun spots and the faculæ are both
photospheric phenomena.

The eruptive prominences are frequent in the sun-spot
belts, and vary in position with them. The evidence so far
also shows periodic variations in their numbers. The quiescent
prominences, on the other hand, cluster in the polar
regions.

The coronal types clearly vary in the eleven-year cycle,
as was explained in the preceding article. Doubtless the
total solar radiation varies to some extent in the same period,
though this has not been verified observationally, but the
time is now ripe for the investigation.

The spectra of sun spots vary with the period of the spots,
but the Fraunhofer lines are singularly invariable.

The great vibrations which so powerfully agitate the
sun extend to the earth and probably to the whole solar
system. It has long been known that both the horizontal
and vertical components of the earth's magnetism vary in
the sun-spot period, and that magnetic disturbances
(``storms'') are most frequent at the times when sun spots
are most numerous. Likewise, auroræ occur most frequently
\index{Aurorae@{Auroræ}}%
at the epochs of great sun-spot activity. In fact, magnetic
storms and auroræ never occur except when there is great
activity in the sun in the form of sun spots or prominences;
but there are frequent disturbances on the sun without
accompanying terrestrial phenomena. The correlation of
these phenomena is shown in \Figref{155}.
%% -----File: 435.png---Folio 405-------

The first suggested explanation of magnetic storms on the
\index{Magnetic storms, periodicity of}%
earth was that the sun induces changes in the earth's magnetic
state by sending out electromagnetic waves. Lord Kelvin
\index[xnames]{Kelvin}%
raised the objection that if the sun were sending out these
waves in every direction, it would give out as much energy
in $8$~hours of an ordinary electric storm as it radiates in light
\begin{figure}[hbt]%[Illustration:]
\Input{435}{png}
\Caption[Curves of magnetic storms, prominences, \DPtypo{faculae}{faculæ}, and sun spots
from 1882 to~1904.]{Fig}{155}
\end{figure}%
and heat in $4$~months. A recent exhaustive discussion of the
data has led Maunder to the conclusion that the source of
\index[xnames]{Maunder}%
the periodic magnetic storms is in the sun, that the magnetic
disturbances are confined to restricted areas on the sun, and
that their influences are propagated out from the sun in
cones which rotate with the sun; that when these cones of
magnetic disturbances strike the earth, magnetic storms are
%% -----File: 436.png---Folio 406-------
induced, and that these magnetic storms have intimate,
though unknown, relations with sun spots. The most
important contribution of this investigation was that there
is much observational evidence to show that the sun is not
to be regarded as surrounded by a polarized magnetic sphere,
but that there are definite and intense stream lines of magnetic
influence, probably connected with the coronal rays,
reaching out principally from the spot zones in directions
which are not necessarily exactly radial. It is a little too
early to formulate a precise theory as to whether these streams
are electrified particles driven off by magnetic forces and
light pressure, or whether they involve the minute corpuscles
of which atoms are composed, or whether they are phenomena
of matter and energy of a character and in a state not yet
recognized by science.


\Section{XVII}{QUESTIONS}

1. The apparent diameter of the sun as seen from the earth is about~$32'$;
what are the apparent thicknesses of the corona, chromosphere,
and reversing layer?

2. The sun's disk is considerably brighter at its center than near
its margin (\Figref{141}); can this phenomenon be explained by the absorption
of light by the reversing layer? By small solid or liquid
particles somewhere above the photosphere?

3. If the smallest spot that can be seen subtends an angle of~$1'$,
what is the diameter of the smallest sun spot that can be seen simply
through a smoked glass?

4. In what direction do sun spots appear to cross the sun's disk
as a consequence of its rotation?

5. Why cannot the corona be observed with the aid of the spectroscope
at any time, just as the prominences are observed?
%% -----File: 437.png---Folio 407-------

\normalsize


\Chapter{XII}{Evolution of the Solar System}

\Section{I}{General Considerations on Evolution}

\Article{240}{The Essence of the Doctrine of Evolution.}---The
\index{Evolution!essence of}%
\index{Theory of evolution}%
fundamental basis on which science rests is the orderliness
of the universe. That it is not a chaos has been confirmed
by an enormous amount of experience, and the principle
that it is orderly is now universally accepted. This principle
is completed in a fundamental respect by the doctrine of
evolution.

According to the fundamental principle of science the
universe was orderly yesterday, is orderly to-day, and will
be orderly to-morrow; according to the doctrine of evolution,
the order of yesterday changed into that of to-day in
a continuous and lawful manner, and the order of to-day
will go over into that of to-morrow continuously and systematically.
That is, the universe is not only systematic
and orderly in space, but also in time. The real essence of
the doctrine of evolution is that it maintains the orderliness
of the universe in time as well as in space.

Evolution may be from the simple and relatively unorganized
to the complex and highly organized, or it may be in
the opposite direction. In fact, evolution generally involves
the two types of changes. For example, the minerals of the
soil and the elements of the atmosphere sometimes combine
and produce a tree having foliage, flowers, and fruit. But
the tree grows, at least partly, on the disintegrating products
of other trees or plants, and in its own trunk the processes
of decay are active. Or, to take a less commonplace example,
with the advancement of civilization men have become
%% -----File: 438.png---Folio 408-------
more sensitive to discords and more and more capable of
appreciating certain types of harmony. There is almost
certainly a corresponding improvement in the structure of
their nervous system. On the other hand, there is degeneration
in the quality of their teeth and hair. The changes
in the two directions are both examples of evolution.

As knowledge increases it is found that everything is continually
changing. Individuals change, institutions change,
languages change, and even the ``eternal hills'' are broken
up and washed away by the elements in a moment of geological
time. Moreover, all these changes are found to be
perfectly orderly. The doctrine of evolution, as defined here,
is so fundamentally sensible and is confirmed by so much
experience that scientists, the world over, accept it with absolute
confidence. There have been, and there doubtless
will continue to be, differences of opinion regarding what
the precise processes of certain particular evolutions may
have been, but there is no disagreement whatever regarding
the fundamental principles.

\Article{241}{The Value of a Theory of Evolution.}---The importance
\index{Evolution!value of}%
\index{Theory of evolution!value of}%
of a general principle is proportional to the number
of known facts it correlates. This is a general proposition
with special applications in science. Since a theory of
evolution is concerned largely with the relations among
the data established by experience, it naturally forces an
attempt at their correlation. Moreover, the relations are
examined in a critical spirit, so that any errors in the data
or misconceptions regarding their relations are apt to be
revealed. Therefore, an attempt to construct a theory of
evolution is of value because it leads to a better understanding
of the material upon which it is being based.

A theory of evolution invariably demands a knowledge
of facts in addition to those upon which it is based. In this
way it stimulates and directs investigation. A great majority
of the investigations which scientific men make are for
the purpose of proving or disproving some theory they have
%% -----File: 439.png---Folio 409-------
tentatively formulated. The true scientist often has pre-conceived %[** TN: Only instance]
notions as to what is true, but he conscientiously
follows the results of experience.

A broad scientific theory involves many secondary theories
depending upon special groups of phenomena. For example,
a theory of the origin and development of the solar system
will involve theories of the sun's heat, of the revolution of
the planets, of the rotation of the planets, of the planetoids,
of the zodiacal light, etc. In the construction of a general
theory of evolution the secondary theories are related to
the whole, and in this way they are subjected to a searching
examination. The criticism of secondary theories, whether
the result is favorable or adverse, constitutes another important
value of the development of a theory of evolution.

The activities of men are largely directed toward satisfying
their intellectual wants, though this fact might be easily
overlooked. For example, they do not ordinarily visit foreign
countries to get more to eat or wear, but to acquire
broader views of the world. The important thing in traveling
is not that a person goes physically to any particular
place, but that he gets the intellectual experiences that
result from going there. Astronomers cannot travel through
the vast regions of space which they explore, but the long
arms of their analysis reach out and gather up the facts
and bring them to their consciousness with a vividness
scarcely surpassed in any experience. As their powerful
instruments and mathematical processes extend their experience
in space, so a theory of evolution, to the extent that it is
complete and sound, extends their experience in time.

Finally, a theory which gives unity to a great variety of
observational data is of rare æsthetic value. It is related
to the catalogue of imperfectly correlated facts upon which
it is based as a finished and magnificent cathedral is to the
unsightly heaps of stone, brick, and wood from which it
was built. In some reflections along this line, near the
close of his popular work on astronomy, Laplace said,
%% -----File: 440.png---Folio 410-------
``Contemplated as one grand whole, astronomy is the most
beautiful monument of the human mind, the noblest record
of its intelligence.''

In view of these considerations it is evident that the evolution
of the solar system is a subject to which the astronomer
naturally gives serious attention. The foremost authorities
of the present time have treated the question in lectures,
in essays, and in books. When new discoveries are made
their bearings on evolutionary theories are at once examined.
Astronomers are rapidly approaching the point of view of the
biologists, who interpret all of their phenomena in terms of
evolutionary doctrines. Yet scarcely a generation ago many
astronomers regarded the consideration of the evolution of
the solar system as a dangerous speculation.

\Article{242}{Outline of the Growth of the Doctrine of Evolution.}---Every
great discovery doubtless has been the culmination
of a long period of preliminary work, and before final success
has been attained generally many men have approximated
to the truth. So it has been with the doctrine of evolution.
The ancient Greeks developed theories that everything had
evolved from fire, or from air and water. These theories
contained the germ of the idea of evolution, but their authors
had not laid securely enough the foundations of science to
enable them to treat successfully the development of the
universe. After the decline of their intellectual activity
the subject of evolution was not considered seriously for
many centuries.

In the eighteenth century geologists were groping for a
satisfactory theory regarding the succession of the life
forms whose fossils were found in the rocks. They seem to
have concluded on the whole that the earth had been subject
to a number of great cataclysms in which all life was
destroyed. They supposed that following each destruction of
life there had been a new creation in which higher forms were
produced. The prevalence of such ideas as these shows with
what difficulty the doctrine of evolution was developed.
%% -----File: 441.png---Folio 411-------

In 1750 Thomas Wright, of Durham, England, published
\index[xnames]{Wright, Thomas}%
a theory of the evolution, not only of the solar system, but
also of all the stars that fill the sky. The chief merit of
this work was that indirectly it gave a straightforward
exposition of the doctrine of evolution. Its chief influence
seems to have been on the young philosopher Kant, into
\index[xnames]{Kant}%
whose hands it fell. Kant at once turned his brilliant mind
to the contemplation of the problems of cosmogony, or the
evolution of the celestial bodies, and in 1755 he published
a remarkable book on the subject. But the world seems not
to have been ripe for the idea of evolution, because neither
the work of Wright nor that of Kant had any important
influence upon science.

In 1796 the great French astronomer and mathematician
Laplace published his celebrated ``Nebular Hypothesis.'' It
\index{Nebular hypothesis}%
\index[xnames]{Laplace}%
was supported by the great name of its author, and it was
relatively simple and easily understood. Moreover, during
the French Revolution the world had acquired a new point
of view and had become more receptive of new ideas. For
these reasons the theory of Laplace soon obtained wide
acceptance among scientific men. It made a profound
impression on geologists because it furnished them with an
account of the early history of the earth. It gave them
astronomical authority for an originally hot and molten
earth which had solidified on cooling. It encouraged them
to interpret geological phenomena by geological principles.
In the early decades of the nineteenth century geologists
largely abandoned the idea that the earth had necessarily
been visited by destructive cataclysms, and adopted the view
that it had undergone a continuous series of great changes
at a roughly uniform rate.

The work of the geologists led naturally to the extension
of the doctrine of evolution to the biological sciences. In
the first place, the belief that the earth was enormously
old had become current. In the second place, there were
unmistakable evidences that the surface of the earth had
%% -----File: 442.png---Folio 412-------
undergone extensive changes. In the third place, the early
rocks contained only fossils of low forms of life, while the
later rocks contained fossils of higher forms of life. In addition,
there were many direct evidences of a purely biological
character that there was an almost continuous series of life
forms from the lowest to the highest.

The principle of biological evolution seems to have been
taking definite shape simultaneously in the minds of Charles
Darwin, Alfred Russel Wallace, and Herbert Spencer.
\index[xnames]{Darwin, Charles}%
\index[xnames]{Spencer}%
\index[xnames]{Wallace, Alfred Russel}%
Darwin and Wallace were naturalists and Spencer was a
philosopher. In 1858 Darwin published his \textit{Origin of
Species}, in which he brought together the results of almost
\index{Origin!of species}%
a lifetime of keen observations and profound reflections.
He gave unanswerable evidence for his conclusion that
during the geological ages, as a consequence of changing
environment, natural selection, survival of the fittest, etc.,
one species of animals gradually changed into another, and
that at the present time all the higher types of animals,
including man, are more or less closely related.

In spite of the fact that the doctrine of evolution is full of
hope for the future progress of the human race, Darwin's
book aroused the bitterest antagonism. While biologists do
not now fully agree with him as to the relative importance
of the various factors involved in biological evolution, nevertheless
they universally accept his fundamental conclusions.
Moreover, the changes in political, social, and religious institutions
are now considered in the light of the same ideas.
That is, the condition of the whole universe at one time
evolves continuously and in obedience to all the factors operating
on it into that which exists at another time.

In brief, the development of the modern doctrine of evolution
is as follows: In the middle of the eighteenth century
its first beginnings were laid in astronomy by Wright and
\index[xnames]{Wright, Thomas}%
Kant. At the end of the century it was given an enormous
\index[xnames]{Kant}%
impulse by the astronomer and mathematician, Laplace.
\index[xnames]{Laplace}%
His theory of the origin of the earth stimulated geologists
%% -----File: 443.png---Folio 413-------
to adopt it in the early decades of the nineteenth century.
By the middle of the century it was being definitely applied
in the biological sciences. In 1858 Darwin published his
\index[xnames]{Darwin, Charles}%
great masterpiece, \textit{The Origin of Species}, which gave the
\index{Origin!of species}%
whole world a new point of view and revolutionized its
methods of thought. The development and adoption of the
doctrine of evolution was the greatest achievement of the
nineteenth century.


\Section{XVIII}{QUESTIONS}

1. Is the erosion of the chasm below Niagara Falls an example
of an evolution? Is the clearing away of the forests and the preparation
of the land for cultivation? Is an explosion of dynamite?

2. Would the direct creation of men and lower animals be an
example of evolution?

3. Do the changes in scientific ideas constitute an evolution?

4. Are religious ideas undergoing an evolution?

5. Will the doctrine of evolution undergo an evolution?

6. The universe in our vicinity at the present time is believed
to be orderly; is it reasonable to suppose that in remote regions or
at remote times it was not orderly?

7. Why was the doctrine of evolution first clearly understood in
astronomy?

8. According to the doctrine of evolution, will two identical
conditions of the universe lead to identical results? Is it probable
that the universe is twice in exactly the same state?

\normalsize


\Section{II}{Data of the Problem of Evolution of the Solar System}

\Article{243}{General Evidences of Orderly Development.}---There
are certain obvious evidences that the solar system
has undergone an orderly evolution. For example, the
planets all revolve around the sun in nearly the same plane
and in the same direction. There are in addition over $800$~planetoids
which have similar motions. Moreover, the sun
and the four planets whose surface markings are distinctly
visible rotate in the same direction. So great a uniformity
can scarcely be the result of chance.
%% -----File: 444.png---Folio 414-------

In order to treat the matter numerically, suppose there are
800 bodies whose planes of motion do not differ from the
plane of the earth's orbit by more than~$18°$, and whose
directions of motion are the same as that of the earth. Since
the inclination of an orbit could be anything from $0°$ to~$180°$,
the chance that it would lie between $0°$ and~$18°$ is~$\frac{1}{10}$. The
probability that the planes of the orbits of two bodies would
be less than~$18°$ is~$\left(\frac{1}{10}\right)^2$. And the probability that the
same would be true for~$800$ bodies is only~$\left(\frac{1}{10}\right)^{800}$, or unity
divided by $1$~followed by $800$~ciphers. This probability is
so small that we are forced to the conclusion that the arrangement
of the planets in the solar system is not accidental.
Both Kant and Laplace made use of this line of reasoning.
\index[xnames]{Kant}%
\index[xnames]{Laplace}%

A planet may revolve around the sun in an orbit of any
eccentricity from $0$ to~$1$. Of the more than $800$~planets
and planetoids, the orbits of~$624$ have eccentricities less than~$0.2$,
the orbits of all except~$26$ have eccentricities less than~$0.3$,
and the orbit of only one has an eccentricity greater
than~$0.5$. These remarkable facts imply that some systematic
cause has been at work which has produced planetary
orbits of low eccentricity. And both the positions of
the planes and the small eccentricities of the orbits of the
planets prove conclusively that the solar system, in all its
history, has not been subject to any important external disturbance,
such as a closely passing star.

\Article{244}{Distribution of Mass in the Solar System.}---Nearly
all the matter of the solar system is concentrated in the sun.
In fact, all the planets together contain less than one seventh
of one per cent of the mass of the entire system. Although
the mass of Jupiter is more than $2.5$~times that of all the
other planets combined, it is less than one thousandth that
of the sun.

It is important to know whether the masses of the sun
and planets are now changing. There is certainly at present
no appreciable transfer of matter from one body to
another. The sun may be losing some particles by ejecting
%% -----File: 445.png---Folio 415-------
them from its surface in an electrified condition, and a very
small percentage of the ejected particles may strike the
planets, but it is very improbable that the process has had
important effects on the distribution of mass in the solar
system, even in the enormous intervals of time required for
its evolution.

The mass of the earth is slowly increasing by the meteoric
material which it sweeps up in its journey around the sun.
It is not unreasonable to suppose that the other planets,
and possibly the sun, are growing similarly. This growth,
at least in the case of the earth, is too slow at present to
have a very important bearing on the evolution of the
whole system. But if the meteors are permanent members
of the solar system, the more they are swept up by the
planets the more infrequent they become and the smaller the
number a planet encounters in a day. Consequently, the
acquisition of meteoric material by collision may once have
been a much more important factor in the evolution of
the planets than it is at the present time. In fact, so far
as general considerations go, appreciable fractions of the
masses of the planets may have been obtained from meteoric
material. But it is improbable that the great sun has
grown sensibly in this way.

It follows from this discussion that probably the remote
antecedent of the solar system consisted of an overwhelming
central mass and a very small quantity of matter distributed
somewhat irregularly out from it to an enormous
distance. At any rate, if this were not the original distribution
of matter, the conditions must have been such that the
central condensation resulted in harmony with the laws
of dynamics. The ever-increasing distances between the
planets is shown in Figs.\ \Fref{96}~and~\Fref{97}. The relatively small
masses of the planets and their enormous distances from one
another are among the most remarkable facts that need to
be taken into account when considering their origin and
evolution.
%% -----File: 446.png---Folio 416-------

An additional fact which must be noted is that the terrestrial
planets contain the heaviest known substances. The
sun also contains heavy elements (\Artref{234}), though the
spectral lines of the very heaviest have not been found.
The constitution of the large planets is not so well known,
though it may be inferred from their low densities and moderate
temperatures that they contain largely only the light
elements. Any hypothesis as to the origin of the planets,
in order to be satisfactory, must make provision for this
distribution of the elements.

\Article{245}{Distribution of Moment of Momentum.}---In attempting
\index{Moment of momentum!of solar system}%
to go back to the origin of the solar system it is
natural to consider its mass and distribution of mass because
matter is indestructible. For a similar reason, the distribution
of the moment of momentum of the system among its
various members is of fundamental importance. That is,
if the solar system has undergone its evolution free from
exterior disturbances, its total moment of momentum is
now exactly equal to what it was at the beginning and at
every stage of its development.

As has been stated, the small mutual inclinations of the
orbits of the planets and the small eccentricities of their
orbits both prove that the solar system has been subject
to no important exterior influences since the planets were
formed. Hence any hypothetical antecedent of the system
must be assigned the quantity of moment of momentum it
now possesses. Although this fact is perfectly clear, it was
overlooked by Kant and was not given adequate consideration
\index[xnames]{Kant}%
by Laplace and his followers.
\index[xnames]{Laplace}%

In \Tableref{XII} the mass and moment of momentum is
given for the sun and each of the eight planets in such units
that the sums are unity. The moment of momentum of the
sun depends upon its law of density. In the computation it
was assumed that the mass is concentrated toward the interior
according to a law of increase of density formulated
by Laplace. The rotations of the planets contribute so
%% -----File: 447.png---Folio 417-------
little to the final results that it is not important what law of
density is used for them.

\begin{table}[hbt]
\begin{center}
\Caption{Table}{XII}
\index{Moment of momentum!of solar system}%
%\caption[Masses and Moments of Momentum]{}
\settowidth{\TmpLen}{\textsc{Moment of}}
\begin{tabular}{|l|@{\quad}c|@{\quad}c@{\quad}|}
\hline
\TEntry{2\TmpLen}{\TFontsize\THead Body} &
\TEntry{\TmpLen}{\TFontsize\THead Mass} &
\TEntry{\TmpLen}{\medskip\TFontsize\THead Moment of Momentum\medskip} \\
\hline
\Strut
Sun \MyDotFill     & $0.9986590$ & $0.027423$ \\
Mercury \MyDotFill & $0.0000001$ & $0.000017$ \\
Venus \MyDotFill   & $0.0000025$ & $0.000576$ \\
Earth \MyDotFill   & $0.0000030$ & $0.000827$ \\
Mars \MyDotFill    & $0.0000003$ & $0.000112$ \\
Jupiter \MyDotFill & $0.0009558$ & $0.599273$ \\
Saturn \MyDotFill  & $0.0002852$ & $0.241924$ \\
Uranus \MyDotFill  & $0.0000430$ & $0.052845$ \\
Neptune \MyDotFill & $0.0000511$ & $0.077003$\rule[-1.5ex]{0pt}{0pt} \\
\hline
\rule{0pt}{3ex}%
\quad Total  & $1.0000000$ & $1.000000$\rule[-2ex]{0pt}{0pt} \\
\hline
\end{tabular}
\end{center}
\end{table}

It is seen from this table that although the mass of the sun
is $700$~times as great as that of all the planets combined,
its moment of momentum is only a little over $\frac{1}{40}$~that of the
planets. Or, considering the material interior to the orbit
of Saturn, it is found that while Jupiter contains only $\frac{1}{10}$~of
one per cent, or~$\frac{1}{1000}$, of the entire mass, it possesses
more than $95$~per cent of the moment of momentum.

One at once inquires whether the distribution of moment
of momentum is now being changed. The mutual attractions
of the planets produce some changes in the distribution
of moment of momentum, but they are of no importance
whatever in connection with the problem under consideration.
The tides which a planet generates in the sun reduce the
moment of momentum of the sun and increase that of the
planet. But here again the results are inappreciable even
for thousands of millions of years. The earth encounters
meteoric matter in its revolution around the sun, and it is
probable that the other planets are subject to similar disturbances.
The result of the resistance by meteors is to
reduce the moment of momentum of the planets. At present
%% -----File: 448.png---Folio 418-------
the effects of meteors on the motion of the earth are
inappreciable, but it is not certain that they were not once
important. However, whether or not they have ever been
of importance, they cannot relieve the inequalities in the
table, for they are decreasing the moment of momentum of
the planets, which are still relatively very large. In fact,
there have been no known influences at work which could
have sensibly modified the distribution of the moment of
momentum of the system since the sun and planets have
been separate bodies.

It remains to inquire whether the sun and planets may not
once have been parts of one mass with a distribution of
moment of momentum quite different from that found at
present. Since the planets are not receding from the sun,
the only possibility is that the sun and the planets were
formerly so expanded that the material of which they are
composed was more or less intermingled.

According to the contraction theory of the heat of the sun,
the sun's dimensions were formerly greater than they are
at present. Indeed, the sun has been supposed to have
once filled all the space now occupied by the planets. Followed
backward in time, the sun is found to be larger and
larger, rotating more and more slowly because its moment
of momentum remained constant during contraction, and
more and more nearly spherical because a rotating body
becomes more oblate with contraction. It follows from the
table that if the planets which are interior to Jupiter were
added to the sun they would not have an important effect
on its moment of momentum.

Now suppose the sun was once expanded out to the orbit
of Jupiter; its radius was more than $1000$~times its present
radius, its volume was more than $1000^3 = 1,000,000,000$~times
its present volume, and its density was correspondingly
less. Even if it was not condensed toward the center,
the density at its periphery was then less than one millionth
of that of the earth's atmosphere at sea level. It follows from
%% -----File: 449.png---Folio 419-------
the fact that the moment of momentum was necessarily
constant, that its period of rotation must have been about
$70,000$~years. But Jupiter's period of revolution is about
$12$~years. Now, therefore, either Jupiter was then quite
independent of the general solar mass; or, if not, in some
unknown way this extremely tenuous material must have
imparted to that minute fraction of itself which later became
Jupiter enough moment of momentum to reduce the period
of this part from $70,000$~years to $12$~years. More specifically,
it is seen from the table that Jupiter, which contains one
tenth of one per cent of the mass of the solar system within
the orbit of Saturn, carries over $95$~per cent of the moment
of momentum. It is incredible that this extreme distribution
of moment of momentum could have developed from an
approximately uniform distribution, especially in a mass
of such low density, and no one has been able to formulate
a plausible explanation of it. Consequently, it must be
concluded that the distribution of moment of momentum
in the solar system has not changed appreciably since it has
been free from important exterior forces.

\Article{246}{The Energy of the Solar System.}---In considering
\index{Energy!of solar system}%
the energy of the solar system, the discussion must include
its kinetic energy, heat energy, potential energy, and subatomic
energy.

The kinetic energy of a body is its energy of motion
including translation, rotation, and internal currents. The
kinetic energy of the solar system consists of its energy of
translation and of the internal motions of its parts. The
former cannot have changed except by the action of exterior
forces. Moreover, its value is not accurately known, and
it has no relation to the remaining energy of the system so
long as no other celestial body is encountered. Therefore
it will be given no further consideration in this connection.
The mutual attractions of the planets change their translatory
motions, but in such a way that the sum of their kinetic
and potential energies remains constant.
%% -----File: 450.png---Folio 420-------

The sun, planets, and satellites raise tides in one another.
In these tides there is some friction in which kinetic energy
\index{Tidal!evolution}%
degenerates into heat energy, which is radiated away into
space. In this way the solar system is losing energy. The
heat energy from all other sources is likewise being lost by
radiation.

The potential energy of a system is equal to the work
which may be done upon it, in virtue of the relative positions
of its parts, by the forces to which it is subject. For example,
a body $100$~feet above the surface of the earth is subject to
the attraction of the earth. The earth would do a certain
amount of work upon the body in causing it to fall from an
altitude of $100$~feet to its surface. This work equals the
potential energy of the body in its original position. In
the case of the translations of the planets, as has been stated,
the sum of their kinetic and potential energies is constant.
But if the sun or a planet contracts, the potential energy of
its expanded condition is transformed into heat (\Artref{216}),
which is at least partly lost by radiation. In this way the
total energy of the system decreases, and the diminution may
be large in amount.

There is certainly a large amount of subatomic energy in
uranium, radium, and probably in all other elements. In the
case of the radioactive substances this energy is slowly transformed
into heat, which is dissipated by radiation. As has
been suggested (\Artref{219}), the subatomic energies may be
liberated in great quantities under the extreme conditions
of pressure and temperature which prevail in the interior
of the sun.

Since the solar system is losing energy in several ways and
acquiring only inappreciable amounts from the outside, as,
for example, the radiant energy received from the stars, it
originally had more energy than at present, and this condition
must be satisfied by all hypotheses respecting its
evolution.
%% -----File: 451.png---Folio 421-------


\Section{XIX}{QUESTIONS}

1. What is the probability that when $3$~coins are tossed up they
will all fall heads up? What is the probability that in a throw of
$4$~dice there will be $4$~aces up? If $100$~coins were found heads up,
could it reasonably be supposed that the arrangement was accidental?
How would its probability compare with that that the
positions of the orbits of the planets and planetoids are accidental?

2. Suppose a star should pass near the solar system in the plane
of the orbits of the planets; would it disturb the positions of the
planes, or the eccentricities, of their orbits?

3. How many tons of meteors would have to strike the earth
daily in order to double its mass in $200,000,000$~years? How many
would daily strike each square mile of its surface?

4. What is the definition of moment of momentum? How
does it differ from momentum? Is it manifested in various forms
like energy? Does the loss of energy of a body by radiation change
its moment of momentum?

5. The mass of the earth is $1.2$~times that of Venus (\Tableref{XII});
why is its moment of momentum more than $1.2$~times that of
Venus?

6. Could the total energy of the solar system have been infinite
at the start? Can the system have existed in approximately its
present condition for an infinite time?

7. When carbon and oxygen unite chemically, heat is produced;
is this heat energy developed at the expense of the kinetic, potential,
heat, or subatomic energies of the original materials?

\normalsize


\Section{III}{The Planetesimal Hypothesis\protect\footnotemark}
\index{Hypothesis!planetesimal}%
\index{Planetesimal!hypothesis}%
\footnotetext{The Planetesimal Hypothesis was developed by Professor T.~C. Chamberlin
\index[xnames]{Chamberlin}%
and the author in 1900 and the following years.}

\Article{247}{Brief Outline of the Planetesimal Hypothesis.}---The
fundamental conditions imposed by the distribution of
mass and moment of momentum in the solar system, together
with many supplementary considerations, have led to the
planetesimal hypothesis. According to this hypothesis, the
remote ancestor of the solar system was a more or less condensed
and well-defined central sun, having slow rotation,
surrounded by a vast swarm of somewhat irregularly scattered
secondary bodies, or planetesimals (little planets),
%% -----File: 452.png---Folio 422-------
which all revolved in elliptical orbits about the central mass
in the same general direction. This organization evidently
satisfies the data of the problem. Moreover, the spiral
nebulæ (\Artref{302})\DPnote{** TN: Square brackets in original.} offer numerous examples of matter which
is apparently in this state.

According to the planetesimal hypothesis, our present
\index{Planetesimal!organization}%
sun developed from the central parent mass and possibly
some outlying parts which fell in upon it because they had
small motions of translation. The revolving scattered material
contained nuclei of various dimensions which, in their
motions about the central sun, swept up the remaining
scattered material and gradually grew into planets whose
masses depend upon the original masses of the nuclei and
the amount of matter in the regions through which they
passed. The angles between the planes of the orbits were
gradually reduced by the collisions, and at the same time
the eccentric orbits became more nearly circular. In the
process of growth the planetary nuclei acquired their forward
rotations.

\Article{248}{Examples of Planetesimal Organization.}---The
planetoids afford a trace of the former planetesimal condition
of the solar system. The average inclination and the
average eccentricity of their orbits are considerably larger
than the corresponding quantities for the planets. If the
region which they occupy had been swept by a dominating
nucleus, they would have combined with it in a planet occupying
approximately the mean position of the planes of their
orbits and having a small eccentricity (\Artref{252}).

Another example of planetesimal organization is furnished
by the particles of which the rings of Saturn are
composed. One might at first thought conclude that they
would have formed one or more satellites if dominating nuclei
had been revolving around the planet in the zone which
they occupy. But they are very close to Saturn, and a satellite
revolving at their distance would be subject to the strains
of the tides produced by the planet. As has been stated
%% -----File: 453.png---Folio 423-------
(\Artref{183}), Roche showed that a fluid satellite could not revolve
\index[xnames]{Roche}%
within $2.44$~radii of a planet without being broken
up, unless its density were greater than that of the planet.
Since the rings of Saturn are within this limit, it follows
that they could not have formed a satellite, and that a
large nucleus revolving among them, instead of sweeping
them up, would itself have been reduced to the planetesimal
condition, unless it was solid and strong enough to withstand
great tidal strains.

The examples of planetesimal organization which have
been given may not be very convincing. But we may inquire
whether there are not numerous examples in the heavens,
beyond the solar system, confirmatory of the planetesimal
theory. The answer is in the affirmative. There are tens
of thousands of spiral nebulæ that are almost certainly in
the planetesimal condition, though on a tremendous scale.
They consist of central sunlike nuclei which are generally
well defined, and arms of widespreading, scattered material.
Their arms in most cases probably contain large masses, but
they are small in comparison with the central suns. Their
great numbers imply that they are in general semi-permanent
in character. Consequently, the material of which
the arms are composed cannot in general be moving along
them, either in toward or out from the central nucleus, for
under these circumstances they would condense into suns
or dissipate into space, and in either case lose their peculiar
characteristics. Besides this, matter subject to the law
of gravitation could not move along the arms of spirals. It
is therefore believed that in a spiral nebula the arms are
composed of material which, instead of proceeding along
them, moves across them around the central nucleus as
a focus. The spirals owe their coils to the fact that the
inner parts revolve faster than the outer parts. As a
rule they radiate white light, which indicates that they are
at least partly in a solid or liquid state. When a spiral is
seen edgewise to the earth there is a dark band through its
%% -----File: 454.png---Folio 424-------
center, doubtless produced by dark, opaque material revolving
at its periphery.

While a few spiral nebulæ have been known for a long
time, their great numbers were not suspected until Keeler
\index[xnames]{Keeler}%
began to photograph them with the Crossley reflector at the
Lick Observatory. In a paper published in 1900 shortly
\index{Lick Observatory}%
before his death, he said:

``1. Many thousands of unrecorded nebulæ exist in the
sky. A conservative estimate places the number within the
reach of the Crossley reflector at about~$120,000$. The number
of nebulæ in our catalogues is but a small fraction of this.

``2. These nebulæ exhibit all gradations of apparent size
from the great nebula in Andromeda down to an object
which is hardly distinguishable from a faint star disk.

``3. Most of these nebulæ have a spiral structure\ldots.
While I must leave to others an estimate of the importance
of these conclusions, it seems to me that they have a very
direct bearing on many, if not all, questions concerning the
cosmogony. If, for example, the spiral is the form normally
assumed by a contracting nebulous mass, the idea at once
suggests itself that the solar system has been evolved from a
spiral nebula, while the photographs show that the spiral
is not, as a rule, characterized by the simplicity attributed to
the contracting mass in the nebular (Laplacian) hypothesis.
This is a question which has already been taken up by
Chamberlin and Moulton of the University of Chicago.''
\index[xnames]{Chamberlin}%

While the spirals are almost certainly examples of planetesimal
organization, those which have been photographed
are enormously larger than the parent of the solar system
unless, indeed, there are many undiscovered planets beyond
the orbit of Neptune. But, as Keeler remarked, there is no
lower limit to the apparent dimensions of the spiral nebulæ,
and it is possible that many of them are actually of very
moderate size.

\Article{249}{Suggested Origin of Spiral Nebulæ.}---Although the
\index{Origin!of spiral nebulæ}%
\index{Spiral nebulae@{Spiral nebulæ}!origin of}%
validity of the planetesimal theory does not hang upon any
%% -----File: 455.png---Folio 425-------
hypothesis as to the origin of spiral nebulæ, yet, if the solar
system has evolved from a spiral nebula, the theory of its
origin will not be regarded as complete and fully satisfactory
until the mode of generation of these nebulæ has been
explained. The best suggestion regarding their genesis,
which is due primarily to Chamberlin, is as follows:
\index[xnames]{Chamberlin}%

There are several hundreds of millions of stars in the
heavens and they are moving with respect to one another with
an average velocity of about $600,000,000$~miles per year.
While their motions are by no means %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{455}{png}
\Caption[Deflection of ejected material
by a passing star.]{Fig}{156}
\end{wrapfigure}
entirely at random, yet
there are millions of them
moving in essentially
every direction. It is inevitable
that in the course
of time every star will pass
near some other star. If
two stars should collide,
the energy of their motion
would largely be changed
into heat and the combined
mass would be transformed
into a gaseous
nebula. If they should
simply pass near one another
without striking, an event which would occur many
times more frequently than a collision, a spiral nebula would
probably be formed, as will now be shown.

Consider two stars passing near each other. They both
move about their center of gravity, but no error will be
committed in representing one of them as being at rest and
the other as passing by it. If the stars are equal, their
effects on each other are the same, but in order not to divide
the attention, only the action of~$S'$ on~$S$ will be considered.

Consider~$S'$ when it is at the position~$S_1'$, \Figref{156}. It
raises tides on~$S$, one on the side toward~$S'$ and the other on
the opposite side. The heights of the tides depend upon the
%% -----File: 456.png---Folio 426-------
relative masses of the two suns and their distance apart
compared to the radius of~$S$. An approach within %[Illustration: Break]
\begin{wrapfigure}[27]{\WLoc}{2.75in}
\Input[2.75in]{456}{jpg}
\Caption[Eruptive prominence at three
altitudes.\textit{Photographed by Slocum at
the Yerkes Observatory.}]{Fig}{157}
\index{Prominences}%
\index{Yerkes Observatory}%
\index[xnames]{Slocum}%
\end{wrapfigure}
$10,000,000$~miles
is more than $100$~times as probable as even a grazing
collision. At this distance the tide-raising force of~$S'$ on~$S$
compared to the surface gravity of~$S$ is more than 2000
times the tide-raising
force of the moon on the
earth compared to the
surface gravity of the
earth. The tide-raising
force varies directly as
the radius of the disturbed
body and inversely
as the cube of the
distance of the disturbing
body (\Artref{153}).
Hence, if the nearest
approach were $5,000,000$~miles,
the tide-raising
force would be more than
$16,000$~times greater,
relatively to surface
gravity, than that of the
moon on the earth. This
force would raise tides
approximately $500$~miles
high if the sun were a
homogeneous fluid, and
there would be a corresponding slight constriction of the sun
in a belt midway between the tidal cones. The tides on a
highly heated gaseous body would probably be much higher.

The sun is the seat of violent explosive forces which now
often eject matter in the eruptive prominences to distances
of several hundred thousand miles (\Figref{157}). If the sun
were tidally distorted, the eruptions would be mostly toward
and from the disturbing sun; certainly the ejections would
%% -----File: 457.png---Folio 427-------
reach to greater distances in these directions. Besides this,
after the ejected material had once left the sun, its distance
would be increased still further by the attraction of~$S'$.
Consequently, if~$S'$ were not moving along its orbit, the
ejections toward and from it would be to more remote distances
than they would be in any other direction. In fact,
those toward~$S'$ might even strike it. But $S'$ would be moving
along in its orbit, and, in a short time, it would have
a component of attraction at right angles to the original
direction of motion of the ejected matter. Consequently,
by the time $S'$~had arrived at~$S_2'$, the paths of the ejected
masses would be curved somewhat like those shown in \Figref{156}.
It is easy to see that, for the mass ejected toward~$S'$,
the curvature is in the right direction; a discussion based
on the resolution of the forces involved (\Artref{153}) proves
that, for the mass ejected in the other direction, the indicated
curvature is also correct. Eventually $S'$~would move on in
its orbit so far that it would no longer have sensible attraction
for the ejected masses, and they would be left revolving
around~$S$ in elliptical orbits. If the initial speed of the
ejected material were very great, it might leave~$S$ never to
return.

The critical question is whether matter would be ejected
far enough to produce the large orbits required by the
theory. In order to throw light on this question the \hyperref[Table:XIII]{following
table} has been computed, giving the surface velocities
necessary to cause undisturbed ejected matter to recede
various distances from the surface of the sun.

