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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% Project Gutenberg's A Textbook of General Astronomy, by Charles A. Young%
+%                                                                         %
+% 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.net                          %
+%                                                                         %
+%                                                                         %
+% Title: A Textbook of General Astronomy                                  %
+%        For Colleges and Scientific Schools                              %
+%                                                                         %
+% Author: Charles A. Young                                                %
+%                                                                         %
+% Release Date: August 30, 2011 [EBook #37275]                            %
+%                                                                         %
+% Language: English                                                       %
+%                                                                         %
+% Character set encoding: ISO-8859-1                                      %
+%                                                                         %
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+\begin{document}
+
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+\clubpenalty=1000
+
+\pagestyle{empty}
+\pagenumbering{alph}
+
+\pdfbookmark[0]{Project Gutenberg Boilerplate}{Project Gutenberg Boilerplate}
+
+\begin{verbatim}
+Project Gutenberg's A Textbook of General Astronomy, by Charles A. Young
+
+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.net
+
+
+Title: A Textbook of General Astronomy
+       For Colleges and Scientific Schools
+
+Author: Charles A. Young
+
+Release Date: August 30, 2011 [EBook #37275]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK A TEXTBOOK OF GENERAL ASTRONOMY ***
+\end{verbatim}
+
+\clearpage
+\begin{center}
+\pdfbookmark[0]{A Text-book of General Astronomy}{A Text-book of General Astronomy}
+\small
+Produced by Susan Skinner, Nigel Blower, Brenda Lewis and
+the Online Distributed Proofreading Team at
+http://www.pgdp.net (This file was produced from images
+generously made available by the Bibliothèque nationale
+de France (BnF/Gallica) at http://gallica.bnf.fr,
+illustrations generously made available by The Internet
+Archive/Canadian Libraries)
+\end{center}
+\nbVS
+
+{%
+  \setlength{\parindent}{0pt}
+  {\normalsize\centering\itshape Transcriber's Notes\par}
+  \small
+  A small number of minor typographical errors and inconsistencies
+  have been corrected. See the {\ttfamily\footnotesize DPtypo} command
+  in the \LaTeX\ source for more information.
+}
+\clearpage
+
+\frontmatter
+%% -----File: 001.png---
+%% -----File: 002.png---
+%% -----File: 003.png---
+\pdfbookmark[1]{Frontispiece}{Frontispiece}
+
+\nbVS
+
+\includegraphicsmid{frontispiece}{FRONTISPIECE.
+
+\textsc{The Great Telescope of the Lick Observatory,\\
+        Mt.\ Hamilton, Cal.}
+
+Object-Glass made by A. Clark \& Sons:\\
+Aperture, 36~in.; Focal Length, 56~ft.\ 2~in.\\
+Mounting by Warner \& Swasey.}
+
+\nbVS
+
+\clearpage
+
+%% -----File: 004.png---
+\begin{center}
+\nbVS
+\so{\Large A~{}~TEXT-BOOK}
+
+\nbVS
+{\small OF}
+
+\nbVS
+{\Huge GENERAL ASTRONOMY}
+
+\nbVS
+{\small FOR}
+
+\nbVS
+{\large COLLEGES AND SCIENTIFIC SCHOOLS}
+
+\nbVS[2]
+{\small BY} \\[2ex]
+{\large CHARLES A. YOUNG, \textsc{Ph}.D., LL.D.,}  \\[1ex]
+\textsc{\small Professor of Astronomy in the College of New Jersey (Princeton)}.
+
+\nbVS[2]
+\nbrule
+
+\nbVS[2]
+\textsc{Boston, U.S.A., and London}:  \\
+GINN \& COMPANY, PUBLISHERS.  \\
+1889.
+\end{center}
+\clearpage
+
+%% -----File: 005.png
+\begin{center}
+\nbVS
+
+\textsc{Entered at Stationers' Hall}.
+
+\nbrule
+
+\textsc{Copyright, 1888, by}  \\
+CHARLES A. YOUNG,
+
+\nbrule
+
+\textsc{All Rights Reserved}.
+
+\nbVS
+
+\textsc{Typography by J.~S. Cushing \& Co., Boston, U.S.A.}  \\
+\nbrule[4cm] \\
+\textsc{Presswork by Ginn \& Co., Boston, U.S.A.}
+\end{center}
+%% -----File: 006.png---Folio v-------
+
+\clearpage
+\pdfbookmark[1]{Preface}{Preface}
+\chapter*{PREFACE.}
+\markboth{PREFACE.}{}
+\pagestyle{fancy} % for remainder of document
+\thispagestyle{empty}% just for first page of Preface, ToC, Introduction
+\nbrule\bigskip
+
+\textsc{The} present work is designed as a text-book of Astronomy
+suited to the \textit{general} course in our colleges and schools of
+science, and is meant to supply that amount of information
+upon the subject which may fairly be expected of every
+``liberally educated'' person. While it assumes the previous
+discipline and mental maturity usually corresponding to the
+latter years of the college course, it does not demand the
+peculiar mathematical training and aptitude necessary as the
+basis of a \textit{special} course in the science---only the most elementary
+knowledge of Algebra, Geometry, and Trigonometry
+is required for its reading. Its aim is to give a clear, accurate,
+and justly proportioned presentation of astronomical
+facts, principles, and methods in such a form that they can
+be easily apprehended by the average college student with a
+reasonable amount of effort.
+
+The limitations of time are such in our college course that
+probably it will not be possible in most cases for a class to
+take thoroughly everything in the book. The fine print is to
+be regarded rather as collateral reading, important to anything
+like a complete view of the subject, but not essential to
+the course. Some of the chapters can even be omitted in
+cases where it is found necessary to abridge the course as
+much as possible; \textit{e.g.}, the chapters on Instruments and on
+Perturbations.
+
+While the work is no mere compilation, it makes no claims
+to special originality: information and help have been drawn
+%% -----File: 007.png---Folio vi-------
+from all available sources. The author is under great obligations
+to the astronomical histories of Grant and Wolf, and
+especially to Miss Clerke's admirable ``History of Astronomy in
+the Nineteenth Century.'' Many data also have been drawn
+from Houzeau's valuable ``Vade Mecum de l'Astronomie.''
+
+It has been intended to bring the book well down to date,
+and to indicate to the student the sources of information on
+subjects which are necessarily here treated inadequately on
+account of the limitations of time and space.
+
+Special acknowledgments are due to Professor Langley and
+to his publishers, Messrs.\ Ticknor \&~Co., for the use of a
+number of illustrations from his beautiful book, ``The New
+Astronomy''; and also to D.~Appleton \&~Co.\ for the use of
+several cuts from the author's little book on the Sun. Professor
+Trowbridge of Cambridge kindly provided the original
+negative from which was made the cut illustrating the comparison
+of the spectrum of iron with that of the sun. Warner
+\& Swasey of Cleveland and Fauth \&~Co.\ of Washington have
+also furnished the engravings of a number of astronomical
+instruments.
+
+Professors Todd, Emerson, Upton, and McNeill have given
+most valuable assistance and suggestions in the revision of the
+proof; as indeed, in hardly a less degree, have several others.
+
+The author will consider it a great favor if those who may
+use the book will kindly communicate to him, either directly
+or through the publishers, any errata, in order that they
+may be promptly corrected.
+
+{\footnotesize\textsc{Princeton, N.~J.}, August, 1888.}
+
+\nbrule[4cm]
+
+{\footnotesize\textsc{Note}. In this issue of the book a number of errors which appeared in the
+first impression have been corrected.
+
+\indent\indent April, 1889.}
+%% -----File: 008.png---Folio vii-------
+
+\clearpage
+\pdfbookmark[1]{Table of Contents}{Table of Contents}
+\chapter*{TABLE OF CONTENTS.}
+\markboth{TABLE OF CONTENTS.}{}
+\thispagestyle{empty}% just for first page of Preface, ToC, Introduction
+\nbrule\bigskip
+
+{\small
+\newlength{\nbtocpagewidth}
+\settowidth{\nbtocpagewidth}{\nbtocpages{INDEX}}% Use index to get maximum width needed for page range (mmm-nnn)
+\begin{changemargin}{\parindent}{\nbtocpagewidth}
+\nbtocpagesheading
+
+\nbtocentry{INTRODUCTION}{INTRODUCTION}
+
+\nbtocentry{CHAPTERI}{CHAPTER I.---\textsc{The Doctrine of the Sphere}; Definitions and General
+Considerations}
+
+\nbtocentry{CHAPTERII}{\stretchyspace CHAPTER II.---\textsc{Astronomical Instruments}: the Telescope; Time-Keepers
+and Chronograph; the Transit Instrument and Accessories;
+the Meridian Circle and Reading Microscope; the Altitude and Azimuth
+Instrument; the Equatorial Instrument and Micrometer; the
+Sextant}
+
+\nbtocentry{CHAPTERIII}{CHAPTER III.---\textsc{Corrections to Astronomical Observations}: the Dip
+of the Horizon; Parallax; Semi-Diameter; Refraction; and Twilight}
+
+\nbtocentry{CHAPTERIV}{CHAPTER IV.---\textsc{Problems of Practical Astronomy}: the Determination
+of Latitude, of Time, of Longitude, of a Ship's Place at Sea,
+of Azimuth, and of the Apparent Right Ascension and Declination
+of a Heavenly Body; the Time of Sunrise or Sunset}
+
+\nbtocentry{CHAPTERV}{CHAPTER V.---\textsc{The Earth}: the Approximate Determination of its
+Dimensions and Form; Proofs of its Rotation; Accurate Determination
+of its Dimensions by Geodetic Surveys and Pendulum Observations;
+Determination of its Mass and Density}
+
+\nbtocentry{CHAPTERVI}{CHAPTER VI.---\textsc{The Earth's Orbital Motion}: the Motion of the
+Sun among the Stars; the Equation of Time; Precession; Nutation;
+Aberration; the Calendar}
+
+\nbtocentry{CHAPTERVII}{CHAPTER VII.---\textsc{The Moon}: her Orbital Motion; Distance and Dimensions;
+Mass, Density, and Superficial Gravity; Rotation and
+Librations; Phases; Light and Heat; Physical Condition; Influence
+exerted on the Earth; Surface Structure; and Possible Changes}
+
+\clearpage
+\nbtocpagesheading
+
+\nbtocentry{CHAPTERVIII}{CHAPTER VIII.---\textsc{The Sun}: Distance and Dimensions; Mass and Density;
+Rotation; Solar Eye-Pieces, and Study of the Sun's Surface;
+General Views as to Constitution; Sun Spots: their Appearance,
+Nature, Distribution, and Periodicity; the Spectroscope and the Solar
+Spectrum; Chemical Composition of the Sun; the Chromosphere and
+Prominences; the Corona}
+
+\nbtocentry{CHAPTERIX}{CHAPTER IX.---\textsc{The Sun's Light and Heat}: Comparison of Sunlight
+with Artificial Lights; the Measurement of the Sun's Heat, and Determination
+of the Solar Constant; the Pyrheliometer, Actinometer, and
+Bolometer; the Sun's Temperature; Maintenance of the Sun's Radiation;
+and Conclusions as to its Age and Future Endurance}
+%% -----File: 009.png---Folio viii-------
+
+\nbtocentry{CHAPTERX}{CHAPTER X.---\textsc{Eclipses}; Form and Dimensions of Shadows; Lunar
+Eclipses; Solar Eclipses, Total, Annular, and Partial; Ecliptic Limits,
+and Number of Eclipses in a Year; the Saros; Occultations}
+
+\nbtocentry{CHAPTERXI}{CHAPTER XI.---\textsc{Central Forces}; Equable Description of Areas;
+Areal, Linear, and Angular Velocities; Kepler's Laws and Inferences
+from them; Gravitation demonstrated by the Moon's Motion; Conic
+Sections as Orbits; the Problem of \textit{Two} Bodies; the ``Velocity from
+Infinity'' and its Relation to the Species of Orbit described by a
+Body moving under Gravitation; Intensity of Gravitation}
+
+\nbtocentry{CHAPTERXII}{CHAPTER XII.---\textsc{The Problem of Three Bodies}; Disturbing Forces;
+Lunar Perturbations and the Tides}
+
+\nbtocentry{CHAPTERXIII}{CHAPTER XIII.---\textsc{The Planets: their Motions, Apparent and
+Real}; the Ptolemaic, Tychonic, and Copernican Systems; the Orbits
+and their Elements; Planetary Perturbations}
+
+\nbtocentry{CHAPTERXIV}{CHAPTER XIV.---\textsc{The Planets themselves}; Methods of determining
+their Diameters, Masses, Densities, Times of Rotation, etc.; the ``Terrestrial
+Planets,''---Mercury, Venus, and Mars; the Asteroids; Intra-Mercurial
+Planets and the Zodiacal Light}
+
+\nbtocentry{CHAPTERXV}{CHAPTER XV.---\textsc{The Major Planets},---Jupiter, Saturn, Uranus, and
+Neptune}
+
+\clearpage
+\nbtocpagesheading
+
+\nbtocentry{CHAPTERXVI}{CHAPTER XVI.---\textsc{The Determination of the Sun's Horizontal Parallax
+and Distance}; Oppositions of Mars and Transits of Venus;
+Gravitational Methods; Determination by Means of the Velocity of
+Light}
+
+\nbtocentry{CHAPTERXVII}{CHAPTER XVII.---\textsc{Comets}: their Number, Motions, and Orbits; their
+Constituent Parts and Appearance; their Spectra, Physical Constitution,
+and Probable Origin}
+
+\nbtocentry{CHAPTERXVIII}{CHAPTER XVIII.---\textsc{Meteors}: Aerolites, their Fall and Physical Characteristics;
+Shooting Stars and Meteoric Showers; Connection between
+Meteors and Comets}
+
+\nbtocentry{CHAPTERXIX}{CHAPTER XIX.---\textsc{The Stars}: their Nature and Number; the Constellations;
+Star-Catalogues; Designation of Stars; their Proper Motions,
+and the Motion of the Sun in Space; Stellar Parallax}
+
+\nbtocentry{CHAPTERXX}{CHAPTER XX.---\textsc{The Light of the Stars}; Star Magnitudes and Photometry;
+Variable Stars; Stellar Spectra; Scintillation of the Stars}
+
+\nbtocentry{CHAPTERXXI}{CHAPTER XXI.---\textsc{Aggregations of Stars}: Double and Multiple Stars;
+Clusters; Nebulæ; the Milky Way, and Distribution of Stars in Space;
+Constitution of the Stellar Universe; Cosmogony and the Nebular
+Hypothesis}
+
+\nbtocentry{APPENDIX}{APPENDIX.---\textsc{Tables of Astronomical Data}}
+
+\nbtocentry{INDEX}{INDEX}
+
+\end{changemargin}
+}%end small
+
+%% -----File: 010.png---Folio ix-------
+%% -----File: 011.png---Folio x-------
+
+%% -----File: 012.png---Folio 1-------
+\mainmatter
+
+\clearpage
+\pdfbookmark[1]{Introduction}{Introduction}
+\chapter*{INTRODUCTION.}
+\chslabel{INTRODUCTION}
+\markboth{INTRODUCTION.}{}
+\thispagestyle{empty}
+\nbrule\bigskip
+
+\nbarticle{1.} \nbparatext{Astronomy} (\mytextgreek{>'astron nómos})
+is the science which treats of the
+heavenly bodies. As such bodies we reckon the sun and moon, the
+planets (of which the earth is one) and their satellites, comets and
+meteors, and finally the stars and nebulæ.
+
+We have to consider in Astronomy:---
+\begin{asparaenum}[(\itshape a\/\normalfont)]
+\item The motions of these bodies, both real and apparent, and the
+laws which govern these motions.
+\item Their forms, dimensions, and masses.
+\item Their nature and constitution.
+\item The effects they produce upon each other by their attractions,
+radiations, or by any other ascertainable influence.
+\end{asparaenum}
+
+It was an early, and has been a most persistent, belief that the
+heavenly bodies have a powerful influence upon human affairs, so
+that from a knowledge of their positions and ``aspects'' at critical
+moments (as for instance at the time of a person's birth) one could
+draw up a ``horoscope'' which would indicate the probable future.
+
+The \textit{pseudo-science} which was founded on this belief was named
+Astrology,---the elder sister of Alchemy,---and for centuries Astronomy
+was its handmaid; \textit{i.e.}, astronomical observations and calculations
+were made mainly in order to supply astrological data.
+
+At present the end and object of astronomical study is chiefly
+knowledge pure and simple; so far as now appears, its development
+has less direct bearing upon the material interests of mankind than
+that of any other of the natural sciences. It is not likely that great
+inventions and new arts will grow out of its laws and principles, such
+as are continually arising from physical, chemical, and biological
+discoveries, though of course it would be rash to say that such outgrowths
+are impossible. But the student of Astronomy must expect
+his chief profit to be intellectual, in the widening of the range of
+thought and conception, in the pleasure attending the discovery of
+simple law working out the most complicated results, in the delight
+%% -----File: 013.png---Folio 2-------
+over the beauty and order revealed by the telescope in systems otherwise
+invisible, in the recognition of the essential unity of the material
+universe, and of the kinship between his own mind and the infinite
+Reason that formed all things and is immanent in them.
+
+At the same time it should be said at once that, even from the
+lowest point of view, Astronomy is far from a useless science. The
+\textit{art of navigation} depends for its very possibility upon astronomical
+prediction. Take away from mankind their almanacs, sextants, and
+chronometers, and commerce by sea would practically stop. The
+science also has important applications in the survey of extended
+regions of country, and the establishment of boundaries, to say
+nothing of the accurate determination of time and the arrangement
+of the calendar.
+
+It need hardly be said that Astronomy is not separated from kindred
+sciences by sharp boundaries. It would be impossible, for instance,
+to draw a line between Astronomy on one side and Geology
+and Physical Geography on the other. Many problems relating to
+the formation and constitution of the earth belong alike to all three.
+
+\nbarticle{2.} Astronomy is divided into many branches, some of which, as
+ordinarily recognized, are the following:---
+
+{1.} \nbparatext{Descriptive Astronomy.}---This, as its name implies, is merely
+an orderly statement of astronomical facts and principles.
+
+\subparagraph{2.} \nbparatext{Practical Astronomy.}---This is quite as much an art as a
+science, and treats of the instruments, the methods of observation,
+and the processes of calculation by which astronomical facts are
+ascertained.
+
+\subparagraph{3.} \nbparatext{Theoretical Astronomy}, which treats of the calculations of orbits
+and ephemerides, including the effects of so-called ``perturbations.''
+
+\subparagraph{4.} \nbparatext{Mechanical Astronomy}, which is simply the application of mechanical
+principles to explain astronomical facts (chiefly the planetary
+and lunar motions). It is sometimes called \textit{Gravitational} Astronomy,
+because, with few exceptions, gravitation is the only force sensibly
+concerned in the motions of the heavenly bodies. Until within thirty
+years this branch of the science was generally designated as \textit{Physical
+Astronomy}, but the term is now objectionable because of late it has
+been used by many writers to denote a very different and comparatively
+new branch of the science; viz.,---
+%% -----File: 014.png---Folio 3-------
+
+\subparagraph{5.} \nbparatext{Astronomical Physics, or Astro-physics.}---This treats of the
+physical characteristics of the heavenly bodies, their brightness and
+spectroscopic peculiarities, their temperature and radiation, the nature
+and condition of their atmospheres and surfaces, and all phenomena
+which indicate or depend on their physical condition.
+
+\subparagraph{6.} \nbparatext{Spherical Astronomy.}---This, discarding all consideration of
+absolute dimensions and distances, treats the heavenly bodies simply
+as objects moving on the ``surface of the celestial sphere'': it has to
+do only with angles and directions, and, strictly regarded, is in fact
+merely Spherical Trigonometry applied to Astronomy.
+
+\nbarticle{3.} The above-named branches are not distinct and separate, but
+they overlap in all directions. Spherical Astronomy, for instance,
+finds the demonstration of many of its formulæ in Gravitational
+Astronomy, and their application appears in Theoretical and Practical
+Astronomy. But valuable works exist bearing all the different
+titles indicated above, and it is important for the student to know
+what subjects he may expect to find discussed in each; for this
+reason it has seemed worth while to name and define the several
+branches, although they do not distribute the science between them
+in any strictly logical and mutually exclusive manner.
+
+In the present text-book little regard will be paid to these subdivisions,
+since the object of the work is not to present a complete
+and profound discussion of the subject such as would be demanded
+by a professional astronomer, but only to give so much knowledge of
+the facts and such an understanding of the principles of the science
+as may fairly be considered essential to a liberal education. If this
+result is gained in the reader's case, it may easily happen that he will
+wish for more than he can find in these pages, and then he must have
+recourse to works of a higher order and far more difficult, dealing
+with the subject more in detail and more thoroughly.
+
+\sloppy
+To master the present book no further preparation is necessary
+than a very elementary knowledge of Algebra, Geometry, and Trigonometry,
+and a similar acquaintance with Mechanics and Physics,
+especially Optics. While nothing short of high mathematical attainments
+will enable one to become eminent in the science, yet a perfect
+comprehension of all its fundamental methods and principles, and a
+very satisfactory acquaintance with its main results, is quite within
+the reach of every person of ordinary intelligence, without any more
+extensive training than may be had in our common schools. At the
+%% -----File: 015.png---Folio 4-------
+same time the necessary statements and demonstrations are so much
+facilitated by the use of trigonometrical terms and processes that it
+would be unwise to dispense with them entirely in a work to be used
+by pupils who have already become acquainted with them.
+
+\fussy
+In discussing the different subjects which present themselves, the
+writer will adopt whatever plan appears best fitted to convey to the
+student clear and definite ideas, and to impress them upon the mind.
+Usually it will be best to proceed in the Euclidean order, by first
+stating the fact or principle in question, and then explaining its
+demonstration. But in some cases the inverse process is preferable,
+and the conclusion to be reached will appear gradually unfolding
+itself as the result of the observations upon which it depends, just as
+its discovery came about.
+
+The frequent references to ``Physics'' refer to the ``Elementary
+Text-Book of Physics,'' by Anthony \&~Brackett; 3d edition, 1887.
+Wiley \&~Sons, N.Y.
+\chelabel{INTRODUCTION}
+%% -----File: 016.png---Folio 5-------
+
+\Chapter{I}{The Doctrine of the Sphere}
+\nbchaptercenter{THE ``DOCTRINE OF THE SPHERE,'' DEFINITIONS, AND GENERAL
+CONSIDERATIONS.}
+
+\textsc{Astronomy}, like all the other sciences, has a terminology of its
+own, and uses technical terms in the description of its facts and
+phenomena. In a popular essay it would of course be proper to
+avoid such terms as far as possible, even at the expense of circumlocutions
+and occasional ambiguity; but in a text-book it is desirable
+that the reader should be introduced to the most important of them
+at the very outset, and made sufficiently familiar with them to use
+them intelligently and accurately.
+
+%       *       *       *       *       *
+
+\nbarticle{4.} \nbparatext{The Celestial Sphere.}---To an observer looking up to the
+heavens at night it seems as if the stars were glittering points attached
+to the inner surface of a dome; since we have no direct perception of
+their distance there is no reason to imagine some nearer than others,
+and so we involuntarily think of the surface as \textit{spherical} with ourselves
+in its centre. Or if we sometimes feel that the stars and
+other objects in the sky really differ in distance, we still instinctively
+imagine an immense sphere surrounding and enclosing all. Upon
+this sphere we imagine lines and circles traced, resembling more or
+less the meridians and parallels upon the surface of the earth, and
+by reference to these circles we are able to describe intelligently the
+apparent positions and motions of the heavenly bodies.
+
+This celestial sphere may be regarded in either of two different
+ways, both of which are correct and lead to identical results.
+
+\begin{asparaenum}[(\itshape a\/\normalfont)]
+\item We may imagine it, in the first place, as transparent, and of
+merely finite (though undetermined) dimensions, \textit{but in some way
+so attached to, and connected with, the observer that his eye always
+remains at its centre wherever he goes}. Each observer, in this way
+of viewing it, carries his own sky with him, and is the centre of his
+own heavens.
+
+\item Or, in the second place,---and this is generally the more convenient
+way of regarding the matter,---we may consider the celestial
+%% -----File: 017.png---Folio 6-------
+sphere as mathematically \textit{infinite} in its dimensions; then, let the
+observer go where he will, he cannot sensibly get away from its
+centre. Its radius being ``greater than any assignable quantity,''
+the size of continents, the diameter of the earth, the distance of the
+sun, the orbits of planets and comets, even the spaces between the
+stars, are all insignificant, and the whole visible universe shrinks
+\textit{relatively} to a mere point at its centre. In what follows we shall use
+this conception of the celestial sphere.\footnote
+  {To most persons the sky appears, not a true hemisphere, but a \textit{flattened} vault,
+  as if the horizon were more remote than the zenith. This is a subjective effect
+  due mainly to the intervening objects between us and the horizon. The sun and
+  moon when rising or setting look much larger than when they are higher up, for
+the same reason.}
+\end{asparaenum}
+
+\includegraphicsouter{illo001}{\textsc{Fig.~1.}}
+
+The apparent place of any celestial body will then be the point
+on the celestial sphere where the line drawn from the eye of the
+observer in the direction in which he sees the object, and produced
+indefinitely, pierces the sphere. Thus, in \figref{illo001}{Figure~1}, $A$, $B$, $C$ are
+the apparent places of $a$, $b$, and $c$,
+the observer being at $O$. The apparent
+place of a heavenly body evidently
+depends solely upon its \textit{direction}, and
+is wholly independent of its \textit{distance}
+from the observer.
+
+\nbarticle{5.} \nbparatext{Linear and Angular Dimensions.}---Linear
+dimensions are such as may
+be expressed in \textit{linear} units; \textit{i.e.}, in
+miles, feet, or inches; in metres or
+millimetres. Angular dimensions are
+expressed in \textit{angular} units; \textit{i.e.}, in
+right angles, in radians,\footnote
+  {A \textit{radian} is the angle which is measured by an arc equal in
+  length to radius.
+  Since a circle whose radius is unity has a circumference of $2 \pi$, and contains $360°$,
+  or $21,600'$, or $1,296,000''$, it follows that a radian contains $\left(\dfrac{360}{2\pi}\right)°$
+  or $\left(\dfrac{21600}{2\pi}\right)'$, or
+  $\left(\dfrac{1296000}{2\pi}\right)''$; \textit{i.e.}\ (approximately), a radian
+  $= 57.3°= 3437.7'= 206264.8''$. Hence,
+  to reduce to seconds of arc an angle expressed in radians, we must multiply
+  it by the number $206264.8$; a relation of which we shall have to make frequent
+  use.
+
+  See Halsted's Mensuration, p.~25.}
+or (more commonly in astronomy) in degrees,
+minutes, and seconds. Thus, for instance, the \textit{linear} semi-diameter
+%% -----File: 018.png---Folio 7-------
+of the sun is about 697,000 kilometers (433,000 miles),
+while its \textit{angular} \DPtypo{semidiameter}{semi-diameter} is about $16'$, or a little more than
+a quarter of a degree. Obviously, angular units alone can properly
+be used in describing apparent distances and dimensions in the sky.
