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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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+      \makebox[0pt][c]{\huge AN INTRODUCTION TO ASTRONOMY}%
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+%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{document}
+
+\pagestyle{empty}
+\pagenumbering{Alph}
+\phantomsection
+\pdfbookmark[-1]{Front Matter}{Front Matter}
+
+%%%% PG BOILERPLATE %%%%
+\Pagelabel{PGBoilerplate}
+\phantomsection
+\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate}
+
+\begin{center}
+\begin{minipage}{\textwidth}
+\small
+\begin{PGtext}
+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 ***
+\end{PGtext}
+\end{minipage}
+\end{center}
+
+\clearpage
+
+
+%%%% Credits and transcriber's note %%%%
+\begin{center}
+\begin{minipage}{\textwidth}
+\begin{PGtext}
+Produced by Brenda Lewis, Andrew D. Hwang, Bup, and the
+Online Distributed Proofreading Team at http://www.pgdp.net
+(This file was produced from images generously made
+available by The Internet Archive/American Libraries.)
+\end{PGtext}
+\end{minipage}
+\end{center}
+\vfill
+
+\begin{minipage}{0.85\textwidth}
+\small
+\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note}
+\subsection*{\centering\normalfont\scshape%
+\normalsize\MakeLowercase{\TransNote}}%
+
+\raggedright
+\TransNoteText
+\end{minipage}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\frontmatter
+
+\pagenumbering{roman}
+\pagestyle{empty}
+
+\normalsize
+
+%% -----File: 001.png----------
+% AN INTRODUCTION TO ASTRONOMY
+\HalfTitle
+
+%% -----File: 002.png---Folio i-------
+
+\null\vfill
+\begin{center}
+\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
+\end{center}
+\vfill
+\newpage
+
+%% -----File: 003.png---Folio ii-------
+
+\thispagestyle{empty}
+\begin{sidewaysfigure}[H]
+\centering\Input{003}{jpg} %[Illustration: Fig. 1]
+\Caption[The Lick Observatory, Mount Hamilton, California.]{Fig}{1}
+\label{Fig:frontispiece}
+\end{sidewaysfigure}
+
+%% -----File: 004.png---Folio iii-------
+
+\cleardoublepage
+\begin{center}
+\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}
+\end{center}
+
+%% -----File: 005.png---Folio iv-------
+\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.
+\end{center}
+%% -----File: 006.png---Folio v-------
+
+
+\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.%
+}
+\clearpage
+\fancyhf{}
+
+%% -----File: 009.png---Folio viii-------
+% [Blank Page]
+%% -----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
+\fi
+
+% Printed by the index environment:
+%\vfill
+%\begin{center}
+%\rule{10em}{0.5pt}
+
+%\footnotesize Printed in the United States of America.
+%\end{center}
+%% -----File: 620.png---Folio 590-------
+% [Blank Page]
+%% -----File: 621.png---Folio 591-------
+\pagestyle{empty}
+\phantomsection
+\pdfbookmark[0]{Catalogue}{Catalogue}
+
+\addtolength{\textheight}{0.75in}%
+
+\null\vfill
+\begin{center}
+\setlength{\fboxsep}{12pt}
+\framebox{%
+  \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
+
+%%%% LICENSE %%%%
+\pagenumbering{Alph}
+\pagestyle{fancy}
+\phantomsection
+\pdfbookmark[0]{Project Gutenberg License}{License}
+\fancyhf{}
+\fancyhead[C]{\Heading{Project Gutenberg License}}
+\SetPageNumbers
+
+\begin{PGtext}
+End of Project Gutenberg's An Introduction to Astronomy, by Forest Ray Moulton
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+
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% End of Project Gutenberg's An Introduction to Astronomy, by Forest Ray Moulton
+%                                                                         %
+% *** END OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO ASTRONOMY ***%
+%                                                                         %
+% ***** This file should be named 32000-t.tex or 32000-t.zip *****        %
+% This and all associated files of various formats will be found in:      %
+%         http://www.gutenberg.org/3/2/0/0/32000/                         %
+%                                                                         %
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@@ -0,0 +1,31242 @@
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% 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]                             %
+%                                                                         %
+% Language: English                                                       %
+%                                                                         %
+% Character set encoding: ISO-8859-1                                      %
+%                                                                         %
+% *** START OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO ASTRONOMY ***
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+%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{document}
+
+\pagestyle{empty}
+\pagenumbering{Alph}
+\phantomsection
+\pdfbookmark[-1]{Front Matter}{Front Matter}
+
+%%%% PG BOILERPLATE %%%%
+\Pagelabel{PGBoilerplate}
+\phantomsection
+\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate}
+
+\begin{center}
+\begin{minipage}{\textwidth}
+\small
+\begin{PGtext}
+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]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO ASTRONOMY ***
+\end{PGtext}
+\end{minipage}
+\end{center}
+
+\clearpage
+
+
+%%%% Credits and transcriber's note %%%%
+\begin{center}
+\begin{minipage}{\textwidth}
+\begin{PGtext}
+Produced by Brenda Lewis, Andrew D. Hwang, Bup, and the
+Online Distributed Proofreading Team at http://www.pgdp.net
+(This file was produced from images generously made
+available by The Internet Archive/American Libraries.)