The most remarkable thing shown in the table is that after
a velocity is reached sufficient to cause the ejected matter
to recede a few millions of miles, a small change in the initial
speed produces radically different final results. Since prominences
now ascend to a height of half a million of miles
without the disturbing influence of a visiting sun, it is seen
that the numerical requirements of the hypothesis are not
excessive. Moreover, numerous actual computations of
%% -----File: 458.png---Folio 428-------
hypothetical cases have shown that, on the recession of~$S'$,
the ejected material is usually left revolving around~$S$ in
elliptical orbits.

\begin{table}[hbt]
\begin{center}
\Caption{Table}{XIII}
%\caption[Height of Ascent of Ejected Material from a Star]{}
\TFontsize
\setlength{\tabcolsep}{3pt}
\settowidth{\TmpLen}{\scshape Initial Velocity}
\begin{tabular}{|*{4}{c|}}
\hline
\TEntry{\TmpLen}{\medskip\THead Height of Ascent\medskip} &
\TEntry{\TmpLen}{\THead Initial Velocity} &
\TEntry{\TmpLen}{\THead Height of Ascent} &
\TEntry{\TmpLen}{\THead Initial Velocity} \\
\hline
\Strut
$\phantom{1,}100,000$ mi. &  $\phantom{1}72$ mi.\ per sec. & $\phantom{12}5,000,000$ mi. & $353$ mi.\ per sec. \\
$\phantom{1,}200,000$ mi. &            $121$ mi.\ per sec. & $\phantom{1}10,000,000$ mi. & $368$ mi.\ per sec. \\
$\phantom{1,}300,000$ mi. &            $157$ mi.\ per sec. & $\phantom{1}20,000,000$ mi. & $376$ mi.\ per sec. \\
$\phantom{1,}400,000$ mi. &            $184$ mi.\ per sec. & $\phantom{1}50,000,000$ mi. & $380$ mi.\ per sec. \\
$\phantom{1,}500,000$ mi. &            $206$ mi.\ per sec. &           $100,000,000$ mi. & $382$ mi.\ per sec. \\
          $1,000,000$ mi. &            $268$ mi.\ per sec. &           $500,000,000$ mi. & $383$ mi.\ per sec. \\
          $2,000,000$ mi. &            $316$ mi.\ per sec. &              Infinite       & $384$ mi.\ per sec.\rule[-2ex]{0pt}{0pt} \\
\hline
\end{tabular}
\end{center}
\end{table}

As one star passes another the ejection of material is more
or less continuous. When the visiting star is far away, the
ejections are to moderate distances and the matter returns to
the sun. As the visiting
star approaches,
the ejected materials
recede farther and
their paths become
more curved. At a
certain time the lateral
disturbance of~$S'$
becomes so great that
the ejected material
revolves around~$S$ instead
of falling back
upon it. Let the
orbits for this case be those marked $1$~and~$1'$ in \Figref{158}, the
former being toward~$S'$, and the latter away from it. At a
later time the ejections will be to greater distances and the
materials will have greater lateral motions. Suppose they
are $2$~and~$2'$, and so on for still later ejections until $S'$~recedes
from~$S$.
%% -----File: 459.png---Folio 429-------

Now consider the location of all of the ejected material
at a given time after $S'$ has passed its nearest point to~$S$.
If it has been sent out %[Illustration: Break, moved down]
\begin{wrapfigure}{\WLoc}{3in}
\Input[3in]{458}{png}
\Caption[The origin of a spiral nebula.]{Fig}{158}
\end{wrapfigure}
from $S$ continuously, it will lie along
two continuous curves, represented by the full lines in \Figref{158}.
\begin{figure}[hbt]%[Illustration:]
\Input{459}{jpg}
\Caption[The great spiral nebula in Canes Venatici (M.~51), showing
the two arms. \textit{Photographed by Ritchey at the Yerkes Observatory.}]{Fig}{159}
\index{Canes Venatici, spiral nebula in}%
\index{Nebulae@{Nebulæ}!spiral}%
\index{Spiral nebulae@{Spiral nebulæ}}%
\index{Yerkes Observatory}%
\index[xnames]{Ritchey}%
\end{figure}%
These are the arms of the spiral nebula whose individual
particles move across them in the dotted lines. The
diagram shows an ideal simple case, and \Figref{159} an actual
photograph. But if the approach of $S'$ were close, or if there
were a partial collision, and if the ejected material should go
beyond~$S'$, a very complicated structure would result. The
%% -----File: 460.png---Folio 430-------
arms of the spiral might be very irregular (\Figref{160}), the
particles might cross them at a great variety of angles, and
some of them might continue to recede indefinitely.
\begin{figure}[hbt]%[Illustration:]
\Input{460}{jpg}
\Caption[The great spiral nebula in Triangulum (M.~33). \textit{Photographed
by Ritchey at the Yerkes Observatory.}]{Fig}{160}
\index{Spiral nebulae@{Spiral nebulæ}}%
\index{Yerkes Observatory}%
\index[xnames]{Ritchey}%
\end{figure}

Thus, the suggested explanation of the origin of the spiral
nebulæ rests upon the existence of a great number of stars,
\index{Nebulae@{Nebulæ}!spiral}%
their rapid and somewhat heterogeneous motions which
imply near approaches now and then, their eruptive activities,
and the disturbance of one star by another passing near it.
%% -----File: 461.png---Folio 431-------
All the factors involved are well established---the only question
is that of their quantitative efficiency. Here some
doubts remain. It follows from the number of stars, the
space they occupy, and their motions that, if they were moving
at random, an individual sun would pass near some other
one, on the average, only once in many thousands of millions
of years. Perhaps the mutual gravitation of the stars is
important out on the borders of the great clusters of suns
of which the Milky Way is composed, where it may reasonably
\index{Milky Way}%
be supposed that their relative velocities are small,
and it may be that in these regions close approaches are for
this reason much more frequent. But in any case the demands
of time are very formidable. Besides this, many of the spiral
nebulæ are of such enormous dimensions that it is difficult
to suppose they have been produced by the encounter or
near approach of ordinary suns. It may be stated, however,
that, in the first place, there is no positive knowledge whatever
respecting the masses of spiral nebulæ; and that, in
the second place, near approaches are not confined to single
stars, but may involve multiple stars, clusters, and systems
of stars. The observed spirals may be simply the larger
examples originating from several or many suns.

It should be remembered that, whatever doubts may
remain respecting the validity of this or any other hypothesis,
the spiral nebulæ certainly exist in great numbers, and they
apparently have, on an enormous scale, an organization
similar to that which we have inferred must have been the
antecedent of the solar system. And it may be stated again
that the planetesimal hypothesis rests primarily upon the
evidence now furnished by the solar system, and that it does
not stand or fall with any theory respecting spiral nebulæ.

\Article{250}{The Origin of Planets.}---According to the planetesimal
\index{Evolution!of planets}%
\index{Origin!of planets}%
\index{Planets!evolution of}%
\index{Planets!origin of}%
hypothesis, the parent of the solar system consisted
of a central sun surrounded by a vast swarm of planetesimals
which moved approximately in the same plane
in essentially independent elliptic orbits. Among these
%% -----File: 462.png---Folio 432-------
planetesimals there were nuclei, or local centers of condensation,
which, in their revolutions, swept up the smaller planetesimals
and grew into planets. It is not to be understood
that the original nuclei were solid or even continuous masses.
It is much more probable that in their early stages they were
swarms of smaller masses having about the same motion
with respect to the central sun, and that, under their mutual
attractions and collisions, they gradually condensed into continuous
bodies. Indeed, the condensation may have been
very slow and may have been dependent to an important
extent upon the impacts of other planetesimals.

It seems to be impossible to determine the probable masses
of the original nuclei. If they were less than that of the
moon at present, they could not have retained any atmospheres
under their gravitative control. But as the nuclei
grew, their surface gravities increased, and a time came
when those which have become the larger planets possessed
sufficient gravitative power to prevent the escape of atmospheric
particles. The acquisition of atmospheres was then
inevitable because, in the first place, the materials grinding
together and settling under the weight of accumulating
planetesimals would squeeze out the lighter elements; in
the second place, the pulverizing and heating effects of the
impacts of meteors would liberate gases; and, in the third
place, the growing planets in their courses around the sun
would sweep up directly great numbers of atmospheric
molecules. The extent of the atmospheres of the planets
at all stages of their growth depended primarily on their
surface gravities.

The rate at which the nuclei swept up the planetesimals must
have been excessively slow. This conclusion follows from the
fact that if all the matter in the largest planet were scattered
around the sun in a zone reaching halfway to the adjacent
planets, the resulting planetesimals would be very far
apart, and also from the fact that the orbits of only a fraction
of them would at any one time intersect the orbit of the
%% -----File: 463.png---Folio 433-------
nucleus. It must be remembered that the orbits of the planetesimals
were continually changed by their mutual attractions
and especially by the attractions of the nuclei. Moreover,
the orbits of the nuclei were continually altered by collisions
with the planetesimals and by their perturbations of one
another. Consequently, if the orbits of the nuclei and certain
planetesimals did not originally intersect, they might
very well have done so later. But it does not follow that
they have all been swept up yet, or, indeed, that they all
ever will be swept up. Possibly some of the meteors which
the earth now encounters are the straggling remains of the
original planetesimals.

If the planetesimal theory is correct, the earth is very old
and the sun must have important sources of energy besides
its contraction. Most of the geological processes did not
begin until it became large enough to retain water and an
atmosphere. These same conditions were necessary for even
the beginnings of the development of life, which may have
had a continuous existence from the time the earth was half
its present size.

\Article{251}{The Planes of the Planetary Orbits.}---If the planetesimal
\index{Planetary orbits!planes of}%
hypothesis is true, it must explain the important
features of the solar system. The most striking thing about
the motions of the planets is that they all go around the sun
in the same direction, and the mutual inclinations of the
planes of their orbits are small. However, some deviations
exist, and in general they are greatest in case of the small
masses like Mercury and the planetoids.

It is assumed that the planetesimals all revolved around
the sun in the same direction. This would certainly have
been true if they originated by the close approach of two
suns, as explained in \Artref{249}. But the planes of their orbits
would not be exactly coincident. The plane of motion of
an ejected particle would depend upon its direction of
ejection and the forces to which it was subject. The ejections
would be nearly toward or directly away from the visiting
%% -----File: 464.png---Folio 434-------
sun, but slight deviations would be expected because
the ejecting body might be rotating in any direction, and the
direction of ejection would depend to some extent upon its
rotation.

Consider, therefore, a central body surrounded by an
enormous swarm of planetesimals which move in intersecting
elliptical orbits, some close to the sun and others far away.
The system of planetoids now in the solar system gives
a fair picture of the hypothetical situation, especially if, as
seems very probable, there are countless numbers of small
ones which are invisible from the earth. Suppose, also, that
there exist a number of nuclei revolving at various distances.
They gradually sweep up the smaller masses, and the problem
is to determine what happens to the planes of their orbits.

Consider a nucleus and all the planetesimals which it will
later sweep up. All together they have what may be called
in a rough way an average plane of revolution. This is a
perfectly definite dynamical quantity which Laplace treated
and which he called the ``invariable plane.''

When all the masses have united, the resulting body will
inevitably revolve in this plane. If the nucleus originally
moved in some other plane, the plane of its orbit would continually
change as its mass increased. The same would be
true for every other nucleus. There would be also an average
plane for the whole system. Those nuclei which moved
in regions that were richest in planetesimals, and that grew
the most, would, in general, have final orbits most nearly
coincident with this average plane. It is clear that so far
as the planes of the orbits of the planets are concerned
(see \Tableref{IV}), the consequences of the planetesimal theory
are in perfect harmony with the facts established by
observation.

\Article{252}{The Eccentricities of the Planetary Orbits.}---The
\index{Planetary orbits!eccentricities of}%
orbits of the original planetesimals probably had a considerable
range of eccentricities. This view is supported by the
fact that the eccentricities of the orbits of the planetoids vary
%% -----File: 465.png---Folio 435-------
from nearly zero to about~$0.5$. It is also supported by the
computations of orbits of particles which were assumed to
be ejected from one sun when another was passing it. The
problem is to find whether nearly circular planetary orbits
would be evolved from such a system of planetesimals.

When a nucleus sweeps up a planetesimal, the impact on
the larger body may be in any direction. If the nucleus
overtakes the planetesimals so that they act like a resisting
medium, the eccentricity of its orbit is in general diminished,
as was proved by Euler more than $150$~years ago. But many
\index[xnames]{Euler}%
other kinds of encounters can occur between bodies all
moving in the same direction around the sun. Collisions will
obviously be most numerous between bodies whose orbits
are approximately of the same dimensions; if the orbits of
two bodies differ greatly in size, collision between them is
impossible unless the orbits are very elongated. It is a remarkable
general proposition that if two bodies are moving in
orbits of the same size and shape, but differently placed, and
if they collide in any way, the eccentricity of the orbit of
the combined mass will be smaller than the common eccentricity
of the orbits of the separate parts.\footnote
  {To prove this, suppose a nucleus~$M$ and a planetesimal~$m$ are moving in
  orbits whose major semi-axis and eccentricity are $a_0$~and~$e_0$. Let their
  velocities at the instant preceding collision be $V_0$~and~$v_0$, and their combined
  velocity after collision be~$V$. The kinetic energy of the two bodies at the
  instant preceding collision is $\frac{1}{2}(MV_0^2 + mv_0^2)$. Their kinetic energy after
  their union is~$\frac{1}{2}(M + m)V^2$. The latter will be smaller than the former
  because some energy will have been transformed into heat by the impact of
  the two parts. Therefore $MV_0^2 + mv_0^2 > (M + m)V^2$.

  It is shown in celestial mechanics in the problem of two bodies that in
  elliptic orbits $V^2 = \dfrac{2}{r} - \dfrac{1}{a}$. Hence, the inequality becomes
  \[
    M\left(\frac{2}{r} - \frac{1}{a_0}\right)
  + m\left(\frac{2}{r} - \frac{1}{a_0}\right)
    > (M + m)\left(\frac{2}{r} - \frac{1}{a}\right),
  \]
  where $a$ is the major semi-axis of the combined mass. It follows from this
  inequality that $\dfrac{M + m}{a_0} < \dfrac{M + m}{a}$, whence $a < a_0$. That is, under the circumstances
  of the problem a collision always reduces the major semi-axis of
  the orbit.

  Another principle established in celestial mechanics is that the moment
  of momentum is constant whether there are collisions or not. The orbital
  moment of momentum of a mass~$m$ is $m\sqrt{a(1-e^2)}$, where $e$ is the eccentricity.
  The condition that the moment of momentum before collision
  shall equal that after collision is, therefore,
  \begin{gather*}%[** TN: N.B. Hacks to get \sqrt symbols the same size]
  M\sqrt{\smash[b]{a_0(1-e_0^2)}} + m\sqrt{\smash[b]{a_0(1-e_0^2)}} = (M + m) \sqrt{a(1-e^2)}, \text{ or } \\
  \sqrt{\smash[b]{a_0(1-e_0^2)}} =\sqrt{a(1-e^2)}.
  \end{gather*}
  Since $a_0 > a$, it follows that $\sqrt{(1-e_0^2)}<\sqrt{1-e^2\vphantom{()}}$, and therefore that $e < e_0$.}
%% -----File: 466.png---Folio 436-------

Of course, if two orbits were of exactly the same size, the
periods of the bodies would be the same and collisions would
result either at the first revolution or only after their mutual
attractions had modified their motions. But if they were
of nearly the same size, the conditions for collisions would be
favorable, and in nearly all cases the eccentricity would be
reduced.

It follows from this discussion that, in general, collisions
between planetesimals cause the eccentricities of their orbits
to decrease. Consequently, the more a nucleus grows by
sweeping up planetesimals, the more nearly circular, in general,
its orbit will be. If a nucleus revolves in a region rich in
planetesimals, the result is likely to be a large planet whose
orbit has small eccentricity. These conclusions agree precisely
with what is found in the solar system, for the orbits of
all the large planets are nearly circular, while the orbits of
some of the smaller planets and many of the planetoids are
considerably eccentric.

\Article{253}{The Rotation of the Sun.}---If the central body in
\index{Rotation!of sun}%
\index{Sun!rotation of}%
the planetesimal system rotates in the direction of the motion
of the outlying parts, the final result will be a sun rotating
in the direction of revolution of its planets. But if the
planetesimal organization is the result of the close approach
of two suns, the central mass might originally have been
rotating in any direction. In this case the final outcome
is not quite so obvious.

The only planetesimals which could sensibly affect the
rotation of the central mass are those which fall back upon
it. If the planetesimals originated by the close approach
%% -----File: 467.png---Folio 437-------
of two suns, there would certainly be many which would
return to the central mass. They would not fall straight in
towards its center, but would have a small forward motion
similar in character to that of the remainder of the planetesimals.
The result of the collision would be that the sun
would acquire their moment of momentum. It does not
seem unreasonable that the mass of the central sun might
grow in this way by as much as $10$~per cent. Since the planetesimals
would have enormously more moment of momentum
than equal masses in the central body, they would
substantially determine its direction of rotation. In fact,
if they were moving in orbits whose eccentricity was~$0.9$
and if they just grazed the sun at their perihelion, the mass
necessary to account for the present rotation of the sun, if
it had no rotation originally, would be one fifth of one per
cent of the sun's mass.

Another interesting result remains to be mentioned. The
planetesimals would strike the equatorial region of the sun
in greatest abundance and would give it the most rapid
motion. Unless the inequalities in motion were worn down
by friction the equatorial zone would be rotating fastest, as
is the case with our own sun.

\Article{254}{The Rotations of the Planets.}---The earth, Mars,
\index{Jupiter!rotation of}%
\index{Mars!rotation of}%
\index{Neptune!rotation of}%
\index{Planets!rotations of}%
\index{Rotation!of Jupiter}%
\index{Rotation!of Mars}%
\index{Rotation!of Neptune}%
\index{Rotation!of Saturn}%
\index{Rotation!of Uranus}%
\index{Saturn!rotation of}%
\index{Uranus!rotation of}%
Jupiter, and Saturn rotate in the direction in which the
planets revolve; the surfaces of the other planets have not
been observed well enough to enable astronomers to determine
how they rotate. It has been generally supposed that
the equators of Uranus and Neptune coincide with the
planes of the orbits of their satellites, but the evidence in
support of the supposition is as yet inconclusive.

The earlier theories regarding the origin of the planets all
fail to explain their forward rotations.

Chamberlin has shown that if a planet develops from
\index[xnames]{Chamberlin}%
a planetesimal system it will in general rotate in the direction
of its revolution. Consider a nucleus~$N$, \Figref{161},
which, in its early stages, will probably be simply an immense
%% -----File: 468.png---Folio 438-------
swarm of planetesimals. For simplicity, suppose its orbit
is a circle~$C$ around the sun as a center (if this assumption
were not made, the discussion would not be essentially modified).
The %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.75in}
\Input[2.75in]{468}{png}
\Caption[Development of the forward
rotation of a planet nucleus by the accretion
of planetesimals.]{Fig}{161}
\end{wrapfigure}
planetesimals which can encounter~$N$ are divided
into three classes: (\textit{a})~those whose aphelion points are
inside the circle~$C$; (\textit{b})~those whose perihelion points are
inside~$C$ and whose aphelion points are outside of~$C$; and
(\textit{c})~those whose perihelion
points are outside of~$C$.
They are designated
by (\textit{a}),~(\textit{b}), and~(\textit{c}) respectively
in \Figref{161}.

Consider collisions of
the planetesimals of
class~(\textit{a}) with the nucleus~$N$.
A collision
can occur only when a
planetesimal is near its
aphelion point. At and
near this point the
planetesimal is moving
slower than the nucleus.\footnote
  {Let $V$ and~$v$ represent the velocity of the nucleus and planetesimal
  respectively, and $A$ and~$a$ the semi-axes of their orbits. It is shown in
  celestial mechanics that $V^2 = \rule{0pt}{16pt}\dfrac{2}{r} - \dfrac{1}{A}$, and $v^2 = \dfrac{2}{r} - \dfrac{1}{a}$. Since $a < A$ and $r$ is
  the same in the two equations, it follows that $V^2 > v^2$.}

Hence the nucleus will overtake the planetesimal, and the
collision will be a blow backward on the inner side of the
nucleus. That is, planetesimals of class~(\textit{a}) tend to give the
nucleus a forward rotation.

Planetesimals of class~(\textit{b}) can strike the nucleus so as to
tend to give it a rotation in either direction, or so as not to
have any effect on its rotation. If they are not distributed
in some special way, the collective result of the collision of
many of them will be very small.
%% -----File: 469.png---Folio 439-------

Planetesimals of class~(\textit{c}) move faster than the nucleus
at the time of collision. Therefore they overtake the nucleus
and tend to give it a forward rotation.

It follows from this discussion that two of the three classes
of planetesimals tend to give the nucleus a forward rotation.
The effects are most important at the equator of the planet,
for there they strike farthest from its axis. Hence, the impacts
of planetesimals on the whole tend to make the equators
of fluid planets rotate faster than the higher latitudes, as
is the case with Jupiter and Saturn. The precise final result
depends upon the initial rotation of the nucleus and upon the
distribution of the planetesimals among the three classes.

Obviously the relative numbers of planetesimals in classes
(\textit{a})~and~(\textit{c}) would in general be small. In order to get some
idea of the numbers required to account for the observed
rotations, a numerical example has been treated. It was
assumed that the original earth nucleus had no rotation
and that the planetesimals of class~(\textit{b}) gave it none. It
was assumed that all the planetesimals of classes (\textit{a})~and~(\textit{c})
moved in orbits having the eccentricity~$0.2$ and that they
struck the nucleus $4000$~miles from the center. Then, in
order to account for the present rotation of the earth, it was
found that their total mass must have been about $5.7$~per
cent of that of the whole earth. Whether or not these
results are reasonable cannot be determined without further
quantitative investigations. But it must be insisted that
the results are qualitatively correct, and that not even this
much can be said for any earlier hypothesis regarding the
origin of the planets.

In the preceding discussion the effects of the rotations of
the original nuclei, or swarms of planetesimals out of which
the nuclei condensed, have been ignored. As a matter of
fact, they were probably in rotation around axes essentially
perpendicular to the plane of the system. There seems to
be no conclusive reason why the original rotations should
have been in one direction rather than in the other. The
%% -----File: 470.png---Folio 440-------
observed rotations of the planets seem to indicate that,
for some reason at present unknown, the original nuclei
rotated in the forward direction.

\Article{255}{The Origin of Satellites.}---According to the planetesimal
\index{Satellites!origin of}%
theory, the satellites developed either from small
secondary nuclei which were associated with the larger
planetary nuclei from the beginning, or from neighboring
secondary nuclei which became entangled at a later time in
the outlying parts of the swarms of planetesimals constituting
the nuclei. If the satellites originated in the former way,
their directions of revolution would be the same as those of
rotation of their respective primaries; if in the latter way,
they might revolve originally in any directions around their
primaries.

With the exception of the eighth and ninth satellites of
Jupiter and the ninth satellite of Saturn (and possibly the
satellites of Uranus and Neptune), all the known satellites
revolve in the directions in which their primaries rotate.
This seems to indicate that at least most of the satellites
originated from secondary nuclei which were associated with
their respective primary nuclei from the beginning and partook
of their common motion of rotation. The satellite
nuclei, like the planetary nuclei, swept up the planetesimals
and grew in mass. The craters on the moon may have
been produced by the impact of planetesimals.

With the growth in mass of a planet its attraction for its
satellites increases and this results in a reduction in the
dimensions of their orbits. Suppose the most remote direct
satellites were originally revolving at the greatest distances
at which their primaries could hold them in gravitative control,
and that their orbits have been reduced to their present
dimensions by the growth of the planets. The amount of
reduction in the size of the orbit of a satellite depends upon
the amount of growth of the planet around which it revolves,
and furnishes the basis for computing the increase in the
mass of the planet.
%% -----File: 471.png---Folio 441-------

The three retrograde satellites revolve at great distances
from their respective primaries in orbits which are rather
eccentric and considerably inclined to their respective systems.
Their origin is evidently different from that of the
direct satellites. They may have been neighboring planetesimals
which became entangled in the remote parts of the
planetary swarm. The question arises why they revolve
in the retrograde direction. The answer probably depends
upon the fact that, at a given distance, a retrograde satellite
is much more stable, so far as the disturbance of the sun is
concerned, than a direct one. Consequently, a retrograde
satellite would not be driven by collisions away from the control
of its planet so easily as a direct one. Also, the effects of
collisions with planetesimals and satellitesimals (planetesimals
revolving around planetary nuclei) must be considered.

\Article{256}{The Rings of Saturn.}---The rings of Saturn are
\index{Rings of Saturn}%
\index{Saturn!ring system of}%
swarms of particles revolving in the plane of the planet's
equator. According to the planetesimal theory, they are
the remains of outlying masses in the original nucleus which
were moving so fast that they did not fall toward the center.
Of course, they were subject to encounters with in-falling
planetesimals. These collisions transformed some of their
energy of motion into heat and some of them fell toward,
or perhaps on to, the growing planetary nucleus. It may
be that only a small part of the original ring material now
remains. But when they fell, they retained at least a portion
of their motion of revolution, and the result was that they
struck the planet obliquely in the direction in which it
rotated. This increased its rotation, especially in the plane
of its equator.

There may be and probably are collisions even now
among the particles which constitute the rings of Saturn.
If there are collisions, the energy of motion is being transformed
into heat, and this comes from the energy of the
orbital motions, with the result that the dimensions of the
rings are being decreased. They may ultimately disappear
%% -----File: 472.png---Folio 442-------
for this reason, and it is not impossible that other planets
also once had ring systems.

\Article{257}{The Planetoids.}---The planetoids occupy a zone in
\index{Planetoids!orbits of}%
which there was no predominating nucleus. They probably
have not grown so much relatively as the planets by the
accretion of planetesimals. Hence the ranges in the eccentricities
and inclinations of their orbits give a better idea
of the character of the orbits of the original planetesimals.

Besides the known planetoids, there are probably thousands
of others which are so small that they have not been
seen. There may be others also between the orbits of
Jupiter and Saturn and beyond the orbit of Saturn. At
those vast distances none but large bodies would be visible,
both because they would not be strongly illuminated by
the sun and also because they would always be very remote
from the earth. The planetoid Eros has escaped collision
with Mars only because of the inclination of its orbit. It
is not unreasonable to suppose that there are many other
planetesimals between the orbits of the earth and Mars
which are too small to be visible.

\Article{258}{The Zodiacal Light.}---It is universally agreed that
\index{Light!zodiacal}%
\index{Zodiacal light}%
the zodiacal light is due to a great swarm of small bodies,
or particles, revolving around the sun near the plane of the
earth's orbit. These small bodies are in reality planetesimals
which have not been swept up by the planets either
because of the high inclination of their orbits, or more
probably because their orbits are so nearly circular that they
do not cross the orbits of any of the planets.

\Article{259}{The Comets.}---Recent investigations have shown
\index{Comets!origin of}%
\index{Origin!of comets}%
that it is very probable that comets are permanent members
of the solar system. As they have no intimate relationship
to the planets, the question of their origin presents new
problems and difficulties.

According to the planetesimal theory, the comets are
possibly only the outlying and tenuous fragments of the original
nebula which did not partake of the general rotation
%% -----File: 473.png---Folio 443-------
of the system. If the planetesimal system was produced by
the near approach of two suns, they may have had their
origin, as Chamberlin has suggested, in the dispersion and
\index[xnames]{Chamberlin}%
scattering of earlier planetesimals which revolved in different
planes; or there may have been explosions of lighter gases
in various directions, which, under the disturbing action of
the visiting sun, did not fall back upon our own; or the
comets may be matter which was ejected from the visiting
sun. The differences in the lengths of their orbits and in the
positions of the planes of their orbits may originally have
been much less than at present, for the planets may have disturbed
their motions to almost any extent. The planets
may have captured some comets and greatly enlarged the orbits
of an equal number of others, and they may have entirely
changed the positions of the planes of the cometary orbits.

\Article{260}{The Future of the Solar System.}---The theory has
\index{Sun!past and future of}%
been developed that the planets have grown up from nuclei
by the accretion of scattered planetesimals. They acquired
and retained atmospheres when their masses became great
enough to prevent the escape of gases, molecule by molecule.
Their masses are still increasing, but the process of growth
seems to be essentially finished. Those planets which are
dense and solid like the earth will retain all their essential
characteristics as long as the sun continues to furnish radiant
energy at its present rate. The large rare planets will lose
heat from their interiors and may contract appreciably.
The reason that loss of heat may be important for them and
not for the solid planets is that it can be carried to the surface
rapidly by convection in a gaseous or liquid body, while
in a solid body it is transferred from the interior only by the
excessively slow process of conduction.

The duration of the sun is a very important factor in the
future of the planets. There is no known source of energy
which could supply its present rate of radiation many tens
of millions of years. Yet it is not safe to conclude that the
sun will cool off in a few millions of years because the earth
%% -----File: 474.png---Folio 444-------
gives indisputable evidences (\Artref{219}) that the sun has
radiated more energy than could have been supplied by any
known source. The existence of hundreds of millions of
stars blazing in full glory also suggests strongly that the
lifetime of a sun is very long, for it is not reasonable to suppose
that, if they endured only a comparatively short time,
so many of them would now have such great brilliancy. In
view of these uncertainties it is not safe to set any definite
limit on the future duration of the sun, however probable
its final extinction may be.

If the sun cools off before something destroys the planets,
they will revolve around it cold, lifeless, and invisible, while
it pursues its journey through the trackless infinity of space.
If the radiation of the sun does not sensibly diminish, the
earth, and possibly some of the other planets, will continue
to be suited for the abode of life until they are in some
way destroyed. Whether or not the sun becomes cold, the
planets will be broken into fragments when our sun passes
sufficiently near another star. Their remains may then be
dispersed among the revolving masses of a new planetesimal
system, to become in time parts of new planets. Indeed,
the meteorites which now strike the earth often give evidence
of having once been in the interior of masses of planetary
dimensions, and Chamberlin has suggested that they may
\index[xnames]{Chamberlin}%
be the remains of a family of planets antedating our own.
To such dizzy heights are we led in sober scientific pursuits!

The question of the purpose of all the wonderful things in
the universe is one which ever arises in the human mind.
With sublime egotism men have usually answered that everything
was created for their pleasure and benefit. The sun
was made to give them light by day, and the moon and the
myriads of stars to illuminate their way by night. The
numberless plants and animals of forest and prairie and
sea were supposed to exist primarily for the profit of the
human race. But with the increase of knowledge this conclusion
is seen to be absurd. How infinitesimal a part of
%% -----File: 475.png---Folio 445-------
the solar system and its energy man can use, to say nothing
of that in the hundreds of millions of other systems which
are found in the sky!

How many billions of creatures were born, lived, and died
before man appeared! The precise time of the beginning of
life on the earth and the manner of its origin are lost in the
distant past. In the oldest rocks laid down as sediments
tens of millions of years ago in the Archeozoic era there are
indications of the existence of low forms of life on the earth.
In the Cambrian period trilobites and other lowly creatures
lived in great abundance. In the Ordovician period the
types of low forms greatly increased and the vertebrates
began to appear; in the Silurian, they were firmly established;
in the Devonian, they were still further developed. And
after many other geological periods had passed, the higher
forms of life, including man, appeared. Now man finds
himself a part of this great life stream, not something fundamentally
different from the rest and that for which it exists.
If the earth shall last some millions or tens of millions of
years in the future, as seems likely, the physical and mental
changes which the human race will undergo may be as great
as those through which the animal kingdom has passed during
the long periods of geological time. This is especially probable
if men learn how to direct the processes of their own
evolution. But if they do not, the human race may become
extinct just as many other species of animals have become
extinct. However this may be, it seems certain that its end
will come, for eventually the light of the sun will go out, or
the earth and the other planets will be wrecked by a passing
star, and the question of the purpose of it all, if indeed
there is any purpose in it, still remains unanswered.


\Section{XX}{QUESTIONS}

1. Are the particles which produce the zodiacal light an example
of the planetesimal organization?

2. In the case of one star passing by another, why would their
ejections of material be largely toward or from each other?
%% -----File: 476.png---Folio 446-------

3. Show by a resolution of the forces that the material ejected
both toward and from~$S'$ will describe curves around $S$ in the same
direction.

4. Will the orbit of~$S'$ be changed if it changes the moment of
momentum of the system~$S$? What will be the result in the very
special case where the orbit of~$S'$ relatively to~$S$ is originally a
parabola?

5. In view of \Tableref{XIII}, what fraction of the material ejected
from~$S$ would reasonably be expected (\textit{a})~to fall back on~$S$, (\textit{b})~to
revolve around it in the planetesimal state, (\textit{c})~to escape from its
gravitative control? On the basis of these figures, find what fraction
of~$S$ would need to be ejected altogether in order to provide
material for the planets.

6. Would the eccentricities of the orbits of the material which fell
back upon~$S$ have been large or small? Would most of the collisions
have been grazing, as was assumed in the discussion in \Artref{253}?

7. In view of the kinetic theory of gases, would a gaseous nucleus
as massive as the moon concentrate or dissipate? Would a nucleus
of the materials found in the sun remain gaseous on cooling?

\normalsize


\Section{IV}{Historical Cosmogonies}

\Article{261}{The Hypothesis of Kant.}---The work of Thomas
\index{Hypothesis!of Kant}%
\index[xnames]{Kant}%
Wright, which preceded that of Kant by five years, was
\index[xnames]{Wright, Thomas}%
concerned chiefly with the evolution of the whole sidereal
universe. Wright supposed the Milky Way is made up of
a great number of mutually attracting systems which are
spread out in a great double ring rotating about an axis
perpendicular to its plane. Kant was the first one to undertake
the development of a detailed theory of the evolution of
the solar system on the basis of the law of gravitation.

Kant's interest in cosmogony was aroused by the book
of Wright, which fell into his hands in 1751. He at once
turned his keen and penetrating mind to the question of the
origin of the planets, and wrote a brilliant work on the subject.
On almost every page he gave proof of the intellectual
power which later made him the foremost philosopher of his
time, yet his theories were not without serious imperfections.

Kant postulated that the parent of the solar system was
a vast aggregation of simple elements, without motion and
%% -----File: 477.png---Folio 447-------
subject only to gravitational and chemical forces and the
repulsion of molecules in a gaseous state. Nothing could
have been simpler for a start. The problem was to show how
such a system could develop into a central sun and a family
of widely separated planets.

Kant reasoned that motions among the molecules would
\index[xnames]{Kant}%
be set up by their chemical affinities and mutual attractions.
He stated that the large molecules would draw to themselves
the smaller ones in their immediate neighborhood, and that
with growth their power of growing would continually
increase. He believed that not only would aggregations of
molecules be formed, but that these masses would acquire
motions both because of the attraction of the system as a
whole and also because of their mutual attractions. Kant
called attention to the fact that attraction would be opposed
by gaseous expansion, and he supposed that these repulsive
forces in some obscure way would generate lateral motions
in the small nuclei. At first the nuclei would be moving
in every possible direction, but he assumed that successive
collisions would eliminate all except a few moving in the
same direction in nearly circular orbits.

The beauty and generality of Kant's theory are enticing,
but it involves some obvious and fatal difficulties. In the
first place, the attractive and repulsive forces would not
be competent to set up a general revolution of a system
which was originally at rest. His conclusion in this matter
squarely violates the principle that the moment of momentum
of an isolated system remains constant.

Notwithstanding clear statements by Kant, some writers
have modified his theory by supposing that there was heterogeneous
motion of the original chaos with a predominance in
the direction in which the planets now revolve. But with
this concession to the theory, which makes it dynamically a
different theory, difficulties still remain. It is not at all
clear that in a system of such enormous extent the orbits
of all bodies except those having motion in the dominant
%% -----File: 478.png---Folio 448-------
direction would be destroyed by collisions. There is, indeed,
no apparent reason why, if this were the true history of the
origin of the planets, some planets should not now be found
revolving at right angles to the general plane of the system,
or even in the retrograde direction. This is not impossible,
as is proved by the motions of the comets. Thus it is seen that
if Kant's hypothesis is taken strictly as he gave it, the condition
\index[xnames]{Kant}%
that the moment of momentum of the system should
have its present value is violated, and that if the postulates
are changed so as to relieve this difficulty, others still remain.

Kant's theory has also secondary difficulties of a serious
nature. For example, in a gas the mutual attractions of
the molecules could not draw them together into small nuclei.
Even the moon could not now add to its mass if it should
pass through a gas. To avoid this difficulty one might assume
that there was first condensation into the liquid or solid state.
So many molecules would be involved in the formation of
even the minutest drop that, by an averaging process, their
lateral motions would essentially destroy one another, the
particle would fall toward the center of the whole system,
and no planets would be formed. In order to avoid this
difficulty it is necessary to depart from Kant's ideas and to
assume either that the whole gaseous mass was rotating with
considerable velocity, or that the matter was not in a gaseous
state. If the first of the two assumptions is made, it is found
by a mathematical treatment that the moment of momentum
of the system would be more than $200$~times what it is
at present. Since the moment of momentum would remain
unaltered, the second alternative must be adopted. But
this is directly contrary to the fundamental assumptions of
Kant, and it is hardly permissible to regard a theory as having
preserved its identity after having been modified to this
extent. The condition to which one is forced, viz., that of
discrete particles in orbital revolution in the same direction,
is actually the planetesimal organization.

In successive chapters Kant considered the densities and
%% -----File: 479.png---Folio 449-------
ratios of the masses of the planets, the eccentricities of the
planetary orbits and the origin of comets, the origin of satellites
and the rotation of the planets, etc. He even claimed
to have proved without observational evidence the existence
of life on other planets. In spite of the keenness of his
intellect and his remarkable powers of generalization, his
theory has not had much influence on speculations in cosmogony,
because it is marred by so many serious errors in
the application of physical and dynamical laws.

\Article{262}{The Hypothesis of Laplace.}---The hypothesis of
\index{Hypothesis!of Laplace}%
\index{Laplacian hypothesis}%
\index{Nebular hypothesis}%
Laplace appeared near the end of a splendid popular work
on astronomy which %[Illustration: Break]
he published
in 1796. He advanced it ``with
that distrust which everything
ought to inspire that is not a
result of observation or of calculation.''
How great an advance
over Kant this one sentence
\index[xnames]{Kant}%
indicates!

In outline, the hypothesis of
Laplace was that originally the
solar atmosphere (in later editions
a nebulous envelope), in an
intensely heated %[Illustration: Break]
\begin{wrapfigure}[21]{\WLoc}{2.375in}
\Input[2.375in]{479}{jpg}
\Caption[Laplace.]{Fig}{162}
\index[xnames]{Laplace}%
\end{wrapfigure}
state, extended
out beyond the orbit of the
farthest planet; the whole mass
rotated as a solid in the direction
in which the planets now revolve; the dimensions of
the solar atmosphere were maintained mostly by gaseous
expansion of the highly heated vapors, and only slightly
by the centrifugal acceleration due to the rotation; as the
mass lost heat by radiation, it contracted under the mutual
gravitation of its parts; simultaneously with its contraction,
its rate of rotation necessarily increased because
the moment of momentum remained constant; after the
rotating mass had contracted to certain dimensions the centrifugal
%% -----File: 480.png---Folio 450-------
acceleration at the equator equaled the attraction by
the interior parts and an equatorial ring was left behind, the
remainder continuing to contract; a ring was abandoned
at the distance of each planet; a ring could scarcely have
had absolute uniformity, and, separating at some point, it
united at some other because of the mutual attractions of
its parts and formed a planet; and, finally, the satellites were
formed from rings which were left off by the contracting planets,
Saturn's rings being the only examples still remaining.

The contraction theory of the sun's heat, which was developed
by Helmholtz in 1854, fitted in very well with the Laplacian
\index[xnames]{Helmholtz}%
hypothesis and was considered as supporting it.
Some objections to the Laplacian theory, however, began to
appear. In 1873 Roche, the author of the theorem that a
\index[xnames]{Roche}%
satellite would be broken up by tidal strains if its distance
\index{Roche's limit}%
from its primary should become less than $2.44$~radii of the
latter, seriously undertook to modify the hypothesis of
Laplace so as to relieve it of the difficulties with which it
\index[xnames]{Laplace}%
was beset. His modifications were for the most part improbable
and do not in the least meet later objections. Kirkwood,
\index[xnames]{Kirkwood}%
an American astronomer, criticized the Laplacian
hypothesis and pointed out that the direct rotation of the
planets might be due to the effect of the sun's tides on them
when they were expanded in the gaseous state. In 1884
Faye made very radical modifications of the hypothesis of
\index[xnames]{Faye}%
Laplace for the purpose of avoiding the difficulties in which
it was becoming involved. He supposed that the planets
were formed in the depths of the solar nebula and that those
nearer the sun are actually older than those which are more
remote. About 1878 Darwin began his great work on the
\index[xnames]{Darwin, George H.}%
tides which he regarded as supplementing and strengthening
the hypothesis of Laplace.

It is now generally recognized that the Laplacian hypothesis
fails because it does not meet the most fundamental
requirements of the problem. For example, the density of
the hypothetical solar atmosphere must have varied in harmony
%% -----File: 481.png---Folio 451-------
with the laws of gases. With this distribution of
density, which can be theoretically determined, and the
rotation which is given by the revolution of the planets, it is
an easy matter to compute the moment of momentum possessed
by the hypothetical system when it extended out to
the orbit of Neptune. It turns out to be more than $200$~times
that of the system at present. If the hypothesis of
Laplace were correct, the two would be equal; the discrepancy
\index[xnames]{Laplace}%
is so enormous that the hypothesis must be radically
wrong.