+For instance, one cannot say correctly that the two stars which are
+known as ``the pointers'' are two or five or ten \textit{feet} apart: their
+distance is about five \textit{degrees}.
+
+It is sometimes convenient to speak of ``\textit{angular area},'' the unit
+of which is a ``square degree'' or a ``square minute''; \textit{i.e.}, a small
+square in the sky of which each side is $1°$ or $1'$. Thus we may compare
+the angular area of the constellation Orion with that of Taurus,
+in \textit{square degrees}, just as we might compare Pennsylvania and New
+Jersey in square miles.
+
+\nbarticle{6.} \nbparatext{Relation between the Distance and Apparent Size of Object.}---Suppose
+a globe having a radius $BC$ equal to $r$. As seen from
+the point $A$ (\figref{illo002}{Fig.~2}) its apparent (\textit{i.e.}, \textit{angular}) \DPtypo{semidiameter}{semi-diameter} will
+be $BAC$ or $s$, its distance being $AC$ or $R$.
+
+\includegraphicsmid{illo002}{\textsc{Fig.~2.}}
+
+We have immediately from Trigonometry, since $B$ is a right angle,
+\[
+  \sin s = \frac{r}{R}.
+\]
+
+If, as is usual in Astronomy, the diameter of the object is small
+as compared with its distance, we may write
+\[
+s = \frac{r}{R},
+\]
+which gives $s$ in \textit{radians} (not in degrees or seconds). If we wish it
+in the ordinary angular units,
+\[
+s° = 57.3 \frac{r}{R}, \quad \text{or} \quad s'' = 206264.8\frac{r}{R}.
+\]
+In either form of the equation we see that the apparent diameter
+\textit{varies directly as the linear diameter, and inversely as the distance}.
+%% -----File: 019.png---Folio 8-------
+
+In the case of the moon, $R =$ about 239,000 miles; and $r$, 1081
+miles. Hence $s = \frac{1081}{239000} = \frac{1}{221}$ of a radian, which is a little more
+than $\frac{1}{4}$ of a degree.
+
+\begin{fineprint}
+It may be mentioned here as a rather curious fact that most persons say
+that the moon appears about \textit{a foot in diameter}; at least, this seems to
+be the average estimate. This implies that the surface of the sky appears
+to them only about 110 feet away, since that is the distance at which a disc
+one foot in diameter would have an angular diameter of $\frac{1}{110}$ of a radian, or $\frac{1}{2}°$.
+\end{fineprint}
+
+\nbarticle{7.} \nbparatext{Vanishing Point.}---Any system of parallel lines produced in
+one direction will \textit{appear} to pierce the celestial sphere at a single
+point. They actually pierce it at different points, separated on the
+surface of the sphere by linear distances, equal to the actual distances
+between the lines, but on the infinitely distant surface these linear
+distances, being only finite, become invisible, subtending at the centre
+angles less than anything assignable. The different points, therefore,
+coalesce into a \textit{spot} of apparently infinitesimal size---the so-called
+``vanishing point'' of perspective. Thus the axis of the earth and
+\textit{all lines parallel to this axis} point to the celestial pole.
+
+\nbthought
+
+In order to describe intelligibly the apparent position of an object
+in the sky, it is necessary to have certain points and lines from which
+to reckon. We proceed to define some of these which are most
+frequently used.
+
+\nbarticle{8.} \nbparatext{The Zenith.}---The Zenith is the \textit{point vertically overhead}, \textit{i.e.},
+the point where a plumb-line, produced upwards, would pierce the
+sky: it is determined by the \textit{direction of gravity} where the observer
+stands.
+
+If the earth were exactly spherical, the zenith might also be defined
+as the point where a line drawn \textit{from the centre of the earth upward
+through the observer} meets the sky. But since, as we shall see
+hereafter, the earth is not an exact globe, this second definition indicates
+a point known as the \textit{Geocentric Zenith}, which is not identical
+with the \textit{True} or \textit{Astronomical Zenith}, determined by the direction of
+gravity.
+
+\nbarticle{9.} \nbparatext{The Nadir.}---The Nadir is the point opposite the zenith---under
+foot, of course.
+
+Both zenith and nadir are derived from the Arabic, which language
+has also given us many other astronomical terms.
+%% -----File: 020.png---Folio 9-------
+
+\nbarticle{10.} \nbparatext{Horizon.}---The Hor{\=\i}zon\footnote{%
+  Beware of the common, but vulgar, pronunciation,
+  \textit{Hór{\=\i}zon}.}
+is a great circle of the celestial
+sphere, having the zenith and nadir as its poles: it is therefore
+half-way between them, and 90° from each.
+
+A \textit{horizontal plane}, or the \textit{plane of the horizon}, is a plane perpendicular
+to the direction of gravity, and the horizon may also be correctly
+defined as the intersection of the celestial sphere by this plane.
+
+Many writers make a distinction between the \textit{sensible} and \textit{rational}
+horizons. The plane of the sensible horizon passes through the
+observer; the plane of the rational horizon passes through the centre
+of the earth, parallel to the plane of the sensible horizon: these two
+planes, parallel to each other, and everywhere about 4000 miles
+apart, trace out on the sky the two horizons, the sensible and the
+rational. It is evident, however, that on the infinitely distant surface
+of the celestial sphere, the two traces sensibly coalesce into one single
+great circle, which is the horizon as first defined. In strictness,
+therefore, while we can distinguish between the two horizontal \textit{planes},
+we get but one \textit{horizon circle} in the sky.
+
+\nbarticle{11.} \nbparatext{The Visible Horizon} is the line where sky and earth meet.
+On land it is an irregular line, broken by hills and trees, and of no
+astronomical value; but at sea it is a true circle, and of great importance
+in observation. It is not, however, a \textit{great} circle, but,
+technically speaking, only a \textit{small} circle; depressed below the true
+horizon by an amount depending upon the observer's elevation above
+the water. This depression is called the \textit{Dip of the Horizon}, and will
+be discussed further on.
+
+\nbarticle{12.} \nbparatext{Vertical Circles.}---These are great circles passing through
+the zenith and nadir, and therefore necessarily perpendicular to the
+horizon---\textit{secondaries} to it, to use the technical term.
+
+\textbf{Parallels of Altitude,} or \textbf{Almucantars.}---These are small circles
+parallel to the horizon: the term Almucantar is seldom used.
+
+The points and circles thus far defined are determined entirely by
+the \textit{direction of gravity} at the station occupied by the observer.
+
+\nbthought
+
+\nbarticle{13.} \nbparatext{The Diurnal Rotation of the Heavens.}---If one watches the
+sky for a few hours some night, he will find that, while certain stars
+rise in the east, others set in the west, and nearly all the constellations
+change their places. Watching longer and more closely, it will
+%% -----File: 021.png---Folio 10-------
+appear that the stars move in circles, uniformly, in such a way as
+not to disturb their relative configurations, but as if they were
+attached to the inner surface of a revolving sphere, turning on its
+axis once a day. The path thus daily described by a star is called its
+``\textit{diurnal circle}.''
+
+It is soon evident that in our latitude the visible ``pole'' of this
+sphere---the point about which it turns---is in the north, not quite
+half-way up from the horizon to the zenith, for in that region the stars
+hardly move at all, but keep their places all night long.
+
+\nbarticle{14.} \nbparatext{The Poles.}---The Poles may be defined as the two points in the
+sky, one in the northern hemisphere and one in the southern, where a
+\textit{star's diurnal circle reduces to zero}; \textit{i.e.}, points where, if a star were
+placed, it would suffer no apparent change of place during the whole
+twenty-four hours. The line joining these poles is, of course, the
+\textit{axis} of the celestial sphere, about which it seems to rotate daily.
+
+The exact place of the pole may be found by observing some star
+very near the pole at two times 12 hours apart, and taking the middle
+point between the two observed places of the star.
+
+The definition of the pole just given is independent of any theory
+as to the cause of the apparent rotation of the heavens. If, however,
+%% -----File: 022.png---Folio 11-------
+we admit that it is due to the earth's rotation on its axis, then
+we may define the poles as the \textit{points where the earth's axis produced
+pierces the celestial sphere}.
+
+\nbarticle{15.} \nbparatext{The Pole-star (Polaris).}---The place of the northern pole is
+very conveniently marked by the \textit{Pole-star}, a star of the second magnitude,
+which is now only about $1\frac{1}{4}°$ from the pole: we say \textit{now}, because
+on account of a slow change in the direction of the earth's
+axis, called ``precession'' (to be discussed later), the distance between
+the pole-star and the pole is constantly changing, and has been
+for several centuries gradually decreasing.
+
+The pole-star stands comparatively solitary in the sky, and may
+easily be recognized by means of the so-called ``pointers,''---two
+stars in the ``dipper'' (in the constellation of Ursa Major)---which
+point very nearly to it, as shown in \figref{illo003}{Fig.~3}. The pole is very nearly
+on the line joining Polaris with the star Mizar ($\zeta$~Urs.\ Maj., at the
+bend in the handle of the dipper), and at a distance just about one-quarter
+of the distance between the pointers, which are nearly $5°$
+apart.
+
+The southern pole, unfortunately, is not so marked by any conspicuous
+star.
+
+\includegraphicsmid{illo003}{\textsc{Fig.~3.}---The Pole Star and the Pointers.}
+
+\nbarticle{16.} \nbparatext{The Celestial Equator}, or \nbparatext{Equinoctial Circle}.---This is a great
+circle midway between the two poles, and of course $90°$ from each.
+It may also be defined as the intersection of the plane of the earth's
+equator with the celestial sphere. It derives its name from the fact
+that, at the two dates in the year when the sun crosses this circle---about
+March 20 and Sept.~22---the day and night are equal in length.
+
+\nbthought
+
+\nbarticle{17.} \nbparatext{The Vernal Equinox}, or \nbparatext{First of Aries}.---The Equinox, strictly
+speaking, is the \textit{time when} the sun crosses the equator, but the term
+has come by accommodation to denote also the \textit{point where} it crosses,
+though in strictness it should be called the ``\textit{Equinoctial Point}.''
+This crossing occurs twice a year, once in September and once in
+March, and the \textit{Vernal Equinox is the point on the equator where
+the sun crosses it in the spring}. It is sometimes called the \textit{Greenwich
+of the Celestial Sphere}, because it is used as a reference point
+in the sky, much as Greenwich is on the earth. Its position is not
+marked by any conspicuous star.
+
+Why this point is also called the ``First of Aries'' will appear
+later, when we come to speak of the zodiac and its ``signs.''
+%% -----File: 023.png---Folio 12-------
+
+\nbarticle{18.} \nbparatext{Hour-Circles.}---Hour-circles are great circles of the celestial
+sphere passing through its poles, and consequently perpendicular
+to the celestial equator. They correspond exactly to the meridians
+of the earth, and some writers call them ``Celestial Meridians''; but
+the term is objectionable, as likely to lead to confusion with \textit{the}
+Meridian, to be noted immediately.
+
+\nbthought
+
+\nbarticle{19.} \nbparatext{The Meridian and Prime Vertical.}---\textit{The Meridian is the great
+circle passing through the pole and the zenith.} Since it is a great
+circle, it must necessarily pass through \textit{both} poles, and through the
+nadir as well as the zenith, and must be perpendicular both to the
+equator and to the horizon.
+
+It may also be correctly defined as the \textit{Vertical Circle} which passes
+through the \textit{pole}; or, again, as the \textit{Hour-Circle} which passes through
+the \textit{zenith}, since all vertical circles must pass through the zenith, and
+all hour-circles through the pole.
+
+\textit{The Prime Vertical} is the Vertical Circle (passing through the
+zenith) at right angles to the meridian; hence lying \textit{east and west}
+on the celestial sphere.
+
+\nbarticle{20.} \nbparatext{The Cardinal Points.}---The North and South Points are the
+points on the horizon where it is intersected by the meridian. The
+East and West Points are where it is cut by the prime vertical, and
+also by the equator. The North \textit{Point}, which is on the horizon, must
+not be confounded with the North \textit{Pole}, which is not on the horizon,
+but at an elevation equal (see \artref{Art.}{30}) to the latitude of the observer.
+
+\nbthought
+
+With these circles and points of reference we have now the means
+to describe intelligibly the position of a heavenly body, in several
+different ways.
+
+We may give its \textit{altitude} and \textit{azimuth}, or its \textit{declination} and \textit{hour-angle};
+or, if we know the time, its \textit{declination} and \textit{right ascension}.
+Either of these pairs of co-ordinates, as they are called, will define
+its place in the sky.
+
+\nbarticle{21.} \nbparatext{Altitude and Zenith Distance}\ (\figref{illo004}{Fig.~4}).---The Altitude of a
+heavenly body is \textit{its angular elevation above the horizon}, and is measured
+by the arc of the vertical circle passing through the body, and
+intercepted between it and the horizon.
+%% -----File: 024.png---Folio 13-------
+
+The Zenith Distance of a body is simply its angular distance from
+the zenith, and is the complement of the altitude. Altitude + Zenith
+Distance = $90°$.
+
+\nbarticle{22.} \nbparatext{Azimuth and Amplitude} (\figref{illo004}{Fig.~4}).---The Azimuth of a body
+is the \textit{angle at the zenith, between the meridian and the vertical circle,
+which passes through the body}. It is measured also by the arc of the
+horizon intercepted between the north or south point, and the foot
+of this vertical. The word is of Arabic origin, and has the same
+meaning as the \textit{true bearing} in surveying and navigation.
+
+\includegraphicsmid{illo004}{\textsc{Fig.~4.}---The Horizon and Vertical Circles.\\[1ex]
+\begin{tabular}{@{}l@{\;}|@{\;}l@{}}
+  $O$, the place of the Observer. & $M$, some Star.\\
+  $OZ$, the Observer's Vertical.  & $ZMH$, arc of the Star's Vertical Circle.\\
+  $Z$, the Zenith; $P$, the Pole. & $TMR$, the Star's Almucantar.\\
+  $SENW$, the Horizon.            & Angle $TZM$, or arc $SWNEH$, Star's \textit{Azimuth}.\\
+  $SZPN$, the Meridian.           & Arc $HM$, Star's \textit{Altitude}.\\
+  $EZW$, the Prime Vertical.      & Arc $ZM$, Star's \textit{Zenith Distance}.
+\end{tabular}}
+
+The \textit{Amplitude} of a body is the complement of the azimuth.
+Azimuth + Amplitude = $90°$.
+
+\begin{fineprint}
+There are various ways of reckoning azimuth. Many writers express it
+in the same manner as \textit{the bearing} is expressed in surveying; \textit{i.e.}, so many
+degrees east or west of north or south; N.~$20°$ E., etc. The more usual
+way at present is, however, to reckon it in degrees from the south point clear
+round through the west to the point of beginning: thus an object in the
+SW. would have an azimuth of $45°$; in the NW., $135°$; in the N., $180°$; in
+the NE., $225°$; and in the SE., $315°$. For example, to find a star whose
+azimuth is $260°$, and altitude $60°$, we must face N.~$80°$ E., and then look
+up two-thirds of the way to the zenith. The object in this case has an
+\textit{amplitude} of $10°$ N. of W., and a zenith distance of $30°$. Evidently both
+the azimuth and altitude of a heavenly body are continually changing, except
+in certain very special cases.
+\end{fineprint}
+%% -----File: 025.png---Folio 14-------
+
+In \figref{illo004}{Fig.~4}, $SENW$ represents the horizon, $S$ being the south point,
+and $Z$ the zenith. The angle $SZM$, which numerically equals the
+arc $SH$, is the \textit{Azimuth} of the star $M$; while $EZM$, or $EH$ is its
+\textit{Amplitude}. $MH$ is its \textit{Altitude}, and $ZM$ its \textit{Zenith Distance}.
+
+\nbarticle{23.} \nbparatext{Declination and Polar Distance} (\figref{illo005}{Fig.~5}).---The Declination of
+a heavenly body is its \textit{angular distance north or south of the celestial
+equator}, and is measured by the arc of the hour-circle passing through
+the object, intercepted between it and the equator. It is reckoned
+positive ($+$) north of the celestial equator, and negative ($-$) south
+of it. Evidently it is precisely analogous to the latitude of a place
+on the earth. The \textit{north-polar distance} of a star is its angular distance
+from the North Pole, and is simply the complement of the
+declination. Declination $+$ North-Polar Distance = $90°$.
+
+The declination of a star remains always the same; at least, the
+slow changes that it undergoes need not be considered for our
+present purpose. ``\textit{Parallels of Declination}'' are small circles parallel
+to the celestial equator.
+
+\nbarticle{24.} \nbparatext{The Hour-Angle} (\figref{illo005}{Fig.~5}).---The Hour-Angle of a star is the
+\textit{angle at the pole between the meridian and the hour-circle passing
+through the star}. It may be reckoned in degrees; but it also
+may be, and most commonly is, reckoned in \textit{hours}, \textit{minutes}, and
+\textit{seconds of time}; the hour being equivalent to fifteen degrees, and
+the minute and second of time being equal to fifteen minutes and seconds
+of arc respectively.
+
+Of course the hour-angle of an object is continually changing,
+being zero when the object is on the meridian, one hour, or fifteen
+degrees, when it has moved that amount westward, and so on.
+
+\nbarticle{25.} \nbparatext{Right Ascension} (\figref{illo005}{Fig.~5}).---The Right Ascension of a star
+is \textit{the angle at the pole between the star's hour-circle and the hour-circle}
+(\textit{called the Equinoctial Colure}), \textit{which passes through the vernal
+equinox.}
+
+It may be defined also as the arc of the equator, intercepted
+between the vernal equinox and the foot of the star's hour-circle.
+
+It is always reckoned from the equinox \textit{toward the east}; sometimes
+in degrees, but usually in \textit{hours, minutes, and seconds of time.
+The right ascension, like the declination, remains unchanged by the
+diurnal motion.}
+%% -----File: 026.png---Folio 15-------
+
+\nbarticle{26.} \nbparatext{Sidereal Time} (\figref{illo005}{Fig.~5}).---For many astronomical purposes
+it is convenient to reckon time, not by the sun's position in the sky,
+but by that of the vernal equinox.
+
+\includegraphicsmid{illo005}{\textsc{Fig.~5.}---Hour-Circles, etc.\\[1ex]
+\begin{tabular}{@{}p{.48\textwidth}|p{.48\textwidth}@{}}
+\hangindent=2em  $O$, place of the Observer; $Z$, his Zenith.
+
+\hangindent=2em  $SENW$, the Horizon.
+
+\hangindent=2em  $POP'$, line parallel to the Axis of the Earth.
+
+\hangindent=2em  $P$ and $P'$, the two Poles of the Heavens.
+
+\hangindent=2em  $EQWT$, the Celestial Equator, or Equinoctial.
+
+\hangindent=2em  $X$, the Vernal Equinox, or ``First of Aries.''
+
+&
+\hangindent=2em  $PXP'$ the Equinoctial Colure, or Zero Hour-Circle.
+
+\hangindent=2em  $m$, some Star.
+
+\hangindent=2em  $Ym$, the Star's \textit{Declination}; $Pm$, its \textit{North-polar Distance.}
+
+\hangindent=2em  Angle $mPR$ = arc $QY$, the Star's (eastern)
+         \textit{Hour-Angle}; = $24^\text{h}$ \textit{minus} Star's (western)
+         Hour-Angle.
+
+\hangindent=2em  Angle $XPm$ = arc $XY$, Star's \textit{Right Ascension},
+         Sidereal time at the moment = $24^\text{h}$ \textit{minus}
+         angle $XPQ$.
+\end{tabular}}
+
+\textit{The Sidereal Time} at any moment may be defined as \textit{the hour-angle
+of the vernal equinox}. It is sidereal \textit{noon}, when the equinoctial
+point is on the meridian; 1~o'clock (sidereal) when its hour-angle
+is $15°$; and 23~o'clock when its hour-angle is $345°$, \textit{i.e.}, when the vernal
+equinox is an hour \textit{east} of the meridian; the time being reckoned
+round through the whole 24~hours. On account of the annual motion
+of the sun among the stars, the \textit{Solar Day}, by which time is reckoned
+for ordinary purposes, is about 4~minutes longer than the sidereal
+day. The exact difference is $3^{\text{m}}$ $56^{\text{s}}$.394 (sidereal), or just one day
+in a year; there being $366\frac{1}{4}$ \textit{sidereal} days in the year, as against
+$365\frac{1}{4}$ \textit{solar} days.
+
+\nbarticle{27.} \nbparatext{Observatory Definition of Right Ascension.}---It is evident from
+the above definition of sidereal time, that we may also define the
+Right Ascension of a star as \textit{the sidereal time taken the star crosses
+the meridian}. The Star and the Vernal Equinox are both of them
+%% -----File: 027.png---Folio 16-------
+fixed points in the sky, and do not change their relative position during
+the sky's apparent daily revolution; a given star, therefore,
+always comes to the meridian of any observer the same number of
+hours after the vernal equinox has passed; and this number of hours
+is the sidereal time at the moment of the star's transit, and measures
+its right ascension. In the observatory, this definition of right ascension
+is the most natural and convenient.
+
+It is obvious that the right ascension of a star corresponds in the
+sky exactly with the \textit{longitude} of a place on the earth; terrestrial
+longitude being reckoned from Greenwich, just as right ascension
+is reckoned from the vernal equinox.
+
+N.B.\quad \textit{We shall find hereafter that the \emph{stars} have latitudes and
+longitudes of their own; but unfortunately these \emph{celestial} latitudes and
+longitudes do not correspond to the terrestrial, and great care is necessary
+to prevent confusion}. (See \artref{Art.}{179}.)
+
+\nbarticle{28.} An \textit{armillary sphere}, or some equivalent apparatus, is almost
+essential to enable a beginner to get correct ideas of the points,
+circles, and co-ordinates defined above, but the figures will perhaps
+be of assistance.
+
+The first of them (\figref{illo004}{Fig.~4}) represents the horizon, meridian, and
+prime vertical, and shows how the position of a star is indicated by
+its altitude and azimuth. This framework of circles, depending
+upon the direction of gravity, of course always remains \textit{apparently}
+unchanged in position, as if attached directly to the earth, while the
+sky apparently turns around outside it.
+
+The other figure (\figref{illo005}{Fig.~5}) represents the system of points and
+circles which depend upon the earth's rotation, and are independent
+of the direction of gravity. The vernal equinox and the hour-circles
+apparently revolve with the stars while the pole remains fixed upon
+the meridian, and the equator and parallels of declination, revolving
+truly in their own planes, also appear to be at rest in the sky. But
+the whole system of lines and points represented in the figure (horizon
+and meridian alone excepted) may be considered as attached
+to, or marked out upon, the inner surface of the celestial vault and
+whirling with it.
+
+It need hardly be said that the ``appearances are deceitful''---that
+which is really carried around by the earth's rotation is the
+observer, with his plumb-line and zenith, his horizon and meridian;
+while the stars stand still---at least, their motions in a day are insensible
+as seen from the earth.
+%% -----File: 028.png---Folio 17-------
+
+\begin{fineprint}
+At the poles of the earth, which are, mathematically speaking, ``singular''
+points, the definitions of the Meridian, of North and South, etc., break
+down.
+
+There the pole (celestial) and zenith coincide, and any number of circles
+may be drawn through the two points, which have now become one. The
+horizon and equator coalesce, and the only direction on the earth's surface
+is due south (or north)---east and west have vanished.
+
+A single step of the observer will, however, remedy the confusion: zenith
+and pole will separate, and his meridian will again become determinate.
+\end{fineprint}
+
+\nbarticle{29.} To recapitulate: The \textit{direction of gravity} at the point where
+the observer stands determines the Zenith and Nadir, the Horizon, and
+the Almucantars (parallel to the Horizon), and all the vertical circles.
+One of the verticals, the \textit{Meridian}, is singled out from the rest by
+the circumstance that it passes through the \textit{pole} of the sky, marking
+the North and South Points where it cuts the horizon.
+
+Altitude and Azimuth (or their complements, Zenith Distance
+and Amplitude) are the co-ordinates which designate the position
+of a body by reference to the Zenith and the Meridian.
+
+Similarly, the \textit{direction of the earth's axis} (which is independent
+of the observer's place on the earth) determines the Poles, the
+Equator, the Parallels of Declination, and the Hour-Circles. \textit{Two}
+of these Hour-Circles are singled out as reference lines; one of them,
+the Meridian, which passes through the Zenith, and is a purely
+\textit{local} reference line; the other, the Equinoctial Colure, which passes
+through the Vernal Equinox, a point chosen from its relation to the
+sun's annual motion. Declination and \textit{Hour-Angle} are the co-ordinates
+which refer the place of a star to the Pole and the Meridian;
+while Declination and \textit{Right Ascension} refer it to the Pole and Equinoctial
+Colure. The latter are the co-ordinates usually employed in
+star-catalogues and ephemerides to define the positions of stars and
+planets, and correspond exactly to Latitude and Longitude on the
+earth, by means of which geographical positions are designated.
+
+\nbarticle{30.} \nbparatext{Relation of the Apparent Diurnal Motion of the Sky to the
+Observer's Latitude.}---Evidently the apparent motions of the stars
+will be considerably influenced by the station of the observer, since
+the place of the pole in the sky will depend upon it. The \textit{Altitude}
+of the pole, or its \textit{height in degrees above} the horizon, is always equal
+to the \textit{Latitude} of the observer. Indeed, the German word for latitude
+(astronomical) is \textit{Polhöhe; i.e.}, simply ``Pole-height.''
+%% -----File: 029.png---Folio 18-------
+
+This will be clear from \figref{illo006}{Fig.~6}. The latitude of a place is
+the \textit{angle between its plumb-line and the plane of the equator}; the
+angle $ONQ$ in the figure. [If the earth were truly spherical, $N$
+would coincide with $C$, the centre of the earth. The ordinary
+definition of latitude given in the geographies disregards the slight
+difference.]
+
+\includegraphicsmid{illo006}{\textsc{Fig.~6.}---Relation of Latitude to the Elevation of the Pole.}
+
+Now the angle $H'OP''$ is equal to $ONQ$, because their sides are
+mutually perpendicular; and it is also the \textit{altitude of the pole}, because
+the line $HH'$ is horizontal at $O$, and $OP''$ is parallel to $PP'$, the
+earth's axis, and therefore points to the celestial pole.
+
+This fundamental relation, \textit{that the altitude of the celestial pole is
+the Latitude of the observer}, cannot be too strongly impressed on the
+student's mind. The usual symbol for the latitude of a place is $\phi$.
+
+\nbarticle{31.} \nbparatext{The Right Sphere.}---If the observer is situated at the
+earth's equator, \textit{i.e.}, in latitude zero ($\phi = 0$), the pole will be in the
+horizon, and the equator will pass vertically overhead through the
+zenith.
+
+The stars will rise and set vertically, and their diurnal circles will
+all be bisected by the horizon, so that they will be 12 hours above
+it and 12 below. This aspect of the heavens is called the \textit{Right
+Sphere}.