+\end{PGtext}
+\end{minipage}
+\end{center}
+\vfill
+
+\begin{minipage}{0.85\textwidth}
+\small
+\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note}
+\subsection*{\centering\normalfont\scshape%
+\normalsize\MakeLowercase{\TransNote}}%
+
+\raggedright
+\TransNoteText
+\end{minipage}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\frontmatter
+
+\pagenumbering{roman}
+\pagestyle{empty}
+
+\normalsize
+
+%% -----File: 001.png----------
+% AN INTRODUCTION TO ASTRONOMY
+\HalfTitle
+
+%% -----File: 002.png---Folio i-------
+
+\null\vfill
+\begin{center}
+\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
+\end{center}
+\vfill
+\newpage
+
+%% -----File: 003.png---Folio ii-------
+
+\thispagestyle{empty}
+\begin{sidewaysfigure}[H]
+\centering\Input{003}{jpg} %[Illustration: Fig. 1]
+\Caption[The Lick Observatory, Mount Hamilton, California.]{Fig}{1}
+\label{Fig:frontispiece}
+\end{sidewaysfigure}
+
+%% -----File: 004.png---Folio iii-------
+
+\cleardoublepage
+\begin{center}
+\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}
+\end{center}
+
+%% -----File: 005.png---Folio iv-------
+\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.
+\end{center}
+%% -----File: 006.png---Folio v-------
+
+
+\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.%
+}
+\clearpage
+\fancyhf{}
+
+%% -----File: 009.png---Folio viii-------
+% [Blank Page]
+%% -----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
+\fi
+
+% Printed by the index environment:
+%\vfill
+%\begin{center}
+%\rule{10em}{0.5pt}
+
+%\footnotesize Printed in the United States of America.
+%\end{center}
+%% -----File: 620.png---Folio 590-------
+% [Blank Page]
+%% -----File: 621.png---Folio 591-------
+\pagestyle{empty}
+\phantomsection
+\pdfbookmark[0]{Catalogue}{Catalogue}
+
+\addtolength{\textheight}{0.75in}%
+
+\null\vfill
+\begin{center}
+\setlength{\fboxsep}{12pt}
+\framebox{%
+  \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
+
+%%%% LICENSE %%%%
+\pagenumbering{Alph}
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+\phantomsection
+\pdfbookmark[0]{Project Gutenberg License}{License}
+\fancyhf{}
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+\SetPageNumbers
+
+\begin{PGtext}
+End of Project Gutenberg's An Introduction to Astronomy, by Forest Ray Moulton
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+
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% End of Project Gutenberg's An Introduction to Astronomy, by Forest Ray Moulton
+%                                                                         %
+% *** END OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO ASTRONOMY ***%
+%                                                                         %
+% ***** This file should be named 32000-t.tex or 32000-t.zip *****        %
+% This and all associated files of various formats will be found in:      %
+%         http://www.gutenberg.org/3/2/0/0/32000/                         %
+%                                                                         %
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
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diff --git a/LICENSE.txt b/LICENSE.txt
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--- /dev/null
+++ b/LICENSE.txt
@@ -0,0 +1,11 @@
+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this eBook outside of the United States should confirm copyright
+status under the laws that apply to them.
diff --git a/README.md b/README.md
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #32000 (https://www.gutenberg.org/ebooks/32000)