The details of the Laplacian hypothesis are subject to
equally serious difficulties. For example, it would be impossible
for successive rings to be left off. Kirkwood long ago
\index[xnames]{Kirkwood}%
pointed out that if instability in the equatorial zone once
set in, it would persist, and Chamberlin has shown that the
\index[xnames]{Chamberlin}%
result would be a continuous disk of particles describing
nearly circular orbits. Further, if a ring were left off, it
could not even begin to condense into a planet because both
gaseous expansion and the tidal forces due to the sun would
more than offset the mutual gravitation of its parts. It has
been seen how large and dense\footnote
   {The power of control of a planet on an atmosphere is proportional to
    the product of its density and radius.}
a planet must be in order
to hold an atmosphere; while the ring would be large, its
density would be extremely low and it could not control the
lighter elements. And it has been shown that even if a circular
ring had in some way largely condensed into a planet,
the process could not have completed itself. In order that
a nucleus may gather up scattered materials, it is necessary
that they shall be moving in considerably eccentric orbits.

Since the Laplacian hypothesis fails in the fundamental
requirement of moment of momentum, as well as in a number
of other essential respects, it will be sufficient simply to
enumerate some of the phenomena which are obviously not
in harmony with it:

(1) It does not provide for the planetoids with their
%% -----File: 482.png---Folio 452-------
interlacing orbits, some having high inclinations or eccentricities.

(2) It does not permit of the existence of an object having
such an orbit as that of Eros, which reaches from near
that of the earth out beyond that of Mars.

(3) It implies that a continuous disk of particles, such as
that producing the zodiacal light, cannot exist.

(4) It does not anticipate the considerable eccentricity
and inclination of Mercury's orbit.

(5) It does not agree with the fact that the terrestrial
planets seem to be at least as old as the more remote ones.

(6) It does not permit of there being any retrograde satellites
because the rings abandoned by a contracting nebula
would necessarily all rotate in the same direction.\footnote
  {Attempts have been made, though not successfully, to avoid this difficulty
  by invoking tidal friction (\Artref{264}).}

(7) It implies that the rotation period of each planet shall
be shorter than the shortest period of revolution of its satellites.
This condition is not only violated in the case of the
inner satellite of Mars, but the particles of the inner ring of
Saturn revolve in half the period of the planet's rotation.

\Article{263}{Tidal Forces.}---The sun and moon generate tides
\index{Tide-raising!forces}%
in the oceans that cover the earth. Tides are raised also
in the atmosphere and in the solid earth itself. Similarly,
every celestial body raises tides in every other celestial body.
The first problem which will be considered here will be the
character of the tide-raising forces, and the second will be
the effects of the tides on the rotations and revolution of the
two bodies.

Consider the tide-raising effects of $m$ on~$M$, \Figref{163}.
For simplicity, consider the effects of $m$ on $P$~and~$P'$, two
particles on the surface of~$M$. The problem of the resolution
of the forces is that which was treated in \Artref{153}. Let
$MA$ represent the acceleration of $m$ on $M$ in direction and
amount. Then the acceleration of $m$ on $P$ and $P'$ will be
represented by $PB$ and $P'B'$ respectively. The former is
%% -----File: 483.png---Folio 453-------
greater than~$MA$ because the acceleration varies inversely
as the square of the distance, and $Mm$~is greater than~$Pm$.
For a similar reason $P'B'$~is less than~$MA$. Now resolve
$PB$ into two components, $PC$~and~$PD$, in such a way that
$PC$~shall be equal and parallel to~$MA$. Similarly, resolve
$P'B'$ into~$P'C'$, equal and parallel to~$MA$, and~$P'D'$. Since
$PC$~and~$P'C'$ are equal and parallel to~$MA$, they have no
tendency to displace $P$~and~$P'$ respectively with respect to~$M$.
The remaining components, $PD$~and~$P'D'$, are the tide-raising
\index{Tide-raising!forces}%
accelerations.

Now consider the tide-raising forces all around~$M$. They
are as indicated in \Figref{94}. The forces toward $m$ are slightly
\begin{figure}[hbt]%[Illustration:]
\Input{483}{png}
\Caption[The tide-raising force.]{Fig}{163}
\end{figure}%
greater than those in the opposite direction, while the compressional
forces at $90°$ from these directions are half as great.
It is clear from this figure that if $M$~were not rotating and $m$~were
not revolving around it, there would be a tide on the
side of~$M$ towards~$m$, and a nearly equal tide on the side of~$M$
away from~$m$ (compare \Artref{153}). The motions of the
bodies produce important modifications.

Suppose the rotation of~$M$ and the revolution of~$m$ are in
the same direction and that the period of rotation of~$M$ is
shorter than that of the revolution of~$m$. This is the case
in the earth-moon system. Under these circumstances the
tides $T_1$~and~$T_2$ are carried somewhat ahead of the line~$Mm$,
as represented in \Figref{164}. The more nearly equal the rates
of rotation of~$M$ and revolution of~$m$, the more nearly will
the tides $T_1$~and~$T_2$ be in the line~$Mm$.
%% -----File: 484.png---Folio 454-------

Consider a point on the rotating body~$M$. It will first
pass the line~$Mm$, and then somewhat later it will pass the
tide~$T_1$. The interval is the lag of the tide. In the case
of the earth-moon system a meridian passes eastward across
the moon (the moon seems to pass westward across the
meridian), and somewhat later the meridian passes the tidal
cone and has high tide. The problem is enormously complicated
in the case of the earth by the addition of the tides due
to the sun, by the varying distances of the moon and sun
north or south of the celestial equator, by their changing distances
from the earth, and especially by the irregular contours
of the continents and the varying depths of the oceans.
These modifying factors are so numerous and in some cases
so poorly known that at present it is not possible to predict
entirely in advance of observation either the times or heights
of the tides; but, after a few observational data have established
the way in which the tides at a station depend upon
the unknown factors, predictions become thoroughly reliable,
for the tides vary in perfect harmony with the tidal forces.

\Article{264}{Tidal Evolution.}---The tides are not fixed on the
\index{Tidal!evolution}%
surface of~$M$, \Figref{164}, unless the period of its rotation equals
the period of revolution of~$m$. When the periods are unequal
so that the tides move around the rotating body, some energy
is changed to heat by friction. This energy comes from the
kinetic and potential energies of the system; and, in accordance
with the laws of dynamics, the transformation of
energy takes place in such a way that the total moment of
momentum of the system remains unchanged. Of course,
in general there will be tides on both of the mutually attracting
bodies.

The character of the transformation of energy that takes
place under tidal friction depends upon the dynamical
properties of the system. Suppose that the motions of
rotation and revolution are in the same direction and that
the period of~$M$ is shorter than that of~$m$. It can be shown
that under these circumstances the periods of both $M$~and~$m$
%% -----File: 485.png---Folio 455-------
and their distance apart are increased. The reason that the
period of rotation of $M$ is increased is that $m$ has a component
of attraction back on both $T_1$ and~$T_2$, \Figref{164}, as can be
shown by resolving the forces as they were resolved in \Figref{163}.
If $m$ pulls $T_1$ and $T_2$ backward, it follows from the
reaction of forces that $T_1$ and~$T_2$ pull~$m$ forward. The result
of a forward component on $m$ is to increase the size of its
orbit and to lengthen its period.

If $m$ is near~$M$, the effect of the tides on the period of revolution
of $m$ is greater than their effect on the period of rotation
of~$M$. If the bodies are far apart, the result is the opposite.

Suppose the two bodies are initially close together and that
the period of rotation of $M$ is only a little shorter than the
\begin{figure}[hbt]%[Illustration:]
\Input{485}{png}
\Caption[Tidal cones and the lag of the tides.]{Fig}{164}
\index{Lag of tides}%
\index{Tidal!cones}%
\index{Tides!lag of}%
\end{figure}%
period of revolution of~$m$. The friction of the tides will
lengthen both periods and increase the difference between
them. If nothing else interferes, this will continue until a
certain distance between the bodies has been reached. After
that, the effect on the period of rotation of $M$ will be greater
than that on the period of revolution of~$m$. Consequently,
although both periods will continue to increase in length,
they will approach equality. Eventually, the periods of
rotation and revolution will be equal, the tides will remain
fixed on~$M$, and there will be no further tidal friction or
tidal evolution.

The most important case from a practical point of view
has been considered, but there are two others. In the first
the bodies move in the same direction, but the period of
%% -----File: 486.png---Folio 456-------
rotation of~$M$ is longer than that of revolution of~$m$. Under
these circumstances both periods are decreased, the relative
amounts depending on the distance of the bodies from each
other. If the bodies are initially far apart, the effect will be
greater on the period of rotation of~$M$ than on the period of
revolution of~$m$, and the two periods will approach equality.
But if the bodies are near together, the effect will be relatively
greater on the period of~$m$, the periods will not approach
equality, and the bodies will ultimately collide. In the
second case the rotation of~$M$ is in the direction opposite to
that of the revolution of~$m$. Under these circumstances
$M$~rotates faster and faster, the distance of~$m$ continually
decreases, and the inevitable outcome is the collision and
union of the two bodies.

The rate at which tidal friction takes place depends upon
the physical properties of the bodies. If they are perfect
fluids so that there is no degeneration of energy, there is no
tidal evolution. Likewise if they are perfectly elastic, there
is no tidal friction.

The rate of tidal friction also depends upon the difference
in the periods of the two bodies. If the difference between
the periods is small, the tides $T_1$~and~$T_2$, \Figref{164}, are almost
in the line~$Mm$, and it is obvious that the backward components
are small and the rate of tidal friction is very slow.
Suppose the periods are approaching equality. The smaller
their difference the slower is their rate of change, and they
never become exactly equal but approach equality as the
time becomes infinitely great.

\Article{265}{Effects of the Tides on the Motions of the Moon.}---The
\index{Tides!effects of, on moon}%
most striking thing in the earth-moon system is that
the moon's periods of rotation and revolution are equal.
It is extremely improbable that this unique relation is accidental.
The only explanation of it heretofore advanced is
that the moon's period of rotation has been brought into
equality with its period of revolution by the tides generated
in it by the earth.
%% -----File: 487.png---Folio 457-------

The tidal force exerted by the earth on the moon is about
$20$~times the tidal force exerted by the moon on the earth.
The amount of tidal friction is proportional to the square of
the tidal force. Therefore, if the physical properties of the
earth and moon were the same and if their periods of rotation
were equal, the effectiveness of the tides generated by the
earth on the moon in changing the moment of momentum
of the moon would be $400$~times that of the tides generated
by the moon on the earth in changing the moment of momentum
of the earth. Since the moment of momentum of a body
is proportional to the product of its mass and the square of its
radius, a given change in the moment of momentum of the
moon alters its rate of rotation $1200$~times as much as the
same change in moment of momentum alters the rate of
rotation of the earth. Consequently, taking the two factors
together, if the earth and moon were physically alike
and had the same period of rotation, tidal friction would
change the period of rotation of the moon $400 × 1200 =
480,000$~times as fast as it would change the period of rotation
of the earth.

The results which have been obtained seem to be very
favorable to the theory that the tides have caused the
moon to present one side toward the earth, but some serious
difficulties remain. It can be shown that, considering the
tidal interactions of the earth and moon and the effect of
the sun's tides on the moon, the present condition of the
earth-moon system is not one of equilibrium. The tides
raised by the earth on the moon have no effect under present
circumstances on the rotation and revolution of the moon.
The tides raised by the moon on the earth increase the length
of the month but do not affect the rotation of the moon.
The tides raised by the sun on the moon increase the moon's
period of rotation but do not affect its revolution. Consequently
the moon's periods of rotation and revolution are
both increasing, and it is infinitely improbable that all the
factors on which these effects depend are so related that
%% -----File: 488.png---Folio 458-------
they are exactly equal. Even if they were equal at one time,
they would become unequal with a changed distance of the
moon from the earth. That is, the present is not a fixed state
of equilibrium, and the consideration of the tides does not
remove the difficulties. It seems probable from this line of
thought that some influence so far not considered has caused
the moon always to present the same face toward the earth.

\Article{266}{Effects of the Tides on the Motions of the Earth.}---The
\index{Tides!effects of, on earth}%
theory of the tidal evolution of the earth-moon system,
on the basis of certain assumptions regarding the physical
condition of the earth, was elaborated by Sir George Darwin
\index[xnames]{Darwin, George H.}%
in a splendid series of investigations. While the experiment
of Michelson and Gale (\Artref{25}) proves that his assumptions
\index[xnames]{Gale}%
\index[xnames]{Michelson}%
are not satisfied, at least at the present time, the possible
sequence of events which he worked out is interesting.

Since the tides are increasing the lengths of both the
day and the month, both of these periods were formerly
shorter and the moon was nearer the earth. On the basis
of his assumptions, Darwin traced the day back until it was
only four or five of our present hours. At that time the
moon was revolving close to the earth in a period almost
equally short. This led him to the conclusion that at an
earlier stage the earth and moon were one body, that they
divided into two parts because of the rapid rotation of the
combined mass, and that they have attained their present
state as a consequence of tidal friction. The same reasoning
leads to the conclusion that in the future they will continue
to separate and that the day will continually increase
in length.

The critical question is whether the physical properties
of the earth are such that the rate at which tidal evolution
takes place makes it an appreciable factor in the history of
the earth. Darwin supposed the main effects were due to
bodily tides in the earth which he assumed to be viscous.
Since it is highly elastic, they cannot at present be important,
but it has generally been assumed that, whatever its present
%% -----File: 489.png---Folio 459-------
condition may be, it was formerly viscous. There is absolutely
no evidence to support the assumption, and if its
present properties of solidity and elasticity are a consequence
of the pressure in its interior, the assumption seems
very improbable. As Poisson and Lord Kelvin showed,
\index[xnames]{Kelvin}%
\index[xnames]{Poisson}%
the temperature of the interior of the earth cannot have
fallen appreciably in several hundreds of millions of years by
the conduction of heat to its surface. Since the temperature
of the interior of the earth cannot have changed appreciably,
there seems to be no good ground for assuming that
the earth was once viscous.

Since there cannot now be an important degeneration of
energy in the bodily tides of the earth, tidal evolution must
depend at present almost entirely upon the tides in the
ocean and the atmosphere. The latter may be neglected
without important error. The oceanic tides are so irregular
that it is difficult to determine their effects on the rotation
of the earth. But MacMillan has made liberal estimates of
\index[xnames]{MacMillan}%
the unknown factors, and has found that at present the
length of the day cannot be increasing at a rate of more
than one minute in $30,000,000$~years.

It is possible to determine observationally the present rate
of tidal evolution. The day and the month are increasing
in length, but the effect on the day is the greater. Consequently,
if the length of the month is measured in days, as
is done practically, it will seem to be decreasing in length.
It is found from observations that the moon is getting ahead
of its predicted place from $4$ to $6$~seconds of arc in $100$~years.
That is, in $1240$ revolutions the moon gets ahead of its predicted
place about $\frac{1}{400}$ of its diameter. On the basis of these
facts and the assumption that the increase in the length of
the month is due to the tidal interactions of the earth and
moon, the proper discussion shows that at the present time
the length of the day is increasing as a consequence of all the
factors affecting the rotation of the earth at the rate of one
minute in $900,000,000$~years.
%% -----File: 490.png---Folio 460-------

It is evident that tidal evolution is not an important factor
in the earth-moon system for any period short of several
hundred millions of years. Either the theory of tidal evolution
as elaborated by Darwin must be abandoned as not
\index[xnames]{Darwin, George H.}%
representing what has actually taken place, or a condition
for the earth's interior totally different from that which exists
at present must be arbitrarily assumed.

\Article{267}{Tidal Evolution of the Planets.}---There is perhaps
\index{Tidal!evolution}%
some slight evidence that Mercury and Venus always keep
the same side toward the sun, and this condition has been
ascribed to the effects of tides which the sun may have raised
in them. The tidal force exerted by the sun on Mercury is
about $2.5$~times as great as that of the moon on the earth.
In view of the fact that the moon's tides on the earth certainly
do not have appreciable effects, it does not seem probable
that the sun's tides have radically changed the rotations
of Mercury and Venus. Besides this, it must be remembered
that the condition of equality of periods of rotation and
revolution would be attained in any case only after an
infinite time.

The tidal action of the sun on the more remote planets is
much less than that on the earth. No other satellite has
relatively as great effects on its primary as the moon has on
the earth. Consequently, we are forced to the conclusion
that in the solar system tidal evolution has not been an important
factor for a period of at least several hundreds of
millions of years.


\Section{XXI}{QUESTIONS}

1. According to Kant's theory, why should the sun rotate in the
direction the planets revolve?

2. Is the assumption of Laplace that the original nebula was
highly heated in harmony with the present temperature of the sun
and Lane's law? Why did Laplace make the assumption?

3. Why did Laplace assume that the original nebula was rotating
as a solid?

4. To what extent does the contraction theory of the sun's heat
%% -----File: 491.png---Folio 461-------
support the Laplacian hypothesis? Is it opposed to the planetesimal
hypothesis and Kant's hypothesis?

5. In what way does the Laplacian hypothesis fail to meet the
requirements of moment of momentum?

6. On the basis of Lane's law, what was the temperature of the
surface of the sun if it extended to the orbit of the earth? How do
you account for the presence of refractory materials in the earth,
under the Laplacian hypothesis?

7. Explain carefully in what respects the seven things mentioned
at the end of \Artref{262} are opposed to the Laplacian hypothesis.

8. What should be the present shape of the sun if the Laplacian
hypothesis were true?

9. In the case of the earth and moon, what is the magnitude of
the tidal force at the point on the side of the earth toward the moon
compared to the whole attraction of the moon? Compared to the
attraction of the earth?

10. The tides produced on the earth by the moon decrease the
earth's moment of momentum; what becomes of that which the
earth loses, and what changes in the system does it cause?

11. Show that when $M$~rotates faster than $m$~revolves, the
attractions of~$m$ for both $T_1$~and~$T_2$ tend to decrease the rate of
rotation of~$M$.

12. Suppose the rate of rotation of the earth is constant and that
in a century the moon gets $6''$~ahead of the place it would occupy
if its rate of revolution were constant. How long would be required
for its period to decrease $1$~per cent?

\normalsize

%% -----File: 492.png---Folio 462-------
\begin{figure}[hbtp]%[Illustration]
\centering\Input{492}{jpg}
\Caption[Milky Way in Aquila. \textit{Photographed by Barnard at the Yerkes
Observatory, August~27, 1905.}]{Fig}{165}
\index{Milky Way}%
\index{Yerkes Observatory}%
\index[xnames]{Barnard}%
\end{figure}
%% -----File: 493.png---Folio 463-------


\Chapter{XIII}{The Sidereal Universe}

\Section{I}{The Apparent Distribution of the Stars}

\Article{268}{On the Problems of the Sidereal Universe.}---The
\index{Distribution!of stars}%
\index{Gravitation!law of}%
\index{Law!of gravitation}%
\index{Stars!distribution of}%
invention of the telescope and the discovery of the law of
gravitation were followed by a long period of successes in
unraveling the mysteries of the solar system. The positions
of the sun, moon, and planets were measured with extraordinary
precision, and the law of gravitation accounted
for the numerous peculiarities of their motions. What had
been mysterious and inexplicable became familiar and thoroughly
understood. Telescopes of continually increasing
power enabled astronomers to measure accurately the distances
and diameters of these bodies and to learn much of
their surface conditions. At last the invention of the spectroscope
\index{Spectroscope}%
enabled them to determine the chemical constitution
of the sun.

There is great pleasure now in working in a science whose
data are exact and whose laws are firmly established. The
certainty of the results satisfies the human instinct for final
truth. But there was also pleasure of a different kind for
those pioneers who first explored the planetary system with
good instruments and showed by mathematical processes
that its members are obedient to law. For them every
observation and every calculation was an adventure. They
were continually in fear that their dreams of knowing the
order prevailing in the universe would be shattered; they
were continually elated by having their theories confirmed.

The pioneer days of discovery in the solar system are past.
Not that great discoveries do not remain to be made, but
%% -----File: 494.png---Folio 464-------
they will henceforth fit into a large body of organized facts.
From now on the romance and excitement of astronomical
adventure will be largely furnished by the explorations of the
sidereal universe. Astronomers have become accustomed
to the fact that the sun is a million times as large as the earth,
and familiarity has dulled their amazement at its terrific
activities; from now on they must deal with millions of
stars, some of them much larger and thousands of times
more luminous than the sun. They have measured and at
least partially grasped the enormous dimensions of the solar
system; from now on they must conceive of and deal with
distances millions of times as great. They have observed
the differences in characteristics exhibited by eight planets;
from now on they will be face to face with the infinite diversity
presented by the stars. They have definitely accepted
the doctrine that the solar system has undergone a great
evolution whose details are yet much in doubt; the corresponding
question for hundreds of millions of other systems
is looming up more indistinctly through the fogs of uncertainties
which still surround them. It might be supposed
that astronomers would be tempted to lay down the arms
of their understanding before the transcendental and appallingly
difficult problems presented by the sidereal system.
Instead, with all the weapons at their command, they are
making more vigorous, persistent, and successful attacks than
ever before. Astronomers of all the leading countries are
united and coöperate in this campaign; they employ telescopes
of many kinds, spectroscopes, photographic plates,
measuring machines, and powerful mathematical processes in
their attempts to penetrate the unknown.

\Article{269}{The Number of Stars of Various Magnitudes.}---The
\index{Magnitudes of stars}% [** TN: Move up one page]
\index{Number of stars}%
\index{Stars!number of}%
simplest and most easily determined thing about the
stars is their number. Of course the number that can be
seen depends upon the power of the instrument with which
the observations are made. If the stars extend infinitely
in every direction with approximately equal distances from
%% -----File: 495.png---Folio 465-------
one another, the number of them revealed by a telescope will
be proportional to the space it brings within visual range.
On the other hand, if the stars are less densely distributed at
a great distance in any direction, then the number of faint
distant stars seen in that direction will fall short of being
proportional to the space penetrated by the instrument.
For this reason it is important to find the number of stars of
each magnitude down to the limits of range of the most
powerful telescopes.

Consider first what the apparent distribution in magnitude
would be if stars of every actual size and luminosity were
scattered uniformly throughout space. Take a large enough
volume so that accidental groupings will not sensibly affect
the results. For example, suppose there are $5000$~stars in
the first six magnitudes and compute the number there should
be, under the hypothesis, in the first seven magnitudes.
% [** TN: Original uses centered ellipses; using \ldots for consistency]
The sixth-magnitude stars are $2.512\,\ldots$~times as bright
as the seventh-magnitude stars. Since the magnitudes of
stars of any given absolute brightness are directly proportional
to the squares of their distances, it follows that stars
of the seventh magnitude are $\sqrt{2.512}\,\ldots = 1.585\,\ldots$~times as
distant as corresponding stars of the sixth magnitude.
Therefore the volume occupied by stars out to the seventh
magnitude, inclusive, is $(1.585\,\ldots)^3 = 3.98\,\ldots$~times that
occupied by the first six magnitudes. Hence, if the stars
were uniformly distributed and the light of the remote ones
were in no way obstructed, there would be $3.98\,\ldots$~times as
many stars in the first seven magnitudes as in the first six
magnitudes, or nearly $20,000$~stars. The ratio is the same
for the total number of stars up to any two successive
magnitudes because the particular magnitudes do not
enter into its computation. And it can be shown easily
that the ratio of the number of stars of any magnitude to the
number of the next magnitude brighter is also $3.98\,\ldots$.

It remains to examine the results furnished by the observations.
The stars are so extremely numerous that only a
%% -----File: 496.png---Folio 466-------
small fraction of the total number within reach of modern
instruments has been counted. But an approximation to the
solution of the problem of determining the number of stars
\index{Number of stars}%
\index{Stars!number of}%
has been obtained by counting sample regions of known size
in many parts of the sky, and then multiplying the result
by the number necessary to include the whole celestial sphere.
By far the most extensive work of this kind has been carried
out by Chapman and Melotte of the Royal Observatory at
\index[xnames]{Chapman}%
\index[xnames]{Melotte}%
Greenwich. They made use of stars down to magnitude
$17.5$, where $4,000,000$ of them send to the earth only a little
more light than one star of the first magnitude. Their
results are given in the \hyperref[Table:XIV]{following table}.\DPnote{** TN: Change ":" to "."}\footnote
  {The numbers in the first of this table disagree with those in \Tableref{II}
  because here, in the first line, for example, the number is that of stars from
  magnitude $5.0$ to~$6.0$, while in \Tableref{II} the corresponding number is that of
  stars whose magnitudes are $4.5$ to~$5.5$.}
\begin{table}[hbt]
%\caption[Numbers of stars in magnitudes $5$ to~$17$]{}
\begin{center}
\Caption{Table}{XIV}
\begin{tabular}{|*{2}{c|}|*{2}{c|}}
\hline
\Strut
\TFontsize\THF Magnitude & \TFontsize\THF Number of Stars &
\TFontsize\THF Magnitude & \TFontsize\THF Number of Stars \\
\hline
\Strut
$\Z5$ to $\Z6$ & $\Z\Z 2,026$ & $11$ to $12$ & $\phantom{00,}\, 961,000$ \\
$\Z6$ to $\Z7$ & $\Z\Z 7,095$ & $12$ to $13$ & $\Z  2,023,000$ \\
$\Z7$ to $\Z8$ & $\Z  22,550$ & $13$ to $14$ & $\Z  3,964,000$ \\
$\Z8$ to $\Z9$ & $\Z  65,040$ & $14$ to $15$ & $\Z  7,824,000$ \\
$\Z9$ to  $10$ & $   172,400$ & $15$ to $16$ & $   14,040,000$ \\
 $10$ to  $11$ & $   426,200$ & $16$ to $17$ & $   25,390,000$ \\
\hline
\end{tabular}
\end{center}
\end{table}

The ratio of the number of stars of a given magnitude to
the number of stars one magnitude fainter is $3.5$ at the
beginning of the table, and it continually decreases to $1.8$
at the end. Therefore, not only is the ratio for every
interval of one magnitude less than the $3.98$ corresponding
to uniform distribution of the stars, but it falls off about $50$
per cent in $12$ magnitudes.

What conclusions can be drawn from the facts given by
the table? It is certain that the stars cannot be uniformly
distributed to indefinite distances unless there is something
%% -----File: 497.png---Folio 467-------
which prevents their light from coming to us. If there were
a sufficient number of dark stars and planets, the light from
remote luminous stars would be shut off; but the number of
non-luminous bodies required to account for the black sky
would be millions of times the number of bright ones. In
spite of the fact that certain variable stars (\Artref{288}) prove
the existence of relatively dark bodies, and that analogy
with the planets would lead to the conclusion that there
are many non-luminous bodies of secondary dimensions,
it seems extremely improbable that they are sufficiently
numerous to explain the observed phenomena. But if the
obscure matter were finely divided, as in meteoric dust, a
given mass of it would be a much more effective screen,\footnote
  {The effectiveness of opaque matter of given total mass in cutting off
  light is inversely proportional to the radius of its separate parts.}
and the total mass requirements would not be so severe.
Finely divided material would not only absorb light, but it
\index{Absorption of light}%
would scatter the blue light and cause distant stars to appear
redder than nearer stars of the same character.

There are certain phenomena which give slight support
to the hypothesis that there is some scattering of light of
this nature, but they are not conclusive. One of them is
directly related to the question in hand. Kapteyn found
from an investigation of stars down to the fourteenth magnitude,
part of the data being furnished by the visual observations
of Sir John Herschel, that the number of stars of
\index[xnames]{Herschel, John}%
the fainter magnitudes is much greater than is given in the
table of Chapman and Melotte. The faintest stars used in
\index[xnames]{Chapman}%
\index[xnames]{Melotte}%
the construction of their table are obtained from the Franklin-Adams
photographic charts of Greenwich. Turner has
\index[xnames]{Turner}%
suggested that, because of the scattering of light, the remote
faint stars may be deficient in the blue end of the spectrum,
to which photographic plates are most sensitive, and consequently
that a considerable part of the stars belonging visually
to a certain magnitude belong photographically to a
fainter magnitude. In spite of these possible indications of
%% -----File: 498.png---Folio 468-------
scattered particles, it seems extremely improbable that the
falling off of the star ratio from $3.98$ to~$1.8$ is due appreciably
to this cause.

The most obvious, though not necessary, conclusion which
\index{Number of stars}%
\index{Stars!number of}%
has generally been drawn from the table is that the stars are
limited in number and that they occupy a limited portion of
space. In the first seventeen magnitudes there are in round
numbers $55,000,000$~stars. Chapman and Melotte derived
\index[xnames]{Chapman}%
\index[xnames]{Melotte}%
a simple formula which represented the numbers closely
for these magnitudes, and then, under the assumption that
the same formula holds indefinitely beyond, they determined
the magnitude for which there are as many stars
brighter as there are fainter, and computed the total number
of stars altogether. By this process they concluded that the
median magnitude lies between $22.5$~and~$24.3$, which are
several magnitudes beyond the reach of existing instruments,
and that the number of stars of all magnitudes is
between $770,000,000$ and $1,800,000,000$. It is obvious that
such an extrapolation is hazardous, and they did not lay
any particular stress on the results. In fact, the data
given by the observations can be as exactly represented by
many other less simple formulæ which will give totally different
results for the fainter magnitudes.

There is an even simpler line of reasoning which has led
many astronomers to the conclusion that the material universe
is limited. Since the stars of any magnitude are $2.512$~times
fainter than those of the next preceding magnitude,
and, under the hypothesis of uniform distribution, $3.98$~times
more numerous, it follows that if the star density did
not diminish as the distance increases, the stars of each
magnitude would give us $3.98 ÷ 2.512 = 1.58$~times as much
light as those of the next magnitude brighter. Consequently,
the first $20$~magnitudes would give $17,000$~times as much light
as the first-magnitude stars, the first $100$~magnitudes would
give $168,000,000,000,000,000,000$ times as much light, and so
on. If there were no limit to the number of magnitudes and
%% -----File: 499.png---Folio 469-------
no absorbing material, there would be no limit, except for the
mutual eclipsing of the stars, to the amount of light received
from all of them. The sky would be everywhere ablaze
with the average brightness of a star, perhaps equal to that
of the sun. The stars in one hemisphere would give us more
than $90,000$~times as much light as the sun, but actually
the sun gives us $15,000,000$ times as much light as all the stars
together. Therefore, unless much light is absorbed, the
hypothesis of uniform distribution of the stars to infinity
is radically false.

Is it necessary, therefore, to conclude that the number of
stars is limited and that they occupy only a finite part of
space? By no means; simply that they cannot be distributed
with approximate uniformity throughout infinite
space. It was pointed out by Lambert long ago that, just
as the solar system is a single unit in a galaxy of several hundred
million stars, so the Galaxy may be but a single one out
of an enormous number of galaxies separated by distances
which are very great in comparison with their dimensions,
and that these galaxies may form larger units, or super-galaxies,
and so on without limit. There is nothing in such
an organization which is inconsistent with the facts established
by observation, for it is possible to build up infinite
systems of stars in this way which would give us only a
finite amount of light. Hence the conclusion to be adopted
is that the sun is in the midst of an aggregation of at least
several hundred millions of stars which form a sort of system,
and that beyond and far distant from this system there may
be other somewhat similar systems in great numbers, which
may be units in larger systems, and so on without limit.

It is conceivable that the ether is not infinitely extensive,
but that it surrounds the stars of the sidereal system (and
other stellar systems if there are such) as the atmospheres
surround the planets. Light could not come to us from
beyond its borders, however many stars might exist there,
as sound cannot come to the earth from other bodies beyond
%% -----File: 500.png---Folio 470-------
the limits of its atmosphere. It must be understood that this
is merely a suggestion entirely without any observational basis.

\Article{270}{The Apparent Distribution of the Stars.}---The
\index{Distribution!of stars}%
\index{Stars!distribution of}%
brighter stars are quite irregularly distributed over the sky,
but a careful examination of the fainter of even those which
can be seen with the unaided eye shows that they are considerably
more numerous in and near the Milky Way than
\index{Galaxy}%
\index{Milky Way}%
elsewhere. When those stars which can be seen only with
the help of a telescope are included, the condensation toward
the Milky Way is still more pronounced.

Precise numbers for all the stars are known only to the
ninth magnitude; but the star counts of the Herschels, and
\index[xnames]{Herschel, John}%
\index[xnames]{Herschel, William}%
especially the work of Chapman and Melotte, go much
\index[xnames]{Chapman}%
\index[xnames]{Melotte}%
further and give what are very probably approximately
correct results down to the seventeenth magnitude. Since
the stars are apparently condensed toward the Milky Way,
it is natural to use its plane as the fundamental plane of
reference. According to E.~C. Pickering the north pole of
\index[xnames]{Pickering, E. C.}%
the Galaxy is in right ascension~$190°$ and its declination is~$+ 28°$.
The Milky Way is very irregular in outline, and it
is difficult to locate its center; but its median line is possibly
not quite a great circle, from which it follows that the sun
is somewhat out of the plane near which the stars are congregated.

Let the center of the Milky Way be the circle from which
galactic latitudes are counted. Chapman and Melotte
divided the sky up into eight zones, the first including the
belt of galactic latitude $0°$ to~$±10°$, the second the two belts
from $±10°$ to~$±20°$, the third the two belts from $±20°$ to~$±30°$,
the fourth from $±30°$ to~$±40°$, the fifth from $±40°$
to~$±50°$, the sixth from $±50°$ to~$±60°$, the seventh from
$±60°$ to~$±70°$, and the eighth the regions from $±70°$ to~$±90°$
around the galactic poles. With the belts numbered
in this order they found for the average number of stars in
each magnitude in $10$~square degrees the results given in
\Tableref{XV}.
%% -----File: 501.png---Folio 471-------

\begin{table}[hbt]
\begin{center}
\Caption{Table}{XV}
%\caption[Distribution of Stars by Magnitude]{}
\TFontsize
\setlength{\tabcolsep}{2pt}
\settowidth{\TmpLen}{~Latitude~}
\makebox[0pt][c]{%
\begin{tabular}{|*{9}{@{}c@{}|}}
\hline
\Strut%
  \textsc{Zone}
& I & II & III & IV & V & VI & VII & VIII\rule[-2ex]{0pt}{0pt} \\
\hline
&&&&&&&&\\[-1em]\hline %[** -1em to bring lines close together - may need tweaking]
\begin{tabular}{@{}c@{}}
  \TEntry{\TmpLen}{\medskip\centering Galactic\\~Latitude~\medskip}\\ \hline
  \TEntry{\TmpLen}{\medskip\centering\scshape Mag.\medskip}
\end{tabular}&
\TEntry{\TmpLen}{\centering $   0$ to $±10°$} &
\TEntry{\TmpLen}{\centering $±10°$ to $±20°$} &
\TEntry{\TmpLen}{\centering $±20°$ to $±30°$} &
\TEntry{\TmpLen}{\centering $±30°$ to $±40°$} &
\TEntry{\TmpLen}{\centering $±40°$ to $±50°$} &
\TEntry{\TmpLen}{\centering $±50°$ to $±60°$} &
\TEntry{\TmpLen}{\centering $±60°$ to $±70°$} &
\TEntry{\TmpLen}{\centering $±70°$ to $±90°$} \\
\hline
\rule{0pt}{3.5ex}%
1 to 5  &\phantom{12,345}\llap{0.27} & \phantom{12,345}\llap{0.23}  &
         \phantom{12,345}\llap{0.15} &  \phantom{1,234}\llap{0.11}  &
          \phantom{1,234}\llap{0.11} &  \phantom{1,234}\llap{0.11}  &
          \phantom{1,234}\llap{0.13} &  \phantom{1,234}\llap{0.13}  \\
6       &\phantom{12,34}\llap{0.7}\phantom{5}  & \phantom{12,34}\llap{0.7}\phantom{5} &
         \phantom{12,34}\llap{0.5}\phantom{5}  &  \phantom{1,23}\llap{0.4}\phantom{4} &
          \phantom{1,23}\llap{0.3}\phantom{4}  &  \phantom{1,23}\llap{0.3}\phantom{4} &
          \phantom{1,23}\llap{0.3}\phantom{4}  &  \phantom{1,23}\llap{0.3}\phantom{4} \\
7       &\phantom{12,34}\llap{2.6}\phantom{5}  & \phantom{12,34}\llap{2.3}\phantom{5} &
         \phantom{12,34}\llap{1.8}\phantom{5}  &  \phantom{1,23}\llap{1.5}\phantom{4}  &
          \phantom{1,23}\llap{1.2}\phantom{4}  &  \phantom{1,23}\llap{1.1}\phantom{4}  &
          \phantom{1,23}\llap{1.1}\phantom{4}  &  \phantom{1,23}\llap{1.1}\phantom{4}  \\
8       &\phantom{12,34}\llap{8.0}\phantom{5}  & \phantom{12,34}\llap{7.0}\phantom{5}  &
         \phantom{12,34}\llap{6.1}\phantom{5}  &  \phantom{1,23}\llap{4.8}\phantom{4} &
          \phantom{1,23}\llap{3.8}\phantom{4}  &  \phantom{1,23}\llap{3.4}\phantom{4}  &
          \phantom{1,23}\llap{3.2}\phantom{4}  &  \phantom{1,23}\llap{3.1}\phantom{4}  \\
9       &\phantom{12,3}24  & \phantom{12,3}21  &  \phantom{12,3}18  & \phantom{1,2}14 &
  \phantom{1,2}10  & \phantom{1,2}10 & \phantom{1,23}9 & \phantom{1,23}8  \\
10      &\phantom{12,3}62  & \phantom{12,3}55  & \phantom{12,3}50  & \phantom{1,2}38 &
  \phantom{1,2}28  & \phantom{1,2}26 & \phantom{1,2}22 & \phantom{1,2}20  \\
11      &\phantom{12,}157  & \phantom{12,}135  & \phantom{12,}123  & \phantom{1,}93  &
  \phantom{1,2}63  & \phantom{1,2}62 & \phantom{1,2}52 & \phantom{1,2}47  \\
12      &\phantom{12,}363  & \phantom{12,}311  & \phantom{12,}280  & \phantom{1,}199 &
  \phantom{1,}136  & \phantom{1,}141 & \phantom{1,}115 & \phantom{1,}100  \\
13      &\phantom{12,}798  & \phantom{12,}658  & \phantom{12,}569  & \phantom{1,}409 &
  \phantom{1,}276  & \phantom{1,}295 & \phantom{1,}240 & \phantom{1,}205  \\
14      &\phantom{1}1,642  & \phantom{1}1,354  & \phantom{1}1,142  & \phantom{1,}770 &
   \phantom{2,}531 & \phantom{1,}572 & \phantom{1,}482 & \phantom{1,}392  \\
15      &\phantom{1}3,253  & \phantom{1}2,650  & \phantom{1}2,080  & 1,390  &
   \phantom{1,}940 &           1,050 & \phantom{1,}916 & \phantom{1,}773  \\
16      &\phantom{1}6,150  & \phantom{1}4,936  & \phantom{1}3,680  & 2,340  &
             1,680 &           1,830 &           1,630 &           1,400  \\
17      &11,540            & \phantom{1}9,170  & \phantom{1}6,350  & 3,980  &
             2,870 &           3,100 &           2,990 &           2,610\rule[-1.5ex]{0pt}{0pt}  \\
\hline
\rule{0pt}{3ex}%
Total   &24,000  &19,300  &14,300  & 9,240  & 6,540  & 7,090  & 6,460  & 5,560\rule[-2ex]{0pt}{0pt}  \\
\hline
\end{tabular}}
\end{center}
\end{table}

Three things follow from this table: (\textit{a})~Stars of all
magnitudes down to the seventeenth are more numerous in
the plane of the Milky Way than near its poles. Since the
only reasonable supposition is that the nearer stars are distributed
more or less uniformly with no special relations to
the Milky Way, it follows from the fact the bright stars
are condensed near the Milky Way that some of them are
very distant. That is, the stars differ greatly in absolute
luminosity, a conclusion confirmed by direct evidence.
(\textit{b})~The decrease in the number of stars is on the average
gradual from the Milky Way to its poles, showing that the
sun is actually in the midst of the clouds of stars on which the
table is based. (\textit{c})~The relative condensation in the plane
of the Milky Way is greater, the fainter the stars. This
proves that the stars are not only much more numerous
near the plane of the Milky Way, but also that they extend
to much greater distances in this plane than in the direction
%% -----File: 502.png---Folio 472-------
\begin{figure}[hbtp]%[Illustration]
\centering\Input{502}{jpg}
\Caption[Great star clouds in Sagittarius. \textit{Photographed by Barnard at
the Yerkes Observatory.}]{Fig}{166}
\index[xnames]{Barnard}%
\end{figure}%
%% -----File: 503.png---Folio 473-------
of its poles. The counts of stars by Kapteyn, based in part
\index[xnames]{Kapteyn}%
on the visual observations of Sir~John Herschel, give still
\index[xnames]{Herschel, John}%
greater relative condensation in the plane of the Milky Way,
\index{Milky Way}%
and still more strongly confirm this conclusion.