+
+\nbarticle{32.} \nbparatext{The Parallel Sphere.}---If the observer is at the pole of the
+earth ($\phi = 90°$), then the celestial pole will be in the zenith, and
+the equator will coincide with the horizon. If he is at the \textit{North}
+Pole, all stars north of the celestial equator will remain permanently
+%% -----File: 030.png---Folio 19-------
+above the horizon, never rising or falling at all, but sailing around
+on circles of altitude (or \textit{Almucantars}) parallel to the horizon.
+Stars in the Southern Hemisphere, on the other hand, would never
+rise to view. As the sun and the moon move in such a way that
+during half the time they are alternately north and south of the
+equator, they will be half the time above the horizon and half the
+time below it. The moon would be visible for about a fortnight at a
+time, and the sun for six months.
+
+\includegraphicsmid{illo007}{\textsc{Fig.~7.}---The Oblique Sphere and Diurnal Circles.}
+
+\nbarticle{33.} \nbparatext{The Oblique Sphere} (\figref{illo007}{Fig.~7}).---At any station between the
+pole and equator the stars will move in circles oblique to the horizon,
+$SENW$ in the figure. Those whose distance from the elevated pole
+is less than the latitude of the place will, of course, never sink below
+the horizon,---the radius of the ``\textit{Circle of Perpetual Apparition},''
+as it is called (the shaded cap around $P$ in the figure), being just
+equal to the height of the pole, and becoming larger as the latitude
+increases. On the other hand, stars within the same distance of the
+depressed pole will lie within the ``\textit{Circle of Perpetual Occultation},''
+and will never rise above the horizon.
+
+A star exactly on the celestial equator will have its diurnal circle
+$EQWQ'$ bisected by the horizon, and will be above the horizon just
+as long as below it. A star north of the equator (if the North Pole
+is the elevated one) will have more than half of its diurnal circle
+above the horizon, and will be visible more than half the time; as, for
+instance, a star at $A$: and of course the reverse will be true of stars
+%% -----File: 031.png---Folio 20-------
+on the other side of the equator.\footnote
+  {A Celestial Globe will be of great assistance in studying these diurnal circles.
+  The north pole of the globe must be elevated to an angle equal to the latitude of
+  the observer, which can be done by means of the degrees marked on the brass
+  meridian. It will then at once be easily seen what stars never set, which ones
+  never rise, and during what part of the 24 hours any heavenly body at a known
+  distance from the equator is above or below the horizon.}
+Whenever the sun is north of
+the equator, the day will therefore be longer than the night for all
+stations in northern latitude: how much longer will depend both on
+the latitude of the place and the sun's distance from the celestial
+equator.
+\chelabel{CHAPTERI}
+
+%% -----File: 032.png---Folio 21-------
+
+\Chapter{II}{Astronomical Instruments}
+\nbchaptercenter{ASTRONOMICAL INSTRUMENTS.}
+
+\nbarticle{34.} \textsc{Astronomical} observations are of various kinds: sometimes
+we desire to ascertain the apparent distance between two bodies at a
+given time; sometimes the position which a body occupies at a given
+time, or the moment it arrives at a given circle of the sky, usually
+the meridian. Sometimes we wish merely to examine its surface, to
+measure its light, or to investigate its spectrum; and for all these
+purposes special instruments have been devised.
+
+We propose in this chapter to describe very briefly a few of the
+most important.
+
+\nbarticle{35.} \nbparatext{Telescopes in General.}---Telescopes are of two kinds, refracting
+and reflecting. The former were first invented, and are much
+more used, but the largest instruments ever made are reflectors. In
+both the fundamental principle is the same. The large lens, or mirror,
+of the instrument forms at its focus a \textit{real image} of the object
+looked at, and this image is then examined and magnified by the eye-piece,
+which in principle is only a magnifying-glass.
+
+\begin{fineprint}
+In the form of telescope, however, introduced by Galileo,\footnote
+  {In strictness, Galileo did not invent the telescope. Its \textit{first} invention
+  seems to have been in 1608, by Lipperhey,
+  a spectacle-maker of Middleburg,
+  in Holland; though the honor has also been claimed for two or three other
+  Dutch opticians. Galileo, in his ``Nuncius Sydereus,''
+  published in March,
+  1610, himself says that he had heard of the Dutch instruments in 1609, and
+  by so hearing was led to construct his own, which, however, far excelled in
+  power any that had been made previously; and he was the first to apply
+  the telescope to Astronomy. See Grant's ``History of Astronomy,'' pp.~514
+  and seqq.}
+and still used
+as the ``opera-glass,'' the rays from the object-glass are intercepted by a concave
+lens which performs the office of an eye-piece \textit{before} they meet at the
+focus to form the ``real image.'' But on account of the smallness of the
+field of view, and other objections, this form of telescope is never used when
+any considerable power is needed.
+\end{fineprint}
+%% -----File: 033.png---Folio 22-------
+
+\nbarticle{36.} \nbparatext{Simple Refracting Telescope.}---This consists essentially as
+shown in the figure (\figref{illo008}{Fig.~8}), of a tube containing two lenses: a single
+convex lens, $A$, called the object-glass; and another, of smaller size
+and short focus, $B$, called the eye-piece. Recalling the principles of
+lenses the student will see that if the instrument be directed at a distant
+object, the moon, for instance, all the rays, $a_0a_1a_2$, which fall
+upon the object-glass from a point at the \textit{top} of the moon, will be
+collected at $a$ in the focal plane, at the \textit{bottom} of the image. Similarly
+rays from the \textit{bottom} of the moon will go to $b$ at the \textit{top} of the
+image; moreover, since the rays that pass through the optical centre
+of the lens, $o$, are undeviated,\footnote
+  {In this explanation, we use the approximate theory of lenses (in which their
+  thickness is neglected), as given in the elementary text-books. The more exact
+  theory of Gauss and later writers would require some slight modifications in our
+  statements, but none of any material importance. For a thorough discussion,
+  see Jamin, ``\textit{Traité de Physique},'' or Encyc.\ Britannica,---Optics.}
+the angle $a_0ob_0$ will equal $boa$; or, in
+other words, if the focal length of the lens be five feet, for instance,
+then the image of the moon, seen from a distance of five feet, will
+appear just as large as the moon itself does in the sky,---it will
+subtend the same angle. If we look at it from a smaller distance,
+say from a distance of one foot, the image will look larger than the
+moon; and in fact, without using an eye-piece at all, a person with
+normal eyes can obtain considerable magnifying power from the
+object-glass of a large telescope. With a lens of ten feet focal
+length, such as is ordinarily used in an 8-inch telescope, one can
+easily see the mountains on the moon and the satellites of Jupiter,
+by taking out the eye-piece, and putting the eye in the line of vision
+some eight or ten inches back of the eye-piece hole.
+
+\includegraphicsmid{illo008}{\textsc{Fig.~8.}---Path of the Rays in the Astronomical Telescope.}
+
+The image is a \textit{real} one; \textit{i.e.}, the rays that come from different
+points in the object \textit{actually meet} at corresponding points in the image,
+so that if a photographic plate were inserted at $ab$, and properly
+exposed, a picture would be obtained.
+
+If we look at the image with the naked eye, we cannot come nearer
+%% -----File: 034.png---Folio 23-------
+to the image (unless near-sighted) than eight or ten inches, and so
+cannot get any great magnifying power; but if we use a magnifying-glass,
+we can approach much closer.
+
+\nbarticle{37.} \nbparatext{Magnifying Power.}---If the eye-piece $B$ is set at a distance
+from the image equal to its principal focal distance, then any pencil of
+rays from any point of the image will, after passing the lens, be converted
+into a parallel beam, and will appear to the eye to come from
+a point at an infinite distance, as if from an object in the sky. The
+rays which came from the top of the moon, for instance, and are collected
+at $a$ in the image, will reach the eye as a beam \textit{parallel to the
+line $ac$, which connects $a$ with the optical centre of the eye-piece}. Similarly
+with the rays which meet at $b$. The observer, therefore, will
+see the \textit{top} of the moon's disc in the direction $ck$, and the \textit{bottom} in
+the direction $cl$. It will appear to him \textit{inverted}, and greatly magnified;
+its apparent diameter, as seen by the naked eye and measured
+by the angle $aob$ (or its equal $b_0oa_0$); having been increased to $acb$.
+Since both these angles are subtended by the same line $ab$, and are
+\textit{small} (the \figref{illo008}{figure}, of course, is much out of proportion), they must
+be inversely proportional to the distance $ob$ and $cb$; \textit{i.e.}, $boa : bca =
+cb : ob$; or, putting this into words: The ratio between the natural
+apparent diameter of the object, and its diameter as seen through the
+telescope, \textit{is equal to the ratio between the focal lengths of the eye-lens
+and object-glass}. This ratio is called the \textit{magnifying power}
+of the telescope, and is therefore given by the simple formula
+$M = \dfrac{F}{f}$, where $F$ is the focal length of the object-glass and $f$ that of
+eye-piece,\footnote
+  {A magnifying power of 1 is no magnifying power at all. Object and image
+  subtend equal angles. A magnifying power denoted by a fraction, say $\frac{1}{4}$ would
+  be a \textit{minifying} power, making the object look \textit{smaller}, as when we look at an object
+  through the wrong end of a spy-glass.}
+while $M$ is the magnifying power.
+
+If, for example, the object-glass have a focal length of thirty feet,
+and the eye-piece of one inch, the magnifying power will be 360; the
+power may be changed at pleasure by substituting different eye-pieces,
+of which every large telescope has an extensive stock.
+
+\nbarticle{38.} \nbparatext{Brightness of Image.}---Since all the rays from a star which
+fall upon the large object-glass are transmitted to the observer's eye
+(neglecting the losses by absorption and reflection), he obviously
+%% -----File: 035.png---Folio 24-------
+receives a quantity of light much greater than he would naturally get:
+as many times greater as the area of the object-lens is greater than
+that of the pupil of the eye. If we estimate this latter as having a
+diameter of one-fifth of an inch, then a 1-inch telescope would increase
+the light twenty-five times, a 10-inch instrument 2500 times,
+and the great Lick telescope, of thirty-six inches' aperture, 32,400
+times, the amount being proportional to the \textit{square} of the diameter
+of the lens.
+
+It must not be supposed, however, that the apparent brightness of
+an object like the moon, or a planet which shows a disc, is increased
+in any such ratio, since the eye-piece spreads out the light to cover a
+vastly more extensive angular area, according to its magnifying
+power; in fact, it can be shown that no optical arrangement can
+show an \textit{extended surface} brighter than it appears to the naked
+eye. But the \textit{total quantity} of light utilized is greatly increased
+by the telescope, and in consequence, multitudes of stars, far too
+faint to be visible to the unassisted eye, are revealed; and, what is
+practically very important, \textit{the brighter stars are easily seen by day}
+with the telescope.
+
+\nbarticle{39.} \nbparatext{Distinctness of Image.}---This depends upon the exactness
+with which the lens gathers to a single \textit{point} in the focal image all
+the rays which emanate from the corresponding point in the object.
+A single lens, with spherical surfaces, cannot do this very perfectly,
+the ``aberrations'' being of two kinds, the \textit{spherical} aberration and
+the \textit{chromatic}. The former could be corrected, if it were worth while,
+by slightly modifying the form of the lens-surfaces; but the latter,
+which is far more troublesome, cannot be cured in any such way.
+The violet rays are more refrangible than the red, and come to a
+focus nearer the lens; so that the image of a star formed by such
+a lens can never be a luminous point, but is a round patch of light
+of different color at centre and edge.
+
+\begin{fineprint}%comment as 032.png
+\nbarticle{40.} \nbparatext{Long Telescopes.}---By making the diameter of the lens very
+small as compared with its focal length, the aberration becomes less conspicuous;
+and refractors were used, about 1660, having a length of more than
+100 feet and a diameter of five or six inches. The object-glass was mounted
+at the top of a high pole and the eye-piece was on a separate stand below.
+With such an ``aerial telescope,'' of six inches aperture and 120 feet focus,
+Huyghens discovered the rings of Saturn. His object-glass still exists, and
+is preserved in the collection of the Royal Society in London.
+\end{fineprint}
+%% -----File: 036.png---Folio 25-------
+
+\nbarticle{41.} \nbparatext{The Achromatic Telescope.}---The chromatic aberration of a
+lens, as has been said, cannot be cured by any modification of the lens
+itself; but it was discovered in England about 1760 that it can be
+nearly corrected by making the object-glass of \textit{two} (or more) lenses,
+of \textit{different kinds of glass}, one of the lenses being convex and the
+other concave. The convex lens is usually made of \textit{crown} glass, the
+concave of \textit{flint} glass. At the same time, by properly choosing the
+curves, the \textit{spherical} aberration can also be destroyed, so that such a
+compound object-glass comes reasonably near to fulfilling the condition,
+that it should gather to a mathematical point in the image all
+the rays that reach the object-glass from a single point in the object.
+
+\begin{wrapfigure}{r}{0pt}
+\footnotesize
+\renewcommand\arraystretch{0.2}
+\begin{tabular}[t]{c} 1\\ \includegfx{illo009a}\\ \textit{Clark}   \end{tabular}\:
+\begin{tabular}[t]{c} 2\\ \includegfx{illo009b}\\ \textit{Gauss}   \end{tabular}\:
+\begin{tabular}[t]{c} 3\\ \includegfx{illo009c}\\ \textit{Littrow} \end{tabular}
+\caption*{\textsc{Fig.~9.}---Different Forms of the Achromatic Object-glass.}
+\end{wrapfigure}
+
+\begin{fineprint}
+These object-glasses admit of considerable variety of forms. Formerly
+they were generally made, as in \figref{illo009c}{Fig.~9}, No.~3, having the two lenses close
+together, and the adjacent surfaces of the same, or nearly the same, curvature.
+In small object-glasses the lenses are often cemented together with
+Canada balsam or some other transparent medium. At present some of the
+best makers separate the two lenses by a considerable distance, so as to
+admit a free circulation of air between them; in the Pulkowa and Princeton
+object-glasses, constructed by
+Clark, the lenses are seven inches
+apart, and in the Lick telescope six
+and a half inches; as in No.~1. In
+a form devised by Gauss (No.~2),
+which has some advantages, but is
+difficult of construction, the curves
+are very deep, and both the lenses are of watch-glass form---concave on one
+side and convex on the other. In all these forms the crown glass is outside;
+Steinheil, Hastings, and others have constructed lenses with the \textit{flint-glass}
+lens outside. Object-glasses are sometimes made with \textit{three} lenses instead
+of two; a slightly better correction of aberrations can be obtained in this
+way, but the gain is too small to pay for the extra expense and loss of light.
+\end{fineprint}
+
+\nbarticle{42.} \nbparatext{Secondary Spectrum.}---It is not, however, possible with the
+kinds of glass at present available to secure a perfect correction of the
+color. Our best achromatic lenses bring the yellowish green rays to
+a focus \textit{nearer the lens} than they do the red and violet. In consequence,
+the image of a bright star is surrounded by a purple halo,
+which is not very noticeable in a good telescope of small size, but
+is very conspicuous and troublesome in a large instrument.
+
+\begin{fineprint}
+This imperfection of achromatism makes it unsatisfactory to use an ordinary
+lens (\textit{visually} corrected) for astronomical photography. To fit it to
+make good photographs, it must either be specially corrected for the rays
+%% -----File: 037.png---Folio 26-------
+that are most effective in photography, the blue and violet (in which case it
+will be almost worthless visually), or else a subsidiary lens, known as a ``photographic
+corrector,'' may be provided, which can be put on in front of the
+object-glass when needed. A new form of object-glass, devised independently
+by Pickering in this country and Stokes in England, avoids the necessity
+of a third lens by making the crown-glass lens of such a form that when
+put close to the flint lens, with the \textit{flatter side out}, it makes a perfect object-glass
+for visual purposes; but by simply reversing the crown lens, with the
+more convex side outward, and separating the lenses an inch or two, it becomes
+a photographic object-glass. A 13-inch object-glass of this construction
+at Cambridge performs admirably.
+
+Much is hoped from the new kind of glass now being made at Jena. In
+combination with crown glass it produces lenses almost free from chromatic
+aberration, and if it can be produced in homogeneous pieces of sufficient
+size, it will revolutionize the art of telescope making.
+\end{fineprint}
+
+\nbarticle{43.} \nbparatext{Diffraction and Spurious Disc.}---Even if a lens were perfect
+as regards the correction of aberrations, the ``wave'' nature of light
+prevents the image of a luminous point from being also a point; the
+image must \textit{necessarily} consist of a central \textit{disc}, brightest in the centre
+and fading to darkness at the edge, and this is surrounded by a
+series of bright rings, of which, however, only the smallest one is
+generally easily seen. The size of this disc-and-ring system can be
+calculated from the known wave-lengths of light and the dimensions
+of the lens, and the results agree very precisely with observation.
+The diameter of the ``spurious disc'' \textit{varies inversely} with the aperture
+of the telescope. According to Dawes, it is about $4''.5$ for a
+1-inch telescope; and consequently $1''$ for a $4\frac{1}{2}$-inch instrument, $0''.5$
+for a 9-inch, and so on.
+
+\begin{fineprint}
+This circumstance has much to do with the superiority of large instruments
+in showing minute details. No increase of magnifying power on a
+small telescope can exhibit things as sharply as the same power on the larger
+one; provided, of course, that the larger object-glass is equally perfect in
+workmanship, and the \textit{air} in good optical condition.
+
+If the telescope is a good one, and if the air is perfectly steady,---which
+unfortunately is seldom the case,---the apparent disc of a star should be
+perfectly round and well defined, without wings or tails of any kind, having
+around it from one to three bright rings, separated by distances somewhat
+greater than the diameter of the disc. If, however, the magnifying power
+is more than about 50 to the inch of aperture, the edge of the disc will begin
+to appear hazy. There is seldom any advantage in the use of a magnifying
+power exceeding 75 to the inch, and for most purposes powers ranging from
+20 to 40 to the inch are most satisfactory.
+\end{fineprint}
+%% -----File: 038.png---Folio 27-------
+
+\nbarticle{44.} \nbparatext{Eye-Pieces.}---For many purposes, as for instance the examination
+of close double stars, there is no better eye-piece than the simple
+convex lens; but it performs well only when the object is exactly in
+the centre of the field. Usually it is best to employ for the \textit{eye-piece}
+a combination of two or more lenses which will give a more extensive
+field of view.
+
+Eye-pieces belong to two classes, the \textit{positive} and the \textit{negative}. The
+former, which are much more generally useful, act as simple magnifying-glasses,
+and can be used as hand magnifiers if desired. The focal
+image formed by the object-glass lies \textit{outside} of the eye-piece.
+
+In the \textit{negative} eye-pieces, on the other hand, the rays from the
+object-glass are intercepted before they come to the focus, and the
+image is formed between the lenses of the eye-piece. Such an eye-piece
+cannot be used as a hand magnifier.
+
+\begin{fineprint}
+\nbarticle{45.} The simplest and most common forms of these eye-pieces are the
+Ramsden (positive) and
+Huyghenian (negative).
+Each is composed of two
+plano-convex lenses, but
+the arrangement and
+curves differ, as shown
+in \figref{illo010a}{Fig.~10}. The former
+gives a very flat field of
+view, but is not achromatic;
+the latter is more
+nearly achromatic, and
+possibly defines a little better just at the centre of the field; but the fact
+that it is a \textit{negative} eye-piece greatly restricts its usefulness. In the Ramsden
+eye-piece the focal lengths of the two component lenses, both of which
+have their flat sides out, are about equal to each other, and their distance is
+about one-third of the sum of the focal lengths. In the Huyghenian the
+curved sides of the lenses are both turned towards the object-glass; the
+focal distance of the field lens should be exactly \textit{three} times that of the lens
+next the eye, and the distance between the lenses one-half the sum of the
+focal lengths. The peculiarity of the Steinheil ``monocentric'' eye-piece
+which is a triple achromatic positive lens, consisting of a central convex
+lens of crown glass, with a concave meniscus of flint glass cemented to each
+side, is that \textit{the curves are all struck from the same centre}, the thickness of the
+lenses being so computed as to produce the needed corrections. It is free
+from all internal reflections, which in other eye-pieces often produce ``ghosts,''
+as they are called.
+
+\begin{wrapfigure}{r}{0pt}
+\footnotesize \itshape
+\begin{tabular}[b]{c}Ramsden   \\(Positive)               \\\includegfx{illo010a}\end{tabular}\quad
+\begin{tabular}[b]{c}Huyghenian\\(Negative)               \\\includegfx{illo010b}\end{tabular}\quad
+\begin{tabular}[b]{c}Steinheil \\`Monocentric'\\(Positive)\\\includegfx{illo010c}\end{tabular}
+\caption*{\textsc{Fig.~10.}---Various Forms of Telescope Eye-piece.}
+\end{wrapfigure}
+
+There are numerous other forms of eye-piece, each with its own advantages
+and disadvantages. The \textit{erecting} eye-piece, used in spy-glasses, is
+%% -----File: 039.png---Folio 28-------
+essentially a compound microscope, and gives erect vision by again inverting
+the already inverted image formed by the object-glass.
+
+It is obvious that in a telescope of any size the object-glass is the most
+important and expensive part of the instrument. Its cost varies from a few
+hundred dollars to many thousands, while the eye-pieces generally cost only
+from \$5 to \$20 apiece.
+\end{fineprint}
+
+\nbarticle{46.} \nbparatext{Reticle.}---When a telescope is used for \textit{pointing}, as in most
+astronomical instruments, it must be provided with a \textit{reticle} of some
+sort. This is usually a metallic frame with \textit{spider lines} stretched
+across it, placed, not near the object-glass itself (as is often supposed),
+but at the \textit{focus} of the object-glass, where the image is
+formed, as at $a\:b$ in \figref{illo008}{Fig.~8}.
+
+\begin{fineprint}
+It is usually so arranged that it can be moved in or out a little to get it
+exactly into the focal plane, and then, when the eye-piece (positive) is adjusted
+for the observer's eye to give distinct vision of the object, the ``wires,''
+as they are called, will also be equally distinct. As spider-threads are very
+fragile, and likely to get broken or displaced, it is often better to substitute
+a thin plate of glass with lines ruled upon it and blackened. Of course,
+provision must be made for illuminating either the field of view or the
+threads themselves, in order to make them visible in darkness.
+\end{fineprint}
+
+\nbarticle{47.} \nbparatext{The Reflecting Telescope.}---When the chromatic aberration
+of lenses came to be understood through the optical discovery of
+the dispersion of light by Newton, the reflecting telescope was
+invented, and held its place as the instrument for star-gazing until
+well into the present century, when large achromatics began to be
+made. There are several varieties of reflecting telescope, all agreeing
+in the substitution of a large concave mirror in place of the object-glass
+of the refractor, but differing in the way in which they get at
+the image formed by this mirror at its focus in order to examine it
+with the eye-piece.
+
+\begin{fineprint}
+\nbarticle{48.} In the Herschellian form, which is the simplest, but only suited to
+very large instruments, the mirror is \textit{tipped} a little, so as to throw the image
+to the side of the tube, and the observer stands with his back to the object
+and looks down into the tube. If the telescope is as much as two or three
+feet in diameter, his head will not intercept enough light to do much harm,---not
+nearly so much as would be lost by the second reflection necessary in
+the other forms of the instrument. But the inclination of the mirror, and
+the heat from the observer's person, are fatal to any very accurate definition,
+and unfit this form of instrument for anything but the observation of nebulæ
+and objects which mainly require light-gathering power.
+%% -----File: 040.png---Folio 29-------
+
+\begin{wrapfigure}{r}{0pt}
+\footnotesize
+\begin{tabular}{@{}m{1em}@{}m{7.1cm}@{}}
+1 & \includegfx{illo011a}\\
+2 & \includegfx{illo011b}\\
+3 & \includegfx{illo011c}\\
+\end{tabular}
+\caption*{\textsc{Fig.~11.}---Different Forms of Reflecting Telescope.\\[0.8ex]
+1.~The~Herschellian;\; 2.~The~Newtonian;\; 3.~The~Gregorian.}
+\end{wrapfigure}
+
+\sloppy
+In the Newtonian telescope, a small plane reflector standing at an angle
+of 45° is placed in the centre of the tube, so as to intercept the rays reflected
+by the large mirror a little before they come to their focus, and throw them
+to the side of the tube, where the eye-piece is placed.
+
+
+In the Gregorian form (which was the first invented), the large mirror is
+pierced through its centre, and the rays from it are reflected through the
+hole by a small \textit{concave} mirror, placed a little outside of the principal focus
+at the mouth of the tube. With this instrument one looks directly at the
+stars as with a refractor, and the image is erect.
+
+\fussy
+The Cassegrainian form is very similar, except that the small concave
+mirror of the Gregorian is replaced by a \textit{convex} mirror, placed a little inside
+the focus of the large mirror, which makes the instrument a little shorter,
+and gives a flatter field
+of view.
+
+Formerly the great
+mirror was always made
+of a composition of copper
+and tin (two parts
+of copper to one of tin)
+known as ``speculum
+metal.'' At present it is
+usually made of glass
+\textit{silvered} on the front surface,
+by a chemical process
+which deposits the
+metal in a thin, brilliant
+film. These silver-on-glass
+reflectors, when new,
+reflect much more light
+than the old specula, but
+the film does not retain its polish so long. It is, however, a comparatively
+simple matter to renew the film when necessary.
+
+The largest telescopes ever made have been reflectors. At the head of the
+list stands the enormous instrument of Lord Rosse, constructed in 1842, with
+a mirror six feet in diameter and sixty feet focal length. Next in order are
+a number of instruments of four feet aperture, first among which is the great
+telescope of the elder Herschel, built in 1789, followed by the telescope
+erected by Lassell at Malta in 1860, the Melbourne reflector by Grubb in 1870,
+and the still more recent silver-on-glass reflector of the Paris observatory,
+which, however, has proved a failure, owing to defective support of the mirror.
+
+\nbarticle{49.} \nbparatext{Relative Advantages of Refractors and Reflectors.}---There has
+been a good deal of discussion on this point, and each construction has its
+partisans.
+
+In favor of the reflectors we may mention,---
+
+First. \textit{Ease of construction and consequent cheapness.} The concave mirror
+%% -----File: 041.png---Folio 30-------
+has but one surface to figure and polish, while an object-glass has four.
+Moreover, as the light goes \textit{through} an object-glass, it is evident that the
+glass employed must be perfectly clear and of uniform density through and
+through; while in the case of the mirror, the light does not penetrate the
+material at all. This makes it vastly easier to get the material for a large
+mirror than for a large lens.
+
+Second (and immediately connected with the preceding). \textit{The possibility
+of making reflectors much larger than refractors.} Lord Rosse's great reflector
+is six feet in diameter, while the Lick telescope, the largest of all refractors,
+is only three.
+
+Third. \textit{Perfect achromatism.} This is unquestionably a very great advantage,
+especially in photographic and spectroscopic work.
+
+But, on the whole, the advantages are generally considered to lie with the
+refractors.