\Article{271}{The Form and Structure of the Milky Way.}---Before
attempting to arrive at a more precise conclusion regarding
the distribution of the stars in space, it is desirable to obtain
a better idea of the form and properties of the Milky Way.

As has been stated, the center of the Milky Way is nearly
a great circle around the celestial sphere. Its greatest
northerly declination ($45°$~to~$65°$) is at right ascension zero
in the constellation Cassiopeia, where it is about $20°$~wide.
\index{Cassiopeia}%
It extends from this point southeastward across Perseus
\index{Perseus}%
with very irregular outlines (\Mapref{I}, \Artref{82}), and narrows
down where it crosses the borders of Taurus to a width
\index{Taurus}%
of about~$5°$. It then bulges wider in Monoceros and across
\index{Monoceros}%
the northeast corner of Canis Major. Farther south in
\index{Canis Major}%
Argo, with its several divisions, it becomes as much as $30°$
\index{Argo}%
wide, but its borders are irregular, it is broken through by
vacant lanes, one of which in its center is called the ``coal
sack,'' and at right ascension about $9$~hours and declination
$45°$~south a dark gap stretches almost across it. After
reaching its most southerly point in Crux it stretches out in
\index{Crux}%
irregular outline through Centaurus, part of Musca, Circinus,
\index{Centaurus}%
\index{Circinus}%
\index{Musca}%
Norma, and then north again into Ara, Lupus, and Scorpius.
\index{Ara}%
\index{Lupus}%
\index{Norma}%
\index{Scorpius}%
In Scorpius and in Sagittarius to the east are some of the most
\index{Sagittarius}%
remarkable star clouds in the heavens, \Figref{166}. Barnard's
\index[xnames]{Barnard}%
photographs of these regions show countless suns massed
in banks, with intervening dark lanes, the whole often
enveloped by a soft nebulous haze (see \Figref{167}). Northeast
of Scorpius lie Ophiuchus, Serpens, and Aquila. From
\index{Aquila}%
\index{Ophiuchus}%
\index{Serpens}%
Aquila and Ophiuchus northward through Vulpecula and
\index{Vulpecula}%
Cygnus to Cepheus, the Milky Way is divided longitudinally
\index{Cepheus}%
\index{Cygnus}%
by a rift of varying width and form. This bifurcation, which
extends through more than $50°$~of its length, is one of its
most remarkable features. In Cepheus the two branches
%% -----File: 504.png---Folio 474-------
join and reach on into Cassiopeia, where the description of
\index{Cassiopeia}%
the Milky Way began.

It is obvious that the stars do not form any simple system.
It seems probable that the Galaxy is composed of a large
\index{Galaxy}%
\begin{figure}[hbt]%[Illustration:]
\Input{504}{jpg}
\Caption[The region of Rho Ophiuchi. \textit{Photographed by Barnard.}]{Fig}{167}
\index[xnames]{Barnard}%
\end{figure}%
number of star clouds, each with peculiarities of its own,
but having relations to the whole mass of stars. Since the
Milky Way is roughly in the form of a great discus, or
``grindstone'' as Herschel called it, the prevailing motions
\index[xnames]{Herschel, William}%
must be in its plane in order to have preserved its shape.
%% -----File: 505.png---Folio 475-------
This does not mean that the relative velocities would need to
be great enough to be easily observed; they would, in fact,
be very slight as seen from the enormous distances separating
the stars from the earth.


\Section{XXII}{QUESTIONS}

1. Prove that the magnitudes of stars of equal absolute brightness
are proportional to the squares of their distances.

2. Prove that, under the hypothesis of the second paragraph
of \Artref{269}, the ratio of the number of stars of any magnitude
to the number of the next magnitude brighter is~$3.98$.

3. If there are $2000$~stars of magnitude $5$~to~$6$, and if the ratio
for successive magnitudes were~$3.98$, how many stars would there
be of magnitude $16$~to~$17$?

4. Prove that the effectiveness of a given mass in screening
off light is inversely proportional to the radius of the particles into
which it is divided.

5. Show in detail how it follows from \Tableref{XV} and the assumption
under~(\textit{a}) that some of the bright stars are very distant.
How many of the $20$~first-magnitude stars have parallaxes greater
than~$0''.2$ (see \Tableref{XVI})?

6. At what distance, expressed in parsecs (\Artref{272}), would the
sun be a first-magnitude star? A sixth-magnitude star? If Canopus
has a parallax of~$0''.005$, how does its absolute brightness compare
with that of the sun?

7. Prove that the area of one hemisphere of the sky is $92,000$
times the apparent area of the sun.

8. Prove in detail that conclusion~(\textit{b}) of \Artref{270} follows from
\Tableref{XV}.

9. At what time of the year does the portion of the Milky Way
which is divided by a longitudinal rift pass the meridian at 8~\PM?
If possible, observe it.

10. Draw a diagram and show that the fact that the central
line of the Milky Way is not quite a great circle proves that the
solar system is not in the center of the disk of stars of which the
Milky Way is composed.

11. The fact that the Milky Way is very oblate implies that it
has large moment of momentum about an axis perpendicular to
its plane. What inference do you draw respecting the general
motions of stars in exactly opposite parts of the Milky Way?

12. If all visible objects belong to the Galaxy, is it possible to
prove the rotation of the Milky Way by observations of the stars?
%% -----File: 506.png---Folio 476-------

13. What observational evidence disproves the hypothesis that
there are infinitely many galaxies distributed with approximate
uniformity, but separated from one another by distances which
are enormous compared to their dimensions?

\normalsize


\Section{II}{Distances and Motions of the Stars}

\Article{272}{Direct Parallaxes of the Nearest Stars.}---One of
\index{Distance!of stars}%
\index{Parallax!determination of}%
\index{Stars!distances of}%
\index{Stars!parallaxes of}%
the proofs that the earth revolves around the sun is that the
apparent directions of the nearest stars vary with the position
of the earth in its orbit (\Artref{51}). The difference in
direction of a star as seen from two points separated from
each other by the mean distance from the earth to the sun
is the parallax of the star; or, in other terms, the parallax
of the star is the angle subtended by the mean radius of the
earth's orbit as seen from the star (\Figref{35}). If the parallax
were one second of arc, the distance of the star would be
$206,265$ times\footnote
  {This number is the number of seconds in the arc of a circle which equals
  its radius in length.}
the mean distance from the earth to the sun.
This distance, which is a very convenient unit in discussing
the distances of the stars, is called the \textit{parsec}, and for most
\index{Parsec, definition of}%
practical purposes it may be taken equal to $200,000$ astronomical
units, or $20,000,000,000,000$ miles. It is the distance
that light travels in about $3.3$~years.

The stars are so remote that the problem of measuring their
parallaxes is one of great practical difficulty. Alpha Centauri,
\index{Alpha Centauri}%
the nearest known star, has a parallax of only~$0''.75$.
That is, its difference in direction as seen from two points on
the earth's orbit, separated by the distance from the earth
to the sun, is the same as the difference in direction of an
object at the distance of $10.8$~miles when seen first with one
eye and then with the other. Not only is the difference in
the apparent position of a star very small as seen from different
parts of the earth's orbit, but it can be determined
only from observations separated by a number of months
%% -----File: 507.png---Folio 477-------
during which climatic conditions and the instruments may
have appreciably changed.

The best direct means of determining the parallax of a
star is by comparing, at various times of the year, its apparent
position with the positions of more distant stars. Let $S$,
\Figref{168}, represent a star whose parallax is required, and $S'$
a much more distant star. When the earth is at~$E_1$ the
angular distance between them is~$\angle SE_1S'$; when the earth
is at~$E_2$, it is~$SE_2S'$. The parallax of~$S$ is~$\angle E_1SE_2$; the
parallax of~$S'$ is~$E_1S'E_2$, which will be negligible if $S'$ is sufficiently
remote. It easily follows from the geometry of the
figure that the parallax of~$S$ minus the parallax of~$S'$ equals
the difference of the measured angles $SE_1S'$~and~$SE_2S'$.
\begin{figure}[hbt]%[Illustration:]
\Input{507}{png}
\Caption[Determination of parallax from apparent changes in relative
positions of stars.]{Fig}{168}
\end{figure}%
Hence, if the parallax of $S'$ is inappreciable, the parallax of~$S$
can be found.

In practice the position of~$S$ is measured with respect to
a number of comparison stars. At present the work is done
almost entirely by photography. Plates of a star and the
surrounding region are secured at different times of the year,
and the distances between the stars are measured under a
microscope on a machine designed for the purpose. The
scale of the photograph is proportional to the focal length
of the telescope, and consequently for this purpose only
large and excellent instruments are of value.

With present means of measurement, a parallax of~$0''.02$
or less cannot be determined with sufficient accuracy to be
of much value; in fact, the probable error in one of~$0''.05$
is large. The great distances of the stars can be inferred
%% -----File: 508.png---Folio 478-------
from the fact that only about $100$ are known whose parallaxes
come within the wider of these limits.

The distances of stars whose parallaxes are $0''.2$ or greater
can be measured with an error not exceeding about $25$~per~cent
of the quantity to be determined. There are at present
$19$ such stars known, $9$ of which are too faint to be seen without
optical aid. These stars are given in \Tableref{XVI}. When
the distance of a star of known magnitude has been determined,
the total amount of light it radiates, or its luminosity,
as compared with the sun can be computed. The luminosity
of each of the nineteen stars is given in the fifth column.

\begin{table}[hbt]
%\caption[Table of nineteen nearest stars]{}
\begin{center}
\Caption{Table}{XVI}
\TFontsize%
\setlength{\tabcolsep}{2pt}
\makebox[0pt][c]{%
\begin{tabular}{|>{\,}l|*{6}{c|}}
\hline
\settowidth{\TmpLen}{Goombridge~34}%
\TEntry{\TmpLen}{\THead Star}
  & \settowidth{\TmpLen}{\textsc{Paral-}}%
    \TEntry{\TmpLen}{\medskip\THead Mag- \\ nitude\medskip}
  & \settowidth{\TmpLen}{\textsc{Paral-}}%
    \TEntry{\TmpLen}{\THead Paral- \\ lax}
  & \settowidth{\TmpLen}{\textsc{(Parsecs)}}%
    \TEntry{\TmpLen}{\THead Distance \\ (Parsecs)}
  & \settowidth{\TmpLen}{\textsc{Luminosity}}%
    \TEntry{\TmpLen}{\THead Luminosity \\ (Sun $=1$)}
  & \settowidth{\TmpLen}{\footnotesize (\textsc{Sun } $=1$)}%
    \TEntry{\TmpLen}{\THead Mass \\ (Sun $=1$)}
  & \settowidth{\TmpLen}{\THead(Mi.\ per Sec.)}%
    \TEntry{\TmpLen}{\THead Velocity \\ (Mi.\ per Sec.)} \\
\hline
&& $''$ &&&& \\
$\alpha$~Centauri  &  $0.3$          & $0.76$ & $1.32$ &  $2.0\Z\Z$          & $1.9$ &  $20$    \\
Lalande 21,185     &  $7.6$          & $0.40$ & $2.50$ &  $0.009$        &   ?   &  $35$\rlap{$+$} \\
Sirius             & \llap{$-$}$1.6$ & $0.38$ & $2.63$ & \llap{$4$}$8.0\Z\Z$ & $3.4$ &  $11$    \\
$\tau$~Ceti        &  $3.6$          & $0.33$ & $3.00$ &  $0.50\Z$         &   ?   &  $20$    \\ %[** distance maybe was 3,00]
Procyon            &  $0.5$          & $0.32$ & $3.13$ &  $9.7\Z\Z$          & $1.3$ &  $12$    \\
C.~Z.~5\textsuperscript{h}~243 & $8.3$& $0.32$& $3.13$ &  $0.007$        &   ?   &  \llap{$1$}$70$ \\
$\epsilon$~Eridani &  $3.3$          & $0.31$ & $3.23$ &  $0.79\Z$         &   ?   &  $14$    \\
61 Cygni           &  $5.6$          & $0.31$ & $3.23$ &  $0.10\Z$         &   ?   &  $63$    \\
Lacaille 9352      &  $7.4$          & $0.29$ & $3.45$ &  $0.019$        &   ?   &  $72$    \\
Pos.\ Med.\ 2164   &  $8.8$          & $0.29$ & $3.45$ &  $0.006$        &   ?   &  $23$\rlap{$+$} \\
$\epsilon$~Indi    &  $4.7$          & $0.28$ & $3.57$ &  $0.25\Z$         &   ?   &  $54$    \\
Groombridge 34     &  $8.2$          & $0.28$ & $3.57$ &  $0.010$        &   ?   &  $30$\rlap{$+$} \\
OA(N.) 17,415      &  $9.3$          & $0.27$ & $3.70$ &  $0.004$        &   ?   &  $14$\rlap{$+$} \\
Krueger 60         &  $9.2$          & $0.26$ & $3.85$ &  $0.005$        &   ?   &  $11$\rlap{$+$} \\
Altair             &  $0.9$          & $0.24$ & $4.17$ & \llap{$1$}$2.3\Z\Z$ &   ?   &  $22$    \\
$\eta$~Cassiopeiæ  &  $3.6$          & $0.20$ & $5.00$ &  $1.4\Z\Z$          & $1.0$ &  $20$    \\
$\sigma$~Draconis  &  $4.8$          & $0.20$ & $5.00$ &  $0.5\Z\Z$          &   ?   &  $30$    \\
Lalande 21,258     &  $8.9$          & $0.20$ & $5.00$ &  $0.011$        &   ?   &  $66$\rlap{$+$} \\
OA(N.) 11,677      &  $9.2$          & $0.20$ & $5.00$ &  $0.008$        &   ?   &  $45$\rlap{$+$} \\
\hline
\end{tabular}}
\end{center}
\end{table}

The $19$~stars of \Tableref{XVI} together with our sun occupy a
sphere whose radius is $5$~parsecs. If they were uniformly
distributed in this space, the distance between adjacent
stars would be about $3.7$~parsecs, or $12.2$~light years. In
view of the fact that a number of stars in the list are far
%% -----File: 509.png---Folio 479-------
below the limits of visibility without optical aid, it is reasonable
to suppose that there may be a considerable number of
others within $5$~parsecs of the sun which are as yet undiscovered.

It should not be supposed that attempts have been made
to measure the parallaxes of all stars brighter than the
ninth, or even the sixth, magnitude. The process is excessively
laborious, and only those stars are selected which are
believed to be within measurable distance, or which are
objects of especial interest for other reasons. A star with a
given motion across the line of sight will apparently move
faster the nearer it is to the observer. Consequently,
those stars will be nearest on the average whose \textit{proper
motions}, as they are called, are greatest. As a rule only those
\index{Proper motion of stars}%
\index{Stars!proper motions of}%
stars are examined for parallax which have been found to
have large proper motions.

{\stretchyspace%
Under the hypotheses that the stars are uniformly distributed
throughout the space occupied by the Galaxy and}
\index{Galaxy}%
that their density is the same as it is in the vicinity of the
sun, the extent of the stellar universe can be computed.
Suppose the space occupied by the stars is spherical in shape
and that there are $500,000,000$ of them. Then it turns out
that, under the hypotheses adopted, the radius of this sphere
is $1500$ parsecs, or $5000$ light-years. Since the Galaxy is very
much flattened, the distance to its poles is probably only a
few hundred parsecs while the borders of its periphery are
probably several thousand parsecs from its center.

One very interesting and important conclusion follows from
\Tableref{XVI}, and that is that the luminosities of the stars
vary enormously. For example, Sirius radiates $12,000$
\index{Sirius}%
times as much light as OA(N.)~17,415. These differences in
luminosity may be due to the fact that some stars are larger
than others, or at least partly to the fact that some are
intrinsically more brilliant than others. Probably both
factors are important. Some stars are certainly much more
massive than others, and the table gives examples of stars
%% -----File: 510.png---Folio 480-------
whose masses differ very much less than their luminosities.
For example, while the mass of Sirius is only $3.4$~times that
\index{Sirius}%
of the sun, its luminosity is $48$~times as great. But Sirius is a
double star and presents in its own system a still more
remarkable contrast. The mass of the brighter component
is approximately twice that of the fainter one, but in luminosity
it is at least $5000$~times greater. There are other stars,
such as Rigel and Canopus, which, though they are so remote
\index{Canopus}%
\index{Rigel}%
that no evidence of their having measurable parallaxes has
been found, shine with the greatest brilliancy. Their luminosity
must be at least several thousand times that of the sun.
In fact, the average luminosity of the stars visible to the
unaided eye probably exceeds that of the sun several hundred
fold. It must not be assumed from this that the
luminosity of the sun is below the average, for it is exceeded
in luminosity by only five of the $19$~stars in the table.

In order to determine the velocity of a star its motion both
\index{Motion!of stars}%
\index{Stars!motions of}%
along and across the line of sight must be found. The proper
motions of all the stars in \Tableref{XVI} are known, but the
radial velocities of six\DPnote{** TN: [sic], table contains seven "+"s.} of them are unknown; in these cases
a plus sign is placed after the number giving the velocity
because the radial component is not known. It follows from
the table that the less luminous stars move with much
higher velocities than the brighter ones. The average speed
of those five stars whose luminosities exceed the sun is $17$~miles
per~second, while the average speed of the six whose
luminosities are less than $0.01$ that of the sun is more than
$50$~miles per~second. Since the more luminous stars are
almost certainly the more massive, it follows that the more
massive stars move more slowly than the smaller ones.

One may inquire to what extent reliance can be put in
conclusions based on only $19$~stars. When compared to
hundreds of millions the number is ridiculously small, but all
the conclusions which have been stated are strongly supported
by the evidence furnished by the much more numerous stars
having smaller and less accurately determined parallaxes.
%% -----File: 511.png---Folio 481-------

\Article{273}{Distances of the Stars from Proper Motions and
Radial Velocities.}---The parallaxes of possibly $100$~stars have
\index{Motion!of stars}%
\index{Stars!motions of}%
\index{Stars!proper motions of}%
\index{Stars!radial velocities of}%
been determined by direct means with considerable accuracy.
Probably not over~$1000$ are within reach of present instruments
and methods. Are astronomers doomed to remain
in ignorance as to the distances of all the other stars which
fill the sky? By no means. There are several indirect
methods of finding the average distances of classes of stars.

Consider all the stars of a large class, say the stars of the
sixth magnitude. Suppose they are moving at random;
that is, that they do not tend to move in any particular
direction, or with any particular speed. Suppose both their
proper motions and their radial velocities have been determined
by observation. Under these hypotheses as many
stars will be approaching as receding, and the velocities of
approach will average the same as those of recession. Also,
the proper motions will be as numerous and as large in one
direction as in the opposite. The extent to which these
conditions are fulfilled is a measure of the accuracy of the
assumptions.

Whatever the individual motions of the class of stars
under consideration, they will have an average speed of motion
which may be represented by~$V$. The average component
of motion toward or from the observer will be~$\frac{1}{2}V$, as can
be shown by a mathematical discussion. This is the average
radial velocity as determined by the spectroscope, and
is therefore known. The average component at right angles
to the line of sight is found by a mathematical discussion
to be $0.7854 V$. This quantity is therefore also known
because $V$ has been given by spectroscopic observations.

Now consider the proper motions. They are expressed
in angle, and they depend upon the distances of the stars
and the speed with which they move across the line of sight.
Since both the linear speed across the line of sight and the
angular velocity, or proper motion, have been found, the
distances of the stars can be computed.
%% -----File: 512.png---Folio 482-------

The hypotheses on which this discussion has been made
are not exactly fulfilled, and the necessary modifications of
the proposed method must now be considered.

\Article{274}{Motion of the Sun with Respect to the Stars.}---Since
\index{Motion!of sun}%
\index{Sun!motion of}%
the stars are in motion, it is reasonable to suppose that
the sun is moving among them. Such was found to be
the case by Sir William Herschel more than a century ago.
\index[xnames]{Herschel, William}%
He proved by observations extending over many years that
the apparent distances between the stars in the direction
of the constellation Hercules are increasing, on the
\index{Hercules}%
average, and that they are decreasing in the exactly opposite
part of the sky. He interpreted this as meaning that
the sun is moving toward the constellation Hercules, and
it is obvious that this would explain the observed phenomena;
for, as objects are approached, they subtend
larger angles. While Herschel's observations gave the
direction of motion of the sun, they did not give its
speed, which could be found by this method only if the
distances of the stars were known. Since the distances of
only a few stars can be measured directly, there is little hope
of determining the motion of the sun in this way with any
considerable degree of accuracy.

The spectroscope has been used to determine both the
direction of the sun's motion and also the rate at which it
moves. Instead of finding as many stars approaching as
receding in every part of the sky, as was assumed in the discussion
in \Artref{273}, it has been found that the stars in the
direction of the constellation Hercules on the whole are
relatively approaching the sun, while those in the opposite
direction are relatively receding. This means that with
respect to the stars which were observed the sun is moving
toward Hercules.

The best determination of the direction of the sun's motion
from proper motions of the stars is by Lewis Boss, who based
\index[xnames]{Boss, Lewis}%
his discussion on the $6188$~stars in his catalogue. The best
\index{Catalogues of stars}%
\index{Stars!catalogues of}%
spectroscopic determination is by W.~W. Campbell, who
\index[xnames]{Campbell}%
%% -----File: 513.png---Folio 483-------
based his discussion on the radial velocities of $1193$~stars
measured at the Lick Observatory and its branch in South
\index{Lick Observatory}%
America. The results of these determinations are as follows:
\begin{center}
\settowidth{\TmpLen}{\scshape Ascension}%
\setlength{\tabcolsep}{3pt}
\begin{tabular}{|l|c|c|c|}
\hline
& \TEntry{\TmpLen}{\medskip\TFontsize\THead Right \\ Ascension\medskip}
& \TFontsize\THF Declination & \TFontsize\THF Speed \\
\hline
\Strut
Solar Apex (Boss)     & $270°.5 ± 1°.5$ & $+34°.3 ± 1°.3$ &      ?           \\
Solar Apex (Campbell) & $268°.5 ± 2°.0$ & $+25°.3 ± 1°.8$ & $12$~mi.\ per~sec. \\
\hline
\end{tabular}
\index[xnames]{Campbell}%
\end{center}
The agreement of these results in right ascension is remarkable,
and the disagreement in declination is small considering
the difference in the methods and the stars used.

The number of stars used by Boss in his determination of
\index[xnames]{Boss, Lewis}%
the direction of the motion of the sun is so great that he could
\index{Motion!of sun}%
\index{Sun!motion of}%
divide them up into separate groups and make the discussion
for each one separately. He took the stars of various galactic
latitudes and obtained essentially the same result for
each group. Dyson and Thackeray found from another (the
\index[xnames]{Dyson}%
\index[xnames]{Thackeray}%
Groombridge) list of $3707$~stars that the declination of the
apex of the sun's way increases from $+16°$ for the brightest
stars to $+43°$ for those from magnitude $8.0$ to~$8.9$. This
was confirmed by Comstock, who found even a greater declination
\index[xnames]{Comstock}%
for the apex of the sun's way as determined from still
fainter stars, but the result must be accepted with reserve
until it is confirmed by a discussion depending on a much
larger and better distributed list of stars. The spectra of the
stars are divided into a number of classes (\Artref{295}), and it
was found both by Boss and by Dyson and Thackeray that
the declination of the apex of the sun's way is about $12°$
greater when determined from stars of Secchi's second type
than it is when determined from stars of the first type. But
the results altogether indicate that the sun is moving, relatively
to the few thousand brightest stars, toward a point
whose right ascension is about $270°$ and whose declination is
about $34°$, and that the speed of relative motion is about $12$~miles
per~second.
%% -----File: 514.png---Folio 484-------

The motion of the sun with respect to the stars evidently
\index{Motion!of sun}%
\index{Sun!motion of}%
requires some modification of the process described in \Artref{273}.
There is, however, no real difficulty, because the effect
of the sun's motion can be avoided by considering only those
components of the proper motions of the stars which are at
right angles to the line of the sun's way.

Campbell made a determination of the mean parallaxes of
\index{Distance!of stars}%
\index{Stars!distances of}%
\index[xnames]{Campbell}%
the stars down to magnitude~$5.5$ by the method of this
article. The brighter stars were not sufficiently numerous to
give very reliable results. He found that the mean parallax
of stars of magnitudes $4.51$ to~$5.50$ is~$0''.0125$, corresponding
to a distance of $80$~parsecs. This volume is $4096$~times
that occupied by the $20$~nearest stars, and if the stars were
uniformly distributed throughout it, the total number of
them down to magnitude~$5.50$ would be~$81,920$, which is
much in excess of the number actually observed.

\Article{275}{Distances of the Stars from the Motion of the Sun.}---The
parallaxes of only a comparatively small number of stars
can be measured directly because their distances are so enormously
great compared to the diameter of the earth's orbit.
If the orbit of the earth were as large as that of Neptune, the
problem would be much easier because of the larger base line
which could be used. But the sun's motion can be made to
afford an indefinitely large base line in statistical discussions,
as will now be shown.

Suppose first that all of the stars of the observable sidereal
universe except the sun are relatively at rest. The motion
of the sun among them will give them an apparent displacement,
or proper motion, in the direction opposite to that in
which it is moving. The farther a star is away the smaller
this proper motion will be. If a star is so far away that no
displacement due to the sun's motion can be observed in one
year, then $10$~years, $100$~years, or any other necessary number
of years may be used. Eventually the effect of the sun's
motion will be observable. Since the sun travels about $4$~astronomical
units per year, it follows that the parallax of a
%% -----File: 515.png---Folio 485-------
star is one fourth of that part of its annual proper motion
which is due to the motion of the sun.

The false hypothesis that all the stars except the sun are
relatively at rest has greatly simplified the problem. As a
matter of fact, the stars are moving with respect to one another
in various directions and with various speeds, and the
proper motion of a star is due both to its own motion and also
to the motion of the sun with respect to the system. Since
the actual motion of any particular star is in general unknown,
it is necessary to take the average motions of many,
and then the results will be consistent, for the motion of the
sun is defined with respect to the many. For any class of
stars the average proper motion perpendicular to the direction
of the sun's motion will be zero, while the average proper
motion in the direction of the sun's motion will depend only
on their distance and the speed of the sun.

This statistical study of the stars was taken up about $20$~years
ago by Kapteyn, of Groningen, who pursued it with
\index[xnames]{Kapteyn}%
rare skill and great industry. A number of other astronomers
have also made important contributions to the subject. It
is interesting to note the different kinds of work which contribute
to the final results. In the first place, the proper
motions of the stars are involved. They are obtained from
two or more determinations of apparent position separated
by considerable intervals. In fact, the longer the intervals
the more accurately are the proper motions determined. In
the second place, the spectroscope is of fundamental importance
because it furnishes the motion of the sun with respect
to the stars. Since certain classes of stars may be moving as
a whole with respect to other classes (\Artref{278}), it follows that
the spectroscopic determination of the motion of the sun
should depend upon all those stars whose distances are
sought from their proper motions. At present the radial
velocities of stars fainter than the sixth magnitude can be
obtained only by costly long exposures, and the practical
limits do not reach beyond the eighth magnitude. On the
%% -----File: 516.png---Folio 486-------
other hand, the determination of the proper motions of stars
many magnitudes fainter offers no observational difficulties.

\Article{276}{Kapteyn's Results Regarding the Distances of the
Stars.}---As will be seen in \Artref{295}, most of the stars are of
\index{Distance!of stars}%
\index{Spectra of stars}%
\index{Stars!distances of}%
\index{Stars!spectra of}%
\index[xnames]{Kapteyn}%
two principal spectral types. Type~I, of which Sirius and
\index{Sirius}%
Vega are conspicuous examples, are white or bluish white.
\index{Vega}%
Their spectra are characterized by absorption lines due to
hydrogen in their atmospheres. They are intensely hot and
probably always of large mass. Type~II are the yellowish
stars, of which the sun, Capella, and Arcturus are examples.
\index{Arcturus}%
\index{Capella}%
The atmospheres of these stars contain many metals.

Kapteyn derived formulæ giving the mean parallaxes of
all stars of each magnitude, and also the mean distances of
stars of each spectral type separately. \Tableref{XVII} gives
Kapteyn's results transformed from parallax to parsecs and
using Campbell's more recent determination of the rate of
\index[xnames]{Campbell}%
motion of the sun.

\begin{table}[hbt]
\settowidth{\TmpLen}{\textsc{Spectral}}%
\begin{center}
\Caption{Table}{XVII}
%\caption[Distances in Parsecs\protect\footnotemark]
\begin{tabular}{|*{4}{c|}}
\hline
\textsc{Magnitude} &
\textsc{All Stars} &
\TEntry{\TmpLen}{\medskip\THead Spectral\\ Type~I\medskip} &
\TEntry{\TmpLen}{\THead Spectral\\ Type~II} \\
\hline
\Strut
$\Z1$ & $\Z24.2$ & $\ZZ39.4$ & $\Z16.8$ \\
$\Z2$ & $\Z31.0$ & $\ZZ50.5$ & $\Z21.6$ \\
$\Z3$ & $\Z39.7$ & $\ZZ64.7$ & $\Z27.6$ \\
$\Z4$ & $\Z50.9$ & $\ZZ82.9$ & $\Z35.4$ \\
$\Z5$ & $\Z65.3$ & $\Z106.3$ & $\Z45.4$ \\
$\Z6$ & $\Z83.7$ & $\Z136.3$ & $\Z58.2$ \\
$\Z7$ & $ 107.3$ & $\Z174.7$ & $\Z74.7$ \\
$\Z8$ & $ 137.5$ & $\Z224.0$ & $\Z95.7$ \\
$\Z9$ & $ 176.3$ & $\Z287.2$ & $ 122.7$ \\
$ 10$ & $ 226.1$ & $\Z368.3$ & $ 157.4$ \\
$ 11$ & $ 289.8$ & $\Z472.1$ & $ 201.7$ \\
$ 12$ & $ 371.6$ & $\Z605.3$ & $ 258.6$ \\
$ 13$ & $ 476.4$ & $\Z776.0$ & $ 331.6$ \\
$ 14$ & $ 610.8$ & $\Z994.9$ & $ 425.2$ \\
$ 15$ & $ 783.0$ & $ 1275.5$ & $ 545.0$ \\
\hline
\end{tabular}
\end{center}
\end{table}%
\footnotetext
  {One parsec equals $200,000$ astronomical units, or in round numbers
  $20,000,000,000,000$ miles.}
%% -----File: 517.png---Folio 487-------

It must be remembered that \Tableref{XVII} gives mean results
derived from the proper motions and radial velocities of
many stars. The results may be in error for the first few
magnitudes because there are not enough bright stars to
make the statistical method reliable. They may also be in
error for the fainter stars because these stars were not used in
deriving the formulæ by which the computations were made.

If the table is correct, the sun is far below the average of
the stars in brilliancy. According to the measures of Wollaston,
\index[xnames]{Wollaston}%
Bond, and Zöllner its magnitude on the stellar basis
\index[xnames]{Bond}%
\index[xnames]{Zollner@{Zöllner}}%
is~$-26.7$, or it gives us $120,000,000,000$ times as much light
as a first-magnitude star. Since the light received from
a body varies inversely as the square of its distance, at the
mean distance of the first-magnitude stars the sun would
send us only $0.005$ as much light as comes from a first-magnitude
star. That is, the first-magnitude stars average about
$200$~times as brilliant as the sun. It must not be concluded
from this that the stars of all magnitudes average so much
more brilliant than the sun, for those of the first magnitude
are a group selected because of their great brilliancy.

\Article{277}{Distances of Moving Groups of Stars.}---If the two
\index{Distance!of stars}%
\index{Motion!of stars}%
\index{Stars!distances of}%
\index{Stars!groups of}%
components of a double star are found to be moving in the
same direction and with the same apparent speed, the conclusion
to be drawn is that they are relatively close together
in space and that they are physically connected; for, if they
were simply in the same direction from the earth without
being related, their apparent motions would almost certainly
differ either in speed or direction. While the conclusion
might be erroneous in the case of only two stars, it could
hardly fail to be true if many stars were involved.

The study of the proper motions of the stars has shown
that there are several groups which have sensibly identical
proper motions; or rather, as the result of perspective,
there are many stars which apparently move with the same
speed toward a common point in the sky. These groups
are widely scattered and many of their members would not
%% -----File: 518.png---Folio 488-------
be suspected of being associated with the others except for
the equality of their motions. For example, Sirius belongs
\index{Sirius}%
to a group which includes five of the stars in the Big Dipper.
\index{Big Dipper}%

The best-known group of stars of the type under consideration
comprises part of the Hyades cluster, in the constellation
\index{Hyades}%
Taurus, and some neighboring stars scattered over an area
\index{Taurus}%
about $15°$ in diameter. This group, which includes $39$~known
stars, was exhaustively discussed by Lewis Boss. The stars,
in their proper motions, all seem to move along the arcs
of great circles. Boss found that the great circles of all
\index[xnames]{Boss, Lewis}%
the stars of the
Taurus stream
intersect in a
common point
whose right ascension
and declination
are, for
the position of the
equinox in 1875,
$6$~h.\ $7.2$~m.\ and~$+6°\,56'$.
It can be
shown that this
means that the
stars of the group
are moving in
lines parallel to the line from the observer to the point of
intersection of the circles. That is, their direction of motion
is defined in this way, and since the stars cover a considerable
area in the sky the point toward which they are moving
is very well determined.

It will now be shown that if, in addition to the data already
in hand, the radial velocity of one of the stars of the group
can be obtained, then the actual motions, the distances, and
the luminosities of all of them can be determined. Let~$O$,
\Figref{169}, be the position of the observer and
\begin{wrapfigure}[17]{\WLoc}{3in}%[Illustration: Move down]
\Input[3in]{518}{png}
\Caption[Components of motion in moving groups
of stars.]{Fig}{169}
\end{wrapfigure}
$OP$~the direction
of motion of the stars of the group. Let $S$ be one of the
%% -----File: 519.png---Folio 489-------
stars which is moving in the known direction~$SA$ with an
unknown speed. Suppose the component~$SB$ is measured
by the spectroscope. Then, since the angle~$ASB$, which
equals the angle~$POS$, is known, the whole component~$SA$
and the proper-motion component~$SC$ can be computed.
That is, the actual distance~$SC$ is found and the proper
motion to which it gives rise was already known. Therefore
the distance~$OS$ can be computed. Since all the stars
of the group must have the same total motion~$SA$, for otherwise
they would not remain long associated, the distances of
all the members can be determined from their respective
proper motions. Of course, it is practically advantageous to
measure the radial velocities of many, or all, of the members
of the group. When the distance of a star of known magnitude
has been found, its absolute luminosity can be computed.

By these methods Boss found that the Taurus group is a
\index{Taurus}%
\index[xnames]{Boss, Lewis}%
globular cluster whose center is distant about $40$~parsecs
from the earth. Since its apparent diameter is about $15°$,
its actual diameter is about $10$~parsecs. There is a slight condensation
toward the center of the cluster, but in the group
as a whole the star density is only a little greater than it is
in the vicinity of the sun. The distances between the stars
of the group are so great that foreign stars could pass through
it without having their motions appreciably disturbed. In
fact, in the motion of the cluster it certainly sweeps past
other stars and there are probably several strangers now
within its borders. Boss found that $800,000$ years ago the
cluster was half its present distance and its apparent size was
twice that at present. In $65,000,000$ years it will have
receded until it will appear from the earth to be a compact
group one third of a degree in diameter, made up of stars of
the ninth magnitude and fainter.

All the $39$~stars of the Taurus cluster are much greater in
light-giving power than the sun. The luminosities of even
the five smallest are from five to ten times that of the sun,
%% -----File: 520.png---Folio 490-------
while the largest are $100$~times greater in light-giving power
than our own luminary. Their masses are probably much
greater than that of the sun.

The Ursa Major group of $13$~stars is another wonderful
\index{Ursa Major}%
system. It is in the form of a disk whose thickness is only
$4$~or $5$~parsecs while its diameter is $50$~parsecs. The distances
of the members of this group from the sun vary from
$2.6$~parsecs, in the case of Sirius, to $22$~parsecs for the stars of
the Big Dipper, and over $40$~parsecs in the case of Beta
\index{Big Dipper}%
Aurigæ. The luminosities of the stars vary from $7$ to more
\index{Beta Aurigae@{Beta Aurigæ}}%
than $400$~times that of the sun.

There is another fairly well-established group in Perseus
\index{Perseus}%
which was discovered almost simultaneously by Kapteyn,
\index[xnames]{Kapteyn}%
Benjamin Boss, and Eddington. There are several other
\index[xnames]{Boss, Benjamin}%
\index[xnames]{Eddington}%
probable groups in which the proper motions are so small
that the results have not been established beyond all question.
In a universe of many stars it is inevitable that there
should be many accidental parallelisms and equalities of
motion. Stars are at present regarded as forming a related
group only if there is something quite distinctive about their
positions or motions.

\Article{278}{Star-Streams.}---In 1904 Kapteyn announced a very
\index{Star!streams}%
important discovery respecting the motions of the stars.
He found that, instead of moving at random, most of the
stars belong to two great streams having well-defined directions
of motion. Stars in all parts of the sky, of all magnitudes
so far as the proper motions are known, and of all
spectral types, partake of these motions. The phenomena
do not seem to be local, so to speak, as was true in case of
the groups considered in \Artref{277}. Yet it would be going
too far to conclude that all the stars in the clouds which make
up the Milky Way belong to these streams, for the discussion
\index{Milky Way}%
was based on only a few thousands of stars, while there are
hundreds of millions in the sky. It seems probable that the
Galaxy is made up of a great many of these streams. There
\index{Galaxy}%
is, in fact, some reason to believe that there is a third drift
%% -----File: 521.png---Folio 491-------
containing stars of the so-called Orion type. But the evidence
\index{Orion}%
for the existence of the two streams discovered by
Kapteyn is conclusive, and his results have been verified by
\index[xnames]{Kapteyn}%
several other astronomers. And in connection with the
larger problems of the Milky Way, it is interesting to note
\index{Milky Way}%
that both streams are moving parallel to its plane.

With respect to the sun as an origin the points toward
which the stars are moving are:

\begin{tabular}{ll}
Apex of Drift~I:  & Right Ascension, $90°$;  \\
                  & Declination, $-15°$.     \\
Apex of Drift~II: & Right Ascension, $288°$; \\
                  & Declination, $-64°$.     \\
\end{tabular}

If the motion of the sun is eliminated and the stars are
considered only with reference to one another, the two
streams necessarily move in opposite directions. With this
reference, the vertices of the two drifts according to Eddington's
\index[xnames]{Eddington}%
discussion of the stars in Boss's catalogue are:
\index[xnames]{Boss, Lewis}%
\begin{center}
\begin{tabular}{l}
Right Ascensions, $94°$, $274°$; \\
Declinations, $+12°$, $-12°$.    \\
\end{tabular}
\end{center}

About $60$~per~cent of the stars on which the discussion was
based belong to Drift~I and $40$~per~cent to Drift~II\@. They are
intermingled in space so that one set of stars is passing
through the other. Their relative velocity is about $24$~miles
per~second, or about $8$~astronomical units per~year.

\Article{279}{On the Dynamics of the Stellar System.}---The
\index{Dynamics of stellar system}%
stars are at least several hundred millions in number, they
occupy an enormous space, and they are moving with respect
to one another with velocities averaging about $20$~miles per
second. In the two centuries during which their proper
motions have been observed, they have in all cases moved in
sensibly straight lines with uniform velocities. Likewise,
spectroscopic determinations of motion in the line of sight
give no evidence of anything but uniform rectilinear motion.
These statements require modification, however, in the case
of the binary stars (\Artref{283}).
%% -----File: 522.png---Folio 492-------

There is no doubt that the paths of the stars eventually
curve, but the time covered by our observations is as yet far
too short for us to detect these deviations. It compares
with the vast intervals required for the stars to move across
the sidereal universe as one tenth of a second compares with
the period of the earth's revolution around the sun.

The first question that springs to the mind is whether the
stars travel in sensibly fixed and closed orbits similar to those
of the planets, or move on indefinitely throughout the region
occupied by the stars without ever retracing any parts of
their paths. Since observations cannot at present answer
this question, the reply must be based on dynamical considerations.
There is clearly no central mass among the stars and
there is no center about which they seem to be distributed
with anything approaching symmetry. Moreover, their
motions give no hint that they are moving, even temporarily,
around some central mass or point.

The conclusion is inevitable that the stars describe more
or less irregular paths, in the course of time probably extending
into all parts of the sidereal system. In fact, the Galaxy
\index{Galaxy}%
was likened by Kelvin to a great gas in which the stars correspond
\index{Gases!kinetic theory of}%
\index{Kinetic theory of gases}%
\index[xnames]{Kelvin}%
to the molecules. When they are far apart their
mutual attractions are inappreciable, just as molecules do
not interfere with the motions of one another except at the
times of collisions. If two stars should collide they would
probably coalesce, the heat generated by their impact changing
them into the nebulous state. This would be quite different
from an elastic rebound of molecules. But actual collisions
would be excessively rare and near approaches would
be relatively much more frequent. A near approach is
dynamically equivalent to an oblique impact of perfectly
elastic bodies, as is illustrated in \Figref{170}. In this figure
$C$~is the center of gravity around which as a focus the two
masses (assumed equal) describe hyperbolas. It is easy to
see that the motion before and after near approach is similar
to that of two elastic spheres colliding a little to the right of
%% -----File: 523.png---Folio 493-------
their respective centers. Consequently there are some good
grounds for comparing the sidereal system to a vast mass of
gas.

There are, however, fundamental differences between a
gas and the stellar system. In a gas the collisions are the
important events in the history of a molecule, and are the
only appreciable factors which influence its motion. In the
stellar system the near approaches of a given star to some
other one are excessively rare,
and the attraction of the whole
system is the primary factor
determining the motion of the
individual star. Or, more
particularly, a molecule in a
vessel of ordinary gas has
thousands of millions of collisions
with other molecules
per second, while the attraction
of the whole mass has no
appreciable effect on its motion.
But in the sidereal
system, a star will in general
travel several times from one
of its visible borders to the
opposite one without once
passing near enough to another
star to have its motion radically altered by the latter,
while its motion is controlled by the attraction of the whole
mass of stars.

It is difficult to realize the great distances which separate
the stars and how feeble are the forces with which they
attract one another. If %[Illustration: Break, moved down]
\begin{wrapfigure}{\WLoc}{2.5in}
\Input[2.5in]{523}{png}
\Caption[Near approach of two
stars is similar to an oblique collision
of elastic bodies.]{Fig}{170}
\end{wrapfigure}
the earth were at rest, it would fall
toward the sun less than one eighth of an inch the first second.
The distance of the relatively near star Sirius is $500,000$
\index{Sirius}%
times as great; and in spite of the fact that its mass is $3.4$~times
that of the sun, in a whole year it would give the sun
%% -----File: 524.png---Folio 494-------
a velocity of only $0.00007$ of an inch per second. Only
after $900,000,000$ years at the present distance would the
relative velocity of the two amount to one mile per second.
Long before such an immense time shall have elapsed the
sun and Sirius will be far separated in space.
\index{Sirius}%

Now consider a group of stars, such as the cluster in Taurus,
\index{Taurus}%
traveling through the stellar system. So far as their mutual
interactions on one another are concerned the result is the
same as though they were not moving with respect to the
other stars. In their motion through space they are subject
as a whole to the changing attractions of the other stars,
and individually to possible close approaches. These factors
may be considered separately.

The Taurus cluster consists of $39$ (possibly more) stars
which occupy a space whose diameter is roughly $10$~parsecs.
From the high luminosity of the individual members of the
group it is reasonable to suppose that they have large masses,
and it will be supposed that they average $10$~times the sun
in mass. It will be assumed that their motions are such
that they are neither simply falling together nor scattering
more widely in space, and that they are distributed uniformly
throughout the volume which they occupy. That is, it is
assumed that there is a balance (speaking roughly) between
the gravitational forces among them and the centrifugal
forces due to their relative motions. With these data and
assumptions their maximum velocities with respect to the
center of gravity of the group, and the time required for one
of them to move from one border of the group to the opposite,
can be computed.

It is found that the velocities of the stars of the group with
respect to their center of gravity will always be less than
$0.4$~of a mile per second, and this maximum will be approached
only very infrequently. If their masses are comparable to
that of the sun instead of being $10$~times as great, the velocities
relative to their center of mass will always be less than
$0.13$~of a mile per second. Consequently, the internal motions
%% -----File: 525.png---Folio 495-------
of the group due to the mutual attractions of its members
will always be small, and the fact that at present the stars
are moving in sensibly parallel lines with the same speed does
not in the least justify the conclusion that the members of
the cluster are in any sense young. It is also found that the
time required for a star to move from one side of the group
to the other under the attraction of all the stars in it is
$25,000,000$ years. At present it does not seem safe to put
any time limits on the life of a star, and consequently it
may be supposed, at least tentatively, that the cluster has
been in existence long enough for the stars of which it is
composed to have made many excursions across it. The
mutual interactions of the stars have a tendency to make
the cluster uniformly spherical with the stars of greatest
mass somewhat condensed toward the center. The approximate
sphericity of the group is in harmony with the hypothesis
that it is very old.

It remains to consider the effect on the cluster of its passage
through star-strewn space. The result depends, of
course, upon the star density of the region which it traverses.
It has been seen that there are $20$~known stars within $5$~parsecs
of the earth. It is not unreasonable to suppose
that there are $10$~other stars within the same distance of the
earth which are at present unknown. Under the assumption
that the stars are scattered uniformly with a density such
that there are~$30$ within a sphere whose radius is $5$~parsecs,
it is found that, on the average, the cluster will have to pass
over a distance of $5700$ parsecs in order that at least one of
its $39$~members shall pass another star within $1000$~times
the distance from the earth to the sun. Since the cluster
moves at the rate of about $16$~miles per second with respect
to the stars now surrounding it, about $40,000$ years will be
required for it to describe one parsec; and to pass over
$5700$ parsecs will require more than $200$~million years. But
$5700$ parsecs is probably far beyond the limits of the visible
universe, and before the cluster shall have traversed any
%% -----File: 526.png---Folio 496-------
considerable fraction of this distance the attraction of the
great mass of stars in the Galaxy will have radically altered,
\index{Galaxy}%
and possibly reversed, its motion.

While the stars of the cluster pass close to other stars only
after very long intervals, they are continually subject to
slight disturbing forces which affect them somewhat unequally.
This results in a slight tendency to scatter the
members of the group. One might be tempted to conclude
from the fact that it is still very coherent that its age should
be counted in hundreds of millions of years at the most.
But it is impossible to determine how many stars once belonging
to it have been torn from it by near approaches to
other stars, or how many of the smaller original stars have
been thrown to its borders by its internal interactions and
then removed by the differential attractions of exterior
bodies, or how much more condensed it may formerly have
been. In short, no certain conclusions respecting the age of
one of these moving clusters can be drawn from the properties
of the motion of their members at present.

It is now possible to pass to the consideration of the whole
sidereal system. The star-streams discovered by Kapteyn
\index[xnames]{Kapteyn}%
and the form of the Galaxy suggest that it is made up largely
of many vast star clouds which move at least approximately
in the plane of the Milky Way. There is a general tendency
\index{Milky Way}%
for the mutual interactions of the members of each star
cloud to reduce it to the spherical or symmetrically oblate
form. Moreover, the stars of smaller mass gradually acquire
greater velocities at the expense of the larger stars, just as
in a mixture of gases of molecules of different weights the
lighter ones on the average move faster than the heavier
ones. The fact that the individual star clouds are not
spherical would argue that they have not had time to acquire
the symmetrical form of equilibrium, if it were not for the
fact that their passage through and near to other star clouds\DPnote{[** TN: Hyphenated here in original, but not elsewhere.]}
may occasionally introduce great irregularities.

But all the star clouds which together constitute the Milky
%% -----File: 527.png---Folio 497-------
Way may be considered as being simply a much larger system.
If it remains isolated from all other systems, it will
similarly tend toward a symmetrical form. Its irregularities
point toward the conclusion that its age is not indefinitely
great; and this would be a necessary conclusion if there were
not the possibility, or perhaps even probability, of the existence
of other galaxies beyond our own near which, or through
which, ours passes after intervals of time of a higher order
of magnitude than any so far considered. These families of
galaxies may be units in still larger systems, and so on without
limit. Therefore it is impossible to conclude from the
irregularities in the star clouds or galaxies that they have
not been of infinite duration. It should be added at once
that most astronomers believe, chiefly on the basis of the
finite amount of energy of the stars, that they have not
existed for an infinite time.

While it has not been possible to answer the more ambitious
questions which have been raised, there remain others
which are not without interest. For example, suppose that
throughout the whole region occupied by the stars they are
as numerous as they are near the sun; that is, that there are
$20$ or~$30$ in a sphere whose radius is $5$~parsecs. Suppose,
further, that there is equilibrium between the attractive and
centrifugal forces. So far as these assumptions approximate
the truth, there is a relation between the dimensions of the
whole stellar system and the mean velocity of stars at its
center, for the velocities depend upon the star density and
the extent of the region which they occupy. Inasmuch as
the star density in the neighborhood of the sun and the
velocities of the stars have been determined by observations,
the extent of the whole system can be computed.

The solar system, which is far from the borders of the
Galaxy, will be supposed to be approximately at its center.
\index{Galaxy}%
The mean velocity of the stars near the sun is about $22$~miles
per~second. This fact and the assumptions which have been
made imply that the radius of the Galaxy is about $1100$
%% -----File: 528.png---Folio 498-------
parsecs and that the total number of stars in it is $260,000,000$.
Although the assumptions are not in exact harmony with
the facts, it is believed that these results are of the correct
order of magnitude. And under the same assumptions the
time required for a star to pass from one side of the system
to the opposite is approximately $200,000,000$ years. Since
this is probably less than the age of the earth, our sun may
have traveled in geological times more than once far toward
the boundaries of the stellar system.

Whatever may have been the history of any particular
star, these results, though they may be appreciably in error
numerically, imply that the stars have undergone considerable
mixing. So far as can be determined at present this
process will continue in the future, the star clouds which
form the Milky Way will become more and more uniform
\index{Milky Way}%
and the motions of the stars more and more chaotic, the stars
of smaller mass will acquire higher velocities than the larger
ones, at rare intervals every star will pass near some other
star, and possibly at intervals of time of a higher order our
Galaxy will encounter other galaxies and again be deformed
\index{Galaxy}%
and made irregular by them.

\Article{280}{Runaway Stars.}---Since the average radial velocity
\index{Runaway stars}%
\index{Stars!groups of}%[** TN: Move up one page]
\index{Stars!runaway}%
of a large group of stars is one half the average of their entire
motions, the spectroscope furnishes the average speed with
which the stars move. The average velocity of the stars
near the sun is about $1.8$~times the velocity of the sun, or
$22$~miles per~second. This is $7.5$~astronomical units per year,
or one parsec in about $27,000$~years.

The stars, however, do not all move with even approximately
the same velocity. The variations in their speeds
are evidenced both by their proper motions and by their
radial velocities. The star having the largest known proper
motion,\footnote
  {Professor Barnard has just (June,~1916) found an eleventh-magnitude
\index[xnames]{Barnard}%
  star in Ophiuchus whose annual proper motion is over~$10''$; its parallax
  has not yet been measured.}
\index{Proper motion of stars}%
namely, $8''.7$ per year, is the sixth in \Tableref{XVI},
%% -----File: 529.png---Folio 499-------
and by astronomers is known as C.~Z. 5~h.~243,\DPnote{** TN: [sic] No superscript, cf. Table XVI.} or No.~243
in the fifth hour of right ascension in the Cordoba Zone
Catalogue. It was discovered by Kapteyn in 1897 from the
\index{Catalogues of stars}%
\index{Stars!catalogues of}%
\index[xnames]{Kapteyn}%
measurement of plates taken by Gill and Innes at the Cape
\index[xnames]{Gill}%
\index[xnames]{Innes}%
Observatory, in South Africa. Its actual velocity is $170$~miles
per second, or nearly $8$~times the average velocity of
the stars. The star known as 1830 Groombridge has a
proper motion of $7''$ per year. Its parallax, which is not
yet accurately known, can scarcely exceed~$0''.1$ and its
velocity probably exceeds $200$~miles per second. The star
61~Cygni is another one in \Tableref{XVI} which moves at a high
speed, though its velocity is exceeded by the velocities of
quite a number of other known stars.

The stars having high velocities are called ``runaway
stars'' because, unless they pass very near other stars in
their journey through space, they will escape, like molecules
from a planet, from the gravitative control of the stars which
constitute the Galaxy, and will recede from them forever.
\index{Galaxy}%
This conclusion is inevitable unless the total mass of the
sidereal system is much greater than has hitherto been supposed.
Even if the extravagant assumption is made that
there are $1,000,000,000$ stars, each as massive as the sun, in
a spherical space whose radius is $1000$~parsecs, it is found that
a star moving through its center with a speed exceeding $72$~miles
per second will entirely escape from the system unless,
in its journey toward the surface, it passes near at least one
other star in a particularly favorable way so that its velocity
is much reduced. Since the probability of such a near approach
is very small, we are forced to the conclusion that these
stars with high velocities are only temporary members of our
Galaxy. The only alternative is that the mass of the system
is at least $10$~times as great as has been estimated.

If the total mass of the stellar system is greatly in excess
of the estimates which have been made, the resulting attractive
forces are greater than the centrifugal forces due to the
average motions of the stars, and, therefore, the stars must
%% -----File: 530.png---Folio 500-------
be on the whole falling together. That is, either the runaway
stars will actually escape from the Galaxy entirely, or
\index{Galaxy}%
the stellar system will necessarily become more and more
concentrated under the mutual gravitation of its parts.

The question of the origin of runaway stars at once arises.
Either they have come in from beyond our Galaxy, perhaps
from a distant one, or their high velocities have been developed
within our stellar system. The first alternative is
certainly possible though it may appear at first to be improbable,
especially in view of the enormous time required
for a star to go from one sidereal system to another. But
these stars will, in most cases, permanently leave our Galaxy,
and there is no apparent reason why stars might not equally
well leave other galaxies.

The second alternative is also possible, for if a large star
and a small star pass near each other the velocity of the small
one may be greatly increased. A series of favorable close
approaches might easily produce the high velocities which
are observed. The process is closely analogous to the development
of high velocities in exceptional cases in a mixture
of gases, the light molecules acquiring the highest velocities.
The difficulty in the case of the stars is that the intervals
between close approaches are so long that the process demands
startling lengths of time. Perhaps astronomers in
the remote future will be able to determine from their
greater knowledge regarding the masses and the velocities
of the stars something respecting the length of time during
which the stars of the stellar system have been subject to
their mutual attractions.

\Article{281}{Globular Star Clusters.}---Perhaps the most wonderful
\index{Clusters of stars}%
\index{Globular star clusters}%
\index{Star!clusters}%
\index{Stars!clusters of}%
objects in the heavens are the dense globular star
clusters. They cover portions of the sky generally less than
$30'$ in diameter, that is, less than the apparent diameter of
the moon. The brightest of them appear to the unaided
eye as faint fuzzy stars, but a large telescope shows that they
are made up of thousands of stars. The most splendid of
%% -----File: 531.png---Folio 501-------
these objects in the northern sky is the great Hercules cluster
\index{Hercules}%
(\Figref{171}), also known to astronomers as Messier~13, in which
\index[xnames]{Messier}%
\begin{figure}[hbt]%[Illustration:]
\Input{531}{jpg}
\Caption[The great globular star cluster in Hercules (M.~13). \textit{Photographed
by Ritchey with the $40$-inch telescope of the Yerkes Observatory.}]{Fig}{171}
\index{Yerkes Observatory}%
\index[xnames]{Ritchey}%
\end{figure}%
Ritchey's photograph, taken with the great $60$-inch reflector
of the Mt.~Wilson Solar Observatory, shows more than $50,000$
\index{Mount Wilson Solar Observatory}%
\index{Solar!Observatory}%
stars. The great cluster Omega Centauri, in the southern
\index{Omega Centauri}%
heavens, is even a more wonderful aggregation of suns.
%% -----File: 532.png---Folio 502-------

The individual stars in most of the globular clusters are
very faint, ranging from about the twelfth magnitude down
to the limits of visibility with the instrument employed.
If we knew the distance of a cluster, we could determine the
luminosity of its members compared to the sun. Then we
could answer the question whether the stars in the clusters
are great suns like our own, but which appear faint and
crowded together only because of their immense distance
from us, or whether they are examples of an evolution in
which the mass is distributed among a very large number of
relatively small bodies. It is not possible to measure directly
the parallaxes of the globular clusters, and their probable
distances can be inferred only from their proper motions.
Unfortunately, we do not yet have any positive data bearing
on the problem except that their positions in the sky are
sensibly fixed. This can only mean that they are very distant,
for there are more than $100$~clusters known, and it is
improbable that all of them should be moving in the same
direction as the sun and with the same speed. It seems to
be clear from their apparent fixity on the sky that their distance
is at least $100$~parsecs and it is much more probable
that it is $1000$ parsecs. At the distance of $100$~parsecs the
sun would be a ninth-magnitude star, while at $1000$ parsecs
it would be of the fourteenth magnitude. If the clusters
are at the smaller distance, their members are much less
luminous than the sun; if at the greater, they are comparable
with the sun.

The problem may also be considered in the reverse order.
\index{Sun!magnitude of}%
That is, if there are any reasons for assuming that the individual
stars in the clusters are comparable to the sun in
luminosity, or related to it in any definite way, then their
distances can be computed. The stars in the clusters are
individually so faint that their spectra cannot be studied;
but valuable information concerning the character of the
light they radiate can be obtained by photographing them
first with plates sensitive to the blue and then to the red
%% -----File: 533.png---Folio 503-------
end of the spectrum. Such work has been carried out at
the Solar Observatory and Shapley finds evidence that the
\index{Mount Wilson Solar Observatory}%
\index{Solar!Observatory}%
\index[xnames]{Shapley}%
stars in the Hercules cluster are like the giant red and yellow
\index{Hercules}%
stars, such as Antares and Arcturus, which are enormously
\index{Antares}%
\index{Arcturus}%
more luminous than the sun. If this conclusion is correct,
the distance of the Hercules cluster is of the order of $10,000$
parsecs. Perhaps a reasonable summary of present information
would be that globular clusters are almost certainly
distant much more than $100$~parsecs, and that their distances
probably range from $1000$ to $10,000$ parsecs.

The actual dimensions of the clusters are appalling. The
distance across one whose apparent diameter is~$30'$ is $\frac{1}{115}$ of
its distance from the earth, or probably of the order of at
least $10$~parsecs. If $50,000$ stars were distributed uniformly
throughout a sphere of these dimensions, the average distance
between adjacent stars would be more than $0.4$~parsec, or
more than $80,000$ times the distance from the earth to the
sun. It is seen from this that, although the globular clusters
are somewhat condensed toward their centers, the actual
distances between the stars of which they are composed are
enormous. There is abundance of room in them for almost
indefinite motion without collision, and there is no apparent
reason why the individual stars should not have planets
revolving around them.

Dynamically, the globular clusters are much simpler than
the Galaxy. They seem to have arrived at an approximately
\index{Galaxy}%
fixed state of symmetrical distribution, though, of
course, the individual stars are in ceaseless motion through
them. The regularity of their arrangement implies that the
process of mixing has been in operation an enormous time,
unless indeed they started in this remarkable state. It is
not difficult to get at least an approximate idea of the time
required for a star to move from the borders to the center of
a globular cluster. The distribution of mass in a cluster is
somewhere between condensation entirely at the center and
uniform density. In the first case the force varies inversely
%% -----File: 534.png---Folio 504-------
as the square of the distance from the center, and in the
second, it varies directly as the distance from the center.
In a cluster whose radius is $5$~parsecs and which contains
$50,000$ stars, each having the mass of the sun, the time required
for a star to move from the surface to the center in
the first case is nearly $800,000$ years, and in the second is
$1,100,000$ years. The actual time is of the order of $1,000,000$
years. Since thousands of these excursions would be necessary
to reduce a group of stars with considerable irregularities
in distribution to the symmetrical forms observed, the
age of these systems must be enormous. Only a thousand
excursions from the periphery to the center and back would
require $1,000,000,000$ years. It is improbable that this
number is too large (it may be many times too small), and
it follows that either the stars exist an enormous time as
luminous bodies, or much of the dynamical evolution of the
clusters was completed before the star stage, if, indeed,
there has been such a preceding stage. And it follows further
from the symmetry of the clusters that for at least hundreds
of millions of years they have not passed near other clusters.

No rapid motions of stars in the globular clusters are to be
expected. With $50,000$ stars, each equal to the sun in mass,
distributed uniformly throughout a sphere whose radius is
$5$~parsecs, the velocity of a permanent member of the group
at its center would be only about $4$~miles per second. Since
the actual clusters have strong central condensations, the
velocity for the ideal case would be considerably exceeded
by stars near their centers. Suppose they move at $10$~miles
per second at right angles to the line of sight. At a distance
of $1000$ parsecs they would move with respect to the center
of the cluster only one second of arc in $300$~years. Of course,
if the assumptions as to the distance or masses are wrong,
the result will be wrong, and, besides, a certain small number
of the stars, especially those of smallest mass, will have
motions in excess of the mean velocities. But it is improbable
that relative motions of the members of star clusters
%% -----File: 535.png---Folio 505-------
will be large enough in any case to be observable inside of
several decades.


\Section{XXIII}{QUESTIONS}

1. Prove that, in \Figref{168}, %[** TN: Not breaking line]
$\angle E_1SE_2 - \angle E_1S'E_2 = \angle SE_1S' - \angle SE_2S'$.

2. Suppose there are $30$~stars within $5$~parsecs of the sun; what is
the average distance between adjacent stars?

3. Draw a diagram to prove that Herschel's observations, \Artref{274},
\index[xnames]{Herschel, William}%
are explained by the conclusion which he drew. If this conclusion
is denied, what other must be accepted?

4. If an angle of $1''.0$ can be measured with an error not exceeding
$10$~per~cent, how small a parallax can be determined with this degree
of accuracy by the method of \Artref{275} in $100$~years?

5. Show by a diagram that if two stars are moving in parallel
lines, then the great circles in which they apparently move, as seen
from the earth, intersect in a point whose direction from the earth is
the direction in which the stars move (\Artref{277}).

6. Since the velocity of our sun is somewhat below the average of
the velocities so far measured, what are the probabilities of the relation
of its mass to the masses of the observed stars?

7. If the radius of the Galaxy is $1100$ parsecs (end of \Artref{279}),
how long would it take the sun at its present speed to pass from the
center of the sidereal system to its borders?

8. If the velocity of the star 1830 Groombridge is $200$~miles per
second and remains constant, how long will be required for it to
recede to a distance from which our Galaxy will appear as a hazy
patch of light $1°$ in diameter?

9. If there are many galaxies, and if the distances between them
compare to their dimensions like the distances between the stars
compare to the dimensions of the stars, how long will be required for
1830 Groombridge to go from our Galaxy to another?

\normalsize


\Section{III}{The Stars}

\Article{282}{Double Stars.}---A few double stars have been known
\index{Double stars}%
\index{Stars!double}%
almost since the invention of the telescope, but William
Herschel was the first astronomer to search for them systematically
and to measure the distances and the directions
of their components from one another. His purpose in measuring
them was to determine the parallax of the nearest ones
%% -----File: 536.png---Folio 506-------
(\Artref{272}), for he assumed, perhaps unconsciously, that the
sun is a typical star, and that when two stars are apparently
in about the same direction from the earth, one is simply
farther away than the other.

Herschel found a large number of double stars whose components
were apparently separated by a few seconds of arc
at the most. A discussion of the probability of there being
such a large number of stars so nearly in lines passing through
the earth would have shown him that their apparent proximity
could not be accidental. He reached the same result
in a few years, for his observations showed him in a considerable
number of cases that the two components were
revolving around their center of gravity. That is, instead
of all stars consisting of single primary bodies accompanied
by families of planets, there are many which are twin suns
of approximately equal mass and dimension. So far as we
know, they may or may not have planetary attendants, for
such small objects shining entirely by reflected light would
be beyond the range of our telescopes even if they were a
thousand times more powerful than any yet constructed.

The names that stand out most prominently in the double-star
astronomy of the nineteenth century are William
Struve, Dawes, John Herschel, and Burnham. In Burnham's
\index[xnames]{Burnham}%
\index[xnames]{Dawes}%
\index[xnames]{Herschel, John}%
\index[xnames]{Struve, William}%
great catalogue of double stars the observations and
descriptions of about $13,000$ of these objects are given. New
ones are constantly being discovered, though the northern
heavens have now been very thoroughly examined with
powerful telescopes. At the Lick Observatory a survey of
the whole heavens to at least $-14°$~declination was begun
by Hussey and Aitken and completed by Aitken. All old
\index[xnames]{Aitken}%
\index[xnames]{Hussey}%
pairs with a separation not exceeding $5''$ of arc were observed,
and $4300$ new pairs were discovered within the same limits.
On using a definition of double star which excludes all wider
pairs except in the case of bright stars, Aitken found that
there are $5400$ of these objects not fainter than the ninth
magnitude north of the celestial equator. This means that
%% -----File: 537.png---Folio 507-------
at least one star in~$18$ of those not fainter than the ninth
magnitude is a double which is visible with the $36$-inch
telescope of the Lick Observatory. Of these stars, $2206$
\index{Lick Observatory}%
have an apparent angular separation not greater than $1''$,
and only~$200$ are separated by more than~$5''$. A very interesting
fact is that, compared to the whole number of stars
of the same brightness, double stars seem to be somewhat
more numerous in the Milky Way than near its poles.
\index{Milky Way}%
Moreover, the average separation of the stars of the spectral
class to which the sun belongs is considerably greater than in
those of the so-called earlier types which include the blue stars.

There are doubtless some cases in which the components
of a double star are at different distances and simply in nearly
the same direction from the observer. But in general they
form physical systems which revolve around their centers of
gravity in harmony with the law of gravitation, and these
pairs are called \textit{binaries}. According to the law of probability,
essentially all of the $5400$~double stars in Aitken's list must
\index[xnames]{Aitken}%
be binaries, for only very rarely would two stars be accidentally
so nearly in the same direction from us.

\Article{283}{The Orbits of Binary Stars.}---The stars in all cases
\index{Binary stars}%
\index{Binary stars!orbits of}%
\index{Orbits!of binary stars}%
\index{Stars!binary}%
are so remote from us that the components of a binary system
cannot be seen as separate stars unless they are a great
distance apart. But when the components of a binary pair
are far from each other, their period of revolution is long,
and observations must therefore extend over many years
in order to furnish data for the computation of their orbits.
Those binary stars which were first discovered and which
have been longest under observation are not very close
together, and, while in many cases it is now certain from
direct observational evidence that they form physical systems,
there are only $40$~or~$50$ in which the observed arcs are
long enough to define the orbits with any degree of precision.
In 1896 See published the orbits of~$40$ of the best-known
\index[xnames]{See}%
binary stars.

The periods of known visual binary stars range from $5.7$~years,
%% -----File: 538.png---Folio 508-------
for Delta Aquilæ, to hundreds and probably thousands
\index{Delta Aquilae@{Delta Aquilæ}}%
of years. The planes of their orbits are inclined at all
angles to the line joining them with the earth, so that, as a
rule, we see their orbits in projection. Indeed, the orbit of
\index[xnames]{See}%
42~Comæ Berenices is sensibly edgewise to us. One of the
most interesting things about the orbits of binaries is that
they are generally considerably eccentric. In the $40$~orbits
in See's list the average eccentricity was~$0.48$, or twelve times
that of the planetary orbits. The orbit of the binary star
Gamma Virginis has an eccentricity of~$0.9$, and therefore the
\index{Gamma Virginis}%
greatest distance of the two members of this pair from each
other is $19$~times their least distance.

\Article{284}{Masses of Binary Stars.}---The masses of those
\index{Binary stars!masses of}%
\index{Masses!of stars}%
\index{Stars!masses of}%
planets which have satellites are found from the periods and
distances of their respective satellites (\Artref{154}). The
masses of Mercury and Venus are found from their attractions
for other bodies, especially comets. The masses of
celestial bodies are found only from their attraction for other
bodies. It is evident, therefore, that the mass of a single star
remote from all other visible bodies cannot be found. But
when the dimensions of the orbit and the period of revolution
of a binary pair are known, their combined mass can be
computed just as the mass of a planet is computed.

The periods of binary stars are determined by direct
observations of their apparent positions. The dimensions
of the orbit of a binary pair can be determined from their
apparent distance apart and their distance from the earth.
The chief difficulty lies in the problem of finding their parallax,
for only a small number of stars are within measurable distance
from the sun.

Those binary stars whose periods and distances are known
with sufficient approximation to make the mass determinations
of value are given in \Tableref{XVIII}. The masses of all
those whose parallaxes are less than~$0''.2$ are subject to some
uncertainty, and the probable error is great if the parallaxes
are less than~$0''.1$.
%% -----File: 539.png---Folio 509-------
\begin{table}[hbt]
\begin{center}
\Caption{Table}{XVIII}
%\caption[Binary stars whose masses are known]{}
\index{Masses!of stars}%
\index{Stars!masses of}%
\begin{tabular}{|l|*{5}{c|}}
\hline
\settowidth{\TmpLen}{Bradley~2388}%
\TEntry{\TmpLen}{\TFontsize\THead Star}
  & \settowidth{\TmpLen}{\textsc{allax}}% [** F2: cf. 0508.png]
    \TEntry{\TmpLen}{\TFontsize\THead Par- \\ allax}
  & \settowidth{\TmpLen}{\textsc{Period}}%
    \TEntry{\TmpLen}{\TFontsize\THead Period}
  & \settowidth{\TmpLen}{\textsc{Semi-}}%
    \TEntry{\TmpLen}{\TFontsize\THead Semi- \\ Axis}
  & \settowidth{\TmpLen}{\textsc{bined}}%
    \TEntry{\TmpLen}{\medskip\TFontsize\THead Com- \\ bined \\ Mass\medskip}
  & \settowidth{\TmpLen}{\textsc{Luminos-}}%
    \TEntry{\TmpLen}{\TFontsize\THead Luminos- \\ ity} \\
\hline
\Strut& $''$ &&&& \\
$\alpha$~Centauri    & $0.76$ &  $81.2$                       & $23.3$  & $1.9$ &  $2.0$ \\
Sirius               & $0.38$ &  $48.8$                       & $20.0$  & $3.4$ & \llap{$4$}$8.0$ \\
Procyon              & $0.32$ &  $39.0$                       & $10.4$  & $0.7$ &  $9.7$ \\
$\eta$~Cassiopeiæ    & $0.20$ & \llap{$3$}$00.$\rlap{(?)}$\Z$ & $47.4$  & $1.2$ &  $1.4$ \\
70 Ophiuchi          & $0.17$ &  $88.4$                       & $26.8$  & $2.5$ &  $1.2$ \\
$o_2$~Eridani        & $0.17$ & \llap{$1$}$80.0$              & $28.2$  & $0.7$ &  $0.8$ \\
Bradley 2388         & $0.13$ &  $45.8$                       & $\Z8.2$ & $0.3$ &  $1.0$ \\
85 Pegasi            & $0.11$ &  $26.3$                       & $\Z7.7$ & $0.7$ &  $0.8$ \\
$\zeta$~Herculis     & $0.10$ &  $34.5$                       & $13.5$  & $2.1$ & \llap{$1$}$1.4$ \\
$\kappa$~Pegasi      & $0.08$ &  $11.4$                       & $\Z3.7$ & $0.4$ &  $3.1$ \\
$\mu_2$~Boötis       & $0.05$ & \llap{$2$}$00.$\rlap{(?)}$\Z$ & $21.5$  & $0.2$ &  $0.7$ \\
\hline
\end{tabular}
\end{center}
\end{table}
In this table the periods are given in years, the semi-axes in
terms of the earth's distance from the sun, the combined
mass in terms of the sun's mass, and the luminosity in terms
of the sun's luminosity at the same distance.

Perhaps the most interesting thing brought out by the
table is that the masses of all of these stars are comparable
to that of the sun, and, with the exception of Sirius, their
luminosities do not differ greatly from that of the sun.
But there are not enough pairs of stars in the table to justify
any very positive general conclusion.

If the orbits of each of the two components of a binary
star with respect to their center of gravity are known, their
separate masses can be computed. The problem of determining
the orbits of two stars with respect to the center
of mass of their system is very difficult because their motions
with respect to neighboring stars, or fixed reference lines,
must be measured. In only a few cases are the results at
present reliable. The discussions of Lewis Boss led him to
\index[xnames]{Boss, Lewis}%
the conclusion that probably in all cases the brighter star
is the more massive, a result which is contrary to that which
was sometimes found in earlier investigations.
%% -----File: 540.png---Folio 510-------

\Article{285}{Spectroscopic Binary Stars.}---The spectroscope has
\index{Binary stars!spectroscopic}%
\index{Spectroscopic binaries}%
contributed very important results to the study of binary
stars. Its application depends upon the fact that it enables
the observer to determine whether a source of light is approaching
or receding (\Artref{226}). Suppose the plane of
motion of a binary system passes through the earth, as is
represented in \Figref{172}. When the stars are in the positions
$A$ and~$B$, one is receding from, and the other is approaching
toward, the earth. If they have similar spectra, the spectrum
of the combined pair will consist of double lines (\Figref{173}),
\begin{figure}[hbt]%[Illustration:]
\Input{540}{png}
\Caption[Orbit of a spectroscopic binary star.]{Fig}{172}
\end{figure}%
for the lines from one will be shifted toward the red
while the lines from the other will be displaced toward the
violet. When the stars have made a quarter of a revolution
around their center of gravity~$O$ and have arrived at~$A'$
and~$B'$, the lines will not be displaced because the stars are
neither approaching toward nor receding from the observer.
After another quarter of a revolution they will be double
again because $A$ will be approaching and $B$~receding.

The data furnished in this way by the spectroscope are
very important because, in the first place, the separation of
the lines determines the relative velocity of the stars in their
orbits. This is true whether the system as a whole is stationary
%% -----File: 541.png---Folio 511-------
with respect to the earth, as has so far been tacitly
assumed, or is moving in the line of sight. The period is
also given. The period and velocity furnish the dimensions
of the orbit and consequently the total mass of the binary
system.

If the two stars of the binary are very unequal in luminosity,
the spectrum of the fainter one will not be obtained,
but the spectral lines of the brighter one will be shifted
alternately toward the red and violet ends of the spectrum.
\begin{figure}[hbt]%[Illustration:]
\Input{541}{jpg}
\Caption[Spectrum of Mizar, showing double lines above and single lines
below (period $20.5$~days). (\textit{Frost; Yerkes Observatory.})]{Fig}{173}
\index{Mizar!spectrum of}%
\index{Yerkes Observatory}%
\index[xnames]{Frost}%
\end{figure}%
The period is given in this case, but only the velocity of the
brighter star with respect to the center of gravity of the
system is known. Since the orbit of one star with respect
to the other is necessarily larger than the orbit of the brighter
one with respect to the center of gravity of the two, the mass
computed in this case will always be too small.

It has so far been assumed that the plane of motion of
the binary star passes through the earth. This condition
is realized only very exceptionally, and indeed is not necessary
for the application of the method. If the plane of
motion does not pass exactly through the earth, the measured
radial velocity is only a fraction of the whole velocity,
%% -----File: 542.png---Folio 512-------
and the size of the orbit and mass of the system based on it
are both too small. Since the planes of the orbits of binary
stars may have any relation to the observer, the measured
radial velocities are in general smaller than the actual
velocities; on the average the former are $0.63$ of the
latter. On the average the calculated masses are about $60$~per~cent
of the true masses.

The spectroscope is particularly valuable in the study of
binary stars because it is not necessary that they should be
near enough to appear as visual binaries. The only requisite
is that they shall be bright enough (above the eighth
magnitude with present instruments) to enable astronomers
to photograph their spectra in a reasonable time. With
very few exceptions the spectroscopic binaries so far known
are not also visual binaries. A second advantage of the
spectroscope is that it furnishes at the same time lower
limits for the orbital dimensions and masses of the stars.

The first known spectroscopic binary was discovered by
E.~C. Pickering at the Harvard Observatory, in 1889, when
\index{Harvard College Observatory}%
\index[xnames]{Pickering, E. C.}%
it was found that the spectrum of Mizar ($\zeta$~Ursæ Majoris)
\index{Mizar}%
consisted of alternately double and single lines (\Figref{173}).
Mizar is a visual double star, but the double lines belong to
a single component of the visual pair. The visual pair probably
are revolving around their center of gravity, but their
distance apart is so great that their period of revolution is
very long and their motions are too slow to be measured
with the spectroscope.

The first spectroscopic binary in which one of the components
is dark was discovered by Vogel, at Potsdam, in
\index[xnames]{Vogel}%
1889. He found that the lines in the spectrum of Algol,
the well-known variable star, shift alternately toward the
red and blue ends of the spectrum with the same period as
that of its variability ($2$~d.\ $20$~h.\ $49$~m.). This confirmed
the theory that this star varies in brightness because a relatively
dark one revolves around it and partially eclipses it
at each revolution. The star Mu Orionis has the short period
\index{Mu Orionis}%
%% -----File: 543.png---Folio 513-------
of $4.45$~days, and the displacements of its spectral lines are
considerable (\Figref{174}).

In 1898 only $13$~spectroscopic binary stars were known.
By 1905 the number had increased to $140$~pairs, $6$~of which
were also visual binaries. When Campbell published his
\index[xnames]{Campbell}%
second catalogue of spectroscopic binaries in 1910, there were
$306$~known pairs. In $19$~cases the spectra of both stars had
been measured, and from the absolute displacements of each
set of lines their relative masses had been determined. With
one possible exception the brighter stars of the systems are
\begin{figure}[hbt]%[Illustration:]
\Input{543}{jpg}
\Caption[Spectra of Mu Orionis (\textit{Frost; Yerkes Observatory}).]{Fig}{174}
\index{Mu Orionis!spectrum of}%
\index{Yerkes Observatory}%
\index[xnames]{Frost}%
\end{figure}%
the more massive. The larger stars are generally less than
twice as massive as the smaller. Of course, the difference is
probably much greater in those cases where the spectrum of
the smaller star is too faint to be observed.

\Article{286}{Interesting Spectroscopic Binaries.}---\textit{Mizar.} As
\index{Mizar!spectrum of}%
has been stated, the brighter component of Mizar was the
first spectroscopic binary discovered. The later work of
Vogel showed that its period is about $20.5$~days, from which
\index[xnames]{Vogel}%
it follows in connection with the dimensions of its orbit
($22,000,000$ miles between the two components) that the
mass of the system is at least four times that of the sun.
The spectra of both stars are present, and their equal displacement
%% -----File: 544.png---Folio 514-------
proves that the masses of the two components are
sensibly equal. The center of gravity of the system is approaching
the solar system at the rate of about $9$~miles per
second. In 1908 Frost and Lee found that the other component
\index[xnames]{Frost}%
\index[xnames]{Lee}%
of Mizar is also a spectroscopic binary of the type
in which the spectrum of only one star of the pair is visible.
In 1908 Frost announced that Alcor is a spectroscopic binary
\index{Alcor}%
of short period in which both spectra are observable. Therefore
Mizar is a visual double each of whose components is a
spectroscopic binary, and the neighboring Alcor is also a
binary.

\textit{Spica.} One of the earliest known spectroscopic binaries
\index{Spica}%
is the first-magnitude star Spica whose spectral lines were
found to vary by Vogel in 1890. The spectrum of the
fainter component has also been observed. The period of
the pair is $4$~days, their mean distance from each other is
about $11,000,000$ miles, and their masses (neglecting the possible
reduction due to the inclination of their orbit) are
respectively $9.6$ and $5.8$~times that of the sun. This system
is receding from the sun at about $1.2$~miles per second.