+
+In their favor we mention:---
+
+First. \textit{Great superiority in light.} No mirror (unless, perhaps, a \textit{freshly
+polished} silver-on-glass film) reflects much
+more than three-quarters of the incident
+light; while a good (single) lens transmits
+over 95 per cent. In a good refractor
+about 82 per cent of the light
+reaches the eye, after passing through
+the four lenses of the object-glass and
+eye-piece. In a Newtonian reflector, in
+average condition, the percentage seldom
+exceeds 50 per cent, and more
+frequently is lower than higher.
+
+Second. \textit{Better definition.}---Any slight
+error at a point in the surface of a glass
+lens, whether caused by faulty workmanship
+or by distortion, affects the direction
+of the ray passing through it only \textit{one-third} as much as the same error
+on the surface of a mirror would do.
+
+\includegraphicsouter{illo012}{\textsc{Fig.~12.}---Effect of Surface Errors in a Mirror and in a Lens.}
+
+If, for instance, in \figref{illo012}{Fig.~12}, an element of the surface at $P$ is turned out
+of its proper direction, $aa'$, by a small angle, so as to take the direction $bb'$,
+then the \textit{reflected} ray will be sent to $f$, and its deviation will be \textit{twice} the
+angle $aPb$. But since the index of refraction of glass is about 1.5 the
+change in the direction of the \textit{refracted} ray from $R$ to $r$ will only be about
+\textit{two-thirds} of $aPb$.
+
+Moreover, so far as distortions are concerned, when a lens bends a little
+by its own weight, \textit{both sides are affected in a nearly compensatory manner},
+while in a mirror there is no such compensation. As a consequence, mirrors
+very seldom indeed give any such definition as lenses do. The least fault
+of workmanship, the least distortion by their own weight, the slightest
+difference of temperature, between front and back, will absolutely ruin the
+image, while a lens would be but slightly affected in its performance by
+the same circumstances.
+%% -----File: 042.png---Folio 31-------
+
+Third. \textit{Permanence.} The lens, once made, and fairly taken care of,
+suffers no deterioration from age; but the metallic speculum or the silver
+film soon tarnishes, and must be repolished every few years. This alone
+is decisive in most cases, and relegates the reflector mainly to the use of
+these who are themselves able to construct their own instruments.
+
+To these considerations we may add that a refractor, though more expensive
+than a reflector of similar power, is not only more permanent, and less
+likely to have its performance affected by accidental circumstances, but is
+lighter and more convenient to use.
+\end{fineprint}
+
+\sloppy
+\nbarticle{50.} \nbparatext{Time-Keepers and Time-Recorders.}---\textit{The Clock, Chronometer,
+and Chronograph.}---Modern practical astronomy owes its development
+as much to the clock and chronometer as to the telescope. The
+ancients possessed no accurate instruments for the measurement of
+time, and until within 200 years, the only reasonably precise method
+of fixing the time of an important observation, as, for instance, of
+an eclipse, was by noting the \textit{altitude} of the sun, or of some known
+star at or very near the moment.
+
+\fussy
+It is true that the Arabian astronomer Ibn Jounis had made some
+use of the pendulum about the year 1000 \textsc{a.d.}, more than 500 years
+before Galileo introduced it to Europeans. But it was not until
+nearly a century after Galileo's discovery that Huyghens applied it
+to the construction of clocks (in 1657).
+
+So far as the principles of construction are concerned, there is no
+difference between an astronomical clock and any other. As a matter
+of convenience, however, the astronomical clock is almost invariably
+made to beat seconds (rarely half-seconds), and has a conspicuous
+second-hand, while the hour-hand makes but \textit{one} revolution a day,
+instead of two, as usual, and the face is marked for twenty-four hours
+instead of twelve. Of course it is constructed with extreme care in
+all respects.
+
+\begin{fineprint}
+\textit{The Escapement}, or ``\textit{Scapement},'' is often of the form known as the
+``Graham Dead-beat''; but it is also frequently one of the numerous ``gravity'' escapements
+which have been invented by ingenious mechanicians. The office of
+the escapement is to be ``unlocked'' by the pendulum at each vibration, so
+as to permit the wheel-work to advance one step, marking a second (or sometimes
+two seconds), upon the clock-face; while, at the same time, the escapement
+gives the pendulum a slight impulse, just equal to the resistance it has
+suffered in performing the unlocking. The work done by the pendulum in
+``unlocking'' the train, \textit{and the corresponding impulse, ought to be perfectly
+constant}, in spite of all changes in the condition of the train of wheels; and
+it is \textit{desirable}, though not \textit{essential}, that this work should be as
+\textit{small} as possible.
+\end{fineprint}
+%% -----File: 043.png---Folio 32-------
+\nbarticle{51.} The pendulum itself is usually suspended by a flat spring, and
+great pains should be taken to have the support extremely firm: this
+is often neglected, and the clock then cannot perform well.
+
+\sloppy
+\textit{Compensation for Temperature.}---In order to keep perfect time,
+the pendulum must be a ``compensation pendulum''; \textit{i.e.}, constructed in such a way
+that changes of temperature will
+not change its length.
+
+\fussy
+An uncompensated pendulum, with steel rod,
+changes its daily rate about one-third of a second
+for each degree of temperature (centigrade).
+A wooden pendulum rod is much less affected
+by temperature, but is very apt to be disturbed
+by changes of \textit{moisture}.
+
+\begin{fineprint}
+\includegraphicsouter{illo013}{\textsc{Fig.~13.}\\
+Compensation Pendulums.\\
+1.~Graham's Pendulum.\\
+2.~Zinc-Steel Pendulum.}
+Graham's mercurial pendulum (\figref{illo013}{Fig.~13}) is the
+one most commonly used. It consists simply of a
+jar (usually steel), three or four inches in diameter,
+and about eight inches high, containing forty or fifty
+pounds of mercury, and suspended at the end of a
+steel rod. When the temperature rises, the rod
+lengthens (which would make the clock go slower);
+but, at the same time, the mercury expands, from
+the bottom upwards, just enough to compensate.
+This pendulum will perform well only when not
+exposed to \textit{rapid changes} of temperature. Under
+rapid changes the compensation \textit{lags}. If, for instance,
+it grows warm quickly, the rod will expand
+before the mercury does; so that, \textit{while the mercury is
+growing warmer}, the clock will run slow, though after
+it has become warm the rate may be all right.
+
+A compensation pendulum, constructed on the
+principle of the old gridiron pendulum of Harrison,
+but of zinc and steel instead of brass and steel, is
+now much used. The compensation is not so easily
+adjusted as in the mercurial pendulum, but when properly made the mechanism
+acts well, and bears rapid alterations of temperature much better than
+the mercurial pendulum. The heavy pendulum-bob, a lead cylinder, is hung
+at the end of a steel rod, which is suspended from the top of a zinc tube,
+and hangs through the centre of it. This tube is itself supported \textit{at the bottom}
+by three or four steel rods which hang from a piece attached to the pendulum
+spring. The standard clock at Greenwich has a pendulum of this kind.
+\end{fineprint}
+
+\nbarticle{52.} \nbparatext{Effect of Atmospheric Pressure.}---In consequence of the
+buoyancy of the air, and its resistance to motion, a pendulum swings
+%% -----File: 044.png---Folio 33-------
+a little more slowly than it would \textit{in vacuo}, and every change in the
+density of the air affects its rate more or less. With mercurial
+pendulums, of ordinary construction, the ``\textit{barometric coefficient},''
+as it is called, is about one-third of a second for an inch of the
+barometer; \textit{i.e.}, an increase of atmospheric density which would
+raise the barometer one inch would make the clock \textit{lose} about one-third
+of a second daily. It varies considerably, however, with different
+pendulums.
+
+\begin{fineprint}
+It is not very usual to take any notice of this slight disturbance; but
+when the extremest accuracy of time-keeping is aimed at, the clock is either
+sealed in an air-tight case from which the air is partially exhausted (as at
+Berlin), or else some special mechanism, controlled by a barometer, is devised
+to compensate for the barometric changes, as at Greenwich. In the
+Greenwich clock a magnet is raised or lowered by the rise or fall of the
+mercury in a barometer attached to the clock-case. When the magnet rises,
+it approaches a bit of iron two or three inches above it, fixed to the bottom
+of the pendulum, and the increase of attraction accelerates the rate just
+enough to balance the retardation due to the air's increased density and
+viscosity. There are several other contrivances for the same purpose.
+\end{fineprint}
+
+\nbarticle{53.} \nbparatext{Error and Rate.}---The ``\textit{error},'' or ``\textit{correction}'' of a clock
+is the amount that \textit{must be added} to the indication of the clock-face
+at any moment in order to give the \textit{true time}; it is, therefore, \textit{plus}
+($+$) when the clock is \textit{slow}, and \textit{minus} ($-$) when it is \textit{fast}. The
+\textit{rate} of a clock is the amount of its \textit{daily gain or loss}; \textit{plus} ($+$) when
+the clock is \textit{losing}. Sometimes the \textit{hourly rate} is used, but ``\textit{hourly}''
+is then always specified.
+
+A \textit{perfect} clock is one that has a \textit{constant rate}, whether that rate
+be large or small. It is desirable, for convenience' sake, that both
+error and rate should be small; but this is a mere matter of adjustment
+by the user of the clock, who adjusts the error by setting the
+hands, and the rate by raising or lowering the pendulum-bob.
+
+\begin{fineprint}
+The final adjustment of rate is often obtained by first setting the pendulum-bob
+so that the clock will run slow a second or two daily, and then
+putting on the top of the bob little weights of a gramme or two, which will
+accelerate the motion. They can be dropped into place or knocked off without
+stopping the clock or perceptibly disturbing it.
+
+The very best clocks will run three or four years without being stopped
+for cleaning, and will retain their rate without a change of more than one-fifth
+of a second, one way or the other, during the whole time. But this is
+%% -----File: 045.png---Folio 34-------
+exceptional performance. In a run as long as that, most clocks would
+be liable to change their rate as much as half a second or more, and to do
+it somewhat irregularly.
+\end{fineprint}
+
+\sloppy
+\nbarticle{54.} \nbparatext{The Chronometer.}---The pendulum-clock not being portable,
+it is necessary to provide time-keepers that are. The chronometer is
+merely a carefully made watch, with a balance wheel compensated
+to run, as nearly as possible, at the same rate in different temperatures,
+and with a peculiar escapement, which, though unsuited to
+watches exposed to ordinary rough usage, gives better results than
+any other when treated carefully.
+
+\fussy
+\begin{fineprint}
+The \textit{box-chronometer} used on ship-board is usually about twice the diameter
+of a common pocket watch, and is mounted on gimbals, so as to keep horizontal
+at all times, notwithstanding the motion of the vessel. It usually
+beats half-seconds. It is not possible to secure in the chronometer-balance
+as perfect a temperature correction as in the pendulum. For this and other
+%% -----File: 046.png---Folio 35-------
+reasons the best chronometers cannot quite compete with the best clocks in
+precision of time-keeping; but they are sufficiently accurate for most purposes,
+and of course are vastly more convenient for field operations. They
+are simply indispensable at sea. \textit{Never turn the hands of a chronometer
+backward.}
+\end{fineprint}
+
+\includegraphicsmid{illo014}{\textsc{Fig.~14.}---A Chronograph by Warner and Swasey.}
+
+\nbarticle{55.}  Before the invention of the telegraph it was customary to note
+time merely ``by eye and ear.'' The observer, keeping his time-piece
+near him, listened to the clock-beats, and estimated as closely
+as he could, in seconds and \textit{tenths} of seconds, the moment when the
+phenomenon he was watching occurred---the moment, for instance,
+when a star passed across a wire in the reticle of his telescope.
+At present the record is usually made by simply pressing a ``key''
+in the hand of the observer, and this, by a telegraphic connection,
+makes a mark upon a strip or sheet of paper, which is moved at a
+uniform rate by clock-work, and graduated by seconds-signals from
+the clock or chronometer.
+
+\nbarticle{56.} \nbparatext{The Chronograph.}---This is the instrument which carries the
+marking-pen and moves the paper on which the time-record is made.
+The paper is wrapped upon a cylinder, six or seven inches in diameter,
+and fifteen or sixteen inches long. This cylinder is made to revolve
+once a minute, by clock-work, while the pen rests lightly upon
+the paper and is slowly drawn along by a screw-motion, so that it
+marks a continuous spiral. The pen is carried on the armature of
+an electro-magnet, which every other second (or sometimes every
+second) receives a momentary current from the clock, causing it to
+make a mark like those which break the lines in the \figref{illo015}{figure} annexed.
+
+\includegraphicsmid{illo015}{\textsc{Fig.~15.}---Part of a Chronograph Record.}
+
+The beginning of a new minute (the 60th sec.) is indicated either
+by a double mark as shown, or by the omission of a mark. When
+the observer touches his key he also sends a current through the
+magnet, and thus interpolates a mark of his own on the record, as
+at $X$ in the \figref{illo015}{figure}: the \textit{beginning} of the mark is the instant noted---in
+this case 54.9$^{\text{s}}$. Of course the minutes when the chronograph was
+started and stopped are noted by the observer on the sheet, and so
+enable him to identify the minutes and seconds all through the record.
+%% -----File: 047.png---Folio 36-------
+
+\begin{fineprint}
+Many European observatories use chronographs in which the record is
+made upon a long fillet of paper, instead of a sheet on a cylinder. The
+instrument is lighter and cheaper than the American form, but much less
+convenient.
+
+The regulator of the clock-work must be a ``continuous'' regulator, working
+continuously, and not by beats like a clock-escapement. There are
+various forms, most of which are centrifugal governors, acting either by
+friction (like the one in the \figref{illo014}{figure}) or by the resistance of the air; or else
+``spring-governors,'' in which the motion of a train, with a pretty heavy
+fly-wheel, is slightly checked at regular intervals by a pendulum.
+
+\sloppy
+\nbarticle{57.} \nbparatext{Clock-Breaks.}---The arrangements by which the clock is made to
+send regular electric signals are also various. One of the earliest and simplest
+is a fine platinum wire attached to the pendulum, which swings through a
+drop of mercury at each vibration. All of the arrangements, however, in
+which the pendulum itself has to make the electric contact are objectionable,
+and for clocks using the Graham dead-beat scapement no absolutely satisfactory
+means of giving the signals has yet been devised. Clocks with the
+gravity escapements have a decided advantage in this respect. Their wheel-work
+has no direct action in driving the pendulum, and so may be made to
+do any reasonable amount of outside work in the way of ``key-manipulation''
+without affecting the clock-rate in the least. Usually a wheel on the
+axis of the scape-wheel is made to give the electric signals by touching a
+light spring with one of its teeth every other second.
+
+\fussy
+Chronometers are now also fitted up in the same way, to be used with the
+chronograph.
+
+The signals sent are sometimes ``breaks'' in a continuous current, and
+sometimes ``makes'' in an open circuit. Usage varies in this respect, and
+each method has its advantages. The break-circuit system is a little simpler
+in its connections, and possibly the signals are a little more sharp, but it
+involves a much greater consumption of battery material, as the current is
+always circulating, except during the momentary breaks.
+\end{fineprint}
+
+\nbarticle{58.} \nbparatext{Meridian Observations.}---A large proportion of all astronomical
+observations are made at the time when the heavenly body
+observed is crossing the meridian, or very near it. At that moment
+the effects of refraction and parallax (to be discussed hereafter) are
+a minimum, and as they act only in a vertical plane, they do not have
+any influence on the \textit{time} at which the body crosses the meridian.
+
+\nbarticle{59.} \nbparatext{The Transit Instrument} is the instrument used, in connection
+with a clock or chronometer, and often with a chronograph also, to
+observe the time of a star's ``transit'' across the meridian.
+
+If the error of the (sidereal) clock is known at the moment, this
+observation will determine the right ascension of the body, which, it
+%% -----File: 048.png---Folio 37-------
+will be remembered, is simply \textit{the sidereal time at which it crosses the
+meridian}; \textit{i.e.}, the number of hours, minutes, and seconds by which it
+follows the vernal equinox.
+
+\textit{Vice versa}, if the right ascension is known, the error or correction
+of the clock will be determined.
+
+\sloppy
+The instrument (\figref{illo016}{Fig.~16}) consists essentially of a telescope mounted
+upon a stiff axis perpendicular to the telescope tube. This axis is
+placed horizontal, east and west, and turns on \textit{pivots} at its extremities,
+in Y-bearings upon the top of two fixed piers or pillars. A
+small graduated circle is attached, to facilitate ``setting'' the telescope
+at any designated altitude or declination.
+
+\fussy
+\includegraphicsmid{illo016}{\textsc{Fig.~16.}---The Transit Instrument (Schematic).}
+
+The telescope carries at the eye-end, in the focal plane of the
+object-glass, a reticle of some odd number of vertical wires,---five
+or more,---one of which is always in the centre, and the others
+are usually placed at equal distances on each side of it. One or
+two wires also cross the field horizontally.
+
+If the pivots are true, and the instrument accurately adjusted, it
+is evident that the \textit{central vertical wire will always follow the meridian
+as the instrument is turned}; and the instant when a star crosses this
+wire will be the true moment of the star's meridian transit. The
+%% -----File: 049.png---Folio 38-------
+object in having a number of wires is, of course, simply to gain
+accuracy by taking the mean of a number of observations instead of
+depending upon a single one.
+
+In order to ``\textit{level}'' the axis properly, a delicate spirit-level is an
+essential adjunct; it is usual, also, (and important) to provide a convenient
+``\textit{reversing apparatus},'' by which the instrument can be
+turned half round, making the eastern and
+western pivots change places.
+
+\includegraphicsouter{illo017}{\textsc{Fig.~17.}---Reticle of the Transit Instrument.}
+
+The instrument must be thoroughly stiff
+and rigid, without loose joints or shaky
+screws; and the two pivots must be \textit{accurately
+round}, \textit{precisely in line with each other},
+\textit{free from taper}, and precisely \textit{of the same
+size}; all of which conditions may be summed
+up by saying that they \textit{must be portions of
+one and the same geometrical cylinder}.
+
+\begin{fineprint}
+The proper construction and grinding of
+these pivots, which are usually of hard bell metal (sometimes of steel),
+taxes the art of the most skilful mechanician. The level, also, is a delicate
+instrument, and difficult to construct.
+
+Provision is made, of course, for illuminating the field of view at night
+so as to make the reticle wires visible. Usually one (or both) of the pivots
+is pierced, and a lamp throws light through the opening upon a small mirror
+in the centre of the tube, which reflects it down upon the reticle.
+
+The Y's are used instead of round bearings, in order to prevent any
+\textit{rolling} or \textit{shake} of the pivots as the instrument turns.
+
+Fig.~18 shows a modern transit instrument (portable) as actually constructed
+by Fauth \& Co.
+\end{fineprint}
+
+Another form of the instrument is much used, which is often
+designated as the ``Broken Transit.'' A reflector in the central cube
+throws the rays coming from the object-glass, out at right angles
+through one end of the axis, where the eye-piece is placed; so that
+the observer does not have to change his position at all for different
+stars, but simply looks straight forward horizontally. It is very
+convenient and rapid in actual work, but the observations require a
+considerable correction for \textit{flexure of the axis}.
+
+\nbarticle{60.} \begin{fineprint}\textit{Adjustments.}---(1) Focus and verticality of wires. (2)~Collimation.
+(3)~Level. (4)~Azimuth.
+
+First. The first thing to do after the instrument is set on its supports and
+the axis roughly levelled, is \textit{to adjust the reticle}. The eye-piece is drawn out
+%% -----File: 050.png---Folio 39-------
+or pushed in until the wires appear perfectly sharp, and then the instrument
+is directed to a star or to some \textit{distant} object (not less than a mile away),
+and without disturbing the eye-piece, the sliding-tube, which carries the
+reticle, is drawn out or pushed in until the object is also distinct at the
+%% -----File: 051.png---Folio 40-------
+same time with the wires. If this adjustment is correctly made, motion of
+the eye in front of the eye-piece will not produce any apparent displacement
+of the object in the field, with reference to the wires. To test the verticality
+of the wires, the telescope is moved up and down a little, while looking at the
+object; if the axis is level and the wires vertical, the wire will not move off
+from the object sideways. There are screws provided to turn the reticle a
+little, so as to effect this adjustment.
+
+\includegraphicsmid{illo018}{\textsc{Fig.~18.}---A 3-inch Transit, with reversing apparatus. Fauth \& Co.}
+
+When the wires have been thus adjusted for focus and verticality, the
+reticle-slide should be tightly clamped and never disturbed again. The \textit{eye-piece}
+can be moved in and out at pleasure, to secure distinct vision for different
+eyes, but it is \textit{essential that the distance between the object-glass and the
+reticle remain constant}.
+
+Second. \textit{Collimation.} The line joining the optical centre of the object-glass
+with the middle wire of the reticle is called the ``\textit{line of collimation},''
+and this line must be made exactly perpendicular to the axis of rotation
+by moving the reticle slightly to one side or the other by means of the
+adjusting screws provided for the purpose. The simplest way of effecting
+the adjustment is to point the instrument on some well-defined distant
+object, like a nail-head or a joint in brickwork, and then carefully
+to ``reverse'' the instrument without disturbing the stand. If the middle
+wire, after reversal, points just as it did before, the ``collimation'' is correct;
+if not, the middle wire must be moved \textit{half way} towards the object by the
+screws.
+
+\medskip
+
+\textbf{Collimator.}---It is not always easy to find a distant object on which to
+make this adjustment, and a ``\textit{collimator}'' may be substituted with advantage.
+This is simply a telescope mounted horizontally on a pier in front of the
+transit instrument, so that when the transit telescope is horizontal, it can
+look straight into the collimator, which ought to be of about the same size
+as the transit itself.
+
+In the focus of the collimator object-glass are placed two wires forming
+an X, and thus placed they can be seen by a telescope looking into the collimator
+just as distinctly as if they were at an infinite distance and really celestial
+objects. The instrument furnishes us a mark \textit{optically} celestial, but
+\textit{mechanically} within reach of our finger-ends for illumination, adjustment,
+etc. If the pier on which it is mounted is firm, the collimator cross is in all
+respects as good as a star, and much more convenient.
+
+Third. \textit{Level.} The adjustment for \textit{level} is made by setting a striding
+level on the pivots of the axis, reading the level, then reversing the level
+(not the transit) and reading it again. If the pivots are round and of the
+same size, the difference between the level-readings direct and reversed will
+indicate the amount by which one pivot is higher than the other. One of
+the Y's is made so that it can be raised and lowered slightly by means of a
+screw, and this gives the means of making the axis horizontal. If the
+pivots are not of the same size (and they never are \textit{absolutely}), the astronomer
+must determine and allow for the difference.
+%% -----File: 052.png---Folio 41-------
+
+Fourth. \textit{Azimuth.} In order that the instrument may indicate the meridian
+truly, its axis must lie exactly east and west; \textit{i.e.}, its \textit{azimuth} must be $90°$.
+This adjustment must be made by means of observations upon the stars, and
+is an excellent example of the method of successive approximations, which
+is so characteristic of astronomical investigation. (\textit{a}) After adjusting carefully
+the focus and collimation of the instrument, we set it north and south
+\textit{by guess}, and level it as precisely as possible. By looking at the pole star,
+and remembering how the pole itself lies with reference to it, one can easily
+set the instrument \textit{pretty nearly}; \textit{i.e.}, within half a degree or so. The middle
+wire will now describe in the sky a vertical circle, which crosses the meridian
+at the zenith, and lies very near the meridian for a considerable distance
+each side of the zenith.
+
+(\textit{b}) We must next get an ``approximate'' time; \textit{i.e.}, set our clock or
+chronometer \textit{nearly} right. To do this, we select from the list of several
+hundred stars in the Nautical Almanac (which is to be regarded in about
+the same light with the clock and the spirit level, as an indispensable accessory
+to the transit) a star which is about to cross the meridian \textit{near the zenith}.
+The difference between the right ascension of the star as given in the
+Almanac, and the time shown by the clock-face, will be \textit{very nearly} the
+error of the clock at the time of the observation: not \textit{exactly}, unless the declination
+of the star is such that it passes \textit{exactly} through the zenith, but
+\textit{very nearly}, since the star crosses the meridian near the zenith. We now
+have the time within a second or two.
+
+(\textit{c}) Next turn down the telescope upon some Almanac star, which is
+soon to cross the meridian within $10°$ of the pole. It will appear to move
+very slowly. A little before the time it should reach the meridian, move the
+whole frame of the instrument until the middle wire points upon it, and
+then, by means of the ``Azimuth Screw,'' which gives a slight horizontal
+motion to one of the Y's, \textit{follow the star} until the \textit{indicated moment of its transit};
+\textit{i.e.}, until the clock (corrected for clock error) shows on its face the star's
+right ascension. If the clock correction had been known with absolute exactness,
+the instrument would now be \textit{truly} in the meridian; as the clock error,
+however, is only approximate, the instrument will only be approximately in
+the meridian; but---and this is the essential point---it will be \textit{very much
+more nearly so} than at the beginning of the operation. The supposed incorrectness,
+amounting perhaps to one or two seconds, in the time at which the
+instrument was set on the circumpolar star will, on account of the slow motion
+of the star, make almost no perceptible difference in the direction given
+to the axis.
+
+A repetition of the operation may possibly be needed to secure all the
+desired precision. The accuracy of this azimuth adjustment can then be
+verified by three successive ``culminations'' or transits of the pole star, or
+any other circumpolar. The interval occupied in passing from the upper to
+the lower culmination on the west side of the meridian ought, of course,
+to be exactly equal to the time on the eastern side; \textit{i.e.}, twelve sidereal
+hours.
+\end{fineprint}
+%% -----File: 053.png---Folio 42-------
+
+\nbarticle{61.} The final test of \textit{all} the adjustments, and of the accurate going
+of the clock, is obtained by observing a number of Almanac stars of
+widely different declination. If they all indicate \textit{identically} the same
+clock correction, the instrument is in adjustment; if not, and if the
+differences are not very great, it is possible to deduce from the
+observations themselves the true clock error, and the adjustment
+errors of the instrument.
+
+\begin{fineprint}
+\includegraphicsouter{illo019}{\textsc{Fig.~19.}---The Meridian Circle (Schematic).}
+It is to be added, in this connection, that the astronomer can never assume
+that \textit{adjustments are perfect}: even if once perfect, they would not stay so, on
+account of changes of temperature and other causes. Nor are observations
+ever absolutely accurate. The problem is, from observations more or less
+\textit{inaccurate} but \textit{honest}, with instruments more or less \textit{maladjusted} but \textit{firm}, to
+find the result that would have been obtained by a perfect observation with
+a perfect and perfectly adjusted instrument. It can be more nearly done
+than one might suppose. But the discussion of the subject belongs to
+Practical Astronomy, and cannot be entered into here.
+\end{fineprint}
+
+\nbarticle{62.} \nbparatext{Prime Vertical Instrument.}---For certain purposes, a Transit
+Instrument, provided with an apparatus for rapid reversal, is turned
+quarter-way round and mounted with the axis \textit{north} and \textit{south}, so
+that the plane of rotation lies \textit{east} and \textit{west}, instead of in the meridian.
+It is then called a Prime Vertical Transit.