\textit{Capella.} The first-magnitude star Capella is a spectroscopic
\index{Capella}%
binary, the spectra of both stars being visible, in which
the period is $104$~days and the mean distance (possibly much
reduced by the inclination of the plane of the orbit) about
$50,000,000$ miles. With these data the masses of this pair
are found to be at least $1.2$ and~$0.9$ that of the sun. This
orbit has a very small eccentricity. These stars are receding
from the solar system at the rate of nearly $20$~miles
per second. The parallax of Capella has been investigated
with the utmost care by Elkin, who found for it $0''.09$, corresponding
\index[xnames]{Elkin}%
to a distance of $11$~parsecs. At that distance
the sun would be only $\frac{1}{70}$ as bright as Capella, or approximately
of the fifth magnitude. Since the spectrum of
Capella is almost exactly the same as that of the sun, which
naturally leads to the conclusion that the temperature and
surface brightness of Capella are approximately equal to
%% -----File: 545.png---Folio 515-------
those of the sun, it seems probable that the orbit of the pair
is so inclined that the computed masses are much too small.

\textit{Polaris.} The pole star has two darker companions discovered
\index{Polaris}%
spectroscopically by Campbell in 1889. One is very
\index[xnames]{Campbell}%
close to the bright star and revolves around it in a period of
a little less than $4$~days, while the second companion is much
more distant and requires about $12$~years to complete a
revolution. These stars are all quite distinct from the faint
telescopic companion to Polaris.

\textit{Alpha Centauri.} Alpha Centauri is at the same time a
\index{Alpha Centauri}%
visual and a spectroscopic binary. Moreover, its parallax
has been very accurately determined by direct means, so
that the actual distance of the components from each other
and their masses can be determined (\Tableref{XVIII}). Since
the same results can be determined spectroscopically, their
comparison affords a valuable check on the accuracy of the
results. The spectroscopic data were obtained by Wright
\index[xnames]{Wright, W. H.}%
at the branch of the Lick Observatory in South America,
\index{Lick Observatory}%
and the results obtained from them agree almost exactly
with those based on other methods. But the spectroscope
gives the additional fact, which cannot be determined otherwise,
that Alpha Centauri is approaching the sun at the rate
of $13.8$~miles per~second.

\Article{287}{Variable Stars.}---A star whose brightness changes
\index{Stars!variable}%
is said to be a variable. The first known variable, Omicron
Ceti, was discovered by Fabricius in 1596. The variability
\index{Omicron Ceti}%
\index[xnames]{Fabricius}%
of Algol was definitely announced by Goodricke in 1783,
\index{Algol}%
\index[xnames]{Goodricke}%
though it seems to have been noticed a century earlier.
The following year he recorded the variability of Beta Lyræ.
\index{Beta Lyrae@{Beta Lyræ}}%
But variable stars were not discovered in any considerable
numbers until toward the close of the nineteenth century.
Now more than $3000$ of these objects are known in addition
to those which have been found in considerable numbers in
some of the globular star clusters. Some of them vary regularly
and periodically, with periods ranging from less than a
day to more than two years; others vary irregularly without
%% -----File: 546.png---Folio 516-------
any apparent rule or order. Some flash out brilliantly
for a short time and then sink back more slowly into permanent
oblivion. It is certain that the brightness of every
star varies slowly because of its changing distance from the
sun, if for no other reason, but there is no observational
evidence of a change for this reason.

Variable stars are classified according to the peculiarities
of their light changes, and the principal types are enumerated
in the following articles. It must be remembered, however,
that variable stars are strange objects which present numerous
exceptions to all rules.

\Article{288}{Eclipsing Variables.}---If the plane of the orbit of a
\index{Eclipsing variables}%
\index{Variable stars!eclipsing}%
binary %[Illustration: Break]
\begin{wrapfigure}[16]{\WLoc}{3.375in}
\Input[3.375in]{546}{png}
\Caption[Light curve of typical eclipsing variable
star.]{Fig}{175}
\end{wrapfigure}
{\stretchyspace%
pair passes very nearly through the earth, the stars
partially or totally
eclipse each
other every time
they are in a line
with the earth.
If one of the two
is a dark star and
nearly as large as
the bright one, it
is clear that the
light received
from the pair will
remain constant
except when the brighter star is eclipsed. As the dark star
begins to eclipse the brighter one, the light diminishes very}
rapidly until the time of greatest obscuration, after which as
a rule the star rapidly regains its normal brightness. However,
in some cases the dark star is very large so that the
eclipse persists for a considerable time, and then the variable
remains at minimum for a few minutes or possibly a few
hours.

The variability in the brightness of a star is represented
by a curve. In \Figref{175} the curve for a typical eclipsing
%% -----File: 547.png---Folio 517-------
variable is given. The time is marked off along the horizontal
axis and the brightness of the star is proportional to
the distance of the curve above this axis. The parts marked
$a$ give the brightness when the star shines undimmed by an
eclipse, the points $b$ are where the light begins to wane as
the eclipse commences, and the points $c$ indicate the brightness
at the moment of greatest obscuration. If the fainter
star is somewhat luminous instead of being entirely dark,
there will be a secondary and less pronounced minimum.

The typical eclipsing variable in which one component is
dark is Algol (Beta Persei), whose light curve is essentially
\index{Algol}%
the same as that given in \Figref{175}. About $100$~stars of this
type are known, and they are often called Algol variables.
They are characterized by the shortness of their periods,
many of which are less than $5$~days and only $12$ of which
are longer than $10$~days, and by the regularity of their light
curves. Doubtless the explanation of their short periods
is that when the two stars are far apart they do not eclipse
one another, even partially, unless the plane of their motion
passes very exactly through the earth.

Eclipsing variables are necessarily spectroscopic binary
stars. It increases our confidence in both the methods and
the interpretations to find that the data obtained in the
two distinct ways are perfectly in accord. It is not to be
inferred from this that the data are coextensive. The spectroscope
furnishes the velocity and therefore the dimensions
and mass of the system, especially when both stars are luminous.
From the duration of the eclipses the dimensions of
the stars can be found. Since their masses are known, their
densities can then be computed. It has been found by
Russell, Shapley, and other astronomers that the mean density
\index{Density!of stars}%
\index{Stars!density of}%
\index[xnames]{Russell}%
\index[xnames]{Shapley}%
of the variable stars for which there are sufficient observational
data is about one eighth that of the sun. This is a
remarkable result in view of the fact that usually one of the
pair is very dark, and, according to current doctrine, in a
condensed state approaching extinction. It should be added
%% -----File: 548.png---Folio 518-------
that in the case where there is a single minimum the result
depends upon an assumption as to the relative densities of
the components, and consequently may be considerably in
error.

The period of Algol is $2$~d.\ $20$~h.\ $48$~m.\ $55$~s. It is normally
\index{Algol}%
a star of the second magnitude, but at the time of eclipse it
loses five sixths of its light. In 1889 Vogel discovered that
\index[xnames]{Vogel}%
it is a spectroscopic binary. He found that the combined
mass of the system is two thirds that of the sun, the bright
star has twice the mass of the darker one, the distance between
their centers is about $3,000,000$ miles, the diameters
of the stars are about $1,000,000$ and $800,000$ miles, and their
density is about one fourth that of the sun. Schlesinger
\index[xnames]{Schlesinger}%
found that for the similar system Delta Libræ the density is
\index{Delta Librae@{Delta Libræ}}%
also one fourth that of the sun.

There are several variations from the normal Algol
variable. In one the stars are of unequal size and both
bright. Then each eclipses the other, but the loss of light
is different in the two eclipses, and the light curve has two
minima of different depths. There are often irregularities
which have not yet been explained. Sometimes the periods
increase slightly for a number of years and then decrease
again, showing possibly the presence of a third body. Sometimes
the minima as determined photographically do not
occur at the times found by visual observations.

\Article{289}{Variable Stars of the Beta Lyræ Type.}---Variable
\index{Beta Lyrae@{Beta Lyræ}}%
\index{Variable stars!of Beta Lyræ type}%
stars of the Beta Lyræ type are closely related to those
which have been %[Illustration: Break]
\begin{wrapfigure}[16]{\WLoc}{3.375in}
\Input[3.375in]{549}{png}
\Caption[Light curve of a variable star of the Beta Lyræ type.]{Fig}{176}
\end{wrapfigure}
{\stretchyspace%
considered; in fact, the distinction between
the two classes seems to be disappearing. Their light varies
continuously from maximum to minimum and back to maximum
again. The maxima are all equal, but as a rule there
are two unequal minima. The standard star of this class is
Beta Lyræ (\Figref{176}), which is one of the earliest known
variables and gives the class its name.}

The explanation of the Beta Lyræ variables is that they
consist of two stars revolving in such small orbits compared
%% -----File: 549.png---Folio 519-------
to their dimensions that the intervals in which neither obscures
the other are very short. While this explanation
satisfies the phenomena in a general way, there are many
troubles in connection with the details. For example, about
a dozen minor variations in the light curve of Beta Lyræ
have been detected, or at least strongly suspected. Moreover,
the spectroscopic
data are often
puzzling. But,
on the whole,
astronomers are
satisfied that the
eclipse explanation
is the true
one, and the gap
between the light
curves of Algol
and Beta Lyræ is
gradually being
filled. In fact, Shapley includes many stars of the Beta Lyræ
\index[xnames]{Shapley}%
type among eclipsing variables of the Algol type.

\Article{290}{Variable Stars of the Delta Cephei Type.}---The star
\index{Variable stars!of Delta Cephei type}%
Delta Cephei has given its name to a third class of variables.
In these stars the light curves are periodic with periods ranging
from a few hours to $45$~days. But that which particularly
characterizes these stars is that they increase very
rapidly in brightness from minimum to maximum, and then
decline much more slowly with many minor irregularities
modifying the gradual diminution in brightness. The characteristics
of their light curves are given in \Figref{177}. There
are a few, however, known as the Geminids after Alpha
Geminorum, whose light curves are nearly symmetrical with
\index{Alpha Geminorum}%
respect to their maxima.

The explanation of the Cepheid variables has been a very
puzzling problem. Clearly their light changes are not ordinary
eclipse phenomena, but their spectral lines shift periodically
%% -----File: 550.png---Folio 520-------
with the periods of their %[Illustration: Break]
\begin{wrapfigure}[15]{\WLoc}{3.125in}
\Input[3.125in]{550}{png}
\Caption[Light curve of a variable star of the
Delta Cephei type.]{Fig}{177}
\index{Delta Cephei}%
\end{wrapfigure}
light variations. The natural
conclusion has been that they are spectroscopic binaries and
that the changes in light are abnormal eclipse phenomena.
While the light changes and spectral shifts agree in period,
they absolutely disagree in phase. That is, interpreting
the spectroscopic data in the ordinary way, these stars are
brightest when the principal stars are approaching the
observer and faintest when they are receding, instead of
having their minima when they are eclipsed. Evidently
there are inconsistencies in the interpretations, and it is
questionable whether eclipses have anything whatever to do
with the light variations
of these
stars. A number
of other explanations
have been
suggested, the
most plausible of
which is that the
light variations
are due to internal
oscillations
of the stars
produced perhaps
by collisions with masses of planetary dimensions. It
has been found that very moderate oscillations would account
for the variations in the rates of radiation. According to
this hypothesis, the shifts of the spectral lines are produced
partly by internal motions of the stars and partly by the
effects of alterations in pressure of the radiating parts.

\Article{291}{Variable Stars of Long Period.}---A majority of
\index{Long period variables}%
\index{Variable stars!long period}%
variable stars belong to the class whose periods range from
50 to several hundred days. They are not periodic in the
strict use of the term which is applicable to the Algol variables,
yet their light varies in an approximately periodic manner.
But the intervals between maxima, or between minima, are
%% -----File: 551.png---Folio 521-------
subject to some irregularities, and their luminosities at corresponding
phases are by no means always the same.

The best-known star of this class is Omicron Ceti, the
\index{Omicron Ceti}%
first known variable. It has been observed through more
than $300$~of its cycles, %[Illustration: Break]
\begin{wrapfigure}[16]{\WLoc}{3.25in}
\Input[3.25in]{551}{png}
\Caption[Light curve of variable star of long
period.]{Fig}{178}
\end{wrapfigure}
and yet it has not been found possible
to formulate any law describing accurately its light variations.
Its maxima and its minima are subject to as great
irregularities as the intervals between corresponding phases.
In 1779 William Herschel saw it when it was nearly as bright
\index[xnames]{Herschel, William}%
as Aldebaran, while $4$~years later it was not visible even
\index{Aldebaran}%
through his telescope. This means that it was at least $10,000$
times as bright
at its maximum
as at that particular
minimum.
Ordinarily
its maximum is
much below that
observed by Herschel
in 1779,
and its minimum
is considerably
above the limit
of visibility with
his telescope. Omicron Ceti was called \textit{Mira}, the wonderful,
and $300$ years of observation have only added to the mysteries
associated with its peculiar behavior.

The general characteristics of the light curves of variable
stars of long period is a slow, but gradually accelerated,
increase in brightness followed by a much more gradual
decline. The spectroscope shows marked changes in their
spectra, but no evidence of their being spectroscopic
binaries. They are nearly all red and are probably of not
very high temperatures. The cause of their variation
seems to lie within the stars themselves, yet it is difficult
to imagine any internal disturbances which would produce
%% -----File: 552.png---Folio 522-------
the remarkable fluctuations which are observed in many
stars of this class.

\Article{292}{Irregular Variable Stars.}---In addition to the classes
\index{Irregular nebulae@{Irregular nebulæ}!variables}%
\index{Variable stars!cluster}%
\index{Variable stars!irregular}%
of variable stars so far enumerated, there are others whose
variations have no semblance of periodicity. Some flash
out with relatively great brilliancy after intervals usually
counted in years. These stars are generally, if not always,
red. Others unaccountably fade away now and then and
sometimes become invisible through good telescopes, even
though they had been ordinarily visible with the unaided eye.
These stars are sometimes associated, at least apparently,
with faint nebulous masses.

\Article{293}{Cluster Variables.}---A very interesting and important
discovery was made in the last decade of the nineteenth
century by Bailey at the South American branch of
\index[xnames]{Bailey}%
the Harvard Observatory. He found that in the great
\index{Harvard College Observatory}%
globular cluster, Omega Centauri, $125$~stars were variable
\index{Omega Centauri}%
out of the $3000$ which he examined. He and other astronomers
have found similar variables in many other globular
star clusters. In a given cluster the range of variability is
nearly the same, usually a magnitude or two, the character
of the light variation is essentially the same, and the periods
are approximately the same, generally less than $24$~hours.
Their light curves are closely similar to those of the variables
of the Delta Cephei type, and it is really a question whether
\index{Delta Cephei}%
the cluster variables should be considered a separate class.
The brightness increases with great rapidity from their
minimum to a luminosity at maximum from two to six times
as great. Then they diminish in brightness much more
slowly to their minimum, at which they remain nearly
stationary for a few hours at most.

The approximately equal periods and range of variation
of the cluster variables indicate that they are very much
alike in spite of the enormous distances which separate them.
Possibly they were once much more alike and now differ to
some extent because of slightly different courses of evolution
%% -----File: 553.png---Folio 523-------
or present environment. Or, possibly, though not
probably, there is some great common cause for their changes,
a force causing pulsations in scores of stars distributed widely
throughout the clusters. Although nearly $2000$~of these
objects have already been discovered and studied, astronomers
have no idea as to the reasons for their peculiarities.

\Article{294}{Temporary Stars.}---Occasionally stars have been
\index{Stars!temporary}%
\index{Temporary stars}%
observed to blaze forth in parts of the sky (mostly in the
Milky Way) where none had previously been seen, and then
\index{Milky Way}%
to sink away into obscurity in the course of a few weeks or
months. They are characterized by a sudden rise to one
great maximum of brilliancy which, notwithstanding later
temporary increases, is never repeated. One of the most
remarkable of these stars of which there are any records
blazed out in Cassiopeia in 1572 and was for a time as bright
\index{Cassiopeia}%
as Venus. This is the star which attracted the attention of
Tycho Brahe and turned him to astronomy. The interest of
\index[xnames]{Tycho Brahe}%
Kepler also was stimulated by the discovery of a temporary
\index[xnames]{Kepler}%
star in Ophiuchus in 1604. At its maximum it was as brilliant
\index{Ophiuchus}%
as Jupiter. It must not be supposed all temporary
stars are so brilliant, for only a few rise to such splendor.

In recent times the number of temporary stars discovered
has greatly increased, both because more observers are
scanning the sky than ever before, and more especially because
they are now recorded by photography. In the last
$30$~years $19$~of these objects have been discovered, $15$~of
which were found first on the photographic record of the sky
which is being secured at the Harvard College Observatory.
\index{Harvard College Observatory}%
Only $10$~of these stars were discovered from 1572 to 1886,
when the photography of the sky was first systematically
begun at Harvard.

Temporary stars are called \textit{novæ}, or new stars. A description
of one of them will give a good idea of the characteristics
of all of them. One %[Illustration: Break, moved up]
\begin{wrapfigure}[15]{\WLoc}{3.375in}
\Input[3.375in]{554}{png}
\Caption[Light curve of Nova Persei.]{Fig}{179}
\end{wrapfigure}
of the most interesting and best
studied novæ of recent times is the one discovered by Anderson,
\index[xnames]{Anderson}%
February~22, 1901, in Perseus. On the 23d~of February
\index{Perseus}%
%% -----File: 554.png---Folio 524-------
it was brighter than Capella, while an examination of the
\index{Capella}%
photographs of the region taken by Pickering and by Stanley
\index[xnames]{Pickering, E. C.}%
Williams showed that on the~19th it was not brighter than
\index[xnames]{Williams}%
the $12$th~magnitude. In the short space of four days its
rate of radiation had increased more than $20,000$~fold.
Twenty-four hours later it lost one third of its light, and
within a year it had dwindled to the $12$th~magnitude, or near
the limits of visibility with a telescope of considerable power.
Its light curve for the first three months after its maximum
is shown in \Figref{179}.

The changes in
the spectra of the
novæ are as remarkable
as their
changes in luminosity.
Very
early in their development
they
have (at least in
case of those
stars which were observed early) dark-line spectra. Shortly
thereafter bright lines appear. In the case of Nova Aurigæ,
\index{Nova Aurigae@{Nova Aurigæ}}%
discovered in 1892, and the first temporary star whose spectrum
was examined in any detail, the dark lines and bright
lines were both visible at one time. The displacement of the
bright lines showed, on the basis of the Doppler-Fizeau
\index{Doppler-Fizeau law}%
\index[xnames]{Doppler}%
\index[xnames]{Fizeau}%
principle, a velocity away from the earth of over $200$~miles per
second, while the dark lines showed, on the same basis, an
approach toward the earth of more than $300$~miles per~second.
There are abundant grounds for doubting the correctness of
this interpretation, but no satisfactory explanation is at hand.
These phenomena are characteristic of novæ in general. As
they become fainter the dark lines vanish and the bright lines
characteristic of nebulæ appear, except that in the novæ they
are broad while they are narrow in the nebulæ.
%% -----File: 555.png---Folio 525-------

The most interesting thing observed in connection with
Nova Persei was the nebulous matter which was later found
\index{Nova Persei}%
around it. Its existence was first shown on photographs by
Wolf taken August 22~and~23, 1901. Later photographs by
\index[xnames]{Wolf, Max}%
Perrine and Ritchey showed that it was gradually becoming
\index[xnames]{Perrine}%
\index[xnames]{Ritchey}%
visible at increasing distances from the star. It looked as
though the star had ejected luminous matter, but it was
found on computation that, if this were the correct explanation,
the expelled matter must have been leaving the star
\begin{figure}[hbt]%[Illustration:]
\Input{555}{jpg}
\Caption[Nebulosity surrounding Nova Persei on Sept.~20 and Nov.~13,
1901. \textit{Photographed by Ritchey at the Yerkes Observatory.}]{Fig}{180}
\index{Yerkes Observatory}%
\end{figure}%
with about the velocity of light. This, of course, is improbable
if not impossible.

The temporary stars demand explanation. The theory
\index{Meteors}%
\index{Shooting stars}%
was suggested by Kapteyn and W.~E. Wilson, and expounded
\index[xnames]{Kapteyn}%
\index[xnames]{Wilson, W. E.}%
in detail by Seeliger, that there is invisible nebulous or
\index[xnames]{Seeliger}%
meteoric matter lying in various parts of space, particularly
in the region occupied by the Milky Way (there is confirmatory
\index{Milky Way}%
evidence of this hypothesis); that there are dark
or very faint stars (confirmed by phenomena of eclipse
variables); that the dark stars, rushing through the nebulæ,
blaze into incandescence as meteors glow when they enter
the earth's atmosphere; that the heating is only superficial
and quickly dies away, to be partially revived once or twice
by encounters of the stars with stray nebulous wisps; and
%% -----File: 556.png---Folio 526-------
that the nebulous ring observed around Nova Persei became
\index{Nova Persei}%
visible as it was illuminated by the light from the star itself.

The explanation of Kapteyn at first seems plausible, but
\index[xnames]{Kapteyn}%
there are serious objections to it. In the first place, the
photographs of Nova Persei indicate strongly that the expanding
nebulous ring surrounding it was due to something
actually moving out radially from the star. In the second
place, the density of the nebula demanded to account for
the enormous rise in luminosity is impossibly high. In the
third place, the fact that the star stays at its maximum only
a very short time implies a nebula whose thickness is incredibly
small.

Lindemann has developed the hypothesis that novæ are
\index[xnames]{Lindemann}%
produced by collisions of stars with stars. If one star should
encounter another in central collision with the great speed
at which they would move as a consequence of their initial
motion and mutual gravitation, the heat generated would
be enormous. If they were of equal mass and started from
rest, the heat developed would be five sixths of that
which would be generated, according to the principles of
Helmholtz, by the contraction of both of them from infinite
\index[xnames]{Helmholtz}%
expansion. This heat would be developed in a few hours,
or days at the most, and the temperature of the combined
mass would rise enormously. But with increase of temperature
there would be corresponding expansion, which
would result in a diminution of the temperature. If the
stars were originally gaseous, the final temperature after
expansion would be lower than that before collision because
the conditions are the opposite of those in Lane's law (\Artref{216}),
\index{Lane's law}%
\index[xnames]{Lane}%
according to which the temperature of a gaseous star
increases as it loses heat by radiation and contracts. Or,
stated directly, if heat could be applied to a gaseous star by
radiation or otherwise, it would expand and increase its
potential energy at the expense, not only of all the heat supplied,
but also partly at the expense of that which it already
possessed.
%% -----File: 557.png---Folio 527-------

While in a general way the collision theory of the origin
of novæ corresponds with the observations, it is not without
difficulties. Obviously, actual collisions of stars would be
excessively rare phenomena. Lindemann finds that in
\index[xnames]{Lindemann}%
order to account for the observed number of temporary
stars there must be about $4000$~times as many dark stars as
there are bright ones. Such a large number of obscure
masses would radically modify the dynamics of the stellar
system (\Artref{279}); and it is generally regarded as improbable
that so many of them exist.

\Article{295}{The Spectra of the Stars.}---The spectra of the stars
\index{Spectra of stars}%
\index{Stars!spectra of}%
differ as greatly as their colors. They were first classified
in 1863, by Secchi, who divided them into four groups.
\index[xnames]{Secchi}%
\begin{figure}[hbt]%[Illustration:]
\Input{557}{jpg}
\Caption[The spectrum of Sirius (Secchi's Type~I).]{Fig}{181}
\index{Sirius!spectrum of}%
\end{figure}%
While more powerful instruments have shown many new
facts and have made it necessary to add many new subclasses,
the four types described by Secchi still form a general
basis for classification. A more detailed classification, which
is now much used, was devised by E.~C. Pickering, Miss
\index[xnames]{Pickering, E. C.}%
Maury, Mrs.~Fleming, and Miss Cannon in connection with
\index[xnames]{Cannon, Miss}%
\index[xnames]{Fleming, Mrs.}%
\index[xnames]{Maury, Miss}%
the great photographic survey of stellar spectra which is
being made at the Harvard College Observatory.
\index{Harvard College Observatory}%

\textit{Type I\@.} Stars of Secchi's first type are blue or bluish
white. Examples are Sirius, Vega, and all bright stars in
the Big Dipper except the first one. Nearly half of all stars
\index{Big Dipper}%
examined are of this type. Their spectra are brightest
toward the violet end, indicating presumably that they are
at high temperatures. The spectrum of Sirius is shown in
\Figref{181}.
%% -----File: 558.png---Folio 528-------

Type~I, in Secchi's system, includes Types B~and~A of
\index[xnames]{Secchi}%
the Harvard system. Type~B is often called the Orion type
\index{Harvard College Observatory}%
because of the abundance of these stars in Orion, or the
helium type, because the absorption lines are due almost
entirely to helium, while the metallic lines which are characteristic
of the sun's spectrum are absent. The Type~A,
or Sirian stars, are characterized by strong hydrogen absorption
lines in their spectra, and almost complete absence of
metallic lines.

\textit{Type II\@.} The stars of the second type are somewhat
yellowish; they are called solar stars because their spectra are
\begin{figure}[hbt]%[Illustration:]
\Input{558}{jpg}
\Caption[Spectrum of Beta Geminorum (Harvard Class~K). \textit{Photographed
at the Yerkes Observatory.}]{Fig}{182}
\index{Beta Geminorum}%
\index{Yerkes Observatory}%
\end{figure}%
similar to that of the sun. That is, the lines of helium are
absent, the lines of hydrogen are still present, and there
are many fine metallic lines. The stars of the second type
are about as numerous as those of the first type.

Secchi's second type includes three classes of the Harvard
system. Those nearest like the Sirian stars are called Type~F,
or the calcium type. In their spectra the hydrogen lines
are still conspicuous, though somewhat reduced in density,
and two lines, known as H~and~K, due to calcium have
become conspicuous. Following the class~F is the class~G,
of which the sun is a typical member. Then come the stars
of Type~K, of which Beta Geminorum and Arcturus are examples,
\index{Arcturus}%
in which the intensity of the hydrogen lines is reduced
until they are less conspicuous than some of the
%% -----File: 559.png---Folio 529-------
metallic lines. The spectra of these stars are given in Figs.\ \Fref{182}~and~\Fref{183}.

\textit{Type III\@.} Stars of the third type are red, and the two
most conspicuous examples of them are Antares and Betelgeuze.
\index{Antares}%
\index{Betelgeuze}%
Only about $500$~of these stars are known, and many
of them are variable. Their spectra show heavy absorption
bands, due almost entirely to titanium oxide, which are
sharp on their borders toward the violet and which gradually
fade away toward the red. The fact that a compound exists
in these stars indicates that their temperatures are lower
than those of Types I~and~II\@. The same thing is indicated
by their colors in accordance with the first law of spectrum
analysis (\Artref{223}). In all known cases they have very small
proper motions, which means that they are immensely remote
\begin{figure}[hbt]%[Illustration:]
\Input{559}{jpg}
\Caption[Spectrum of Arcturus (Harvard Class~K). \textit{Photographed at the
Yerkes Observatory.}]{Fig}{183}
\index{Arcturus}%
\index{Harvard College Observatory}%
\index{Yerkes Observatory}%
\end{figure}%
from the sun. Hence such brilliant stars as Antares
and Betelgeuze, whose light is largely absorbed, must be
enormous objects. They are almost certainly many thousand
times greater in volume than our own sun.

The stars of Secchi's third type are of Type~M in the
Harvard system. They are divided into two chief subclasses,
Ma~and~Mb; a third subclass~Md includes the long-period
variable stars whose spectra show bright hydrogen
lines in addition to the bands characteristic of the whole type.

\textit{Type IV\@.} The $250$~stars of Secchi's fourth type are all
faint and of a deep red color. Their spectra have heavy
absorption bands, or flutings, sharp on the red side and indefinite
on the violet, being in this respect opposite to the
stars of the third type. The absorption bands in this case
are probably due to carbon compounds. These stars are all
%% -----File: 560.png---Folio 530-------
very remote from the sun, and nothing is known of their
absolute magnitudes, or of their masses and dimensions.

\textit{The Wolf-Rayet Stars.} There is another class of stars,
\index{Wolf-Rayet stars}%
\index[xnames]{Rayet}%
\index[xnames]{Wolf}%
discovered in 1867 by Wolf and Rayet at the Paris Observatory.
They are Type~O, having five subdivisions, in the
Harvard system. Their spectra consist of fairly continuous
\index{Harvard College Observatory}%
backgrounds on which are superimposed many dark lines
and bands, some few of which are due to helium and hydrogen,
but most of them to unknown substances. They contain
in addition many bright lines. The metallic lines of the
solar spectrum are quite unknown in these stars. Of the
more than $100$~stars of this type so far discovered, all are
situated either in the Milky Way or in the Magellanic Clouds
\index{Magellanic clouds}%
\index{Milky Way}%
in the southern heavens, which have most of the characteristics
of the Milky Way.

\Article{296}{Phenomena Associated with Spectral Types.}---A
\index{Spectra of stars}%
large number of phenomena combine to show that the classification
of stars according to their spectra is on a fundamental
basis. The order of arrangement from the simplest
to the most complex spectra is:
\begin{center}
\begin{tabular}{l*{5}{c}}
Secchi's Types: & Wolf-Rayet; & I; & II; & III; & IV. \\
Harvard Types: & O; & B, A; & F, G, K; & M; & N.
\end{tabular}
\end{center}
If the gaseous nebulæ were included, they would be put
ahead of the Wolf-Rayet stars. There is a fairly continuous
sequence of spectra from Type~O to Type~M, but there
is an abrupt break between Types M and N.

The principal phenomena which are associated with the
spectral types and which agree on the whole, in arranging
the stars in the same order, are:

(\textit{a}) The average radial velocities of the stars, determined
largely at the Lick Observatory and its southern branch,
\index{Lick Observatory}%
and discussed by Campbell, are slowest for stars of Type~B
\index[xnames]{Campbell}%
and increase to Type~M\@. The results, as given by Campbell,
with velocities expressed in miles per second, are:
\begin{center}
\begin{tabular}{l*{7}{c}}
Types: & B, & A, & F, & G, & K, & M, & Planetary Nebulæ. \\
Velocities: & $4.0$, & $6.8$, & $8.9$, & $9.3$, & $10.4$, & $10.6$, & $15.7$
\end{tabular}
\end{center}
%% -----File: 561.png---Folio 531-------

(\textit{b}) The average velocities of the stars across the line of
sight, as determined by Lewis Boss, show a similar relation
\index[xnames]{Boss, Lewis}%
to the spectral type. The results are:
\begin{center}
\begin{tabular}{l*{6}{c}}
Types: & B, & A, & F, & G, & K, & M. \\
Velocities: & $3.9$, & $6.3$, & $10.0$, & $11.5$, & $9.4$, & $10.6$.
\end{tabular}
\end{center}

These results together with those depending on the spectroscope
establish the fact that the stars of Types B~and~A
move on the average only about half as fast as those of
Types G,~K, and~M.

(\textit{c}) In Kapteyn's star-stream~I, the B~and~A stars are
\index[xnames]{Kapteyn}%
relatively numerous, the F,~G, and~K stars occur less frequently,
and the red stars are very few in number. In the
star-stream~II, the B~and~A stars are not numerous, the F,~G,
and~K stars occur in relatively great numbers, and the
M~stars are scarce.

(\textit{d}) While there are two great star-streams, there are very
many divergencies from them on the part of individual
stars. The stars of Type~B scarcely show the star-streaming
tendency, those of Type~A conform very closely to the
two streams, and succeeding types show more and more of
heterogeneity of motion.

(\textit{e}) On considering only stars brighter than magnitude~$6.5$
so as not to have the results influenced by the myriads of
remote stars, it is found that the B~stars are $10$~times as
numerous in the Milky Way as near its poles, the A~stars
\index{Milky Way}%
are less strongly condensed in the Milky Way, and finally,
after continuous gradation through the various types, the
M~stars are scattered uniformly over the sky.

(\textit{f}) For a given magnitude the stars of Type~B are more
remote than those of Type~A, which, in turn, are more remote
than those succeeding down to Type~G; then, beyond
Type~G, the distances increase to stars of Type~M, whose
distances are exceeded only by the B~stars. This means,
of course, that the B~stars are most luminous, the A~stars
less luminous, the G~stars least luminous, while the M~stars
are more luminous than any except the B~stars.
%% -----File: 562.png---Folio 532-------

(\textit{g}) The proportion of B~stars which are spectroscopic
binaries is large, the proportion is less for the A~stars and
it decreases through the list of types to~M.

(\textit{h}) Lower limits to the combined masses of spectroscopic
binaries can be determined (\Artref{285}). The average mass
of those of Type~B is about $7.5$~times the average mass of
all other types.

(\textit{i}) The average period of spectroscopic binaries of Type~B
is very short, the average is a little longer for stars of Type~A,
and increases through Types F,~G,~K, and~M.

(\textit{j}) The average eccentricity of the orbits of spectroscopic
binaries is small for stars of Type~B, is larger for stars of
Type~A, and is increasingly larger for stars of the Types F,~G,
and~K, in order.

\Article{297}{Evolution of the Stars.}---All the resources of science
\index{Evolution!of stars}%
\index{Stars!evolution of}%
have been taxed to the utmost in attempting to discover the
present constitution and properties of the sidereal system.
At the best, astronomers have barely begun to explore the
wonders of that part of infinite space which is within the
reach of modern instruments. Moreover, their observational
experience is limited to a moment of time compared with
the immense ages required for appreciable changes to take
place in the heavenly bodies. Hence it may seem presumptuous
for them to attempt to discover the mode, or modes,
of evolution of the stars. Any theories of stellar evolution
that may be developed at the present time are probably no
more than first approximations, and they may be entirely
wrong.

Astronomers almost universally hold that the stars have
contracted from the nebulæ, and most of them believe that
with increasing age they have gone, or are now going, successively
and in order through the spectral types B,~A,~F,
G,~K, and~M. The B~stars are of very high temperature
and are pouring out radiant energy at an extravagant rate.
After they cool somewhat it is supposed that they become
stars of Type~A. Their spectra are supposed to be simple
%% -----File: 563.png---Folio 533-------
because all compounds, and possibly some elements, are
broken up and dissociated at those high temperatures. With
further loss of heat they are supposed to pass successively
through the other spectral types until, at the M~stage, compounds
exist in their atmospheres. Beyond the M~stage their
light diminishes and they finally become, in the course of
time, cold and dark, and they remain in this condition until,
perhaps, they are again reduced to the nebulous state by
collision with other stars. All the forms in the chain from
nebulæ to relatively dark stars are known to exist from
observational evidence. The many other characteristics
which arrange the stars in nearly, or exactly, the same order
are regarded as strongly supporting the theory.

The theory of the evolution of the stars has strong resemblances
\index{Evolution!of stars}%
\index{Laplacian hypothesis}%
\index{Stars!evolution of}%
to the Laplacian theory of the development of the
solar system. This is only natural in view of the general
acceptance of the theory of Laplace almost up to the present
\index[xnames]{Laplace}%
time. As additional facts have been discovered they have
been placed in this scheme, often without inquiring if they
would not fit as well in some other theory.

Laplace started with an intensely heated and widely expanded
solar nebula and he supposed that it has cooled
down to its present temperature. Helmholtz supplemented
\index[xnames]{Helmholtz}%
and corrected this theory by proving that contraction would
develop an enormous amount of heat and greatly retard the
process of cooling. The conclusions of Helmholtz have been
given place in the theory of the evolution of the stars. Lane
\index{Lane's law!paradox}%
\index[xnames]{Lane}%
made a further very important supplement to the work of
Laplace when he proved that if a body in a monatomic gaseous
state contracts, heat is produced in quantities not only
sufficient to make up for that which had been radiated away,
but also sufficient actually to increase its temperature. In
spite of the fact that the results of Lane have been current
for almost fifty years, they have often been ignored in their
application to the evolution of the stars. If the stars of
any type are in a tenuous monatomic gaseous condition and
%% -----File: 564.png---Folio 534-------
contract, their temperature will inevitably rise and continue
to rise until they cease to be entirely gaseous and monatomic.

Consequently, if the stars of the types B, A, F, G, K, M
are in the order of decreasing temperature and are gaseous,
the logical conclusion on the basis of the supplements to
Laplace's theory is that the evolution proceeded in the reverse
\index[xnames]{Laplace}%
order. Of course, the stars may not all be completely
gaseous. This has given rise to the theory, proposed by
Lockyer and amplified and ably supported by Russell, that
\index[xnames]{Lockyer}%
\index[xnames]{Russell}%
the nebulæ contract into tenuous red stars of Type~M which
have low temperatures; with loss of heat they contract,
their temperatures rise, their spectra become simpler until
they reach their climax in Types A~and~B; after this they
cease to be completely gaseous, and with increasing condensation
and liquefaction, their temperatures decline and their
spectra proceed back through the types F,~G, and K~to~M.
The cogency of the arguments on which these conclusions
rest cannot be denied, and many observational data are
quite in harmony with them. But there are also some things
(for example, the high velocities of the nebulæ, \Artref{301})
which have been thought to be strongly opposed to them.
The two theories are alike in starting from nebulæ and ending
with cold and lifeless suns.

\Article{298}{The Tacit Assumptions of the Theories of Stellar
Evolution.}---In every theory there are many more or less
tacit assumptions, some of which may be of great importance.
It has been found by a large amount of experience that
errors more frequently enter through unexpressed hypotheses
than in any other way. This has been particularly true in
mathematics where it is relatively easy to determine precisely
the location of the error that has been made in any
course of reasoning. It follows that one of the best ways of
avoiding errors is to express fully all the hypotheses on
which reasoning is based. And quite aside from this, it is
useful and important to know all the bases on which conclusions
actually rest. Consequently, the tacit and imperfectly
%% -----File: 565.png---Folio 535-------
established assumptions on which the present theories
of stellar evolution are founded will be enumerated; it will
be found that at the present time most of them must remain
simply assumptions.

(\textit{a}) \textit{It is assumed that the evolution of the stars is from nebulæ
to dense bodies and not in the opposite direction.}

The best evidence in support of or against a proposition
is usually observational; when observational evidence is
lacking, we must resort to reasoning based as far as possible
on principles which have been established by experience.

There is as yet no observational evidence that nebulæ or
stars contract; observations have extended over so short a
time that it could not be expected. On the other hand, in
the case of the novæ, stars are observed to acquire the characteristics
of the Wolf-Rayet stars, which border on the
\index{Wolf-Rayet stars}%
\index[xnames]{Rayet}%
\index[xnames]{Wolf}%
planetary nebulæ. Of course, this may be quite exceptional,
but it should not be neglected. Consequently, in this
matter there is no conclusive observational evidence.

The principal known force which tends to produce condensation
is gravitation. In the case of the stars this force
is balanced by the expansive forces due to their high temperatures.
If their heat is produced only by their contraction,
as they lose heat by radiation, they certainly contract.
But the contraction theory is inadequate to explain
the heat which the sun has radiated (\Artref{219}), and it seems
very probable, if not altogether certain, that stars have
other important sources of energy. As has been suggested,
the heat of the sun is probably due in part to the disintegration
of radioactive substances. Perhaps in the extreme
conditions of pressure and temperature prevailing in the
deep interiors of stars the process of disintegration is greatly
accelerated and is going on in all elements. And probably
there are very important sources of energy not now suspected,
just as the subatomic\DPnote{** sub-atomic} energies were not suspected
a few years ago.

Now suppose the amount of energy generated in a star
%% -----File: 566.png---Folio 536-------
in all these ways is greater than that radiated. Then the
star will inevitably expand and its temperature will fall,
because with increased dimensions gravitation cannot balance
so high a temperature. If the process continues, the
star will expand to a nebula, which will necessarily have a
low temperature. In this case the direction of evolution
would be reversed. But as the star expands, the conditions
in its interior are changed, and the production of energy
might be reduced so that it would only equal that radiated.
In this case the star would reach a condition of equilibrium
which would be indefinitely maintained unless the subatomic\DPnote{** sub-atomic}
and other possible sources of energy were ultimately
exhausted, and it seems certain that they would become exhausted.
Then the star would contract if its disintegrated
products still obeyed the law of gravitation, and its evolution
would proceed in the direction assumed in current
theories, though at a greatly retarded rate.

In reaching the conclusions which have been set forth it
has been assumed that the masses of the stars are constant.
It is clear that their masses probably are increased somewhat
by the accretion of meteoric matter and individual molecules,
but, so far as may be judged from the sun, this is not
an important factor. It is quite certain that the sun is
emitting electrified particles in great numbers and with high
velocities. Probably the auroral displays in the earth's atmosphere
are produced by such particles impinging on the
molecules in the tenuous gases at great altitudes. In view
of the considerable light sometimes emitted by auroræ and
the earth's immense distance from the sun, it seems probable
that the sun loses these particles at a rate which makes
the process important. If so, the stars may possibly be disintegrating
into nebulæ. For example, the nebulosities
around the Pleiades (\Figref{184}) may have come out from these
\index{Pleiades}%
stars instead of being gradually drawn in upon them. Besides
this, comets give numerous examples of matter being
dispersed in space.
%% -----File: 567.png---Folio 537-------
\begin{figure}[hbtp]%[Illustration]
\centering\Input{567}{jpg}
\Caption[The Pleiades. These stars are surrounded by nebulous masses
of enormous volume. \textit{Photographed by Ritchey with the two-foot reflector
of the Yerkes Observatory.}]{Fig}{184}
\index{Pleiades}%
\index{Yerkes Observatory}%
\index[xnames]{Ritchey}%
\end{figure}
%% -----File: 568.png---Folio 538-------

It is obvious that we do not know with any high degree
of certainty in which direction stellar evolution is proceeding.
Sound scientific method calls for keeping both of them
in mind until a decision is reached on the basis of unequivocal
evidence. Whichever of the two conclusions may prevail,
the result will be unsatisfactory, for it will indicate a
universe evolving always in one direction, leaving the origin
unexplained. Possibly there are changes in both directions,
and it may be that stellar evolution in some way and on a
stupendous scale is approximately cyclical like most of the
changes which come entirely within the range of our experience.

(\textit{b}) \textit{It is assumed that all stars have approximately the same
chemical constitution; or, if not, that their spectra do not depend
to an important extent upon their chemical constitutions.}
One or the other of these assumptions is made tacitly when
it is supposed that all stars pass in one direction or the other
through several identical spectral types.

The spectroscope proves that the stars contain familiar
elements; it does not prove that they do not contain some
unknown elements, or that the known elements occur in all
stars in the same proportions. The great diversities on the
earth make it natural to conclude that there are important
differences in the millions of stars in the heavens. Moreover,
the different dimensions, densities, and absorption
spectra of the planets lead to the same conclusion. The
hypothesis that the stars are of approximately identical constitution
must be considered improbable until it is supported
by observational evidence.

It is too bold to assume that if the stars are differently
constituted they nevertheless have the same spectra at the
same temperatures. But the assumption actually made is
not quite so bad as it at first seems, for the stellar spectra
from B to~F, and even~G, are classified primarily on the
basis of their hydrogen emission and absorption lines.
Within these classes there is opportunity for great variety,
%% -----File: 569.png---Folio 539-------
and indeed variety is not wanting. There is nothing obviously
unsound in supposing that the character of the hydrogen
spectra of the stars depends upon their temperatures.
But the question is whether a star which has only helium
and hydrogen lines can ever show the strong metallic absorption
lines which are characteristic of stars of Types F~and~G.
Fortunately, there is now direct evidence on this point, for
there are certain variable stars which, at their maxima,
are of spectral Types B~or~A, while, at their minima, they
are of Types F~or~G. There is nothing inherently improbable
in ascribing these changes in luminosity and spectra to
\begin{figure}[hbt]%[Illustration:]
\centering\Input[4in]{569}{png}
\Caption[For a given density, the more massive the star the higher its
temperature.]{Fig}{185}
\index{Stars!temperatures of}%
\index{Temperature!of stars}%
\end{figure}%
changes in temperature, produced, perhaps, by contracting
and expanding oscillations of these stars.

\phantomsection\label{subart:298c}%
(\textit{c}) \textit{It is assumed that, aside from the rate of change, the evolution
of a star does not depend on its mass.} In considering
this point the assumption that the spectrum of a star depends
upon the temperature of its radiating surface, or radiating
layer, should constantly be borne in mind.

It should be recalled in the first place that the known
masses of the stars differ considerably (\Artref{284}), and it is
improbable that the few which are known cover anywhere
nearly the whole range. Consider two stars, $S$~and~$S'$, \Figref{185},
of the same material and equal density but one having
twice the mass of the other, and fasten attention on unit
%% -----File: 570.png---Folio 540-------
volumes at any corresponding points $P$~and~$P'$ in their interiors.
The pressure on the unit volume at~$P$ is greater
than that on the unit volume at~$P'$, both because the column~$PA$
is longer than~$P'A'$ and also because each unit mass in~$PA$
is subject to a greater attraction than that to which the
corresponding mass in~$P'A'$ is subject. To balance the
higher pressure in the larger star the gaseous mass at~$P$
must have a higher temperature than that at~$P'$. Consequently,
if two stars of the same material are of the same
density at corresponding parts and are of unequal masses,
the temperature of the larger star at all points from its center
to its surface is higher than that of the smaller star; and if
the spectrum of a star depends primarily on its temperature,
their spectra are different.

A mathematical discussion shows that if two stars are of
the same material and of equal densities at corresponding
points, their absolute temperatures are as the squares of
their radii. On combining this result with Lane's law that
\index[xnames]{Lane}%
the absolute temperature of a monatomic gaseous star is
inversely as its radius, it is found that the absolute temperatures
of stars of equal volumes and the same material are
proportional to their masses.

The results which have just been reached are very important,
even if they represent the physical facts only approximately,
and they should not be ignored in discussions
of stellar evolution. For the purposes of numerical illustration
suppose the sun is gaseous and consider a star of the
same material and density having a radius twice as great.
Its mass is eight times that of the sun. By the first law, its
temperature is four times that of the sun. Since the rate
of radiation is proportional to the fourth power of the absolute
temperature, its radiation per unit area is $256$~times
that of the sun. Since its radius is twice that of the sun,
its surface is $4$~times greater, and its whole radiation, or
\textit{luminosity}, is $4 × 256 = 1024$~times that of the sun. That
is, two stars of the same material and density, whose masses
%% -----File: 571.png---Folio 541-------
are in the ratio of only $8$~to~$1$, differ in luminosity in the ratio
of $1024$~to~$1$. If a star were eight times more massive than
the sun, it would have a spectrum of Type B~or~A, if these
spectra indicate high temperatures, and it would be a star
comparable to the most brilliant ones found in the heavens.
On the other hand, if it were one eighth as massive as the
sun, it would have a spectrum characteristic of low temperatures
(Type~M?), and would be a feebly luminous body.

Of course, it is not necessary that other stars should have
\index{Density!of stars}%
\index{Stars!density of}%
the same density as the sun. It is known from eclipsing
variables that comparatively few are as dense as the sun,
and that the densities may be as small as one hundredth or
even one thousandth of that of the sun. It can be shown
that the temperature of a gaseous star is proportional to the
cube root of the product of the square of the mass and the
density. Hence, in order that a star having a density one
hundredth that of the sun should be as hot as the sun, its
mass must be about $10$~times greater. But under these
conditions its surface and luminosity would both be about
$100$~times as great as those of the sun. That is, a star nearly
as brilliant as one of the Pleiades might be only one hundredth
\index{Pleiades}%
as dense as the sun if its mass were only $10$~times greater.
A star $10$~times as great in mass and one tenth as dense as
the sun would be $460$~times as luminous.

It can be seen from this incomplete discussion that in
order that a star shall have high temperature and great
luminosity it must have a mass at least as great as that of
the sun; for it is not probable that a much denser body
would be in a gaseous condition. But the luminosity of a
gaseous star is so sensitive a function of its mass that one
10 times more massive than the sun would be a brilliant
object unless its density were exceedingly low; and one only
one tenth as massive as the sun would be relatively faint,
even if it were as dense as the sun. Therefore, it is not
strange that no stars with very small masses have been
found; one as small as one of the planets could not be self-luminous
%% -----File: 572.png---Folio 542-------
while in a gaseous state. On the other hand, no
star many times more massive than the sun has been found.
Perhaps the reason is that the data respecting masses is yet
so meager; perhaps the temperatures in massive stars become
so great that their atoms disintegrate and the remains
fly away into space.

(\textit{d}) \textit{It is assumed that the contraction of nebulæ into stars
began at such a time, or at such times, and that the individual
nebulæ had such masses that there has resulted the present
sidereal system of nebulæ and stars in all stages from hottest to
coldest.} The implications of this assumption are not at once
fully evident; they can be brought out only by a mathematical
discussion whose results alone can be given here.

On the basis of Stefan's law of radiation and the assumption
\index[xnames]{Stefan}%
that the heat of a star is developed entirely by contraction,
it is found that the change of radius is directly proportional
to the product of the time and the square of the mass.
If there are other important sources of heat, and if the
radiation is from a layer of varying depth instead of from
the surface, the law may be much in error. But on the
assumption that this result applies to the sun, it is possible
to compute the time required for it to have contracted from
any given dimensions. According to the contraction theory
its radius is now diminishing at the rate of a mile in $44$~years.
Consequently, on this basis it has contracted from the orbit
of Mercury in $1,500,000,000$ years. At first thought this
would seem to give a long supply of heat to the earth to
meet geological needs; but if the sun ever filled a sphere
as large as the orbit of Mercury and radiated according to
Stefan's law, whatever the source of heat may have been,
its temperature must have been so low that its rate of radiation
could have been only a little more than one seven-thousandth
that at present, a quantity altogether inadequate
to support life on the earth. According to this contraction
theory, $4,400,000$ years ago the radius of the sun
was $100,000$ miles greater than at present, and its rate of
%% -----File: 573.png---Folio 543-------
radiation was only two thirds that which is now observed.
With this rate of radiation the theoretical mean temperature
of the earth, determined by the method used for Mars in
\Artref{172}, comes out $51°$~lower than at present ($60°$~F.), or
$23°$~below freezing.

The second part of the law gives the interesting and unforeseen
result that the more massive a star, the more rapidly
it contracts. Or, if the results are translated over into a
relation between density and time, it is found that if a star
of large mass and one of smaller mass start with the same
density, the density of the large star will increase faster
than that of the smaller one. The rate of change of density
is proportional to the cube root of the fifth power of the mass.
Therefore, if one star has $8$~times the mass of another and
they start contracting from the same density, it will arrive
at some greater density in $\frac{1}{32}$~of the time required by the
smaller star to reach the same density. As applied to the
stellar system, this means that if the stars all started condensing
from nebulæ at the same time, those which have
the largest masses are at present by far the densest and
hottest. The large stars are probably much hotter on the
average than the small ones, but it is doubtful if they are
denser. It must be remembered that these results depend
upon the very questionable assumption that the heat of
stars is due entirely to their contraction.

\Article{299}{The Origin and Evolution of Binary Stars.}---The
\index{Binary stars!evolution of}%
\index{Binary stars!origin of}%
\index{Origin!of binary stars}%
great number of binary stars calls for a consideration of
their origin and evolution. If the stars have condensed
from nebulæ, it is natural to suppose that binary stars have
developed from nebulæ which divided into two parts, or
that the divisions have taken place after the condensing
masses have reached the star stage. It is also conceivable
that stars which originated separately have later united to
form physical systems. Both of these theories will be considered.

Consider first the theory that the binary stars have originated
%% -----File: 574.png---Folio 544-------
by the fission of nebulæ or larger stars. The basis
for the theory is the very reasonable assumption that the
original nebulæ had more or less rotation, possibly quite
irregular in character. In those cases where the amount of
rotation, that is, the moment of momentum, was small, it
is believed that single stars rotating slowly have resulted.
In those cases where the moment of momentum was large,
it is supposed that there has been separation into two parts.

There is some theoretical basis for this conclusion, though
from a practical point of view it has generally been greatly
overestimated. In a brilliant piece of work on figures of
equilibrium of homogeneous fluids rotating as solids, Poincaré,
\index[xnames]{Poincare@{Poincaré}}%
following Maclaurin and Jacobi, showed that for slow
\index[xnames]{Jacobi}%
\index[xnames]{Maclaurin}%
rotation an oblate spheroid is a figure of equilibrium, for
faster rotation an elongated ellipsoid is the corresponding figure,
and for still faster rotations the ellipsoid has a constriction,
suggesting that for still faster rotations the figure would
be two very unequal masses. Now, when a nebula or a star
contracts it rotates more rapidly because the moment of
momentum is constant. Hence it seems reasonable to suppose
that nebulæ and stars follow at least roughly the figures
found by Poincaré for the homogeneous case.

There is one very important point of difference in the problem
treated by Poincaré and that presented by contracting
bodies. Poincaré considered masses all of the \emph{same density},
but having different rates of rotation. In a contracting
nebula or star both the density and the rate of rotation
change. The increase in density tends to sphericity; the
increase in rate of rotation tends to oblateness. The two
effects almost balance each other, but the effect of increasing
rotation prevails by a narrow margin. For example,
if the sun contracts with loss of heat, it will not become so
oblate as Saturn is now until its density is hundreds of times
greater than that of platinum. This does not mean that a
body contracting from a nebula may not divide into two
parts at any stage of its development, but it shows that the
%% -----File: 575.png---Folio 545-------
tendency for fission is very much smaller than has been
supposed.

Suppose a star divides into two parts. Originally the
two components will be rotating so as to keep their same
faces toward each other. But with further contraction they
will rotate more rapidly while their period of revolution remains
unchanged. Then tidal evolution begins, and under
these conditions Darwin has shown that the tides will increase
\index[xnames]{Darwin, George H.}%
the periods of rotation rapidly and the period of revolution
more slowly. Moreover, if the original orbit had any
eccentricity it will be increased. Consequently, as the age
of a binary star having originated by fission increases, its
period of revolution increases and the eccentricity of its
orbit increases.

From an extensive study of the orbits of spectroscopic
and visual binaries, Campbell has found that stars of Types
\index[xnames]{Campbell}%
B and A have short periods and nearly circular orbits, and
that both the periods and the eccentricities increase, on the
average, through the spectral types F, G, K, and~M. One
would be tempted to infer, in accordance with the theory
of the evolution of stars through the spectral types from B
to~M, that binaries of Type~B had recently originated by
fission and that with increasing age they would go through
the various spectral types with periods increasing correspondingly
from a few hours to an average of more than a
century, and the eccentricity from near zero to an average
of about~$0.5$.

But such an inference would be entirely unwarranted and
erroneous, for an ample consideration of the dynamics involved
shows that when a nebula or star divides into two
equal masses, tidal friction in any time however long is not
competent to make the period more than about twice its
original value; if the masses are unequal but comparable,
as in the case of all known binaries, the period may be
lengthened several fold. But it is altogether impossible for
tidal friction to increase the period of a binary star whose
%% -----File: 576.png---Folio 546-------
components have comparable masses from a few hours or
days to the many years found in the case of most visual
binaries.

There is a similar difficulty in the eccentricities of the
orbits of binary stars. Consequently the important facts
brought out in Campbell's discussion do not confirm the
\index[xnames]{Campbell}%
current theory of the evolution of the stars. So far as the
periods are concerned they are in harmony with the hypothesis
that the B~and~A stars are massive, for the greater
the mass, the shorter the period for a given distance between
the stars, but it is highly improbable that the great range of
periods depends upon the masses alone. The dynamical
conditions imply that if visual binaries originated by fission,
the division took place while they were yet in the nebular
stage.

The hypothesis that two independent stars can unite to
form a binary remains to be considered. If two stars are
drawn toward each other by their mutual gravitation, they
may pass near and around each other without any contact,
as a comet passes around the sun; each may collide with
the outlying parts of the other; they may undergo a grazing,
or partial, collision; and, in the extreme case, they may
have a central collision. If they do not collide at all, they
will recede to the distance from which they were drawn
together, and a binary star cannot result. If they suffer a
collision with outlying parts, their velocities will be reduced
and they may not recede to a very great distance from each
other. The character of their orbits after collision will depend
upon the amount of kinetic energy which is transformed
at the time of collision. This energy goes into heat,
and the question arises whether, if sufficient motion is destroyed
to produce a binary, the heat evolved may not reduce
both stars to the nebulous state.

Consider a special example of two stars each in mass
equal to the sun. At a great distance from each other their
relative velocities might %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{3in}
\Input[3in]{577}{png}
\Caption[Reduction of parabolic orbit to
an ellipse by collision of a sun with a planet
of another sun.]{Fig}{186}
\end{wrapfigure}
be anything from zero to several
%% -----File: 577.png---Folio 547-------
hundred miles per second; take the most favorable case where
it is zero. Suppose that at their nearest approach their
distance from each other is as great as that from the earth
to the sun. Under the hypotheses adopted they will have
a relative velocity of about $37$~miles per second. Suppose
they encounter enough resistance from outlying nebulous
or planetesimal matter, or from collision with a planet, to
reduce their most remote
point of recession
after collision to $100$~astronomical
units.
It can be shown that
their velocity must
have been reduced by
$\frac{1}{200}$~of its amount, or
by $0.185$~mile per
second. This would
generate as much heat
as the sun radiates in
about $8$~years. Consequently
the expansive
effect of the heat
generated by the collision
will not be important,
and after the
encounter the stars
will be moving in an orbit whose eccentricity is~$0.98$ and
whose period is about $250$~years. The resistance could have
been produced by collision with a planet whose mass was $\frac{1}{200}$~that
of one of the suns. It follows that if a star passing the
sun should meet Jupiter, something comparable to what has
been given in the example would result. \Figureref{186} shows
the original parabola, the point of collision~$P$, and the
elliptical orbit after collision.

Now let us follow out the history of the star after such a
collision as has been described. If there are no subsequent
%% -----File: 578.png---Folio 548-------
collisions, the stars will continue to describe very elongated
elliptical orbits about their center of gravity. If there are
subsequent collisions with other planets or with any other
material in the vicinity of the stars, their points of nearest
approach will not be appreciably changed unless the collisions
are far from the perihelion point, their points of most
remote recession will be diminished by each collision, and
the result is that both the period and the eccentricity of the
orbit will be decreased as long as the process continues. If
this is the correct theory of the origin of binary stars, those
whose periods and eccentricities are small, are older on the
average, at least as binaries, than those whose periods and
eccentricities are large, and this would suggest that the B~and~A
stars are older than the K~and~M stars. The only
obvious difficulty with the basis of this theory of the origin
of binary stars is that these near approaches and partial
collisions are necessarily extremely infrequent, while binary
stars are very numerous. The seriousness of this difficulty
depends upon the length of time the stars endure, about
which nothing certain is known.

As has been stated in \Artref{294}, a central collision would
produce a temporary star, which would later change into a
nebula.

\Article{300}{The Question of the Infinity of the Physical Universe
in Space and in Time\DPtypo{}{.}}---There are transcendental
\index{Infinity of physical universe}%
questions which, from their nature, can never be answered
with certainty, but which the human mind ever persists in
attacking. Among such questions is that of the infinity of
the physical universe in space and in time.

It has been seen in \Artref{270} that the apparent distribution
of the stars proves that they cannot be scattered uniformly
throughout infinite space. It has also been seen
that there is no observational evidence that galaxies, separated
by distances of a higher order than those between the
stars, may not be units in larger aggregations and so on to
super-galaxies without limit. This may be adopted as a
%% -----File: 579.png---Folio 549-------
working hypothesis. We may then inquire whether there
will be luminous stars through infinite time, or whether they
all will ultimately become extinct.

According to physical laws as they are known at present,
the stars are pouring radiant energy out into the ether at
an extravagant rate and it is not being returned to them in
relatively appreciable amounts. For example, the sun loses
more light and heat by radiation in a second than it will
receive from all the stars in the sky in a million years. It is
inconceivable that a star has an unlimited store of internal
energy. Therefore its energy will ultimately become exhausted
unless a new supply is furnished in some way. One
method by which the internal energy of a star may be increased
is by collision with another star. But after collision
the combined mass would lose its energy similarly until
another restoration by another collision. But by this process
the matter of the universe becomes aggregated in
larger and larger masses, and if it is finite in amount, a
stage will be reached when no more collisions will take place.
Then these final stars will in the course of time radiate away
all their internal energy and remain throughout eternity
dark, cold, and lifeless. At least, such is the teaching of
present-day science if the physical universe is finite, as has
usually been assumed.

But now suppose that there are myriads of galaxies composing
larger and still larger cosmic units, and remember that
there are no observational facts whatever which contradict
this hypothesis. Under this assumption the energy in the
universe is also infinite. It does not follow from this,
however, that it will last an infinite time, for there are, by
hypothesis, infinitely many bodies which are subject to
collisions and which are radiating energy into the ether.
But, on the other hand, if the relative speed of the larger
cosmic units is great enough, there will be enough energy to
last the infinite universe an infinite time. This follows from
the fact that infinities may be of different orders, as the
%% -----File: 580.png---Folio 550-------
mathematicians say. The actual demands in the present
case are not severe. In order that the energy should last
an infinite time it is sufficient that the relative speeds of the
larger cosmic units of all order\DPnote{** [sic]} shall exceed some finite value.

The energy in any particular galaxy might run down, as
in the finite case considered above; but, according to the
present hypothesis, at immense intervals this galaxy would
collide with some other one with speed sufficient to restore
its internal energies if the energy of their relative motions
were thus transformed. It might require only a very small
fraction of the energy of the relative motions. The process
would terminate, however, if there were only a finite number
of galaxies, but by hypothesis the super-galaxies are units
in still larger aggregations. There might be a restoration
of heat energy by interactions of these larger units, and so
on without limit. It is not profitable to pursue the inquiry
further here, but it is not without interest to know that
according to our present understanding of the laws of nature
it is not necessary to conclude that the physical universe
will in a finite time reach the condition of eternal night and
death. This discussion also gives an answer, though perhaps
not the correct one, to the question why the universe has not
already attained a condition of stagnation and death. In
short, it gives a picture of a universe whose life and activity
are without beginning and without end.


\Section{IV}{The Nebulæ}

\Article{301}{Irregular Nebulæ.}---There are many nebulæ in
\index{Irregular nebulae@{Irregular nebulæ}}%
\index{Nebulae@{Nebulæ}!irregular}%
the sky of enormous extent and irregular form. Among the
finest examples of these objects, though by no means the
most extensive, are the veil-like structures which are seen in
the constellation Cygnus, one of which is shown in \Figref{187}.
\index{Cygnus}%
It is altogether probable that they are at least as remote as
the nearer stars. Since they extend across regions occupied
by hundreds of stars, they are of inconceivable magnitude;
%% -----File: 581.png---Folio 551-------
\begin{figure}[hbtp]%[Illustration]
\centering\Input{581}{jpg}
\Caption[Irregular nebula in Cygnus (N.~G.~C.~6960). \textit{Photographed by Ritchey with the two-foot reflector of the Yerkes Observatory.}]{Fig}{187}
\index{Cygnus}%
\index{Yerkes Observatory}%
\index[xnames]{Ritchey}%
\end{figure}%
%% -----File: 582.png---Folio 552-------
certainly a hundred years are required for light to cross them.
They are extremely faint (the long-exposure photographs
being quite misleading) and they are probably very tenuous,
though nothing is actually known regarding their density.
If they are condensing under gravitation, the process must
be going on extremely slowly.

An example of a less widely extended and apparently
much denser nebula is the great nebula in Orion (\Figref{61}),
which is, perhaps, the most wonderful and beautiful object
in the heavens. It fills a space whose apparent diameter
is more than half a degree. This means it is of enormous
volume, for it is as remote as certain stars which are associated
with its denser parts. Its parallax can scarcely be
over~$0''.01$ and it probably is much smaller; if the larger
value is correct, its diameter is $20,000,000$ times that of the
sun and several years would be required for light to travel
from one side of it to the other. The density of the Orion
nebula is altogether unknown, but it is generally regarded
as being very low. If it averages even $\frac{1}{100,000}$ that of the
atmosphere and if it is spherical (?), its total mass is
$100,000,000,000,000$ times that of the sun, and in spite of its
enormous distance, its attraction for the earth is one fourth
that of the sun. If the nebula is rare, it is difficult to account
for its radiation, because it could not have a high temperature
except possibly in its deep interior where pressure of the outlying
parts would prevent expansion. The luminosity of the
nebulæ, like that of the comets, has long been an unexplained
phenomenon.

The form of the Orion nebula suggests whirling motions
\index{Orion nebula}%
of its parts. Relative internal motions were found first
by Bourget, Fabry, and Buisson; Frost and Maney have
\index[xnames]{Bourget}%
\index[xnames]{Buisson}%
\index[xnames]{Fabry}%
\index[xnames]{Frost}%
\index[xnames]{Maney}%
shown by the spectroscope that its northeastern part is
receding from the solar system, while the southwestern part
is approaching at the relative rate of about $6$~miles per second.
It is clear that unless the density is sufficiently great these
motions will cause the nebula to dissipate in space. On the
%% -----File: 583.png---Folio 553-------
assumption that this is simply a motion of rotation, and
neglecting gaseous expansion, it is found that the nebula is
in no danger of disrupting if its average density is greater
than $10^{-22}$~times that of water. At this limiting density its
total mass would about equal that of the sun.

It was supposed in the days of Sir William Herschel that
\index[xnames]{Herschel, William}%
the nebulæ may be galaxies which are so remote that their
individual stars are not distinguishable, even with the
most powerful telescopes. This is certainly not the true
explanation of the irregular nebulæ. In the first place, the
spectra of the brighter ones for which the data are at hand
consist of bright lines, proving on the basis of the first law
of spectrum analysis that they are incandescent gases under
low pressure. The bright lines belong to a hypothetical element
nebulium, found only in nebulæ, and to hydrogen. In
the second place, they are condensed in the zone of the Milky
Way, which indicates they are in some way connected with
it. Campbell and Moore have found that they show the
\index[xnames]{Campbell}%
\index[xnames]{Moore}%
streaming tendencies which are characteristic of the stars.
For these reasons the conclusion is held that they are tenuous
gaseous members of our own Galaxy.
\index{Galaxy}%

A very interesting fact has recently been discovered in
connection with the Magellanic Clouds, two masses of
\index{Magellanic clouds}%
stars in the far southern heavens, having the appearance of
two smaller galaxies which are quite independent of the
Milky Way. R.~E. Wilson, at the South American branch
\index[xnames]{Wilson, R. E.}%
of the Lick Observatory, has found that the radial velocities
\index{Lick Observatory}%
of the nebulæ in the Magellanic clouds which are bright
enough for measurement show rapid recession of all of these
objects, the average speed being over $150$~miles per second.
This suggests that these aggregations of stars have velocities
with respect to our own Galaxy of a higher order than the
average internal velocities, in harmony with the suggestion
in \Artref{300}.

Barnard has recently brought forward strong evidence
\index[xnames]{Barnard}%
for the conclusion that there are relatively dark and opaque
%% -----File: 584.png---Folio 554-------
masses, perhaps nebulous in character, in certain parts of the
Milky Way. He has found regions in which the stars seem
\index{Milky Way}%
to be blotted out by obscure material, as is shown in \Figref{188}.
Probably the apparent breaks in some of the nebulæ,
\index{Nebulae@{Nebulæ}!spiral}%
as, for example, the celebrated Trifid Nebula in Sagittarius
\index{Sagittarius}%
\index{Trifid Nebula}%
(\Figref{189}), are due to obscuring material which cuts off the
light from certain regions. At any rate, it is difficult to see
\begin{figure}[hbt]%[Illustration:]
\Input{584}{jpg}
\Caption[On the left a bright nebula (in Cygnus) and on the right a
dark patch which is probably due to a dark nebula. \textit{Photographed by
Barnard at the Yerkes Observatory.}]{Fig}{188}
\index{Cygnus}%
\index{Yerkes Observatory}%
\index[xnames]{Barnard}%
\end{figure}%
how matter could be in equilibrium in any such forms as the
luminous matter assumes.

\Article{302}{Spiral Nebulæ.}---Spiral nebulæ are more numerous
\index{Spiral nebulae@{Spiral nebulæ}}%
than all other kinds together. According to Keeler's
\index[xnames]{Keeler}%
original estimate there are at least $120,000$ within the reach
of the telescope which he used; there may be five or ten
times the number within reach of the great reflectors of the
Solar Observatory of the Carnegie Institution. They are
\index{Mount Wilson Solar Observatory}%
\index{Solar!Observatory}%
characterized by their great extent (\Figref{190}) and by irregular
arms, generally two in number when they are distinctly defined,
which wind out from centers. They almost invariably
have well-defined centers, apparently of considerable density,
and their arms usually contain a number of conspicuous
local condensations, or nuclei.
%% -----File: 585.png---Folio 555-------


The spiral nebulæ are further characterized by being white,
whereas the large irregular nebulæ have a greenish tinge due
to the green light from nebulium. Most of them are too
faint for detailed spectroscopic study, but some of the
brighter of them have been found to have spectra similar
to the sun's spectrum. This leads to the inference that they
are perhaps partly solid or liquid. On the other hand,
Seares has photographed
\index[xnames]{Seares}%
some of
them through a
screen which cuts
off the blue end
\begin{wrapfigure}{\WLoc}{3in}%[Illustration:]
\Input[3in]{585}{jpg}
\Caption[The Trifid Nebula. The dark lanes
by which it is crossed are probably due to intervening
dark material. \textit{Photographed with the
Crossley reflector of the Lick Observatory.}]{Fig}{189}
\index{Lick Observatory}%
\index{Trifid Nebula}%
\end{wrapfigure}
of the spectrum.
The brightness of
the arms was
much more reduced
than that
of the central
nuclei, indicating
that a considerable
part of their
light is similar to
that from gases.
Moreover, their
transparency implies
that they are
tenuous. Hence,
they seem to be vast swarms of incandescent solid or liquid
particles, perhaps with many larger masses, surrounded by
gaseous materials. There is difficulty in explaining their
luminosity, though Lockyer attempted to account for the
\index[xnames]{Lockyer}%
light of all nebulæ by ascribing it to heat generated by the
collisions of meteorites of which he supposed they are largely
composed. The obscure material in and around nebulæ
may be very abundant. This supposition is confirmed in the
case of spiral nebulæ, for when one is seen edgewise the dark
%% -----File: 586.png---Folio 556-------
material at its periphery eclipses the center and causes an
apparently dark rift through it (\Figref{191}). Another distinguishing
feature of spiral nebulæ is that they are very
\index{Nebulae@{Nebulæ}!spiral}%
\index{Spiral nebulae@{Spiral nebulæ}}%
infrequent in or near the Milky Way.

\begin{figure}[hbt]%[Illustration:]
\Input{586}{jpg}
\Caption[Spiral nebula in Ursa Major (M.~101). \textit{Photographed by Ritchey
at the Yerkes Observatory.}]{Fig}{190}
\index{Ursa Major}%
\index{Yerkes Observatory}%
\index[xnames]{Ritchey}%
\end{figure}

The spiral nebulæ range in magnitude all the way from the
Great Nebula in Andromeda (\Figref{192}), which is about $1°.5$~long
\index{Andromeda!Nebula}%
and $30'$~wide, to minute, faint objects which are barely
discoverable after long exposures with powerful photographic
telescopes. There is no reason to believe there are not others
still smaller. Since the Andromeda nebula is certainly as
%% -----File: 587.png---Folio 557-------
distant as the nearest stars, its volume is enormous; the
smallest ones may be as small as the solar system, though they
would wind up and lose their spiral characteristics in a short
time.

The suggestion has been made (\Artref{249}) that a spiral
nebula may develop when a star is visited closely by another
star, or when a group of stars passes near another group of
stars. There is no apparent difficulty in explaining small
spirals in this way, but the
large ones present a more
serious problem, especially
if we limit ourselves to the
close approach of two single
stars. It is not at all necessary
to do this, for in a
general way the dynamical
principles involved apply to
aggregates of all dimensions
up to galaxies, and even
beyond if there are larger
units in the universe. There
is possibly some evidence
that the Milky Way has a
\index{Milky Way}%
spiral structure.

Although the larger spirals
are enormous in extent, they
may have only moderate masses. However improbable
this may be on the basis %[Illustration: Break, moved down]
\begin{wrapfigure}{\WLoc}{2.5in}
\Input[2.5in]{587}{jpg}
\Caption[Spiral nebula in Andromeda
(H.~V.~19) presenting edge
toward the earth. Central line
eclipsed by obscure material. \textit{Photographed
with the Crossley reflector
of the Lick Observatory.}]{Fig}{191}
\index{Lick Observatory}%
\index{Nebulae@{Nebulæ}!spiral}%
\index{Spiral nebulae@{Spiral nebulæ}}%
\end{wrapfigure}
of their appearance, it must be remembered
that there is no direct evidence whatever at
present regarding their masses, and the source of their luminosity
is quite unknown. It is natural to suppose that
though a spiral of dimensions comparable to the solar system
might be produced by the disruptive forces of a near approach
of two stars, it would not be possible for one a thousand
times larger to be formed in the same way. An examination
of the equations involved shows that, if a certain
%% -----File: 588.png---Folio 558-------
velocity of ejection would cause matter to recede (neglecting
the attraction of the passing sun) to the distance of
Neptune, a velocity one twenty-four-thousandth greater
would cause it to recede $1000$~times farther (\Tableref{XIII}).
Hence the argument against very large spirals being formed
by the near approach of two great suns is not so conclusive
as it might at first seem. They may have been formed,
however, by the passage near one another of two great
groups of stars such as the globular clusters; or they may
have been formed in some other way not yet considered.

The spectra of spiral nebulæ are in harmony with the
suggested mode of their origin. Their distribution demands
consideration. Their apparent distribution may mean that
they are out on the borders of the Galaxy and that they
\index{Galaxy}%
are not seen in the Milky Way because of their great distances
\index{Milky Way}%
in these directions. It would be expected that close approaches
would occur most frequently in the interior of the
Galaxy where the stars move the fastest if they are making
excursions to and fro through it. On the other hand, out on
the borders they would move more slowly and their mutual
attractions would be more efficient in bringing them together.

\begin{figure}[hbtp]%[Illustration: Moved up]
\centering\Input{589}{jpg}
\Caption[Great Nebula in Andromeda. \textit{Photographed by Ritchey with the two-foot reflector of the Yerkes Observatory.}]{Fig}{192}
\index{Andromeda!Nebula}%
\index{Yerkes Observatory}%
\index[xnames]{Ritchey}%
\end{figure}

There is one fact which is opposed to the suggested explanation
of spiral nebulæ, and that is, as Slipher first found,
\index[xnames]{Slipher, V. M.}%
their radial velocities average very great. For example, the
Great Andromeda Nebula is approaching the solar system at
\index{Andromeda!Nebula}%
the rate of $200$~miles per second. Moreover, Slipher found
spectroscopic evidence that it is rotating. Even if the result
is in doubt for this nebula, it is altogether certain in the case
of another spiral which is edgewise to the earth, and which
Slipher investigated in 1913. Among the stars high velocities
are on the whole associated with small masses. If this
is a universal principle, which seems dynamically sound,
the spirals must have smaller masses than any known
class of stars. Or, perhaps, spirals have been formed on the
whole only from stars which passed one another at great
%% -----File: 589.png---Folio 559-------
%% -----File: 590.png---Folio 560-------
speed, and they of course still possess most of their kinetic
energy.

It has been more than once suggested that the spiral nebulæ
are not in reality nebulæ at all, but distant galaxies.
If this is true, it is difficult to explain their distribution with
respect to the Milky Way, or their strong central condensations,
\index{Milky Way}%
or the fact that they are crossed %[Illustration: Break, moved up]
\begin{wrapfigure}{\WLoc}{2in}
\Input[2in]{590}{jpg}
\Caption[The ring nebula
in Lyra. \textit{Photographed by
Sullivan at the Yerkes Observatory
with the 40-inch
telescope.}]{Fig}{193}
\index[xnames]{Sullivan}%
\end{wrapfigure}
by dark streaks when
they are presented edgewise to us. Besides, the results of
Seares' photographs are opposed to this hypothesis.

\Article{303}{Ring Nebulæ.}---A few nebulæ have the form of
\index{Nebulae@{Nebulæ}!ring}%
almost perfect rings, the best example of which is the one
between Beta Lyræ and Gamma
Lyræ (\Figref{193}). This nebula has
a fifteenth-magnitude star near its
center which has been suspected
of being variable. It is probably
associated with the nebula, though
this is not certain. The spectrum
of the ring nebula in Lyra has
\index{Ring nebula in Lyra}%
been examined and it has been
found that hydrogen extends out
considerably beyond the helium.
The origin and development of
these remarkable objects are quite
beyond conjecture at present.

\Article{304}{Planetary Nebulæ.}---The
\index{Nebulae@{Nebulæ}!planetary}%
planetary nebulæ are supposed to
be next to the O-type stars in evolution, and the O-type stars
are supposed to precede the B-type stars. They are in all
cases apparently small in size, usually rather dense, particularly
near their centers, and they have rather well-defined
outlines. They were named by Herschel from their resemblance
\index[xnames]{Herschel, William}%
to faint planetary disks.

The spectra of about $75$~planetary nebulæ have been examined.
Perhaps the most important result of this examination
is that their radial velocities ($24$~miles per second) are
%% -----File: 591.png---Folio 561-------
at least three times those of the stars of Type~B. This is
squarely opposed to the theory that they condense into stars
of Types O~and~B. If this theory is maintained, an explanation
of the greatly decreased velocities is demanded, and
none is at hand. On the other hand, the novæ go first into
planetary nebulæ and then into Wolf-Rayet stars.
\index{Wolf-Rayet stars}%
\index[xnames]{Rayet}%
\index[xnames]{Wolf}%

The central parts of planetary nebulæ give the lines of
nebulium and hydrogen; the outermost parts give the
hydrogen lines alone. That %[Illustration: Break]
\begin{wrapfigure}{\WLoc}{2.25in}
\Input[2.25in]{591}{jpg}
\Caption[A planetary nebula.
\textit{Photographed with the
Crossley reflector at the Lick
Observatory.}]{Fig}{194}
\index{Lick Observatory}%
\end{wrapfigure}
is, hydrogen forms an atmosphere
around the denser nebulium
and hydrogen cores.

The problem of the rotation of
planetary nebulæ is now being
taken up at a number of observatories.
By an adaptation of the
spectroscope first employed by
Keeler on the rings of Saturn, and
\index[xnames]{Keeler}%
used more recently by Slipher at
\index[xnames]{Slipher, V. M.}%
the Lowell Observatory on planets
and spiral nebulæ, Campbell and
\index[xnames]{Campbell}%
Moore have found that two of
\index[xnames]{Moore}%
these remarkable objects are rotating
around axes approximately at
right angles to a plane passing
through the earth and the longer axes of the nebulæ. On
the basis of the observed relative velocities of $3.1$~to $3.7$~miles
per second, and plausible assumptions regarding the distance
of the nebulæ, they found that their masses are between 3 and
$100$~times that of the sun, with periods of rotation between
$600$ and $14,000$ years. With such slow rates of rotation there
is no possibility of these objects ever dividing into two parts
and forming a binary star, in spite of the fact that their
density probably does not exceed one millionth that of our
atmosphere at sea level.
%% -----File: 592.png---Folio 562-------


\Section{XXIV}{QUESTIONS}

1. If $500,000,000$ stars were scattered uniformly over the celestial
sphere, what would be the apparent angular distance between
adjacent stars? If another star were placed at random on the
sky, what would be the probability that it would be within $1''$
of one of these stars?

2. In the part of the sky covered by Aitken's survey of double
stars (north of declination~$-14°$) there are about $200,000$ stars
brighter than the tenth magnitude; what is the average distance
between adjacent members of this list of stars? Aitken found
$5400$~pairs separated by less than~$5''$; what is the probability
that a particular one of these cases is accidental? What is the
probability that they are all accidental? According to the laws
of probability, how many of the $5400$~stars, in a random arrangement,
should be separated less than~$5''$?

3. Suppose the apparent distance between two stars must be
at least~$0''.2$ in order that they may be seen as two distinct stars
with the largest telescopes; suppose the distance of a double star
is $500$~parsecs; what must be the distance, in astronomical units,
between the components in order that they may be seen as separate
stars? If the mass of each star is equal to that of the sun,
what will be their period of revolution (\Artref{154})? If their dimensions
and surface brilliancy are the same as those of the sun, what
will be their magnitude taken together?

4. Suppose the relative velocity of the two components of a
double star must be $5$~miles per second in order that it may be
possible to determine by the spectroscope that the star is a binary;
how near must the components be to each other in order that it
may be possible to find that the star is a binary if their combined
mass is one tenth that of the sun? Equal to that of the sun?
Ten times that of the sun?