+
+\nbarticle{63.} \nbparatext{The Meridian Circle.}---In
+order to determine the \textit{Declination}
+or \textit{Polar Distance} of an
+object, it is necessary to have
+some instrument for measuring
+angles; mere time-observations
+will not suffice. The instrument
+most used for this purpose is the
+\textit{Meridian Circle}, or \textit{Transit Circle},
+which is simply a transit instrument,
+with a graduated circle
+attached to its axis, and revolving
+with the telescope. Sometimes
+there are two circles, one
+at each end of the axis.
+
+\sloppy
+Fig.~19 represents the instrument
+``schematically,'' showing merely the essential parts. \figref{illo020}{Fig.~20}
+is a meridian circle, with a 4-inch telescope, constructed by Fauth
+\& Co.
+%% -----File: 054.png---Folio 43-------
+
+\includegraphicsmid{illo020}{\textsc{Fig.~20.}---Meridian Circle.\\[1.5ex]
+
+\begin{tabular}{@{}p{.48\textwidth}|p{.48\textwidth}@{}}
+\hangindent=2em  $A$, $B$, $C$, $D$, the Reading Microscopes.
+
+\hangindent=2em  $K$, the Graduated Circle.
+
+\hangindent=2em  $H$, the Roughly Graduated Setting Circle.
+
+\hangindent=2em  $I$, the Index Microscope. This is usually, however,
+     placed half way between $A$ and $D$.
+&
+\hangindent=2em  $F$, the Clamp.    $G$, the Tangent Screw.
+
+\hangindent=2em  $LL$, the Level, only placed in position occasionally.
+
+\hangindent=2em  $M$, the Right Ascension Micrometer.
+
+\hangindent=2em  $WW$, Counterpoises, which take part of the weight
+      of the instrument off from the Y's.
+\end{tabular}}
+%% -----File: 055.png---Folio 44-------
+
+\fussy
+\begin{fineprint}
+In observatory instruments the circle is usually from two to four feet in
+diameter; larger circles were once used, but it is found that their weight,
+and the consequent strains and flexures, render them actually less accurate
+than the smaller ones. The utmost resources of mechanical art are exhausted
+in making the graduation as precise as possible and in providing for
+its accurate reading, as well as in securing the maximum firmness and stability
+of every part of the instrument. The actual divisions are usually
+$5'$ apart (in very large instruments sometimes only $2'$), but the circle is
+``read'' to seconds and tenths of seconds of arc by means of \textit{reading microscopes},
+from two to six in number, fixed to the pier of the instrument. In a
+circle of forty inches diameter, $1''$ is a little less than $\frac{1}{10000}$ of an inch,
+($\frac{20}{206265}$ inch), so that the necessity of fine workmanship is obvious.
+\end{fineprint}
+
+\nbarticle{64.} \nbparatext{The Reading Microscope} (\figref{illo021}{Fig.~21}).---This consists essentially
+of a compound microscope, which forms a magnified image of
+the graduation at the focus of its object-glass, where this image is
+viewed by a positive eye-piece. At the
+place where the image is formed a pair of
+parallel spider-lines or a cross is placed,
+movable in the plane of the image by a
+``micrometer screw''; \textit{i.e.}, a fine screw
+with a graduated head, usually divided into
+sixty parts. One revolution of the screw
+carries the wire $1'$ of arc, which makes
+one division of the screw-head $1''$, the
+tenths of seconds being estimated.
+
+\includegraphicsouter{illo021}{\textsc{Fig.~21.}---The Reading Microscope.}
+
+\begin{fineprint}
+The adjustment of the microscope for
+``runs,'' as it is called (that is, to make one
+revolution of the micrometer screw exactly
+equal to $1'$), is effected as follows. By setting
+the wires first on one of the graduation marks
+visible in the field of view, and then on the
+next mark, it is immediately evident whether
+five revolutions of the screw ``run'' over or
+fall short of $5'$ of the graduation. If they \textit{overrun}, it shows that the image
+of the graduation formed by the microscope objective is too small to fit the
+screw, and \textit{vice-versa}. Now, by simply increasing or decreasing the distance
+$A\,B$ between the objective and the micrometer box, the size of the image
+can be altered at will, and the objective is therefore so mounted that this
+can be done. Of course, every change in the length of the microscope tube
+will also require a readjustment of the distance between the ``limb,'' or
+graduated surface, of the circle and the microscope, in order to secure distinct
+vision; but by a few trials the adjustment is easily made sufficiently precise.
+%% -----File: 056.png---Folio 45-------
+
+The reading of the circle is as follows: An extra index-microscope,
+with low power and large field of view, shows by inspection the degrees
+and minutes. The reading-microscopes are only used to give
+the odd seconds, which is done by turning the screw until the parallel
+spider-lines are made to include one of the graduation lines half-way
+between themselves; the head of the screw then shows directly the
+seconds and tenths, to be added to the degrees and minutes shown
+by the index. Thus in \figref{illo022}{Fig.~22}, the reading of the microscope is
+$3'\,22''.1$, the $3'$ being given by the \textit{scale} in the field, the $22''.1$ by the
+screw-head.
+\end{fineprint}
+
+\includegraphicsouter{illo022}{\textsc{Fig.~22.}---Field of View of Reading Microscope.}
+
+\nbarticle{65.} \nbparatext{Method of observing a Star.}---A minute or two before the star
+reaches the meridian the instrument is approximately pointed, so that
+the star will come into the field of view. As soon as it makes its
+appearance, the instrument
+is moved by the slow-motion
+tangent-screw until
+the star is ``bisected'' by
+the fixed horizontal wire
+of the reticle, and the
+star is kept bisected until
+it reaches the middle vertical
+wire which marks the
+meridian. The microscopes are then read, and their mean result is
+the star's ``circle-reading.''
+
+\begin{fineprint}
+Frequently the star is bisected, not by moving the whole instrument, but
+by means of a ``micrometer wire,'' which moves up and down in the field of
+view. The micrometer reading then has to be combined with the reading
+of the microscope, to get the true circle-reading.
+\end{fineprint}
+
+\nbarticle{66.} \nbparatext{Zero Points.}---In determining the declination or meridian
+altitude of a star by means of its circle-reading, it is necessary to
+know the ``\textit{zero point}'' of the circle. For declinations, the ``zero
+point'' is either the polar or the equatorial reading of the circle; \textit{i.e.},
+the reading of the circle when the telescope is pointed at the pole
+or at the equator.
+
+\textit{The ``polar point''} may be found by observing some circumpolar
+star above the pole, and again, twelve hours later, below it. When
+the two circle-readings have been \textit{duly corrected for refraction and
+instrumental errors}, their mean will be the polar point.
+%% -----File: 057.png---Folio 46-------
+
+\begin{fineprint}
+Suppose, for instance, that $\delta$~Ursæ Minoris, at the ``upper culmination,''
+gives a corrected reading of $52°\, 18'\, 25''.3$, while at the lower culmination the
+reading is $45°\, 31'\, 35''.7$, then the mean of these, $48°\, 55'\, 00''.5$, is the polar
+point, and of course the equatorial reading is $138°\, 55'\, 00''.5$,---just $90°$
+greater. The \textit{polar distance} of the star would be the \textit{half-difference} of the
+two readings, or $3°\, 23'\, 24''.8$.
+\end{fineprint}
+
+\nbarticle{67.} \nbparatext{Nadir Point.}---The determination of the polar point requires
+two observations of the same star at an interval of twelve hours. It is
+often difficult to obtain such a pair; moreover, the \textit{refraction} complicates
+the matter, and renders the result less trustworthy. Accordingly
+it is now usual to use the nadir or the horizontal reading as the
+zero, rather than the polar point.
+
+\includegraphicsouter{illo023}{\textsc{Fig.~23.}\\The Collimating Eye-Piece.}
+
+The \textit{nadir point} is determined by pointing the telescope downwards
+to a basin of mercury, moving the telescope until the image
+of the horizontal wire of the reticle, as seen by reflection, coincides
+with the wire itself. Since the reticle is exactly in the principal
+focus of the object-glass, rays of light emitted by any point in the
+reticle will become a parallel beam after passing the lens, and if this
+beam strikes a plane mirror perpendicularly and
+is returned, the rays will come just as if from a
+real object in the sky, and will form an image
+at the focal plane. When, therefore, the image
+of the central wire of the reticle, seen in the
+mercury basin by reflection, coincides with the
+wire itself, we know that the line of collimation
+must be exactly perpendicular to the surface of
+the mercury; \textit{i.e.}, vertical.
+
+\begin{fineprint}
+To make the image visible it is necessary to illuminate the reticle by light
+thrown \textit{towards} the object-glass from behind the wires, instead of light
+coming from the object-glass towards the eye as usual. This peculiar illumination
+is commonly effected by means of Bohnenberger's ``collimating
+eye-piece,'' shown in \figref{illo023}{Fig.~23}. In the simplest form it is merely a common
+Ramsden eye-piece, with a hole in one side, and a thin glass plate inserted
+at an angle of $45°$. A light from one side, entering through the hole, will be
+(partially) reflected towards the wires, and will illuminate them sufficiently.
+
+The \textit{horizontal point} of course differs just $90°$ from the nadir point. It
+may also be found independently by noting the circle-readings of some star
+observed one night directly, and the next night by reflection in mercury; or,
+if the star is a close circumpolar, both observations may be made the same
+evening, one a few minutes before its meridian passage, the other just as
+long after. But the method of the collimating eye-piece is fully as accurate
+and vastly more convenient.
+
+\includegraphicsmid{illo024}{\textsc{Fig.~24.}---Altitude and Azimuth Instrument.}
+%% -----File: 058.png---Folio 47-------
+
+\nbarticle{68.} \nbparatext{Differential Use of the Instrument.}---We now know the places of
+several hundred stars with so much precision that in many cases it is quite
+sufficient to observe one or two of these ``\textit{standard stars}'' in connection with
+the bodies whose places we wish to determine. The difference between the
+declination of the known star and that of any star whose place is to be
+determined, will, of course, be simply the difference of their circle-readings,
+corrected for refraction, etc. The meridian circle is said to be used ``\textit{differentially}''
+when thus treated.
+\end{fineprint}
+
+\nbarticle{69.} \nbparatext{Errors of Graduation, etc.}---If the circle is from a reputable
+maker, and has four or six microscopes, and if the observations are
+carefully made and all the microscopes read each time, results of
+sufficient precision for most purposes may be obtained by merely
+correcting the observations for ``runs'' and refraction. The outstanding
+errors ought not to exceed a second or two. But when the
+\textit{tenths} of a second are in question, the case is different. It will not
+then do for the astronomer to assume the accuracy of the graduation
+of his circle, but he must investigate the \textit{errors of its divisions}, the
+errors of the \textit{micrometer screws} in the microscopes, the \textit{flexure} of the
+telescope, and the effect of differences of temperature in shifting
+the zero points of the circle, by slightly disturbing the position or
+direction of the microscopes. Of course this is not the place to
+enter into such details, but it is an opportunity to impress again upon
+the student the fact that truth and accuracy are only attainable by
+immense painstaking and labor.
+
+\begin{fineprint}
+\nbarticle{70.} \nbparatext{Mural Circle.}---This instrument is in principle the same as the
+meridian circle, which has superseded it. It consists of a circle, carrying a
+telescope mounted on the face of a \textit{wall} of masonry (as its name implies)
+and free to revolve in the plane of the meridian. The wall furnishes a convenient
+support for the microscopes.
+\end{fineprint}
+
+\includegraphicsouter{illo025}{\textsc{Fig.~25.}\\The Equatorial (Schematic).}
+
+\nbarticle{71.} \nbparatext{Altitude and Azimuth Instrument.}---Since the transit instrument
+and meridian circle are confined to the plane of the meridian,
+their usefulness is obviously limited. Meridian observations, when
+they are to be had, are better and more easily used than any others,
+but are not always attainable. We must therefore have instruments
+which will follow an object to any part of the heavens.
+
+\sloppy
+The \textit{altitude and azimuth instrument} is simply a surveyor's theodolite
+on a large scale. It has a horizontal circle turning upon a \textit{vertical
+axis}, and read by verniers or microscopes. Upon this circle, and
+turning with it, are supports which carry the \textit{horizontal axis} of the
+telescope with its vertical circle, also read by microscopes. Obviously
+%% -----File: 059.png---Folio 48-------
+the readings of these two circles, when the instrument is properly
+adjusted and the zero points determined, will give the altitude
+and azimuth of the body pointed on. \figref{illo024}{Fig.~24} represents a small instrument
+of this kind.
+%% -----File: 060.png---Folio 49-------
+
+\includegraphicsmid{illo026}{\textsc{Fig.~26.}---The 23-inch Princeton Telescope.}
+
+\fussy
+\nbarticle{72.} \nbparatext{The Equatorial.}---The essential characteristic of this instrument
+is that its principal axis, \textit{i.e.}, the axis which rests in \textit{fixed} bearings,
+instead of being either horizontal or vertical, is inclined at an angle
+equal to the latitude of the place, and directed towards the pole, thus
+placing it parallel to the earth's axis of rotation. This axis of the
+instrument is called its \textit{polar axis}; and the graduated circle which it
+carries, and which is parallel to the celestial
+equator, is called the \textit{hour-circle}, because
+its reading gives the \textit{hour-angle} of
+the body upon which the telescope happens
+to be pointed. Sometimes, also, it is
+called the Right Ascension Circle. Upon
+this polar-axis are secured the bearings
+of the \textit{declination axis}, which is perpendicular
+to the polar axis, and carries the
+telescope itself and the declination circle.
+
+In the instruments before described, the
+telescope is a mere \textit{pointer}, and wholly
+subsidiary to the circles; in the equatorial
+the telescope is usually the main thing,
+and the circles are subordinate, serving
+only to aid the observer in finding or
+identifying the body upon which the telescope is directed.
+
+Fig.~25 exhibits schematically the ordinary form of equatorial
+mounting, of which there are numerous modifications. \figref{illo026}{Fig.~26} is the
+23-inch Clark telescope at Princeton, and \figref{illo027}{Fig.~27} is the 4-foot
+Melbourne reflector. The frontispiece is the great Lick telescope
+of thirty-six inches diameter.
+
+\includegraphicsmid{illo027}{\textsc{Fig.~27.}---The Melbourne Reflector.}
+
+The advantages of the equatorial mounting for a large telescope
+are very great as regards convenience. In the first place, when the
+telescope is once pointed upon a star or planet, it is only necessary
+to turn the polar axis with a uniform motion in order to ``follow'' the
+star, which otherwise would be carried out of the field of view in a
+few moments by the diurnal motion. This motion, since it is uniform,
+can be, and in all large instruments usually is, given by clock-work,
+with a continuous regulator of some kind, similar to that used
+in the chronograph. The instrument once directed and clamped,
+and the clock-work started, the object will continue apparently immovable
+in the field of view as long as may be desired.
+
+In the next place, it is very easy to find an object, even if invisible
+to the naked eye, like a faint comet or nebula, or a star in the daytime,
+%% -----File: 061.png---Folio 50-------
+provided we know its declination and right ascension, and
+have the sidereal time; for which reason a sidereal clock or chronometer
+is an indispensable adjunct of the equatorial.
+
+\begin{fineprint}
+To find an object, the telescope is turned in declination until the reading
+of the declination circle corresponds to the declination of the object, and
+then the polar axis is turned until the hour-circle of the instrument (not to
+be confounded with an hour-circle in the sky) reads the \textit{hour-angle} of the
+object. This hour-angle, it will be remembered, is simply the difference between
+%% -----File: 062.png---Folio 51-------
+the sidereal time and the right ascension of the object. The hour-angle
+is east if the right ascension exceeds the time; west, if it is less.
+When the telescope is thus set, the object will be found (with a low magnifying
+power) in the field of view, unless it is near the horizon, in which
+case refraction must be taken into account.
+\end{fineprint}
+
+While the instrument cannot give very accurate determinations of
+the positions of bodies by the direct readings of its circles, on account
+of the irregular flexures of its axes, it may do so indirectly; that is,
+it may be used to determine very accurately the \textit{difference} between
+the right ascension and declination of a comet or planet, for instance,
+and that of some neighboring star, whose place has been already
+determined by the meridian circle; and this is one of the most important
+uses of the instrument.
+%% -----File: 063.png---Folio 52-------
+
+\includegraphicsouter[14]{illo028}{\textsc{Fig.~28.}---The Filar Position-Micrometer.}
+
+\nbarticle{73.} \nbparatext{The Micrometer.}---Micro\-meters of various sorts are employed
+for the purpose. The most common and most generally useful is the
+so-called ``\textit{filar position-micrometer},'' \figref{illo028}{Fig.~28}, which is an indispensable
+auxiliary of every good
+telescope.
+
+It is a small instrument, much
+like the upper part of the reading
+microscope, but more complicated.
+It usually contains a
+reticle of fixed wires, two or
+three parallel to each other, and
+crossed at right angles by a
+second set. Then there are two
+or three wires parallel to the first
+set, and movable by an accurately
+made screw with a graduated
+head and a counter, or
+scale, for indicating the number
+of entire revolutions made by
+the screw. The box containing
+these wires, and carrying the eye-piece and screw, can itself be
+turned around in a plane perpendicular to the optical axis of the
+telescope, and set in any desired position; for example, so that the
+movable wires shall be parallel to the celestial equator, while the
+other set run north and south. This ``position angle'' is read on a
+graduated circle, which forms part of the instrument. Means of
+illumination are provided, giving at pleasure either dark wires in a
+bright field, or \textit{vice versa}.
+
+\includegraphicsmid{illo029}{\textsc{Fig.~29.}---Construction of the Micrometer.}
+
+With this instrument one can measure the distance (in seconds of
+arc), and the direction between any two stars which are near enough
+to be seen at once in the same field of view. This range in small
+%% -----File: 064.png---Folio 53-------
+telescopes may reach $30'$ of arc; while in the larger instruments,
+which, with the same eye-pieces have much higher magnifying powers,
+it is necessarily less,---not more than from $5'$ to $10'$.
+
+\begin{fineprint}
+\nbarticle{74.} A new form of equatorial, known as the
+\textit{Equatorial Coudé}, or \textit{Elbowed} Equatorial, has been
+recently introduced at the Paris Observatory. With large instruments
+of the ordinary form a great deal of inconvenience is encountered by
+the observer, in moving about to follow the eye-piece into the
+various positions into which it is forced by the inconsiderateness
+of the heavenly bodies. Moreover, the revolving dome, which is
+usually erected to shelter a great telescope, is an exceedingly
+cumbrous and expensive affair.
+
+In the Equatorial Coudé, \figref{illo030}{Fig.~30}, these difficulties are overcome by
+the use of mirrors. The observer sits always in one fixed position,
+looking obliquely down through the polar axis, which is also the
+telescope tube.
+
+The Paris instrument has an object-glass about ten inches in
+diameter, and performs very satisfactorily. The two reflections,
+however, cause a considerable loss of light, and some injury to the
+definition. The mirrors, and the consequent complications, also add
+heavily to the cost of the instrument. \figref{illo030}{Fig.~30} is from a photograph
+of this instrument.
+\end{fineprint}
+
+\includegraphicsmid{illo030}{\textsc{Fig.~30.}---The Equatorial Coudé.}
+\nbthought
+
+\nbarticle{75.} All the instruments so far described, except the chronometer,
+are \textit{fixed} instruments; of use only when they can be set up
+firmly and carefully adjusted to established positions. Not one of
+them would be of the slightest use on \DPtypo{shipboard}{ship-board}.
+%% -----File: 065.png---Folio 54-------
+
+We have now to describe the instrument which, with the help of
+the chronometer, is the main dependence of the mariner. It is an
+instrument with which the observer measures the angular distance
+between two objects; as, for instance, the sun and the visible horizon,
+not by pointing first on one and then afterwards on the other, but by
+\textit{sighting them both, simultaneously and in apparent coincidence}; which
+can be done even when he has no fixed position or stable footing.
+
+
+\nbarticle{76.} \nbparatext{The Sextant.}---The graduated limb of the sextant is carried
+by a light framework, usually of metal, provided with a suitable handle
+$X$. The arc is about one-sixth of a circle, as the name implies, and
+is usually from five to eight inches radius. It bears a graduation of
+half-degrees, \textit{numbered as whole degrees}, so that it can measure any
+angle less than $120°$.
+
+An ``\textit{index-arm},'' $MN$ in the \figref{illo031}{figure}, is pivoted at the centre of the
+arc, and carries a vernier which slides along the limb, and can be
+fixed at any point by a clamp and delicately moved by the attached
+tangent screw, $T$. The reading of this vernier gives the angle
+measured by the instrument. The best instruments read to $10''$.
+
+Just over the centre of motion, the ``\textit{index-mirror}'' $M$, about
+two inches by one and one-half in size, is fastened securely to the
+index-arm, so as to be perpendicular to the plane of the limb. At
+%% -----File: 066.png---Folio 55-------
+$H$, the ``horizon-glass,'' about an inch wide and of the same height
+as the index-glass, is secured firmly to the frame of the instrument,
+in such position that, when the vernier of the index-arm reads \textit{zero},
+the index-mirror and horizon-glass will be parallel to each other.
+Only \textit{half} of the horizon-glass is silvered, the upper half being left
+transparent. $E$ is a small telescope.
+
+\includegraphicsmid{illo031}{\textsc{Fig.~31.}---The Sextant.}
+
+If the vernier stands \textit{near}, but not \textit{at} zero, the observer looking
+into the telescope will see together in the field of view \textit{two} separate
+images of the object; and if, while still looking, he slides
+the vernier a little, he will see that one of the images remains fixed,
+while the other moves. The fixed image is due to the rays which
+reach the object-glass of the telescope directly, coming through
+the unsilvered half of the horizon-glass; the movable image, on the
+other hand, is produced by rays which have suffered two reflections,---first,
+from the index-mirror to the horizon-glass; and second, at
+the lower half of the horizon-glass. When the two mirrors are
+parallel, and the vernier reads zero, the two images coincide, provided
+the object is at a considerable distance.
+
+If now the vernier does not stand at or near zero, the observer,
+looking at any object directly through the horizon-glass, will see,
+not only that object, but also whatever other object is so situated
+as to send its rays to the telescope by reflection upon the mirrors;
+\textit{and the reading of the vernier will give the angle at the instrument
+between the two objects whose images thus coincide}; the angle
+between the planes of the two mirrors being just half that between
+the objects, and the half-degrees on the limb being numbered as
+whole ones.
+
+\nbarticle{77.} The principal use of the instrument is in measuring the altitude
+of the sun. At sea the observer, holding the instrument with his right
+hand and keeping the plane of the arc vertical, looks \textit{directly} towards
+the visible horizon at the point under the sun, through the horizon-glass
+(whence its name); then by moving the vernier with his
+left hand, he inclines the index-glass upwards until one edge of the
+reflected image of the sun is brought just to touch the horizon-line,
+noting the exact time by the chronometer, if necessary. The reading
+of the vernier, after correcting for the semi-diameter of the sun, the
+dip of the horizon, the refraction, and the parallax (and for the
+``index-error'' of the sextant, if the vernier does not read strictly
+zero when the mirrors are parallel) gives the sun's true altitude at the
+moment.
+%% -----File: 067.png---Folio 56-------
+
+\begin{fineprint}
+\includegraphicsouter{illo032}{\textsc{Fig.~32.}---Principle of the Sextant.}
+
+\nbarticle{78.} On land the \textit{visible} horizon is of no use, and we have recourse to an
+``\textit{artificial horizon},'' as it is called. This is merely a shallow basin of mercury,
+covered, when necessary to protect it from the wind, with a roof made of
+glass plates having their sides \textit{plane} and \textit{parallel}.
+
+In this case we measure the angle between the sun's image reflected in the
+mercury and the sun itself. The reading of the instrument, corrected for
+index-error, gives \textit{twice the sun's apparent altitude}; which apparent altitude,
+corrected as before for refraction and
+parallax, but not for dip of the horizon,
+gives the true altitude. The skilful use
+of the sextant requires steadiness of
+hand and considerable dexterity, and
+from the small size of the telescope the
+angles measured are of course less precise
+than if determined by large fixed
+instruments. But its portability and
+applicability at sea render it absolutely
+invaluable.
+
+\nbarticle{79.} The principle that the true angle
+between the objects whose images coincide
+is twice the angle between the mirrors
+(or between their normals) is easily
+demonstrated as follows (\figref{illo032}{Fig.~32}):---
+
+The ray $SM$ coming from an object, after reflection first at $M$ (the index-mirror),
+and then at $H$ (the horizon-glass), is made to coincide with the
+ray $OH$ coming from the horizon. We must prove that the angle $SEO$, between
+the object and the horizon, as seen from the point $E$ in the instrument,
+is double the angle $Q$, between $MQ$ and $HQ$, which are normals to the mirrors,
+and therefore double $Q'$, which is the angle between the planes of the
+mirrors.
+
+First, from the law of reflection, we have,
+\[
+SMP = HMP, \text{ or } SMH = 2 × PMH.
+\]
+\begin{flalign*}
+\text{\indent Similarly, }&& MHE &= 2 × MHQ. &&\phantom{\text{\indent Similarly, }}
+\end{flalign*}
+
+From the geometric principle that the exterior angle $SMH$ of the triangle
+$HME$ is equal to the sum of the opposite interior angles at $H$ and $E$, we get
+\[
+HEM = SMH - MHE = 2\,PMH - 2\,MHQ = 2(PMH-MHQ).
+\]
+
+Similarly, from the triangle $HMQ$, we have
+\[
+HQM = PMH - MHQ,
+\]
+which is half the value just found for $HEM$, and proves the proposition.
+\end{fineprint}
+%% -----File: 068.png---Folio 57-------
+
+Of course with the sextant, as with all other instruments, it is
+necessary for the observer who aims at the utmost precision to investigate,
+and take into account its errors of graduation, construction
+and adjustment; but their discussion does not belong here.
+
+\nbarticle{80.} Besides the instruments we have described, there are many
+others designed for special work, some of which, as the zenith telescope,
+and heliometer, will be mentioned hereafter as it becomes
+necessary. There is also a whole class of physical instruments,
+photometers, spectroscopes, heat-measuring appliances, and photographic
+apparatus, which will have to be considered in due time.
+
+\sloppy
+But with clock, meridian circle, and equatorial and their usual
+accessories, all the fundamental observations of theoretical and spherical
+astronomy can be supplied. The chronometer and sextant are
+practically the only astronomical instruments of any use at sea.
+\chelabel{CHAPTERII}
+%% -----File: 069.png---Folio 58-------
+
+\fussy
+\Chapter{III}{Corrections to Astronomical Observations}
+\nbchapterhang{\stretchyspace CORRECTIONS TO ASTRONOMICAL OBSERVATIONS, DIP OF THE
+HORIZON, PARALLAX, SEMI-DIAMETER, REFRACTION, AND
+TWILIGHT.}
+
+\nbarticle{81.} \nbparatext{Dip of the Horizon.}---In observations of the altitude of
+a heavenly body at sea, where the measurement is made from
+the \textit{sea-line}, a correction is needed on account of the fact that
+this visible horizon does not coincide with the true astronomical
+horizon (which is $90°$ from the zenith), but
+falls sensibly below it by an amount known
+as the \textit{Dip of the Horizon}. The amount of this dip
+depends upon the size of the earth
+and the height of the observer's eye above
+the sea-level.
+
+\includegraphicsouter{illo033}{\textsc{Fig.~33.}---Dip of the Horizon.}
+
+In \figref{illo033}{Fig.~33}, $C$ is the centre of the earth,
+$AB$ a portion of its level surface, and $O$ the
+observer, at an elevation $h$ above $A$. The
+line $OH$ is truly horizontal, while the tangent
+line, $OB$, corresponds to the line drawn
+from the eye to the visible horizon. The
+angle $HOB$ is the \textit{dip}. This is obviously equal to the angle $OCB$
+at the centre of the earth, if we regard the earth as spherical, as we
+may do with quite sufficient accuracy for the purpose in hand.
+
+From the right-angled triangle $OBC$ we have directly
+\[
+\cos OCB = \frac{BC}{CO}.
+\]
+Putting $R$ for the radius of the earth, and $\Delta$ for the dip, this becomes
+\[
+\cos \Delta  = \frac{R}{R + h}.
+\]
+
+\begin{fineprint}
+This formula is exact, but inconvenient, because it gives the small angle
+$\Delta$ by means of its cosine. Since, however, $1- \cos \Delta = 2 \sin^2 \frac{1}{2} \Delta$, we easily
+obtain the following:---
+\[
+\sin \tfrac{1}{2} \Delta = \sqrt{\frac{h}{2(R + h)}}.
+\]
+%% -----File: 070.png---Folio 59-------
+
+This gives the true depression of the sea horizon, as it would be if the
+line of sight, drawn from the eye to the horizon line, were \textit{straight}. On
+account of refraction it is not straight, however, and the amount of this
+``terrestrial refraction'' is very variable and uncertain. It is usual to
+diminish the dip computed from the formula by one-eighth its whole amount.
+\end{fineprint}
+
+An approximate formula\footnote%
+  {\setlength{\abovedisplayskip}{0.5\abovedisplayskip}
+   \setlength{\belowdisplayskip}{0.5\belowdisplayskip}
+  This approximate formula may be obtained thus:---
+  \begin{align*}
+    &2 \sin^2 (\tfrac{1}{2} \Delta) = \frac{h}{R+h}
+  = \left( \frac{h}{R} \right) + \left( 1 + \frac{h}{R} \right).\\[-2ex]
+  \intertext{\indent But since $\dfrac{h}{R}$ is a very small fraction, it may be neglected in the divisor $\left( 1 + \dfrac{h}{R} \right)$,
+  and the expression becomes simply,}\\[-6ex]
+    &2 \sin^2 \tfrac{1}{2} \Delta = \frac{h}{R}; \text{ whence }
+      \sin   \tfrac{1}{2} \Delta = \sqrt{\frac{h}{2R}}.
+  \intertext{\indent Since $\Delta$ is a very small angle,}\\[-4ex]
+  & \Delta = \sin \Delta = 2 \sin \tfrac{1}{2} \Delta, \quad \text{so that }\\
+  & \Delta \text{ (in} \textit{ radians})
+  = 2 \sqrt{ \frac{h}{2R} }
+  =   \sqrt{ \frac{h}{\frac{1}{2}R} }.\\
+  \intertext{\indent To reduce radians to minutes, we must multiply by 3438, the number of minutes
+  in a radian. Accordingly, }\\[-6ex]
+  & \Delta'\ (\text{in minutes of arc})
+  = 3438 \sqrt{ \frac{h}{\frac{1}{2}R} }.\\[-2ex]
+  \intertext{\indent If we express $h$ in feet, we must also use the same units for $R$. The mean
+  radius of the earth is about 20,884,000 feet, one-half of which is 10,442,000, and
+  the square root of this is 3231; so that the formula becomes }\\[-4ex]
+  & \Delta' = \frac{3438}{3231} \sqrt{h\text{ (feet)}},
+  \end{align*}
+  which is near enough to that given in the text.
+
+  In fact, the refraction makes so much difference that after taking the
+  numerical factor, $\dfrac{3438}{3231}$, as unity, the formula still gives $\Delta' $ about $\frac{1}{20}$ part too large.
+
+  The formula $\Delta' = \sqrt{ 3 h \ (\textit{metres})}$ is yet more nearly correct.
+  } %endfootnote
+for the dip is
+\[
+\Delta \text{ (in minutes of arc) } = \sqrt{h \text{ (feet)}};
+\]
+or, in words, \textit{the square root of the elevation of the eye $($in feet$)$ gives
+the dip in minutes}. This gives a value about $\frac{1}{20}$  part too large.
+
+Since the dip is applicable only to sextant observations made at
+sea, where, from the nature of the instrument, and the rising and
+falling of the observer with the vessel's motion, it is not possible to
+measure altitudes more closely than within about $15''$, there is no
+need of any extreme precision in its calculation.
+%% -----File: 071.png---Folio 60-------
+
+\nbarticle{82.} \nbparatext{Parallax.}---In the most general sense, ``parallax'' is the change
+of a body's direction resulting from the observer's displacement. In
+the restricted and technical sense in which we are to employ it now,
+it may be defined as the difference \textit{between the direction of a body as
+actually observed and the direction it would have if seen from the earth's
+centre.} Thus in the figure, \figref{illo034}{Fig.~34}, where the observer is supposed
+to be at $O$, the position of $P$ in the sky (as seen from $O$) would be
+marked by the point where $OP$ produced would pierce the celestial sphere. Its position as seen from $C$ would be determined in the
+same way by producing $CP$ to which $OX$ is drawn parallel. The
+angle $POX$, therefore, or its equal, $OPC$, is \textit{the parallax of $P$ for
+an observer at $O$}.
+
+\includegraphicsouter{illo034}{\textsc{Fig.~34.}---Diurnal Parallax.}
+
+Obviously, from the \figref{illo034}{figure}, we may also give the following definition
+of the parallax. \textit{It is the angular
+distance} (number of seconds of
+arc) \textit{between the observer's station and
+the centre of the earth's disc, as seen
+from the body observed}. The moon's
+parallax at any moment for me is my
+angular distance from the earth's centre,
+as seen by ``the man in the moon.''
+
+When a body is in the zenith its
+parallax is zero, and it is a maximum
+at the horizon. In all cases it
+\textit{depresses} a body, diminishing the
+altitude \textit{without changing the azimuth.}
+
+The ``law'' of the parallax is, that \textit{it varies as the sine of the zenith distance
+directly, and inversely as the linear distance} (in miles) \textit{of the body.}
+
+This follows easily from the triangle $COP$, where we have
+$PC : OC =\sin COP : \sin CPO$.
+
+Put $D$ for $PC$, the distance of the body from the earth; $R$ for
+the earth's radius, $CO$; $CPO$, the parallax; $\zeta$ for $ZOP$, the apparent
+zenith distance, and remember that the sine of $\zeta$ is equal to the
+sine of its supplement, $COP$: we then have as the translation of
+the above proportion,
+\begin{flalign*}
+&&D : R &= \sin \zeta : \sin p.&&\\
+&\hspace{\parindent}\text{This gives us } &\sin p &= \frac{R}{D} \cdot \sin \zeta;&\phantom{This gives us }\\
+\intertext{or, since $p$ is always a small angle,}
+&&p'' &= 206265'' \frac{R}{D} \cdot \sin \zeta.&&
+\end{flalign*}
+%% -----File: 072.png---Folio 61-------
+
+\nbarticle{83.} \nbparatext{Horizontal Parallax.}---When a body is at the horizon ($P_{h}$ in
+the \figref{illo034}{figure}), then $\zeta$ becomes $90°$, and $\sin \zeta = 1$. In this case the parallax
+reaches its maximum value, which is called \textit{the horizontal parallax}
+of the body. Taking  $p_h$  as the symbol for this, we have
+\[
+\sin p_h = \frac{R}{D};\mbox{ or, nearly enough, }p_{h} = 206265''\,\frac{R}{D}.
+\]
+Comparing this with the formula above, we see that the parallax of
+a body at any zenith distance equals the \textit{horizontal parallax multiplied
+by the sine of the zenith distance}; i.e.,  $p = p_{h} \sin \zeta$.
+
+\textbf{N.B.} A glance at the \figref{illo034}{figure} will show that we may define the
+\textit{horizontal parallax}, $OPC$, of any body, as \textit{the angular semi-diameter
+of the earth seen from that body}. To say, for instance, that the sun's
+horizontal parallax is $8''.8$, amounts to saying that, \textit{seen from the sun},
+the earth's apparent diameter is twice $8''.8$, or $17''.6$.
+
+\nbarticle{84.} \nbparatext{Relation between Horizontal Parallax and Distance.}\allowbreak---Since we have
+\[
+\sin p_{h} =  \frac{R}{D},
+\]
+it follows of course that $D = R ÷ \sin p_h$;
+\begin{flalign*}
+&\text{or, (nearly) }& D &= \frac{206265''}{p_h''} × R. &&\phantom{\text{or, (nearly) }}
+\end{flalign*}
+If the sun's parallax equals $8''.8$,
+\[
+\mbox{its distance} =  \frac{206265}{8.8} × R = 23439\, R.
+\]
+
+\sloppy
+\nbarticle{85.} \nbparatext{Equatorial Parallax.}---Owing to the ``ellipticity'' or ``oblateness''
+of the earth the horizontal parallax of a body varies
+slightly at different places, being a maximum at the equator, where
+the distance of an observer from the earth's centre is greatest. It
+is agreed to take as the standard the \textit{equatorial} horizontal parallax;
+\textit{i.e.}, the earth's \textit{equatorial} semi-diameter as seen from the body.
+
+\fussy
+\nbarticle{86.} \nbparatext{Diurnal Parallax.}---The parallax we have been discussing is
+sometimes called the \textit{diurnal parallax}, because it runs through all its
+possible changes in one day.
+
+\begin{fineprint}
+When the sun, for instance, is rising, its parallax is a maximum, and by
+throwing it down towards the east, increases its apparent right ascension.
+At noon, when the sun is on the meridian, its parallax is a minimum, and
+%% -----File: 073.png---Folio 62-------
+affects only the declination. At sunset it is again a maximum, but
+now throws the sun's apparent place down towards the west. Although
+the sun is invisible while below the horizon, yet the parallax,
+\textit{geometrically considered}, again becomes a minimum at
+midnight, regaining its original value at the next sunrise.
+\end{fineprint}
+
+The qualifier, ``diurnal,'' is seldom used except when it is
+necessary to distinguish between this kind of parallax and the
+\textit{annual} parallax of the fixed stars, which is due to the
+earth's orbital motion. The stars are so far away that they have no
+sensible \textit{diurnal} parallax (the earth is an infinitesimal
+point as seen from them); but some of them do have a slight and
+measurable \textit{annual} parallax, by means of which we can
+roughly determine their distances.\quad (\chapref{CHAPTERXIX}{Chap. XIX.})
+
+\nbarticle{87.} \nbparatext{Smallness of Parallax.}---The horizontal
+\textit{parallax} of even the nearest of the heavenly bodies is
+always small. In the case of the moon the average value is about
+$57'$, varying with her continually changing distance. Excepting now
+and then a stray comet, no other heavenly body ever comes within a
+distance a hundred times as great as hers. Venus and Mars approach
+nearest, but the parallax of neither of them ever reaches $40''$.
+
+\nbarticle{88.} \nbparatext{Semi-Diameter.}---In order to obtain the true
+altitude of an object it is necessary, if the edge, or
+\textit{``limb,''} as it is called, has been observed, to add or
+deduct the apparent semi-diameter of the object. In most cases this
+will be sensibly the same in all parts of the sky, but the moon is
+so near that there is quite a perceptible difference between her
+diameter when in the zenith and in the horizon.
+
+\begin{fineprint}
+A glance at \figref{illo034}{Fig.~34} shows that in the zenith the moon's distance is
+less than at the horizon, by almost exactly the earth's radius---the
+difference
+between the lines $OZ$ and $OP_h$. Now this is very nearly one-sixtieth part
+of the moon's distance, and consequently the moon, on a night when
+its apparent diameter at rising is $30'$, will be $30''$
+\textit{larger} when near the zenith. Since the semi-diameter given
+in the almanac is what would be seen from the \textit{centre of the
+earth}, every measure of the moon's distance from stars or from the
+horizon will require us to take into account this ``augmentation of
+the semi-diameter,'' as it is technically called.
+
+The formula, easily deduced from the \figref{illo034}{figure} by remembering that the
+angle $PCO = \zeta - p$ (zenith distance $-$ parallax), and that the
+apparent and ``almanac'' diameters will be inversely proportional to
+the two distances $OP$ and $CP$, is
+\[
+  \text{apparent semi-diameter } = \text{ almanac s.\ d.} × \frac {\sin \zeta}{\sin \left( \zeta - p \right)}.
+\]
+\end{fineprint}
+%% -----File: 074.png---Folio 63-------
+
+This measurable increase of the moon's angular diameter at high
+altitudes has nothing to do with the purely subjective illusion which
+makes the disc \textit{look} larger to us when \textit{near the horizon}. That it \textit{is} a
+mere illusion may be made evident by simply looking through a dark
+glass just dense enough to hide the horizon and intervening landscape.
+The moon or sun then seems to shrink at once to normal
+dimensions.
+
+\includegraphicsouter{illo035}{\textsc{Fig.~35.}---Atmospheric Refraction.}
+
+\sloppy
+\nbarticle{89.} \nbparatext{Refraction.}---\\
+Rays of light have their direction changed by
+refraction in passing through the air, and as the \textit{direction in which we
+see a body is that in which its light reaches the eye}, it follows that this
+refraction apparently displaces
+the stars and all
+bodies seen through the
+atmosphere. So far as
+the action is regular, the
+effect is to bend the rays
+directly \textit{downwards}, and
+thus to make the objects
+appear \textit{higher} in the sky.
+Refraction \textit{increases the
+altitude} of a celestial object
+\textit{without altering the
+azimuth}. Like parallax,
+it is zero at the zenith
+and a maximum at the
+horizon; but it follows a
+different law. It is entirely independent of the distance of the
+object, and its amount varies (nearly) as the \textit{tangent} of the zenith
+distance---not as \textit{the sine}, as in the case of parallax.
+
+\fussy
+\begin{fineprint}
+\nbarticle{90.}  This approximate law of the refraction is easily proved.
+
+Suppose in \figref{illo035}{Fig.~35} that the observer at $O$ sees a star in the direction $OS$,
+at the zenith distance $ZOS$ or $\zeta$. The light has reached him from $S'$ by a
+path which was straight until the ray met the upper surface of the air at $A$,
+but afterwards curved continually downwards as it passed from rarer to
+denser regions.
+
+We know that the atmosphere is very shallow as compared with the size
+of the earth, and it is exceedingly rare in the upper portions, so that, as
+far as concerns refraction, we may assume that the point $A$, where the first
+perceptible bending of the ray occurs, is not more than fifty miles high,
+and that the \textit{vertical $AZ'$ is sensibly parallel} to $OZ$; consequently, also,
+%% -----File: 075.png---Folio 64-------
+that \textit{all the successive ``strata of equal density'' are parallel to each other and
+to the upper surface of the air}.
+
+[This amounts to neglecting the earth's curvature between $O$ and $B$.]
+
+The \textit{true} zenith distance (as it would be if there were no refraction) is
+$ZDS'$, which equals $Z'AS'$; and since the retraction, $r$, may be defined as
+the difference between the true and apparent zenith distances, this true
+zenith distance will $= \zeta + r$.
+
+Now from optical principles, when a ray of light passes through a
+medium composed of parallel strata, the final direction of the ray is the
+same as if the medium had throughout the density of the last stratum,
+and therefore the final direction, $SO$, will be the same as if all the air, from
+$A$ down, had the same density as at $O$, with the same index of refraction,
+$n$. We may therefore apply the law of refraction directly at $A$, and write
+$\sin Z'A S' = n \sin BAC \left( = ZOS \right)$, or $ \sin \left( \zeta +r \right) = n \sin \zeta$;
+$AC$ being drawn
+parallel to $OS$.
+
+{\allowdisplaybreaks
+Developing the first member, we have
+\begin{flalign*}
+&&&\sin \zeta \cos r + \cos \zeta \sin r = n \sin \zeta.&&&
+\intertext{\indent But $r$ is always a small angle, never exceeding $40'$; we may therefore take
+$ \cos r = 1$. Doing this and transposing the first term, we get}
+&&&\cos \zeta \sin r
+= n \sin \zeta - \sin \zeta = \left( n - 1 \right) \sin \zeta.&&&\\[1ex]
+&\text{\indent Whence,}&&\sin r = (n- 1) \tan \zeta;&&&  \\[1ex]
+&\text{or, }     &&r'' =(n-1) 206265 \tan \zeta \; \text{(nearly)}.&&&
+\intertext{\indent The index of refraction for air, \textit{at zero centigrade} and \textit{a barometric
+pressure of} $760^\text{mm}$, is $1.000294$; whence,}
+&&&r'' = .000294 × 206265 × \tan \zeta = 60''.6 \tan \zeta.&&&
+\end{flalign*}
+}
+
+This equation holds very nearly indeed down to a zenith distance
+of $70°$, but fails as we approach the horizon. For rays coming nearly
+horizontal, the points $A$ and $B$ are so far from $O$ that the normal
+$AZ'$ is no longer practically parallel to $OZ$; and many of the other
+fundamental assumptions on which the formula is based also break
+down.
+
+At the horizon, where $\zeta = 90°$ and $\tan \zeta = \textit{infinity}$, the formula
+would give $\sin r = \textit{infinity}$ also; an absurdity, since no sine can
+exceed unity. The refraction there is really about $37'$, under the
+circumstances of temperature and pressure above indicated.
+\end{fineprint}
+%% -----File: 076.png---Folio 65-------
+
+\nbarticle{91.} \nbparatext{Effect of Temperature and Barometric Pressure.}---The index
+of refraction of air depends of course upon its temperature and pressure.
+As the air grows \textit{warmer}, its refractive power \textit{decreases}; as it grows
+\textit{denser}, the refraction \textit{increases}. Hence, in all precise observations of
+the altitude (or zenith distance), it is necessary to note both the
+thermometer and the barometer, in order to compute the refraction
+with accuracy. For rough work, like ordinary sextant observations,
+it will answer to use the ``mean refraction,'' corresponding to an
+average state of things.
+
+\begin{fineprint}
+\textbf{Tables of Refraction.}---The computation of the refraction is best
+effected by special tables made for the purpose; of these, Bessel's tables are
+the most convenient, best known, and probably even yet the most accurate.
+It must be always borne in mind, however, that from the action of wind and
+other causes the condition of the air along the path of the ray is seldom perfectly
+normal; in consequence, the actual refraction in any given case is liable
+to differ from the computed by as much as one or even two per cent.
+No amount of care in observation can evade this difficulty; the only remedy
+is a sufficient repetition of observations under varying atmospheric conditions.
+Observations at an altitude below $10°$ or $15°$ are never much to be
+trusted.
+
+\textbf{Lateral Refraction.}---When the air is much disturbed, sometimes objects
+are displaced horizontally as well as vertically. Indeed, as a general
+rule, when one looks at a star with a large telescope and high power, it will
+seem to ``dance'' more or less---the effect of the varying refraction which
+continually displaces the image.
+\end{fineprint}
+
+\nbarticle{92.} \nbparatext{Effect on the Time of Sunrise and Sunset.}---The horizontal refraction,
+ranging as it does from $34'$ to $39'$, according to temperature,
+is always somewhat greater than the diameter of either the sun or
+the moon. At the moment, therefore, when the sun's lower limb
+appears to be just rising, the whole disc is really below the plane
+of the horizon; and the \textit{time} of sunrise in our latitudes is thus
+accelerated from two to four minutes, according to the inclination of
+the sun's diurnal circle to the horizon, which inclination varies with
+the time of the year. Of course, sunset is delayed by the same
+amount, and thus the day is lengthened by refraction from four
+to eight minutes, at the expense of the night.
+
+\nbarticle{93.} \nbparatext{Effect on the Form and Size of the Discs of the Sun and Moon.}---Near
+the horizon the refraction changes very rapidly. While under
+ordinary summer temperature it is about $35'$ \textit{at} the horizon, it is
+%% -----File: 077.png---Folio 66-------
+only $29'$ at an elevation of half a degree; so that, as the sun or moon
+rises, the bottom of the disc is lifted $6'$ more than the top, and the
+vertical diameter is thus made apparently about one-fifth part shorter
+than the horizontal. This distorts the disc into the form of an oval,
+flattened on the under side. In cold weather the effect is much more
+marked. As the horizontal diameter is not at all increased by the
+refraction, the apparent \textit{area} of the disc is notably diminished by it;
+so that it is evident that refraction cannot be held in any way responsible
+for the apparent enlargement of the rising luminary.
+
+\nbarticle{94.} \nbparatext{Determination of the Refraction.}---1.~\textit{Physical Method.}
+Theory furnishes the \textit{law} of astronomical refraction, though the
+mathematical expression becomes rather complicated when we attempt
+to make it exact. In order, therefore, to determine the astronomical
+refraction under all possible circumstances, it is only necessary to
+determine the index of refraction of air, and its variations with temperature
+and pressure, by laboratory experiments, and to introduce the
+constants thus obtained into the formulæ. It is difficult, however, to
+make these determinations with the necessary precision. In fact, at
+present our knowledge of the constants of air rests mainly on astronomical
+work.
+
+2.~\textit{By Observations of Circumpolar Stars.} At an observatory whose
+latitude exceeds $45°$ select some star which passes \textit{through the zenith}
+at the upper culmination. (Its declination must equal the latitude of
+the observatory.) It will not be affected by refraction at the zenith,
+while at the lower culmination, twelve hours later, it will. With the
+meridian circle observe its \textit{polar distance} in both positions, determining
+the ``polar point'' of the circle as described on pp.~46--47. If the
+polar point were not itself affected by refraction, the simple difference
+between the two results for the star's polar distance, obtained
+from the upper and lower observations, would be the refraction at
+the lower point.
+
+As a \textit{first approximation}, however, we may neglect the refraction
+at the pole, and thus obtain a \textit{first} approximate lower refraction.
+By means of this we may compute an \textit{approximate} polar refraction,
+and so get a first ``corrected polar point.'' With this compute a
+\textit{second} approximate lower refraction, which will be much more nearly
+right than the first; this will give a \textit{second}  ``corrected polar point'';
+this will in turn give us a \textit{third} approximation to the refraction; and
+so on. But it would never be necessary to go beyond the third, as
+the approximation is very rapid. If the star does not go exactly
+%% -----File: 078.png---Folio 67-------
+through the zenith, it is only necessary to compute each time approximate
+refractions for its upper observation, as well as for the polar
+point.
+
+At present, however, the refraction is so well known that the
+method actually used is to form ``equations of condition'' from the
+observations of the altitude of known stars under varying circumstances,
+and from these to deduce such corrections to the star places
+and refraction constants as will best harmonize the whole mass of
+material.
+
+\begin{fineprint}
+\nbarticle{95.} 3.~\textit{By Observations of the Altitudes of Equatorial Stars made at an Observatory
+near the Equator}. For an observer so situated, stars that are on the
+celestial equator $(\delta = 0)$ will come to the meridian at the zenith, and will rise
+and fall \textit{vertically, with a motion strictly proportional to the time}; the \textit{true} zenith
+distance of the star at any moment being just equal to its \textit{hour-angle} in
+degrees. We have only, then, to observe the \textit{apparent} zenith distance of a
+star with the corresponding time, and the refraction comes out directly.
+
+If the station is not exactly on the equator, and if the star's declination is
+not exactly zero, it is only necessary to know the latitude and declination
+\textit{approximately} in order to get the refraction very accurately; a considerable
+error in either latitude or declination will affect the result but slightly.
+
+4.~The French astronomer Loewy has recently proposed a method which
+promises well. He puts a pair of reflectors, inclined to each other at a convenient
+angle of from $45°$ to $50°$ (a glass wedge with silvered sides), in front
+of the object-glass of an equatorial. This will bring to the eye two rays
+which make a strictly constant angle with each other, and there is no difficulty
+in finding pairs of stars so situated that their images will come into
+the field of view together. Now, were it not for refraction, these images
+would always keep their relative position unchanged, notwithstanding the
+diurnal motion; but on account of the changes in the refraction, as one star
+rises and the other falls, they will shift in the field, and micrometric measures
+will determine the shifting, and so the refraction, with great precision.
+\end{fineprint}
+
+\nbarticle{96.} \nbparatext{Twilight.}---{\footnotesize(Although this subject is outside the main purpose of this
+chapter, which deals with corrections to be applied to astronomical observations, we
+treat it here because, like refraction, it is a purely atmospheric phenomenon, and
+finds no other more convenient place.)}
+
+\smallskip
+Twilight, the illumination of the sky which begins before sunrise,
+and continues after sunset, is caused by the reflection of light to the
+observer from the upper regions of the earth's atmosphere. It is not
+yet certain whether this is due to reflection from foreign matter in the
+air, such as minute crystals of ice and salt, particles of dust of
+various kinds, and infinitesimal drops of water, or whether the pure
+gases themselves have some power of reflecting light. There is no
+%% -----File: 079.png---Folio 68-------
+doubt, however, that air, under the ordinary conditions, possesses
+considerable power of reflection; so that, as long as any air upon
+which the sun is shining is visible to the observer, it will send him
+more or less light, and appear illuminated.
+
+\includegraphicsmid{illo036}{\textsc{Fig.~36.}---Twilight.}
+
+Suppose the atmosphere to have the depth indicated in the \figref{illo036}{figure}.
+Then, if the sun is at $S$, \figref{illo036}{Fig.~36}, it will just have set to an observer at $1$,
+but all the air within his range of vision will still be illuminated. When,
+by the earth's rotation, he has been transported to $2$, he will see the
+``twilight bow'' rising in the east, a faintly reddish arc separating
+the illuminated part of the sky from the darkened part below, which
+lies in the shadow of the earth. When he reaches $3$, the western
+half of the sky alone remains bright, but the arc of separation between
+the light and darkness has become vague and indefinite; when
+he reaches $4$, only a glow remains in the west; and when he comes
+to $5$, night closes in on him. Nothing remains in sight on which
+the sun is shining.
+
+\begin{fineprint}
+\nbarticle{97.} \textit{Duration of Twilight.}---This depends upon the height of the atmosphere,
+and the angle at which the sun's diurnal circle cuts the horizon. It is
+found as a matter of observation, not admitting, however, of much precision,
+that twilight lasts until the sun has sunk about $18°$ below the horizon; that
+is to say, the angle $1\,C\,5$ in the \figref{illo036}{figure} is about $18°$.
+
+The time required to reach this point in latitude $40°$ varies from two
+hours at the longest days in summer, to one hour thirty minutes about Oct.\
+12 and March~1, when it is least. At the winter solstice it is about one
+hour and thirty-five minutes.
+
+In higher latitudes the twilight lasts longer, and the variation is more
+considerable; the date of the minimum also shifts.
+
+Near the equator the duration is shorter, hardly exceeding an hour at the
+%% -----File: 080.png---Folio 69-------
+sea-level; while at high elevations (where the amount of air above the
+observer's level is less) it becomes very brief. At Quito and Lima it is
+said not to last more than twenty minutes. Probably, also, in mountain
+regions the clearness of the air, and its purity contribute to the effect.