5. Suppose the density of the components of a binary star is
equal to that of the sun and that the two components (assumed
spherical) are in contact; what is their period of revolution if
their combined mass is one tenth that of the sun? Equal to that
of the sun? Ten times that of the sun? What are their relative
velocities in the respective cases? What are their temperatures
in the respective cases [\hyperref[subart:298c]{Art.~298~(\textit{c})}]? What are their luminosities
in the respective cases?

6. Suppose the two components of an eclipsing variable are
equal in mass and that their density is that of the sun; what is
the ratio of the time of eclipse to the period of revolution if their
%% -----File: 593.png---Folio 563-------
combined mass is one tenth that of the sun? Equal to that of
the sun? Ten times that of the sun? Solve the problem if their
density is one tenth that of the sun, and also if it is ten times that
of the sun.

7. Which of the ten phenomena of \Artref{296} fail to arrange the
stars strictly in the order B,~A,~F, G,~K,~M? Which of the ten
phenomena are opposed to the hypothesis that the spectral type
of a star depends on its mass? Which of the ten phenomena are
opposed to the hypothesis that the arrangement of stars according
to age is M,~A,~B, A,~F, G,~K,~M (the hypothesis of Lockyer and
Russell)?

8. The apparent areas of the sun and the denser part of the
Orion nebula are about the same, and the sun is about $30$~magnitudes
brighter than the nebula. Suppose the amount of light they radiate
is proportional to the fourth powers of their absolute temperatures.
What is the temperature of the Orion nebula? If its
diameter is $20,000,000$ times that of the sun, what is its mass
(computed from the relation connecting temperature, mass, and
density of a gaseous body)? Under the same assumptions, what
is its mean density? (The student will not fail to remember that
some of the assumptions on which the computation rests are questionable.)

\normalsize

%% -----File: 594.png---Folio 564-------
% [Blank Page]
%% -----File: 595.png---Folio 565-------

\backmatter
\phantomsection
\pdfbookmark[-1]{Back Matter}{Back Matter}

\phantomsection
\pdfbookmark[0]{Index of Names}{Name Index}

\renewcommand{\indexname}{Index of Names}
\printindex[xnames]
\iffalse
Abbott, 268, 350, 351, 380

Adams, J. C.#Adams, 240, 241, 257

Adams, W. S.#Adams, 388, 389

Agenor, 160

Airy, 240

Aitken, 506, 507

Albategnius, 117

Aldrich, 350, 351

Alexander the Great#Alexander, 116

Anderson, 523

Angstrom@{Ångström}#Ångström, 371, 390

Antoniadi, 285, 286

Arcas, 151

Argelander, 139

Aristarchus, 41, 79, 116

Aristotle, 40, 79, 116

Arrhenius, 73, 403


Backhouse, 263

Bacon, Roger#Bacon, 6

Bailey, 522

Baily, 62

Barnard, 260, 263, 278, 285, 289, 290, 291, 293, 299, 300, 301, 302, 303, 305, 308, 312, 325, 327, 331, 333, 335, 337, 402, 462, 472, 473, 474, 498, 553, 554

Bayer, 140

Belopolsky@{Bélopolsky}#Bélopolsky, 166, 272

Benzenberg, 337

Bessel, 165

Biela, 328, 330, 342

Bode, 257

Boltwood, 363

Bond, 297, 487

Boss, Benjamin#Boss, 490

Boss, Lewis, 141, 482, 483, 488, 489, 491, 509, 531

Bouguer, 42

Bourget, 552

Bouvard, 239

Boys, 62

Bradley, 95, 98

Brandes, 337

Braun, 62
%% -----File: 596.png---Folio 566-------

Bredichin, 323, 324

Brooks, 312, 321, 327

Brorsen, 263, 330

Buffham, 307

Buisson, 552

Bunsen, 371

Burnham, 506


Caesar@{Cæsar}#Cæsar, 184

Callisto, 151

Campbell, 279, 482, 483, 484, 486, 513, 515, 530, 545, 546, 553, 561

Cannon, Miss#Cannon, 527

Cassini, G. D.#Cassini, 274, 275, 297, 300

Cassini, J.#Cassini, 41, 271, 302

Cerulli, 272

Challis, 240

Chamberlin, 73, 346, 421, 424, 425, 437, 443, 444, 451

Chandler, 63, 90, 91, 260, 321

Chapman, 466, 467, 468, 470

Clark, 165

Clarke, 42

Clerk-Maxwell, 303, 326

Columbus, 1, 15, 40, 41

Comstock, 483

Condamine, 42

Copernicus, 79, 117, 118

Cornu, 62

Cowell, 335

Croll, 115

Cromellin, 335

Curtis, 166


Dalembert@{D'Alembert}#Alembert, 95

Darwin, Charles#Darwin, 16, 412, 413

Darwin, George H.#Darwin, 59, 63, 450, 458, 460, 545

Darwin, Horace#Horace, 63

Dawes, 506

Delavan, 324

Denning, 338, 340

Deslandres, 398

Devico@{De Vico}#Vico, 330

Doerfel, 313
%% -----File: 597.png---Folio 567-------

Donati, 330

Doppler, 375, 389, 394, 397, 524

Douglas, 291

Dyson, 483


Eddington, 490, 491

Ehlert, 63

Elkin, 514

Ellerman, 400, 401

Encke, 304, 329, 330

Eratosthenes, 41

Euclid, 116

Eudoxus, 40, 116

Euler, 92, 435

Europa, 160

Evans, 285

Evershed, 386


Fabricius, 515

Fabry, 552

Farrington, 344, 345

Faye, 450

Fizeau, 292, 375, 389, 394, 397, 524

Flamsteed, 140

Fleming, Mrs.#Fleming, 527

Forbes, 262

Foucault, 84, 85

Fowle, 350, 351

Fox, 376, 382

Fraunhofer, 390, 403

Frost, 511, 513, 514, 552


Gale, 52, 56, 59, 388, 458

Galileo, 8, 79, 117, 119, 207, 289, 299, 382

Galle, 240

Gauss, 258, 313

Gilbert, 213

Gill, 142, 247, 499

Godin, 42

Goodricke, 515

Gould, 139

Gregory XIII, Pope#Gregory, 184, 185
%----

Hagen, 84

Hale, 285, 385, 386, 388, 389, 398, 400, 401

Hall, 273, 305

Halley, 156, 165, 327, 332, 334, 335, 336, 342

Harding, 258

Hayford, 33, 42

Hecker, 63

Hegel, 257
%% -----File: 598.png---Folio 568-------

Helmert, 42

Helmholtz, 357, 450, 526, 533

Hencke, 258

Henderson, 101

Hera, 151

Herschel, John#John Herschel, 205, 316, 467, 470, 473, 506

Herschel, William#William Herschel, 239, 297, 305, 306, 316, 329, 470, 474, 482, 505, 521, 553, 560

Hill, 242, 297

Hinks, 247

Hipparchus, 79, 94, 117, 141

Holden, 307

Hooke, 274, 275

Hough, G. W.#Hough, 294, 296

Hough, S. S.#Hough, 63

Huggins, 279

Hughes, 149, 157

Hull, 326

Hussey, 506

Huxley, 362

Huyghens, 297, 299


Innes, 499


Jacobi, 544

Jeffreys, 91

Joule, 355

Julius, 396


Kant, 357, 411, 412, 414, 416, 446, 447, 448, 449

Kapteyn, 142, 473, 485, 486, 490, 491, 496, 499, 525, 526, 531

Keeler, 279, 303, 307, 424, 554, 561

Kelvin, 60, 62, 68, 359, 361, 405, 459, 492

Kepler, 7, 9, 117, 119, 229, 230, 231, 313, 523

Kirchhoff, 371, 390

Kirkwood, 260, 304, 450, 451

Kortozzi, 63

Kustner@{Küstner}#Küstner, 63, 90


Lagrange, 233, 234, 238, 241

Lambert, 313

Lane, 357, 358, 526, 533, 540

Langley, 350, 366, 379, 390

Laplace, 45, 233, 238, 239, 302, 313, 320, 411, 412, 414, 416, 449, 450, 451, 533, 534

Lassell, 297, 307

Lebedew, 326
%% -----File: 599.png---Folio 569-------

Lee, 514

Leibnitz, 234

Leverrier, 240, 241, 257, 342

Lexell, 321

Lindemann, 526, 527

Lockyer, 394, 534, 555

Love, 58

Lowell, 262, 270, 272, 283, 284, 285, 286, 287, 298, 304

Ludendorff, 152


Maclaurin, 544

MacMillan, 88, 459

Magellan, 1

Maney, 552

Mascari, 272

Maskelyne, 62

Maunder, 160, 285, 384, 405

Maury, Miss#Maury, 527

Mayer, 355

Medusa, 160

Melotte, 289, 466, 467, 468, 470

Mendeleeff@{Mendeléeff}#Mendeléeff, 369

Messier, 156, 157, 501

Michelson, 52, 56, 59, 292, 369, 458

Milne, 62

Mitchell, 62 %[** TN: Typo for Michell]

Moore, 553, 561

Muller@{Müller}#Müller, 268, 269, 276


Newcomb, 63, 285, 292, 308

Newton, 7, 8, 9, 15, 34, 35, 41, 42, 62, 63, 79, 80, 94, 119, 120, 230, 232, 233, 238, 313, 329, 332, 355, 365, 366, 390

Nichols, 326

Nicholson, 289


Olbers, 258, 323

Olivier, 338, 340

Orloff, 63


Parkhurst, 259

Perrine, 289, 525

Perrotin, 272, 284, 307

Philolaus, 78

Piazzi, 257, 258

Picard, 41, 42

Pickering, E. C.#Pickering, 152, 470, 512, 524, 527

Pickering, W. H.#Pickering, 216, 262, 284, 287,297, 319

Poincare@{Poincaré}#Poincaré, 242, 544

Poisson, 459
%% -----File: 600.png---Folio 570-------

Ptolemy, 79, 117, 118, 139, 141

Pythagoras, 40, 116


Ramsay, 395

Rayet, 530, 535, 561

Rebeur-Paschwitz, 63

Reich, 62

Ritchey, 210, 402, 429, 430, 501, 525, 537, 551, 556, 559

Ritter, 357

Roche, 303, 327, 346, 423, 450

Roemer@{Römer}, 292 %[** Römer in text]

Rowland, 369, 385, 390

Russell, 517, 534

Rutherford, 367


Sampson, 390

Schaeberle, 166

Schiaparelli, 270, 272, 283, 284, 285, 342

Schlesinger, 518

Schroeter@{Schröter}#Schröter, 269, 271

Schuster, 385

Schwabe, 383

Schwarzschild, 326

Schweydar, 58, 63

Seares, 555

Secchi, 527, 528

See, 507, 508

Seeliger, 525

Shapley, 503, 517, 519

Slipher, E. C.#Slipher, 295, 298

Slipher, V. M.#Slipher, 272, 279, 307, 308, 558, 561

Slocum, 397, 426

Smith, 389

Sosigenes, 184

Spencer, 16, 412

Stefan, 280, 354, 358, 542

Stjohn@{St.\ John}, 386, 387, 394

Stromgren@{Strömgren}#Strömgren, 314

Strutt, 363

Struve, William#Struve, 506

Sullivan, 560

Sundman, 242


Tacchini, 272

Tebbutt, 331

Tempel, 342

Thackeray, 483

Thales, 116

Thetis, 151

Thollon, 284

Tisserand, 308
%% -----File: 601.png---Folio 571-------

Titius, 257

Todd, 262

Turner, 467

Tuttle, 342

Tycho Brahe#Tycho, 7, 118, 119, 141, 153,229, 523


Very, 206

Vogel, 279, 512, 513, 518


Wallace, Alfred Russel#Wallace, 412

Wallace, R. J.#Wallace, 161, 215

Weiss, 342

Whewell, 234

Wien, 372

Wilczynski, 390
%% -----File: 602.png---Folio 572-------

Williams, 284, 524

Wilsing, 62, 390

Wilson, R. E.#Wilson, 553

Wilson, W. E.#Wilson, 525

Witt, 247, 260

Wolf, Max#Wolf, 258, 525

Wolf, 530, 535, 561

Wollaston, 390, 487

Wright, Thomas, 411, 412, 446

Wright, W. H.#Wright, 515


Young, C. A.#Young, 307, 391

Young, Thomas#Young, 365


Zeeman, 385

Zeus, 151, 160

Zollner@{Zöllner}#Zöllner, 205, 487
%[**end of Names Index]
\fi
%% -----File: 603.png---Folio 573-------

\cleardoublepage
\phantomsection
\pdfbookmark[0]{General Index}{Index}

\renewcommand{\indexname}{General Index}
\printindex

\iffalse
%[**GENERAL INDEX]

Absorption of light 350, 467

Absorption spectrum 375

Acceleration, definition of#Acceleration 8

Achernar 144

Aerolites@{Aërolites}#Aërolites 343

Age of earth#Age 360

Alcor 151, 514

Aldebaran 139, 144, 521

Algol 140, 159, 160, 166, 515, 517, 518

Almagest 117

Almucantars 124

Alpha Centauri#Centauri 101, 144, 476, 515

Alpha Crucis#Crucis 144

Alpha Geminorum#Geminorum 519

Altair 139, 144

Altitude 124
  of equator 108
  of pole 108

American Ephemeris and Nautical Almanac#Almanac 176, 253

Andromeda 159, 160
  Nebula 158, 556, 558, 559

Andromid meteors#Andromid 340, 341, 342, 346

Angular distances 150

Antares 144, 156, 503, 529

Aphelion point#Aphelion 104

Apogee 197

Aquarid meteors#Aquarid 342

Aquila 473

Ara 473

Arcturus 144, 157, 486, 503, 528, 529

Areas, law of#Areas 104, 229

Argo 473

Ascending node 188, 249

Astronomical unit 227

Atmosphere 64
  absorption of light by 350
  climatic influences of 71
  composition of 64
  height of 66
  mass of 65
  of Jupiter 296
  of Mars 276
  of Mercury and Venus 268
  of Moon 203
%% -----File: 604.png---Folio 574-------
  of Saturn 306
  of Uranus and Neptune 307
  pressure of 65
  refraction by 74
  role@{rôle of in life processes}#rôle 74

Atoms 68

August meteors 342

Auriga 160

Aurorae@{Auroræ}#Auroræ 66, 404

Autumnal equinox 109

Azimuth 124

Base line 30

Beehive (Præsepe)#Beehive 166

Belt of Orion 163, 165

Beta Aurigae@{Beta Aurigæ}#Aurigæ 490

Beta Centauri#Centauri 144

Beta Geminorum#Geminorum 528

Beta Lyrae@{Beta Lyræ}#Lyræ 155, 515, 518

Betelgeuze 144, 162, 165, 523

Biela's comet 328, 330, 342

Big Dipper 77, 139, 140, 149, 151, 153, 160, 488, 490, 527

Binary stars 507
  evolution of 543
  masses of 508
  orbits of 507
  origin of 543
  spectroscopic 510

Bode's law 257

Bolometer 366

Bootes@{Boötes}#Boötes 157

Brooks' comet 318, 321

Brorsen's comet 330

Calendar 184

Canals of Mars 283

Canes Venatici, spiral nebula in#Venatici 429

Canis Major 165, 473

Canis Minor 165

Canopus 144, 480

Capella 144, 160, 486, 514, 524

Carbon dioxide 64
  effects on climate 73
  production of 73
%% -----File: 605.png---Folio 575-------

Cassiopeia 152, 153, 159, 473, 474, 523

Castor 166

Catalogues of stars 141, 482, 499

Celestial sphere 122

Centaurus 473

Center of gravity of earth and moon#Center of gravity 199

Cepheus 473

Ceres, discovery of#Ceres 257

Chemical constitution of sun 393

Chromosphere 378, 394

Circinus 473

Circumpolar star trails#Circumpolar 78

Clusters of stars 500

Comet
  of 1668#Comet 318
  of 1680#Comet 329
  of 1811#Comet 316, 329
  of 1843#Comet 318
  of 1880 and 1882#Comet 318, 331

Comets
  appearance of 311
  capture of 320
  dimensions of 316
  disintegration of 327
  families of 318
  masses of 317
  naming of 313
  orbits of 313
  origin of 322, 442

Comets' tails, theories of#tails 323

Conic sections 234, 313

Conservation of energy 355

Constellations 139
  list of 148

Contraction of sun 356

Coordinates@{Coördinates}#Coördinates 123

Copernican theory 118

Corona, of sun#Corona 379, 401

Corona Borealis 157

Coronium 403

Corpuscles 367

Craters of moon#Craters 211

Crux 473

Cygnus 473, 550, 551, 554

Date, place of change of#Date 181

Day
  astronomical 181
  civil 181
  invariability of 88
  Julian 185
  longest and shortest 173
  mean solar 175
  sidereal 171
  solar 172

Dearborn Observatory 165
%% -----File: 606.png---Folio 576-------

Declination 126

Deduction 9, 10

Deferent 118

Deimos 273

Delavan's comet 324, 325

Delta Aquilae@{Delta Aquilæ}#Delta Aquilæ 508

Delta Cephei 520, 522

Delta Librae@{Delta Libræ}#Delta Libræ 518

Deneb 144

Density
  of earth 45, 46, 48, 50
  of moon 202, 254
  of sun 254
  of stars 517, 541

Deviation
  of falling bodies 82
  of air currents 85
  of rivers 87

Dialogues of Galileo 119

Dimensions
  of comets 316
  of sun, moon, and planets 254

Discovery of Uranus and Neptune 155, 238

Disintegration
  of comets 327
  of matter 363, 364

Distance
  of moon 20, 194
  of planets 249
  of stars 476, 484, 486, 487
  of sun 247

Distribution
  of stars 463, 470
  of sun spots 383
  of time 179

Diurnal circles 109

Donati's comet 330

Doppler-Fizeau law 375, 389, 394, 397, 524

Double stars 505

Dynamics of stellar system#Dynamics 491

Earth
  age of 360
  density of 45, 46, 48, 50
  dimensions of 33
  elasticity of 59
  mass of 45
  oblateness of 31, 34, 35
  pressure in 51
  revolution of 96
  rigidity of 52, 59
  rotation of 77, 82, 84, 85
  sphericity of 27
  temperature in 51

Earthquakes 60

Earth's orbit 103, 104

Eccentricity 104
  of earth's orbit 104, 249
  of planetary orbits 249
%% -----File: 607.png---Folio 577-------

Eccentric motion 118

Echelon spectroscope 369

Eclipses
  of moon 218
  of sun 220
  phenomena of 223
  uses of 220, 224

Eclipsing variables 516

Ecliptic 94, 127
  obliquity of 105
  pole of 106

Elasticity of earth 52, 59

Electrical repulsion 323

Electrons 367

Elements
  in sun 393
  of orbit 248, 249
  table of 249

Eleven-year cycle 404

Ellipse, definition of#Ellipse 103

Elongations of planets#Elongation 227
  dates of#Elongation 256

Encke's comet 329, 330

Energy
  conservation of 355
  from radium 363
  kinetic 356
  of coal 353
  of solar system 419
  of water 352
  of wind 352
  potential 356
  radiant 356
  radiated by sun 353

Epicycle 118

Epsilon Lyrae@{Epsilon Lyræ}#Epsilon Lyræ 154, 155, 239

Equation of time 176

Equator 106, 125
  altitude of 108

Equinoctial colure 125

Equinoxes 94, 109
  autumnal 109
  how to locate 153
  precession of 92, 94, 115
  vernal 109

Eros 260

Escape of atmosphere 69

Eta Cassiopeiae@{Eta Cassiopeiæ}#Eta Cassiopeiæ 153

Evolution 16
  essence of 407
  of planets 431
  of stars 532, 533
  value of 408

Faculae@{Faculæ}#Faculæ 382
  periodicity of 388, 404

Falling bodies, deviations of#Falling 82
%% -----File: 608.png---Folio 578-------

First-magnitude stars 143, 144

Flash spectrum 391

Flocculi 389

Foci 103

Fomalhaut 144

Fossils, occurrence of#Fossils 362

Foucault's pendulum 84

Fraunhofer lines 390

Galaxy 146, 159, 470, 474, 479, 490, 492, 496, 497, 498, 499, 500, 503, 553, 558

Galileo's Dialogues 119

Gamma Virginis 508

Gases
  kinetic theory of 68, 492
  pressure of 69

Gegenschein 262

Gemini 166

Geographical system 122

Glacial epoch 73

Globular star clusters 500

Grating spectroscope 369

Gravitation
  discovery of 230
  importance of law of 231
  law of 9, 230, 463

Gravity
  surface 245
  of planets 254

Halley's comet 327, 332, 334, 335, 336, 342

Harvard College Observatory#Harvard 144, 260, 512, 522, 523, 527, 528, 529, 530

Heat
  from moon 204
  from sun 350
  received by planets 250

Helium 362, 363, 395

Hercules 156, 159, 482, 501, 503

Horizon 123

Hour angle 131

Hour circle 125

Hyades 160, 162, 488

Hydrocyanic acid 64

Hyperbola 235

Hypothesis
  of Kant 446
  of Laplace 449
  planetesimal 421

Inclination of earth's orbit 105
  of planetary orbits 249

Induction 8

Infinity of physical universe 548

Irregular nebulae@{Irregular nebulæ}#Irregular nebulæ 550
  variables 522

Isostasy 42
%% -----File: 609.png---Folio 579-------

Juno, discovery of 258

Jupiter
  atmosphere of 296
  belts of 293
  great red spot on 294
  markings on 293
  physical condition of 296
  rotation of 292, 437
  satellite system of 289
  seasons of 296

Kepler's laws 229

Kinetic energy 356

Kinetic theory of gases 68, 492

Lag of tides 455

Lane's law 358, 526
  paradox 357, 533

Laplacian hypothesis 449, 533

Latitude
  astronomical 40, 123
  celestial 127
  geocentric 40
  geographical 40
  variation of 63, 89

Law
  of areas 104, 229
  of gravitation 9, 230, 463

Laws
  of force 236
  of motion 8, 80
  of spectrum analysis 371

Leap year 184

Leo 157, 340

Leonid meteors 340, 341, 342

Lexell's comet 321

Libration of Mercury 271

Librations of moon 201

Lick Observatory 150, 160, 166, 260
  277, 278, 285, 289, 291, 424, 483
  507, 515, 530, 553, 555, 557, 561

Light
  absorption of 370
  dispersion of 370
  from moon 204
  from sun 349
  nature of 365
  polarized 366
  pressure of 326
  production of 366
  refraction of 74, 370
  velocity of 22, 99, 291, 354
  wave lengths of 349, 366
  zodiacal 262, 328, 442

Longitude 123
  celestial 127

Long period variables 520

Lowell Observatory 272, 279, 285, 295, 308
%% -----File: 610.png---Folio 580-------

Lunar, craters 211
  mountains 207

Lupus 473

Lyra 23, 153, 156

Lyrid meteors 341, 346

Magellanic clouds 530, 553

Magnetic storms, periodicity of#Periodicity 404, 405

Magnitudes of stars 142, 465

Mars
  atmosphere of 276
  canals of 283
  explanation of canals of 285
  polar caps of 277, 278
  rotation of 274, 437
  satellites of 273
  seasons of 277
  temperature of 277
  water on 279

Mass
  of atmosphere 65
  of moon 71, 198
  of sun 254

Masses
  determination of 244
  of planets 254
  of stars 508, 509

Mean distance, definition of#Mean distance 229

Mean solar time 175

Mercury 266
  albedo of 268
  atmosphere of 268
  librations of 271
  markings of 269
  phases of 266
  rotation of 269
  seasons of 270
  transits of 267

Meridian 124

Meteoric showers 339
  matter, resistance of 88

Meteorites 343
  composition of 344
  origin of 345

Meteors 65, 337, 525
  effects of on earth's rotation 88
  effects of on solar system 343
  height of 65, 338
  number of 338

Mile, nautical#nautical 16

Milky Way 22, 146, 160, 431, 462
  470, 473, 490, 491, 496, 498, 507
  523, 525, 530, 531, 554, 557, 558
  560

Mizar 151, 152, 512
  spectrum of 511, 513
%% -----File: 611.png---Folio 581-------

Molecules 68
  size of 68
  velocity of 69

Moment of momentum 88
  of solar system 416, 417

Monoceros 473

Moon 188
  apogee of 197
  apparent motion of 188
  atmosphere of 203
  craters of 211
  density of 202, 254
  dimensions of 196
  distance of 20, 194
  diurnal circles of 192
  eclipses of 218
  effects of on earth 217
  heat received from 204
  librations of 201
  map of 209
  mass of 71, 198, 254
  mountains of 207
  orbit of 188, 197
  perigee of 197
  periods of 189
  phases of 191
  rays and rills of 214
  rotation of 200
  satellites of 220
  surface changes of 216
  surface gravity of 202
  temperature of 205
  velocity of 196

Motion
  of earth 103
  of sun 96, 482, 483, 484
  of stars 145, 480, 481, 487

Mount Wilson Solar Observatory 285, 348, 387, 396, 401, 501, 503, 554

Mountain method of determining density of earth#Mountain 48

Mu Orionis 512
  spectrum of 513

Musca 473

Nadir 124

Naval Observatory 17, 123, 180, 181

Nebulae@{Nebulæ}#Nebulæ
  irregular 550
  planetary 560
  ring 560
  spiral 429, 430, 554, 556, 557

Nebular hypothesis 411, 449

Neptune
  atmosphere of 307
  discovery of 155, 238
%% -----File: 612.png---Folio 582-------
  physical condition of 308
  rotation of 307, 437
  satellite of 306

Nitrogen 64

Nodes, ascending and descending#Nodes 188

Norma 473

Northern Crown 157

Nova Aurigae@{Nova Aurigæ}#Nova Aurigæ 524

Nova Persei 525, 526

Number of stars 145, 464, 466, 468

Nutation 95

Oblate figure 32

Oblateness of earth 31, 34

Obliquity of ecliptic 105

Omega Centauri 501, 522

Omicron Ceti 515, 521

Ophiuchus 473, 523

Opposition
  definition of 228
  of planets, dates of 256

Orbits
  of binary stars 507
  of comets 313
  of planetoids 259
  of planets, elements of 248, 249

Origin
  of binary stars 543
  of comets 322, 442
  of meteorites 345
  of planetoids 259
  of planets 431
  of species 412, 413
  of spiral nebulæ 424

Orion 77, 160, 162, 163, 491

Orion nebula 163, 164, 552

Orionid meteors 341

Oxygen 64

Pallas, discovery of#Pallas 258

Parabola 235

Parallax
  of stars, definition of#stars 100
  determination of 476
  of sun 247

Parallelogram of forces 81

Parsec, definition of#Parsec 476

Pendulum
  Foucault's 84
  horizontal 60, 63

Penumbra
  of earth's shadow 218
  of sun spots 381

Perigee of moon's orbit 197

Perihelion point
  definition of 104
  longitude of 249

Period, of moon
  sidereal#Period 189
  synodical#Period 189

Period of planets 249
%% -----File: 613.png---Folio 583-------

Periodicity of sun spots 383

Perseid meteors 340

Perseus 140, 159, 160, 473, 490, 523

Perturbations 237

Phases
  of Mercury and Venus#Phases 266
  of moon 191

Phobos 273

Photographic chart of sky 141

Photosphere 378, 379

Planetary orbits
  dimensions of 249
  eccentricities of 249, 434
  planes of 249, 433

Planetesimal
  hypothesis 421
  organization 422

Planetoids
  diameters of 260
  orbits of 259, 442
  origin of 259

Planets 226
  dates of elongation of 256
  dates of opposition of 256
  density of 254
  dimensions of 254
  distances of 249
  evolution of 431
  heat received by 250
  inferior 227
  intra-Mercurian 261
  masses of 254
  origin of 431
  periods of 249
  possible undiscovered 261
  rotations of 437
  superior 227
  surface gravity of 254
  synodical periods of 256
  trans-Neptunian 261

Pleiades 22, 139, 160, 161, 162, 536, 537, 541

Pointers 149, 150

Pole 106
  altitude of 108
  of ecliptic 106

Polar caps of Mars#Polar caps 277, 278

Polaris 139, 149, 150, 153, 515

Pollux 144, 166

Potential energy 356

Praesepe@{Præsepe}#Præsepe 166

Precession of equinoxes 92, 94, 115

Principia 232

Prism spectroscope 369

Procyon 144, 165, 166

Prominences 379, 395, 426

Proper motion of stars#Proper motion 146, 479, 498
%% -----File: 614.png---Folio 584-------

Ptolemaic theory 118

Pulkowa 166

Pyramids 23

Quadrature 191, 228

Radial velocity 144, 375, 377

Radiant point of meteors#Radiant point 339, 341

Radioactivity in sun#Radioactivity 363

Radium 362, 363

Rays and rills 214

Reference points and lines 121

Refraction 74, 370

Regulus 144, 159

Reversing layer 378, 390
  constitution of 392

Revolution of earth 96, 98, 100, 101

Rigel 144, 163, 480

Right ascension 126

Rigidity of earth 52, 59

Ring nebula in Lyra 155, 560

Rings of Saturn 299, 441
  constitution of 302
  permanency of 304

Roche's limit 303, 327, 346, 450

Rotation
  of earth 82, 84, 85
  of Jupiter 292, 437
  of Mars 274, 437
  of Mercury 269
  of moon 200
  of Neptune 307, 437
  of Saturn 305, 437
  of sun 388, 436
  of Uranus 307, 437
  of Venus 271

Runaway stars 498

Sagittarius 473, 554

Salinity of the oceans 361

Satellites
  of Jupiter 289
  of Mars 273
  of moon 220
  of Neptune 306
  of Saturn 297
  of Uranus 306
  origin of 440

Saturn
  physical condition of 306
  ring system of 299, 441
  rotation of 305, 437
  satellite system of 297
  seasons of 306
  shape of 39
  surface markings on 305
%% -----File: 615.png---Folio 585-------

Science 1
  imperfections of 10
  methods of 6
  origin of 4
  value of 2

Scientific theories 12
  contributions to, by astronomy 14

Scintillation of stars 76

Scope of astronomy 19

Scorpius 156, 157, 473

Seasons
  cause of 107
  lag of 112
  length of 112
  of Jupiter 296
  of Mars 277
  of Mercury 270
  of Saturn 306
  of Venus 272

Seismograph 62

Serpens 473

Shape of earth 33, 38

Shape of earth's orbit 102

Shooting stars 65, 337, 525

Sidereal
  day 171
  period of moon 189
  period of planets 249
  year 183
  time 171

Siderites 342

Sirius 139, 140, 143, 144, 165, 166
  322, 479, 480, 486, 488, 493, 494
  spectrum of 527

Solar
  days 172
  energy 353
  Observatory 285, 348, 386, 387
  396, 401, 501, 503, 554
  time 172

Solstices 109

Spectra of stars 486, 527, 530

Spectroheliograph 385, 398

Spectroscope 101, 269, 279, 303, 307, 369, 463

Spectroscopic binaries 510

Spectrum
  absorption 375
  analysis 369
  analysis, laws of 371
  flash 391

Sphericity of earth 27

Spheroid, oblate and prolate#Spheroid 38

Spica 144, 153, 514

Spiral nebulae@{Spiral nebulæ}#Spiral nebulæ 429, 430, 554, 556, 557
  origin of 424

Stability
  of solar system 238
  of satellites 299
%% -----File: 616.png---Folio 586-------

Standard time 177

Star
  clusters 500
  streams 490

Stars
  binary 507
  catalogues of 141, 482, 499
  clusters of 500
  density of 517, 541
  distances of 476, 484, 486, 487
  distribution of 463, 470
  double 505
  evolution of 532, 533
  first-magnitude 143, 144
  groups of 487, 499
  masses of 508, 509
  motions of 145, 480, 481
  number of 145, 464, 466, 468
  parallaxes of 476
  proper motions of 146, 479, 481
  radial velocities of 481
  runaway 498
  spectra of 486, 527
  temperatures of 539
  temporary 523
  twinkling of 76
  variable 515
  velocities of 23

Stefan's law 280, 354, 358

Sun
  apparent motion of 96
  constitution of 378, 392, 393
  density of 254
  distance of 247
  eclipses of 220
  heat received from 350
  light and heat of 349
  magnetic field of 385
  magnitude of 143, 502
  mass of 254
  motion of 482, 483, 484
  parallax of 247
  past and future of 360, 443
  radiation of 353
  rotation of 388, 436
  surface gravity of 245, 254
  temperature of 354

Sunlight in all latitudes 111

Sun's eleven-year cycle 404

Sun's heat
  combustion theory of 358
  contraction theory of 356
  meteoric theory of 358
  subatomic\DPnote{** sub-atomic} energy theory of 364

Sun spots
  distribution and periodicity of 383
  motions of 387
%% -----File: 617.png---Folio 587-------
  penumbra of 381
  periodicity of 383
  polarity of 386
  umbrae of@{umbræ of}#umbræ of 381

Superstition 14

Surface gravity
  determination of 245
  of moon 202
  of planets 254
  of sun 245, 254

Sword of Orion 163

Synodical period
  of moon 189
  of planets 256

Tails of comets, theories of#Tails 323

Taurus 160, 162, 473, 488, 489, 494

Tebbutt's comet 331

Tempel's comet of 1866#Tempel's comet 342

Temperature
  of earth 51
  of Mars 277
  of moon 205
  of sun 354
  of stars 539

Temporary stars 523

Theory of evolution 407
  value of 408

Tidal
  bulges 54
  cones 455
  evolution 420, 454, 460
  experiments 56

Tide-raising
  acceleration 54
  forces 243, 452, 453

Tides
  cause of 242
  effects of, on day 88
  effects of, on earth 458
  effects of, on moon 456
  lag of 455

Time
  distribution of 179
  equal intervals of 169, 170
  equation of 176
  local 177
  mean solar 175
  practical measure of 170
  sidereal 171
  solar 172
  standard 177

Torsion balance 46

Total eclipses 222

Transits of Mercury and Venus#Transits 267

Triangulation 29

Trifid Nebula 554, 555

Tropical year 183
%% -----File: 618.png---Folio 588-------

Tuttle's comet 342

Twilight, duration of 67

Twinkling of stars 76

Umbra
  of earth's shadow 218
  of sun spots 381

Uniformity of earth's rotation 87

Uranium 362

Uranus
  atmosphere of 307
  discovery of 239
  physical condition of 308
  rotation of 307, 437
  satellites of 306

Ursa Major 150, 151, 490, 556

Variability
  of Eros 261
  of Japetus 297

Variable stars
  cluster 522
  eclipsing 516
  irregular 522
  of Beta Lyræ type 518
  of Delta Cephei type 519
  long period 520

Variation
  in lengths of days 172
  of latitude 63, 89
  of sun's radiation 351

Vega 23, 139, 144, 154, 156, 486

Velocity
  of escape 69
  of light 22, 99, 291, 354
  of meteors 337
  of molecules 69
  of moon 196
  of sun 23
  of stars 23

Venus
  atmosphere of 268
  markings of 271
  phases of 266
  rotation of 271
  seasons of 272
  transits of 267

Vernal equinox 109

Vertical circles 124

Vesta, discovery of#Vesta 258

Virgo 153

Vulpecula 473

Wave length of light 349, 366

Wien's law 372

Wolf-Rayet stars 530, 535, 561

Xenon 64 % [** TN: Typo Xeon]

Year
  anomalistic 183
  leap 184
%% -----File: 619.png---Folio 589-------
  sidereal 183
  tropical 183

Yerkes Observatory 77, 139, 158
  161, 163, 164, 192, 208, 210, 212
  215, 259, 275, 285, 291, 300, 301
  302, 312, 333, 376, 397, 400, 426
  429, 430, 462, 501, 511, 513, 525
  528, 529, 537, 551, 554, 556, 559

Zenith 124

Zodiacal light 262, 328, 442
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  \centering%
  \noindent\raisebox{-10pt}{\textsc{\Huge T}}%
  \parbox[t]{3.5in}{%
    \textsc{he} following pages contain advertisements of
    books by the same author or on kindred subjects.%
  }
}
\end{center}
\vfill
\clearpage
%% -----File: 622.png---Folio 592-------

\noindent\textbf{\Large An Introduction to Celestial Mechanics}

\begin{center}
  \textsc{By F.~R. MOULTON}

  \small Professor of Astronomy in the University of Chicago
\end{center}

\begin{flushright}
  \textit{437 pp., 8vo, \$3.50}
\end{flushright}

\Stretchout[1.5]%
Intended to give a satisfactory account of many parts
of celestial mechanics rather than an exhaustive treatment
of any special part; to present the work so as to attain
logical sequence, to make it progressively more difficult,
and to give the various subjects the relative prominence
which their scientific and educational importance deserves.
In short, the aim has been to prepare such a book that
one who has had the necessary mathematical training may
obtain from it, in a relatively short time and by the easiest
steps, a broad and just view of the whole subject.

\bigskip
``Composed with remarkable good judgment, and indispensable
to all students of the subject.''---\textit{N.~Y. Post.}

\MacMillan
% [** TN: Macro prints the following:
% THE MACMILLAN COMPANY
% Publishers 64--66 Fifth Avenue New York]
%% -----File: 623.png---Folio 593-------

\begin{center}
  \textsc{By WILLIS I. MILHAM, Ph.D.}

  \footnotesize Field Memorial Professor of Astronomy in Williams College
\end{center}

\noindent\textbf{\Large How to Identify the Stars}
\begin{flushright}
  \textit{Cloth, 12mo, 38 pages, 75 cents}
\end{flushright}

\Stretchout[1]%
The purpose of this little book is to serve as a guide in taking
the first steps in learning the stars and constellations and
also to point the way to the acquisition of further information
on the part of those who desire it. Excellent star maps are
included.

\vfill
\noindent\textbf{\Large Meteorology}
\begin{center}
\begin{minipage}{3.5in}
  \normalsize\scshape A Text-book of the Weather, the Causes
  of its Changes, and Weather Forecasting
  for the Student and General Reader
\end{minipage}
\end{center}

\begin{flushright}
  \textit{Cloth, 8vo, illustrated, 549 pages, \$4.50}
\end{flushright}

This book is essentially a text-book. For this reason, the
marginal comments at the sides of the pages, the questions,
topics for investigation, and practical exercises have been
added. A syllabus of each chapter has been placed at its
beginning, and the book has been divided into numbered sections,
each treating a definite topic. The book is also intended
for the general reader of scientific tastes; for while it
can hardly be called an elementary treatise, it starts at the
beginning and no previous knowledge of meteorology itself is
anywhere assumed. It is assumed, however, that the reader is
familiar with the great general facts of science. References
have been added at the end of each chapter.

\MacMillan
%% -----File: 624.png---Folio 594-------

\noindent\textbf{\Large The Elements of Practical Astronomy}

\begin{center}
  \textsc{By W.~W. CAMPBELL}

  \footnotesize Astronomer in the Lick Observatory
\end{center}

\begin{flushright}
  \textit{Cloth, 8vo, 254 pages, \$2.00}
\end{flushright}

\Stretchout[1.2]
The elements of practical astronomy, with numerous
applications to the problems first requiring solution. It is
suited for use with students who have had an introductory
training in astronomy and mathematics.

\vfill
\noindent\textbf{\Large Elementary Lessons in Astronomy}

\begin{center}
  \textsc{By SIR NORMAN LOCKYER, K.C.B., LL.D.,
   Sc.D., D.Sc., F.R.S.}
\end{center}

\begin{flushright}
  \textit{Cloth, 12mo, 400 pages, \$1.40}
\end{flushright}

\Stretchout[1.2]
Intended to serve as a textbook\DPnote{** Unhyphenated in original} for use in schools, but
will be found useful to the general reader who wishes to
make himself acquainted with the basis and teachings of
one of the most fascinating of the sciences. The aim
throughout the book is to give a connected view of the
whole subject and to supply facts and ideas founded on
the facts, to serve as a basis for subsequent study and discussion.

\MacMillan

\cleardoublepage

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[]\T1/cmtt/m/n/9.2 End of Project Gutenberg's An Introduction to Astronomy, by 
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Output written on 32000-t.pdf (518 pages, 13901425 bytes).
PDF statistics:
 6627 PDF objects out of 7423 (max. 8388607)
 1462 named destinations out of 1728 (max. 131072)
 1420 words of extra memory for PDF output out of 10000 (max. 10000000)