+\end{fineprint}
+
+\nbarticle{98.} \nbparatext{Height of the Atmosphere.}---It is evident from the \figref{illo036}{figure} that
+at the moment twilight ceases, the last visible portion of illuminated
+air is at the top of the atmosphere, and just half-way between the observer
+and the nearest point where the sun is setting. If the whole arc
+$1$,~$5$ is $18°$, $1$,~$3$ is $9°$: then calling the height of the atmosphere $H$ and
+the earth's radius $R$, and neglecting refraction (\textit{i.e.}, supposing the lines
+$1\,m$ and $5\,m$ to be straight), we have from the right-angled triangle
+$1\,C\,m$, $C\,m = 1\,C × \sec 9°$, or $R+H = R × \sec 9°$; whence $H = R\,
+(\sec 9° - 1) = 0.0125\, R$, or almost exactly fifty miles. This must
+be diminished about one-fifth part on account of the curvature of
+the lines $1\,m$ and $5\,m$ by refraction, making the height of the atmosphere
+about forty miles.
+
+The result must not, however, be accepted too confidently. It only
+proves that we get no sensible \textit{twilight illumination} from air at a
+greater height: above that elevation the air is either too rare, or too
+\textit{pure} from foreign particles, to send us any perceptible reflection.
+There is abundant evidence from the phenomena of meteors that the
+atmosphere extends to a height of 100~miles at least, and it cannot
+be asserted positively that it has \textit{any} definite upper limit.
+
+\begin{fineprint}
+\nbarticle{99.} \nbparatext{Aberration.}---There is yet one more correction which has to be
+applied in order to get the true direction of the line which at the instant of
+observation joins the eye of the observer to the star he is pointing at. The
+\textit{aberration of light} is an apparent displacement of the object observed, due
+to the combination of the earth's orbital motion with the progressive motion
+of light. It can be better discussed, however, in a different connection (see
+\chapref{CHAPTERVI}{Chap.~VI.}), and we content ourselves with merely mentioning it here.
+\end{fineprint}
+\chelabel{CHAPTERIII}
+%% -----File: 081.png---Folio 70-------
+
+\Chapter{IV}{Problems of Practical Astronomy}
+\nbchapterhang{PROBLEMS OF PRACTICAL ASTRONOMY, LATITUDE, TIME,
+LONGITUDE, AZIMUTH, AND THE RIGHT ASCENSION AND
+DECLINATION OF A HEAVENLY BODY.}
+
+\sloppy
+\nbarticle{100.} \textsc{There} are certain problems of Practical Astronomy which
+have to be solved in obtaining the fundamental facts from which we
+deduce our knowledge of the earth's form and dimensions, and other
+astronomical data.
+
+\fussy
+The first of these problems is that of the
+
+\section*{LATITUDE.}
+
+The latitude (\textit{astronomical}) of a place (\artref{Art.}{30}) is simply \textit{the altitude
+of the celestial pole $($Polhöhe$)$}, or, what comes to the same thing, as is
+evident from \figref{illo007}{Fig.~7} (\artref{Art.}{33}), it is the \textit{declination of the zenith}. It may
+also be defined, from the mechanical point of view, as \textit{the angle between
+the plane of the earth's equator and the observer's plumb-line or vertical}.
+
+\begin{fineprint}
+Neither of these definitions assumes anything as to the form of the earth,
+and we shall find farther on that this \textit{astronomical} latitude is seldom identical
+with the \textit{geocentric}, nor even with the \textit{geodetic} latitude of a place. It is,
+however, the only kind of latitude which can be \textit{directly} determined from
+astronomical observations, and its determination is one of the most important
+operations of what may be called Economic Astronomy.
+\end{fineprint}
+
+\nbarticle{101.} \nbparatext{Determination of Latitude.}---\nbparasubtext{First:}{By Circumpolars.} The
+most obvious method of determining the latitude is to observe, with the
+meridian circle or some analogous instrument, the altitude of a circumpolar
+star at its upper culmination, and again, twelve hours later, at
+its lower. Each of the observations must be corrected for \textit{refraction},
+and then the \textit{mean of the two corrected altitudes will be the latitude}.
+
+\begin{fineprint}
+This method has the advantage of being an \textit{independent} one; \textit{i.e.}, it does
+not require any data (such as the declination of the stars used) to be accepted
+on the authority of previous observers. But to obtain much accuracy
+it requires considerable time and a large fixed instrument. In low latitudes
+the refraction is also very troublesome.
+\end{fineprint}
+%% -----File: 082.png---Folio 71-------
+
+\nbarticle{102.} \nbparasubtext{Second:}{By the meridian altitude or zenith distance of a
+body of known declination.}
+
+In \figref{illo037}{Fig.~37} the \DPtypo{semicircle}{semi-circle} $AQPB$ is the meridian, $Q$ and $P$ being
+respectively the equator and the pole, and $Z$ the zenith. $QZ$ is
+the \textit{declination of the zenith}, or the observer's latitude ($=PB=\phi$).
+Suppose now that we observe $Zs$ ($=\zeta_s$), the zenith distance of a
+star $s$ (south of the zenith), as it crosses the meridian, and that its
+declination $Qs$ ($=\delta_s$) is known; then evidently $\phi = \delta_s + \zeta_s$.
+
+In the same way, if the star were at $n$, between zenith and pole,
+$\phi = \delta_n - \zeta_n$.
+
+\begin{fineprint}
+If we use the meridian circle, we can always select stars that pass near
+the zenith where the refraction will be small; moreover, we can select them
+in such a way that some will be as much north of the zenith as others are
+south, and thus \textit{eliminate} the refraction errors. But we have to get our star
+declinations out of catalogues made by previous observers, and so the method
+is not an \textit{independent} one.
+\end{fineprint}
+
+\includegraphicsouter{illo037}{\textsc{Fig.~37.}---Determination of Latitude.}
+
+\nbarticle{103.} \nbparatext{At Sea} the latitude is usually obtained by \textit{observing with the sextant
+the sun's maximum altitude}, which
+of course occurs at noon. Since at sea
+it is seldom that one knows beforehand
+precisely the moment of local noon,
+the observer takes care to begin to observe
+the sun's altitude some ten or
+fifteen minutes earlier, repeating his
+observations every minute or two. At
+first the altitude will keep increasing, but immediately after noon it
+will begin to decrease. The observer uses therefore the \textit{maximum}\footnote
+  {On account of the sun's motion in declination, and the northward or southward
+  motion of the ship itself, the sun's maximum altitude is usually attained
+  not \textit{precisely} on the meridian, but a few seconds earlier or later. This requires a
+  slight correction to the deduced latitude, explained in books on Navigation or
+  Practical Astronomy.}
+altitude obtained, which, corrected for refraction, parallax, semi-diameter,
+and dip of the horizon, will give him the true latitude of
+his ship, by the formula $\phi = \delta \pm \zeta$.
+
+\begin{fineprint}
+\sloppy
+\nbarticle{104.} \nbparasubtext{Third:}{By Circum-meridian Altitudes.}---If the observer knows
+his time with reasonable accuracy, he can obtain his latitude from observations
+made when the body is \textit{near} the meridian, with practically the same
+precision as at the moment of meridian passage. It would take us a little
+%% -----File: 083.png---Folio 72-------
+too far to explain the method of reduction, which is given with the necessary
+tables in all works on Practical Astronomy. The great advantage of this
+method is that the observer is not restricted to a single observation at each
+meridian-passage of the sun or of the selected star, but can utilize the half-hours
+preceding and following that moment. The meridian-circle cannot
+be used, as the instrument must be such as to make extra-meridian
+observations possible. Usually the sextant or universal instrument is
+employed. This method is much used in the French and German geodetic
+surveys.
+\end{fineprint}
+
+\nbarticle{105.} \nbparasubtext{Fourth:}{The Zenith Telescope Method.}---{\footnotesize(Sometimes known as
+the American method, because first practically introduced by Captain Talcott of the
+United States Engineers, in a boundary survey in 1845.)}
+
+\includegraphicsouter{illo038}{\textsc{Fig.~38.}---Principle of the Zenith Telescope.}
+
+The essential characteristic of the method is the \textit{micrometric} measurement
+of the \textit{difference} between the
+nearly equal zenith distances of two stars
+which culminate within a few minutes
+of each other, one north and the other
+south of the zenith, and not very far
+from it: such pairs of stars can always
+be found. When the method was first
+introduced, a special instrument, known
+as the zenith telescope, was generally
+employed, but at present a simple transit
+instrument, with declination micrometer,
+and a delicate level attached to the telescope
+tube, is ordinarily used.
+
+The telescope is set at the proper altitude
+for the star which first comes to the
+meridian, and the ``latitude level,'' as it is
+called, is set horizontal; as the star passes
+through the field of view its distance
+north or south of the central wire is measured by the micrometer.
+The instrument is then reversed, and so set by turning the telescope
+up or down (\textit{without, however, disturbing the angle} $\theta$ (\figref{illo038}{Fig.~38}) \textit{between
+the level and telescope}), that the level is again horizontal. After this
+reversal and adjustment, the telescope tube is then evidently elevated
+at exactly the same angle, $\zeta$, as before, but on the opposite side of the
+zenith. As the second star passes through the field, we measure
+with the micrometer \textit{its} distance north or south of the centre of the
+field; the comparison of the two micrometer measures gives the
+\textit{difference} of the two zenith distances.
+%% -----File: 084.png---Folio 73-------
+
+From \figref{illo037}{Fig.~37} we have
+\begin{align*}
+\text{for star \textit{south} of zenith, } &\phi = \delta_s + \zeta_s; \\
+\text{for star \textit{north} of zenith, } &\phi = \delta_n - \zeta_n.
+\end{align*}
+
+Adding the two equations and dividing by $2$, we have
+\[
+  \phi = \left( \frac{\delta_s + \delta_n}{2} \right)
+       + \left( \frac{\zeta_s  -  \zeta_n}{2} \right).
+\]
+
+The star catalogue gives us the declinations of the two stars
+$( \delta_s + \delta_n )$; and the difference of the zenith distances $( \zeta_s - \zeta_n
+)$ is determined
+with great accuracy by the micrometer measures.
+
+\includegraphicsouter{illo039}{\textsc{Fig.~39.}---Latitude by Prime Vertical Transits.}
+
+\begin{fineprint}
+The great advantage of the method consists in its dispensing with a
+\textit{graduated circle}, and in avoiding almost wholly the errors due to \textit{refraction};
+it virtually utilizes the circles of the fixed observatories by which
+the star declinations have been measured, without requiring them to be
+brought into the field. Forty years ago it was not always easy to find
+accurate determinations of the declinations
+of the stars employed, but at present the star
+catalogues have been so extended and improved
+that this difficulty has practically disappeared,
+so that this method of determining the
+latitude is now not only the most convenient
+and rapid, but is quite as precise as any, if the
+level is sufficiently sensitive. Evidently the
+limit of accuracy depends upon the exactness
+with which the level measures the slight, but
+inevitable, difference between the inclinations
+of the instrument when pointed on the two stars.
+
+\nbarticle{106.} \nbparasubtext{Fifth:}{By the Prime Vertical Instrument} {\normalsize(p.~\pageref{fig:illo020}).}---We observe
+simply the moment when a known star passes the prime vertical on the
+eastern side, and again upon the western side. Half the interval will give
+the \textit{hour-angle} of the star when on the prime vertical; \textit{i.e.,} the angle $ZPS$
+in \figref{illo039}{Fig.~39}, where $Z$ is the zenith, $P$ the pole, and $SZS'$ the prime vertical.
+The distance $PS$ of the star from the pole is the complement of the star's
+declination; and $PZ$ is the complement of the observer's latitude. Since
+the prime vertical is perpendicular to the meridian at the zenith, the triangle
+$PZS$ will be right-angled at $Z$, and from Napier's rule of circular
+parts (taking $ZPS$ as the middle part) we shall have
+\begin{flalign*}
+&&\cos ZPS  &= \tan PZ \cot PS,&&\\
+&\text{or}  &\cos t  &= \cot \phi \tan \delta;&&\\
+&\text{whence}  &\tan \phi  &= \tan \delta \sec t.&&\phantom{whence}
+\end{flalign*}
+%% -----File: 085.png---Folio 74-------
+
+If $\delta$ nearly equals $\phi$, $t$  will be small, and a considerable error in the observation
+of $t$ will then produce very little change in its secant or in the computed
+latitude.
+
+The observations are not so convenient and easy as in the case of the
+zenith telescope, and the number of stars available is less; but the method
+presents the great advantage of requiring nothing but an ordinary transit
+instrument, without any special outfit of micrometer and latitude level.
+It also entirely evades the difficulties caused by refraction.
+\end{fineprint}
+
+\includegraphicsouter{illo040}{\textsc{Fig.~40.}---Latitude by the Gnomon.}
+
+\sloppy
+\nbarticle{107.} \nbparasubtext{Sixth:}{By the Gnomon.}---The ancients had no instruments
+such as we have hitherto described, and of course could not use any
+of the preceding methods of finding the latitude. They were, however,
+able to make a very respectable approximation by means of the
+simplest of all astronomical instruments, \textit{the gnomon}. This is merely
+a vertical shaft or column of known height erected on a perfectly
+horizontal plane; and the
+observation consists in noting
+the length of the shadow
+cast at noon at certain times
+of the year.
+
+\fussy
+Suppose, for instance, that
+on the \textit{day of the summer
+solstice}, at noon, the length
+of the shadow is $AC$, \figref{illo040}{Fig.~40}.
+The height $AB$ being given,
+we can easily compute in
+the right-angled triangle the
+angle $ABC$, which equals
+$SBZ$, the sun's zenith distance
+when farthest north. Again observe the length $AD$ of the
+shadow at \textit{noon of the shortest day in winter}, and compute the angle
+$ABD$, which is the sun's corresponding zenith distance when farthest
+south. Now, since the sun travels equal distances north and south
+of the celestial equator, the mean of the two results will give the
+angular distance between the equator and the zenith; \textit{i.e.}, the \textit{declination
+of the zenith}, which (\artref{Art.}{100}) is the latitude of the place.
+
+\begin{fineprint}
+The method is an independent one, like that by the observation of circumpolar
+stars, requiring no data except those which the observer determines
+for himself. Evidently, however, it does not admit of much accuracy, since
+the penumbra at the end of the shadow makes it impossible to measure its
+length very precisely.
+
+It should be noted that the ancients, instead of designating the position
+%% -----File: 086.png---Folio 75-------
+of a place by means of its latitude, used its \textit{climate} instead; the climate
+(from $ \kappa \lambda \acute \iota \mu \alpha $) being the \textit{slope} of the plane of
+the celestial equator, the angle
+$AEB$, which is the complement of the latitude.
+
+It is supposed, indeed known, that many of the Egyptian obelisks were
+erected primarily to serve as gnomons, and were used for that purpose.
+
+\nbarticle{108.}  \nbparatext{Possible Variations of the Latitude.}---It is an interesting question
+whether the position of the earth's axis is \textit{fixed} with reference to its
+mass and surface. Theoretically it is hardly possible that it should be,
+because any change in the arrangement of the matter of the earth, by
+denudation, subsidence, or elevation, would almost necessarily disturb it.
+If so disturbed, the latitudes of places toward which the pole approached
+would be increased, and those on the opposite side would be decreased. At
+present we can only say that if such disturbance has occurred, it must have
+been extremely slight for the last 200 years, not exceeding 40 or 50 feet at
+most; but there are suspicions of a minute and progressive change of the
+latitude of some of the observatories (notably Pulkowa), which have drawn
+attention to the matter, and the subject is under investigation.
+\end{fineprint}
+
+\section*{TIME AND ITS DETERMINATION.}
+
+\nbarticle{109.}  One of the most important problems presented to the astronomer
+is the determination of \textit{Time}. By universal consent the apparent
+rotation of the heavens is made to furnish the standard, and the
+determination of time is effected by ascertaining by observation the
+\textit{hour-angle of the object selected to mark the beginning of the day by
+its transit across the meridian}. In practice, three kinds of time are
+now recognized, viz., \textit{sidereal time, apparent solar time}, and \textit{mean
+solar time}.
+
+\nbarticle{110.} \textit{Sidereal Time}.---As has already been explained (\artref{Art.}{26}), the
+sidereal time at any moment is \textit{the hour-angle of the vernal equinox at
+that moment;} or, what comes to the same thing, though it sounds differently,
+it is \textit{the time marked by a clock which is so set and adjusted
+as to show noon, or $0^\mathrm{h}\: 00^\mathrm{m} \:00^\mathrm{s}$, at each transit of the vernal equinox.}
+The sidereal \textit{day}, thus defined, is the time intervening between two
+successive transits of the same \textit{star;} at least, it is so within the
+hundredth part of a second, though on account of the precession of
+the equinoxes (and the proper motions of the stars) the agreement is
+not absolute, the difference amounting to about one day in twenty-six
+thousand years.
+
+\nbarticle{111.} \textit{Apparent Solar Time}.---Just as sidereal time is the hour-angle
+of the vernal equinox, so at any moment \textit{the apparent solar time is
+the hour-angle of the sun}. It is the time \textit{shown by the sun-dial,} and
+%% -----File: 087.png---Folio 76-------
+its noon is when the sun crosses the meridian. On account of the
+annual eastward motion of the sun among the stars (due to the
+earth's orbital motion), this day is about four minutes longer than
+the sidereal; and moreover, because the sun's motion in right ascension
+is not uniform, the apparent solar days are not all of the same
+length, nor, consequently, its hours, minutes, or seconds. December
+23d is fifty-one seconds longer from (apparent) noon to noon than
+Sept.\ 16th. For this reason, apparent solar time is not satisfactory
+for scientific use, and has long been discarded in favor of mean
+solar time. Until within about a hundred years, however, it was the
+only kind of time commonly employed, and its use in the city of Paris
+was not discontinued until the year 1816.
+
+{\nbarticle{112.} \textit{Mean Solar Time}.---A ``\textit{fictitious sun}'' is therefore imagined,
+which moves \textit{uniformly and in the celestial equator}, completing its
+annual course in exactly the same time as that in which the actual sun
+makes the circuit of the ecliptic. It is mean noon when this ``fictitious
+sun'' crosses the meridian, and at any moment \textit{the hour-angle
+of this ``fictitious sun'' is the mean time} for that moment.}
+
+\begin{fineprint}
+Sidereal time will not answer for business purposes, because its noon (the
+transit of the vernal equinox) occurs at all hours of night and daylight in
+different seasons of the year. Apparent solar time is scientifically unsatisfactory,
+because of the variation in the length of its days and hours. And
+yet we have to live by the sun; its rising and setting, daylight and night,
+control our actions. In mean solar time we find a satisfactory compromise,
+an invariable time unit, and still an agreement with sun-dial time close enough
+for convenience. It is the time now used for all purposes except in certain
+astronomical work. The difference between apparent time and mean time,
+never amounting to more than about a quarter of an hour, is called the equation
+of time, and will be discussed hereafter in connection with the earth's
+orbital motion, \chapref{CHAPTERVI}{Chap.~VI.}
+\end{fineprint}
+
+The nautical almanac furnishes data by means of which the sidereal
+time may be deduced from the corresponding solar, or \textit{vice versa}, by
+a very brief and simple calculation.
+
+\nbarticle{113.} In practice the problem of determining the time always takes
+the form of \textit{ascertaining the error of a time-piece;} that is, the amount
+by which a clock or watch is fast or slow of the time it ought to show.
+The methods most in use by astronomers are the following:---
+
+First. \textit{By means of the transit instrument}. Since the right ascension
+of a star is the sidereal time of its passage across the meridian
+%% -----File: 088.png---Folio 77-------
+(\artref{Art.}{26}), it is obvious that the difference between the right ascension
+of a known star and the time shown by a sidereal clock at the instant
+when the star crosses the middle wire of an accurately adjusted
+transit instrument, is the error of the clock at that moment. Practically,
+it is usual to observe a number of stars (from eight to ten),
+reversing the instrument once at least, so as to eliminate the collimation
+error (\artref{Art.}{60}). With a good instrument a skilled observer can
+determine this clock error or ``correction'' within about one-thirtieth
+of a second of time, provided proper means are taken to ascertain
+and allow for his ``personal equation.''
+
+\nbarticle{114.} \textit{Personal Equation}.---It is found that every observer has his
+own peculiarities of time observation with a transit, and his \textit{``personal
+equation''} is the amount to be added (algebraically) to the time
+observed by him, in order to get the actual moment of transit as it
+would be recorded by some supposable arrangement, which should
+automatically register the moment when the star's image was bisected
+by the wire.
+
+\begin{fineprint}
+This personal equation differs for different observers, but is reasonably
+(though never strictly) constant for one who has had much practice. In the
+case of observations with the chronograph, it is usually less than $ \pm 0^{\text{s}}.2$. It
+can be determined by an apparatus in which an artificial star, resembling
+the real stars as much as possible in appearance, is made to traverse the field
+of view and to telegraph its arrival at certain wires, while the observer notes
+the moments for himself.
+
+One of the most important problems of practical astronomy now awaiting
+solution is the contrivance of some practical method of time observation
+free from this annoying human element, the personal equation, which is
+always more or less uncertain and variable.
+\end{fineprint}
+
+If mean time is wanted, it can be deduced from the sidereal time
+by the data of the almanac.
+
+The sun can also be observed instead of the stars, the moment of
+the sun's transit being that of apparent noon; but this observation,
+for many reasons, is far less accurate and satisfactory than observations
+of the stars.
+
+\nbarticle{115.}  Second. \textit{The method of equal altitudes}.---If we observe with
+a sextant in the forenoon the time shown by the chronometer when the
+sun attains the height indicated by a certain reading of the sextant, and
+then in the afternoon, the time when the sun again reaches the same
+%% -----File: 089.png---Folio 78-------
+altitude, the moment of apparent noon will be half-way between the
+two observed times; provided, of course, that the chronometer runs
+uniformly during the interval, and also provided that proper correction
+is made for the sun's slight motion in declination---a correction
+easily computed.
+
+The advantage of this method is that the errors of graduation of
+the sextant have no effect, nor is it necessary for the observer to
+know his latitude except approximately.
+
+\includegraphicsouter{illo041}{\textsc{Fig.~41.}---Determination of Time by a Single Altitude.}
+
+\textit{Per contra}, there is, of course, danger that the afternoon observations
+may be interfered with by clouds; and, moreover, both observations
+must be made at the same place.
+
+A modification of this method is now coming into extensive use,
+in which two different stars of known right ascension and of nearly
+the same declination are used, at equal altitudes east and west of the
+meridian.
+
+\nbarticle{116.} \nblabel{pg:78}Third. \textit{By a single altitude of the sun, the observer's latitude
+being known}.---This is the method usual at sea. The altitude of the sun
+having been measured with the sextant, and the corresponding time
+shown by the chronometer having been accurately noted, we compute
+the hour-angle of the sun, $P$,
+from the triangle $ZPS$ (Fig.\
+41), and this hour-angle corrected
+for the equation of
+time, gives the true mean
+time at the observed moment.
+The difference between this
+and that shown by the chronometer
+is \textit{the error of the
+chronometer.} In the triangle
+$ZPS$ all three of the sides are given, viz.: $PZ$ is the complement
+of the latitude $\phi$, which is supposed to be known; $PS$ is the complement
+of the declination $\delta$, which is found in the almanac, as is
+also the equation of time; while $ZS$ or $\zeta$, is the complement of the
+sun's altitude, as measured by the sextant, and corrected for semi-diameter,
+refraction, and parallax. The formula is
+\[
+\sin \tfrac{1}{2}P = \left( \cfrac{\sin \tfrac{1}{2} \left[ \zeta + \left( \phi - \delta
+\right) \right] \sin \tfrac {1}{2} \left[ \zeta - \left( \phi - \delta \right) \right] }{
+\cos \phi \cos \delta} \right) ^{\frac {1}{2}}.
+\]
+
+In order to accuracy, it is desirable that the sun should be on the
+prime vertical, or as near it as practicable. It should \textit{not} be near the
+%% -----File: 090.png---Folio 79-------
+meridian. Any slight error in the assumed latitude produces no
+sensible effect upon the result, if the sun is exactly east or west at
+the time the observation is taken. The disadvantage of the method
+is that any error of graduation of the sextant enters into the result
+with its full effect.
+
+\begin{fineprint}
+In some cases a person is so situated that it is necessary to determine his
+time roughly, without instruments; and this can be done within about a
+half a minute by establishing a noon-mark, which is nothing but a line
+drawn exactly north and south, with a plumb-line, or some vertical edge, like
+the edge of a door-frame or window-sash, at its southern extremity. The
+shadow will then always fall upon the meridian line at \textit{apparent} noon.
+\end{fineprint}
+
+\nbarticle{117.} \nbparatext{The Civil and the Astronomical Day.}---The \textit{astronomical} day
+begins at mean noon. The \textit{civil} day begins at midnight, twelve hours
+earlier. Astronomical mean time is reckoned round through the whole
+twenty-four hours, instead of being counted in two series of twelve
+hours each. Thus, 10~\textsc{a.m.} of Wednesday, May~2, \textit{civil reckoning}, is
+Tuesday, May~1, $22^\text{h}$ by \textit{astronomical reckoning}. Beginners need to
+bear this in mind in using the almanac.
+
+\section*{LONGITUDE.}
+
+\nbarticle{118.}  Having now methods of obtaining the true local time, we can
+attack the problem of longitude, which is perhaps the most important
+of all the economic problems of astronomy. The great observatories
+at Greenwich and at Paris were established simply for the purpose
+of furnishing the observations which could be made the basis of the
+accurate determination of longitude at sea.
+
+The longitude of a place on the earth is \textit{the angle at the pole between
+the meridian of Greenwich and the meridian passing through the observer's
+place;} or it is the arc of the equator intercepted between
+these meridians; or, what comes to the same thing, since this arc is
+measured by the time required for the earth to turn sufficiently to
+bring the second meridian into the same position held by the first, it is
+simply \textit{the difference of their local times},---the amount by which the noon
+at Greenwich is earlier or later than at the observer's place. It is now
+usually reckoned in hours, minutes, and seconds, instead of degrees.
+
+Since it is easy for the observer to find his own local time by the
+methods which have been given, the knot of the problem is really
+this: \textit{being at any place, to find the corresponding local time at Greenwich
+without going there}.
+%% -----File: 091.png---Folio 80-------
+
+The methods of finding the longitude may be classed under three
+different heads:
+
+\textbf{First}, By means of signals simultaneously observable at the places
+between which the difference of longitude is to be found.
+
+\textbf{Second}, By making use of the moon as a clock-hand in the sky.
+
+\textbf{Third}, By purely mechanical means, such as chronometers and the
+telegraph.
+
+\nbarticle{119.} Under the first head we may make use of
+
+[A] \textit{A Lunar Eclipse}.---When the moon enters the shadow of the
+earth, the phenomenon is seen at the same moment, no matter where
+the observer may be. By noting, therefore, his own local time at the
+moment, and afterwards comparing it with the time at which the phenomenon
+was observed at Greenwich, he will obtain his longitude
+from Greenwich. Unfortunately, the edge of the earth's shadow is
+so indistinct that the progress of events is very gradual, so that
+sharp observations are impossible.
+
+[B] \textit{Eclipses of the satellites of Jupiter} may be used in the same
+way, with the advantage that they occur very frequently,---almost every
+night, in fact; but the objection to them is the same as to the lunar
+eclipses,---they are not sudden.
+
+[C] \textit{The appearance and disappearance of meteors} may be and has
+been used to determine the difference of longitude between places
+not more than two or three hundred miles apart, and gives very accurate
+results. (Now superseded by the telegraph.)
+
+[D] \textit{Artificial signals}, such as flashes of powder and rockets, can
+be used between two stations not too far distant. Early in the century
+the difference of longitude between the Black Sea and the Atlantic
+was determined by means of a chain of signal stations on the
+mountain tops; so also, later, the difference of longitude between the
+eastern and western extremities of the northern boundary of Mexico.
+This method is now superseded by the telegraph.
+
+\nbarticle{120.} \textsc{Second}, \textit{the moon regarded as a clock}.
+
+Since the moon revolves around the earth once a month, it is, of
+course, continually changing its place among the stars; and as the
+laws of its motion are now well known, and as the place which
+it will occupy is predicted for every hour of every Greenwich day
+three years in advance in the nautical almanac, it is possible to
+deduce the corresponding Greenwich time by any observation which
+will determine the place of the moon among the stars. The almanac
+%% -----File: 092.png---Folio 81-------
+place, however, is the place at which the moon would be seen by an
+observer \textit{at the centre of the earth}, and consequently the actual observations
+are in most cases complicated with very disagreeable
+reductions for parallax before they can be made available.
+
+The simplest lunar method is,
+
+[A]  \textit{That of Moon Culminations}.---We merely observe with a
+transit instrument the time when the moon's bright limb crosses the
+meridian of the place; and immediately after the moon we observe
+one or more stars with the same instrument, to give us the error
+of our clock. As the moon is observed on the meridian, its parallax
+does not affect its right ascension, and accordingly, by a simple
+reference to the almanac, we can ascertain the Greenwich time at
+which the moon had the particular right ascension determined by
+the observation. The method has been very extensively used, and
+would be an admirable one were it not for the effects of personal
+equation.
+
+\begin{fineprint}
+It seldom happens that the personal equation of an observer is the same
+for such an object as the limb of the moon as it is for a star; and since the
+moon's motion among the stars is very slow, the effect of such a difference
+is multiplied by about 30 (roughly the number of days in a month) in its
+effect upon the longitude deduced.
+\end{fineprint}
+
+[B] \textit{Lunar-Distances.}---At sea it is, of course, impossible to
+observe the moon with a transit instrument, but we can observe its
+distance from the stars near its path by means of a sextant. The
+distance observed will not be the same that it would be if the
+observer were at the centre of the earth, but by a mathematical
+process called ``clearing a lunar'' the distance as seen from the
+centre of the earth can be easily deduced, and compared with the
+distance given in the almanac. From this the longitude can be
+determined. Any error, however, in measuring a lunar-distance
+entails an error about thirty times as great in the resulting longitude,
+and the method is at present very little used, the moon having been
+superseded by the chronometer for such purposes.
+
+[C] \textit{Occultations}.---Occasionally, in its passage through the sky,
+the moon over-runs a star, or ``\textit{occults}'' it. The star vanishes instantaneously,
+and, of course, at the moment of its disappearance the
+distance from the centre of the moon to the star is precisely equal
+to the apparent semi-diameter of the moon; we thus have a ``lunar-distance''
+self-measured.
+
+Observations of this kind furnish one of the most accurate methods
+%% -----File: 093.png---Folio 82-------
+of determining the difference of longitude between widely separated
+places, the only difficulty arising from the fact that the edge of the
+moon is not smooth, but more or less mountainous, so that the distance
+of a star from the moon's centre is not always the same at
+the moment of its disappearance.
+
+[D] \textit{In the same way a solar eclipse may be employed by observing
+the moment when the moon's limb touches that of the sun.}
+
+\begin{fineprint}
+It will be noticed that these two last methods (the methods of occultation
+and solar eclipse) do not belong in the same class with the method of lunar
+eclipse, because the phenomena are not seen at the same instant at different
+places, but the calculation of longitude depends upon the determination of
+the moon's place in the sky at the given time, as seen from the earth's
+centre.
+\end{fineprint}
+
+There are still other methods, depending upon measurements of
+the moon's position by observations of its altitude or azimuth. In
+all such cases, however, every error of observation entails a vastly
+greater error in the final results. Lunar methods (excepting occultations)
+are only used when better ones are unavailable.
+
+\nbarticle{121.} Finally we have what may be called the \textit{mechanical methods}
+of determining the longitude.
+
+[A] \textit{By the chronometer}; which is simply an accurate watch that
+has been set to indicate Greenwich time before the ship leaves port.
+In order to find the longitude by the chronometer, the sailor has to
+determine its ``error'' upon local time by an observation of the altitude
+of the sun when near the prime vertical, as indicated on page~\pageref{pg:78}.
+If the chronometer indicates true Greenwich time, \textit{the error deduced
+from the observation will be the longitude}. Usually, however, the indication
+of the chronometer face requires correction for the rate and
+run of the chronometer since leaving port.
+
+\begin{fineprint}
+Chronometers are only imperfect instruments, and it is important, therefore,
+that several of them should be used to check each other. It requires
+three at least, because if only \textit{two} chronometers are carried and they disagree,
+there is nothing to indicate which one is the delinquent.
+
+On very long voyages the errors of chronometers are cumulative, and the
+error is found to accumulate, not merely in proportion to the time, but \textit{more
+nearly in proportion to the square of the time;} \textit{i.e.,} if the error to be feared in
+the use of a chronometer in longitude determinations at the end of a
+week is about two seconds of time, at the end of the month it would be, not
+eight seconds, but about thirty-two seconds.
+
+If, therefore, a ship is to be at sea, without making port, more than three
+%% -----File: 094.png---Folio 83-------
+or four months at a time, the method becomes untrustworthy, and it may be
+necessary to recur to lunar distances; for voyages of less than a month the
+method is now, practically, all that could be desired.
+\end{fineprint}
+
+[B] But the method which, wherever it is applicable, has superseded
+all others, is that of \textit{The Telegraph}. When we wish to find the
+longitude between two stations connected by telegraph, the process
+is usually as follows: The observers at both stations, after ascertaining
+that they both have clear weather, proceed to determine their
+own local time by extensive series of star observations with the
+transit instrument. Then, at a agreed-upon time, the observer at
+Station A ``switches his clock'' into the telegraphic circuit, so that
+its beats are communicated along the line and received upon the chronograph
+of the other, say the western station. After the eastern clock
+has thus sent its signals, say for two minutes, it is switched out of
+the circuit, and the western observer now switches his clock into the
+circuit, and its beats are received upon the eastern chronograph. The
+operation is closed by another series of star observations.
+
+We have now upon each chronograph sheet an accurate comparison
+of the two clocks, showing the amount by which the western clock is
+slow of the eastern. If the transmission of electric signals were
+instantaneous, the difference shown upon the two chronograph sheets
+would agree precisely. Practically, however, there will always be a
+small discrepancy amounting to twice the time occupied in the transmission
+of the signals; but the mean of the two differences will be
+the true difference of longitude of the places after the proper corrections
+have been applied. \textit{Especial care must be taken to determine
+with accuracy, or to eliminate, the personal equations of the observers}.
+
+\begin{fineprint}
+It is customary to make observations of this kind on not less than five
+or six evenings in cases where it is necessary to determine the difference of
+longitude with the highest accuracy. The \textit{astronomical} difference of longitude
+between two places can thus be telegraphically determined within about
+the one-hundredth part of a second of time; \textit{i.e.}, within about ten feet or so,
+in the latitude of the United States.
+
+It may be noted here that the time occupied by the transmission of electric
+signals in longitude operations is not to be taken as the real measure of
+``the velocity of the electric fluid'' upon the wires, as was once supposed.
+The time apparently consumed in the transmission is simply the time required
+for the current at the receiving station (which current probably
+\textit{begins} at the very instant the key is touched at the other end of the line) to
+become \textit{strong enough} to do its work in making the signal; and this time
+depends upon a multitude of circumstances.
+%% -----File: 095.png---Folio 84-------
+
+\nbarticle{122.}  \nbparatext{Local and Standard Time}.---In connection with time and
+longitude determinations, a few words on this subject will be in place. Until
+recently it has always been customary to use only \textit{local} time, each observer
+determining his own time by his own observations. Before the days of the
+telegraph, and while travel was comparatively slow and infrequent, this was
+best; but the telegraph and railway have made such changes that, for many
+reasons, it is better to give up the old system of local times in favor of a
+system of standard time. It facilitates all railway and telegraphic business
+in a remarkable degree, and makes it practically easy for every one to
+keep accurate time, since it can be daily wired from some observatory to
+every telegraph office.
+
+According to the system that is now established in this country, there are
+five such standard times in use,---the colonial, the eastern, the central, the
+mountain, and the Pacific,---which differ from Greenwich time by exactly
+four, five, six, seven, and eight hours respectively, \textit{the minutes and seconds being
+identical everywhere}. At most places only one of these times is employed;
+but in cities where different systems join each other, there are two standard
+times in use, differing from each other by exactly one hour, and from the
+local time by about half an hour. In some such places the local time also
+maintains its place.
+
+In order to determine the standard time by observation, it is only necessary
+to determine the local time by one of the methods given, and correct
+it according to the observer's longitude from Greenwich.
+
+\nbarticle{123.} \nbparatext{Where the Day Begins.}---If we imagine a traveller starting
+from Greenwich on Monday noon, and journeying westward as swiftly as the
+earth turns to the east under his feet, he would, of course, keep the sun exactly
+on the meridian all day long, and have continual noon. But what noon?
+It was Monday when he started, and when he gets back to London, twenty-four
+hours later, it is Tuesday noon there, and there has been no intervening
+sunset. When does Monday noon become Tuesday noon? The convention
+is that \textit{the change of date occurs at the $180$th meridian from Greenwich.}
+Ships crossing this line \textit{from the east} skip one day in so doing. If it is
+Monday forenoon when the ship reaches the line, it becomes Tuesday forenoon
+the moment it passes it, the intervening twenty-four hours being
+dropped from the reckoning on the log-book. \textit{Vice versa}, when a vessel
+crosses the line \textit{from the western side}, it counts the same day twice, passing
+from Tuesday forenoon back to Monday, and having to do its Tuesday over
+again.
+
+This $180$th meridian passes mainly over the ocean, hardly touching land
+anywhere. There is a little irregularity in the date upon the different
+islands near this line. Those which received their earliest European inhabitants
+\textit{via} the Cape of Good Hope have, for the most part, the Asiatic date,
+belonging to the west side of the $180$th meridian; while these that were approached
+\textit{via} Cape Horn have the American date.
+%% -----File: 096.png---Folio 85-------
+
+When Alaska was transferred from Russia to the United States, it was
+necessary to drop one day of the week from the official dates.
+\end{fineprint}
+
+\section*{THE PLACE OF A SHIP AT SEA.}
+
+\nbarticle{124.} The determination of the place of a ship at sea is commercially
+of such importance that, at the risk of a little repetition, we
+collect together here the different methods available for its determination.
+The methods employed are necessarily such that observations
+can be made with the sextant and chronometer, the only
+instruments available under the circumstances.
+
+\medskip
+The \textbf{Latitude} is usually obtained by observations of the sun's
+altitude at noon, according to the method explained in \artref{Art.}{103}.
+
+\medskip
+The \textbf{Longitude} is usually found by determining the error upon local
+time of the chronometer, which carries Greenwich time. The necessary
+observations of the sun's altitude should be made when the
+sun is near the prime vertical, as explained in \artref{Art.}{116}.
+
+In the case of long voyages, or when the chronometer has for any
+reason failed, the longitude may also be obtained by measuring a
+lunar-distance and comparing it with the data of the nautical almanac.
+
+\medskip
+By these methods separate observations are necessary for the latitude
+and for the longitude.
+
+\nbarticle{125.}  \nbparatext{Sumner's Method.}---Recently a new method, first proposed
+by Captain Sumner, of Boston, in 1843, has been coming largely into
+use. In this method, each observation of the sun's altitude, with the
+corresponding chronometer time, is made to define the position of the
+ship upon a certain line, called \textit{the circle of position}. Two such observations
+will, of course, determine the exact place of the vessel at
+one of the intersections of the two circles.
+
+At any moment the sun is vertically over some point upon the
+earth's surface, which may be called the \textit{sub-solar point}. An observer
+there would have the sun directly overhead. Moreover, if at any
+point on the earth an observer measures the altitude of the sun with
+his sextant, \textit{the zenith distance of the sun} (which is the complement
+of this altitude) \textit{will be his distance from the sub-solar point at the
+moment of observation, reckoned in degrees of a great circle}.
+
+If, then, I take a terrestrial globe, and, opening the dividers so as
+to cover an arc equal to this observed zenith distance of the sun,
+put one foot of the dividers upon the sub-solar point, and sweep a
+%% -----File: 097.png---Folio 86-------
+circle on the surface of the globe around that point, the observer
+must be \textit{somewhere on the circumference of that circle}; and moreover,
+if to the observer the sun is in the \textit{southwest}, he himself must be in
+the opposite direction from this sub-solar point; \textit{i.e., northeast} of it.
+In other words, the \textit{azimuth of the sun} at the time of observation
+informs him upon \textit{what part of the circle} he is situated.
+
+Suppose a similar observation made at the same place a few hours
+later. The sub-solar point, and the zenith distance of the sun, will
+have changed; and we shall obtain a new circle of position, with its
+centre at the new sub-solar point. The observer must be at one of
+its two intersections with the first circle---which of the two intersections
+is easily determined from the roughly observed azimuth of
+the sun.
+
+If the ship moves between the two observations, the proper allowance
+must be made for the motion. This is easily done by shifting
+upon the chart that part of the first circle of position where the
+ship was situated, carrying the line forward parallel to itself, by an
+amount just equal to the ship's run between the two observations,
+as shown by the log. The intersection with the second circle then
+gives the ship's place \textit{at the time of the second observation}.
+
+The only problem remaining is to find the position of the ``sub-solar
+point'' at any given moment. Now, the \textit{latitude} of this point is obviously
+the \textit{declination of the sun} (which is found in the almanac).
+If the sun's declination is zero, the sun is vertically over some point
+upon the equator. If its declination is  $+20°$, it is vertically over
+some point on the twentieth parallel of north latitude, etc.
+
+In the next place, its \textit{longitude} is equal to the \textit{Greenwich apparent
+solar time} at the moment of observation; and this is given by the
+chronometer (which keeps Greenwich mean solar time), by simply adding
+or subtracting the equation of time; so that, by looking in his
+almanac and at his chronometer, the observer has the position of the
+sub-solar point immediately given him. (See note, page~\pageref{pg:90}.)
+
+\includegraphicsmid{illo042}{\textsc{Fig.~42.}---Sumner's Method.}
+
+\begin{fineprint}
+Suppose, for example, that on May 20 (the sun's declination being $+20°$),
+at 11~\textsc{a.m.}, Greenwich \textit{apparent} time (\textit{i.e.}, May 19, $23^\text{h}$ by astronomical reckoning),
+according to the chronometer, the sun is observed to have an altitude
+of $40°$ by a ship in the North Atlantic. The sub-solar point will then be
+(\figref{illo042}{Fig.~42}) at a point in Africa having a latitude of $+20°$, and an east longitude
+of $15°$---at $A$ in the \figref{illo042}{figure}. And the radius of the ``circle of position,''
+\textit{i.e.,} the distance from $A$ to $C$---will be $50°$.
+
+Again, a second observation is made three hours later, when the sun's
+altitude is found to be $65°$. The sub-solar point will then be at $B$, latitude
+%% -----File: 098.png---Folio 87-------
+$20°$, longitude $30°$~W., and the radius of the circle of position $BC$ be
+$25°$, $C$ being the ship's place.
+
+Of course it would be impracticable to carry on a vessel a terrestrial
+globe large enough for the accurate working out of the graphical operation
+indicated, but tables are provided, by which the necessary portions of the
+position circles can be easily drawn upon the ordinary charts.
+\end{fineprint}
+
+\nbarticle{126.} The peculiar advantage of the method is, that a single observation
+is used for all it is worth, giving \textit{accurately} the position of a line
+upon which the ship is somewhere situated, and \textit{approximately} (by the
+rough observation of the sun's azimuth) the part of that line upon
+which its place will be found. In approaching the American coast,
+for instance, if an observation be taken in the forenoon when the
+sub-solar point is over the continent of Africa, the ship's position
+circle will lie nearly parallel to the coast, and then a single observation
+will give approximately the distance of the ship from land, which
+may be all the sailor wishes to know. The observations need not be
+taken at any particular time. We are not limited to observations at
+noon, or to the time when the sun is on the prime vertical. It is to
+be noted, however, \textit{that everything depends upon the chronometer}, as
+much as in the ordinary chronometric determination of longitude.
+
+\nbarticle{127.} \nbparatext{Determination of Azimuth.}---A problem, important, though
+not so often encountered as that of latitude and longitude determinations,
+%% -----File: 099.png---Folio 88-------
+is that of determining \textit{the azimuth, or true bearing, of a line upon
+the earth's surface}. The process is this: With a theodolite having
+an accurately graduated horizontal circle the observer points alternately
+upon the pole star and upon a distant
+signal erected for the purpose; the
+signal being an artificial star consisting of
+a small hole in a plate of metal, with a
+bull's-eye lantern or other light behind it.
+It is desirable that it should be at least
+a mile away from the observer, so that
+any small displacement of the instrument
+will be harmless. The theodolite must
+be carefully adjusted for collimation, and
+especial pains must be taken to have the
+telescope perfectly level.
+
+\includegraphicsouter{illo043}{\textsc{Fig.~43.}---Determination of Azimuth.}
+
+The next morning by daylight the observer
+measures the angle or angles between the night-signal and the
+objects whose azimuth is required.
+
+If the pole star were exactly at the pole, the mere difference
+between the two readings of the circle, obtained when the telescope
+is pointed on the star and on the signal, would directly give the
+azimuth of the signal, As this is not the case, however, the \textit{time}
+at which each observation of the pole star is made must be noted,
+and the azimuth of the star must be computed for that moment.
+This can easily be done, as the right ascension and declination of
+this star are given in the almanac for every day of the year.
+
+\begin{fineprint}
+Recurring to the Z.P.S. [zenith-pole-star] triangle, $N$ (\figref{illo043}{Fig.~43}) being the
+north point of the horizon, $P$ the pole, and $NZ$ the meridian, we at once
+see that the side $PS$ is the complement of the star's declination; the side
+$PZ$ is the complement of the observer's latitude (which must be known);
+and the angle at $P$ is the difference between the right ascension of the pole
+star and the sidereal time of the observation; [$(t-a)$ if the star is west of
+the meridian at the time, and $(a-t)$ if it is east.] This will come out in
+hours, of course, and must be reduced to degrees before making the computation.
+We thus have two sides of the triangle, viz., $PS$ and $PZ$, with
+the included angle at $P$, from which to compute the angle $Z$ at the zenith.
+This is the star's azimuth.
+
+The pole star is used because, being so near the pole, any slight error in
+the assumed latitude of the place or in the sidereal time of the observation
+will hardly produce any effect upon the result, especially if the star be
+caught between five and six hours before or after its upper culmination, at
+%% -----File: 100.png---Folio 89-------
+a time when it changes its azimuth very slowly (near $S'$ or $S''$ in the \figref{illo043}{figure}).
+The sun, or any other heavenly body whose position is given in the almanac,
+can also be used as a reference point in the same way, provided sufficient
+pains \DPtypo{is}{are} taken to secure an accurate observation of the time at the instant
+when the pointing is made. The altitude should not exceed thirty degrees
+or so. But the results are usually rough compared with these obtained by
+means of the pole star.
+\end{fineprint}
+
+\section*{DETERMINATION OF THE POSITION OF A HEAVENLY BODY.}
+
+\nbarticle{128.} The position of a heavenly body is defined by its right
+ascension and declination. These quantities may be determined---
+
+\medskip
+\sloppy
+(1) \textbf{By the meridian circle}, provided the body is bright enough to
+be seen by the instrument and comes to the meridian in the night-time.
+If the instrument is in exact adjustment, the \textit{sidereal time
+when the object crosses the middle wire of the reticle of the instrument is
+directly} (according to \artref{Art.}{27}) \textit{the right ascension of the object}.
+
+\fussy
+The reading of the circle of the instrument, corrected for refraction
+and parallax if necessary, gives the \textit{polar distance} of the object, if
+the polar point of the circle has been determined (\artref{Art.}{66}); or it gives
+the \textit{zenith distance} of the object if the nadir point has been determined
+(\artref{Art.}{67}). In either case the \textit{declination} can be immediately
+deduced, being the complement of the polar distance, and equal to
+the latitude of the observer, minus the distance of the star south of
+the zenith. One complete observation, then, with the meridian circle,
+determines both the right ascension and declination of the object.
+
+\medskip
+If a body (a comet, for instance) is too faint to be observed by
+the telescope of the meridian circle, which is seldom very powerful,
+or if it does not come to the meridian during the night, we usually
+accomplish our object---
+
+\nbarticle{129.} (2) \nbparatext{By the Equatorial}, determining the position of the body
+by measuring the \textit{difference of right ascension and declination} between
+it and some neighboring star, whose place is given in a star
+catalogue, and of course has been determined by the meridian circle
+of some observatory.
+
+\begin{fineprint}
+In measuring this difference of right ascension and declination, we usually
+employ a filar micrometer fitted like the reticle of a meridian circle. It carries
+a number of wires which lie north and south in the field of view, and
+these are crossed at right angles by one or more wires which can be moved
+%% -----File: 101.png---Folio 90-------
+by the micrometer screw. \textit{The difference of right ascension} between the star
+and the object to be determined is measured by simply observing with the
+chronograph the transits of the two objects across the north and south
+wires; \textit{the difference of declination}, by bisecting each object with one of the
+micrometer wires as it crosses the middle of the field of view. The observed
+difference must be corrected for refraction and for the motion of
+the body, if it is appreciable.
+
+Other less complicated micrometers are also in use. One of them, called
+the \textit{ring micrometer}, consists merely of an opaque ring supported in the field
+of view either by being cemented to a glass plate or by slender arms of
+metal. The observations are made by noting the transits of the comparison
+star and of the object to be determined across the outer and inner edges of
+the ring. If the radius of the ring is known in seconds of arc, we can
+from these observations deduce the differences both of right ascension and
+declination. The results are less accurate than those given by the wire
+micrometer, but the ring micrometer has the advantage that it can be used
+with any telescope, whether equatorially mounted or not, and requires no
+adjustment.
+
+There are also many other methods of effecting the same object.
+
+\nbarticle{130.} \nbparatext{To Compute the Time of Sunrise or Sunset.}---To solve this problem,
+it is only necessary to work out the Z.P.S. triangle and find the hour-angle
+$P$, having given precisely the same data as in finding the time by a single
+altitude of the sun (\artref{Art.}{116}). $PZ$ is the observer's co-latitude, $PS$ is the
+complement of the sun's declination (given by the almanac); and the true
+distance from the zenith to the centre of the sun at the moment when its
+upper edge is at the horizon is $90°\, 50'$, which is made up of $90°$, $+16'$ (the
+mean semi-diameter of the sun), plus $34'$ (the mean refraction at the horizon).
+The resulting hour-angle $P$, corrected for the equation of time, gives the mean
+time (\textit{local}) at which the sun's upper limb touches the horizon, under the
+average circumstances of temperature and barometric pressure. If it is very
+cold, with the barometer standing high, sunrise will be accelerated, or sunset
+retarded, by a considerable fraction of a minute. If the sun rises or sets
+over the sea-horizon, and the observer's eye is at any considerable elevation
+above the sea-level, the dip of the horizon must also be added to the $90°\, 50'$
+before making the computation.
+
+The beginning and end of twilight may be computed in the same way
+by merely substituting $108°$ for $90°\, 50'$.
+
+\nbarticle{131.} \nblabel{pg:90}\nbparatext{Note to \artref{Art.}{125}.}---In the explanation of Sumner's method it is
+assumed that the earth is a \textit{perfect sphere}. In the actual application of the
+method certain corrections are therefore necessary to take into account the
+earth's ellipticity.
+\end{fineprint}
+\chelabel{CHAPTERIV}
+%% -----File: 102.png---Folio 91-------
+
+\Chapter{V}{The Earth}
+\nbchaptercenter{THE EARTH AS AN ASTRONOMICAL BODY.}
+\nbchaptersubhang{\stretchyspace Approximate Dimensions---Proofs of its Rotation---Accurate
+Determination of its Form and Size by Geodetic Operations
+and Pendulum Observations---Astronomical, Geodetic and Geocentric
+Latitude---Determination of the Earth's Mass and Density.}
+
+\nbarticle{132.} \textsc{Having} discussed the methods of making astronomical observations,
+we are now prepared to consider the earth in its astronomical
+relations; \textit{i.e.}, those facts relating to the earth which are
+ascertained by astronomical methods, and are similar to the facts
+which we shall have to consider in the case of the other planets.
+The facts are broadly these:---
+
+\begin{asparaenum}[1.]
+\item \textit{The earth is a great ball, about $7918$ miles in diameter.}
+
+\item \textit{It rotates on its axis once in twenty-four sidereal hours.}
+
+\item \textit{It is flattened at the poles}, the polar diameter being nearly
+\textit{twenty-seven miles}, or one \textit{two hundred and ninety-fifth part} less than
+the equatorial.
+
+\item \textit{It has a mean density of about five and six-tenths times that of
+water, and a mass represented in tons by six with twenty-one ciphers
+after it $($or six sextillions of tons, according to the French numeration$)$.}
+
+\item \textit{It is flying through space in its orbital motion around the sun,
+with a velocity of about nineteen miles a second}; i.e., \textit{about seventy-five
+times as swiftly as any cannon-ball.}
+\end{asparaenum}
+
+\section*{I.}
+
+\nbarticle{133.} \nbparatext{The Earth's Approximate Form and Size.}---It is not necessary
+to dwell upon the ordinary proofs of its globularity. We merely mention
+them. \begin{inparaenum}[1.] \item It can be circumnavigated. \item The appearance of
+vessels coming in from sea indicates that the surface is everywhere
+convex. \item The fact that the sea-horizon, as seen from an eminence,
+is everywhere depressed to the same extent below the level
+line, shows that the surface is approximately spherical. \item The fact
+that as one goes from the equator toward the north, the elevation of
+%% -----File: 103.png---Folio 92-------
+the pole increases proportionally to the distance from the equator,
+proves the same thing. \item \textit{The shadow of the earth, an seen upon
+the moon at the time of a lunar eclipse, is that which only a sphere
+could cast}.
+\end{inparaenum}
+
+We may add as to the smoothness and globularity of the earth,
+that if the earth be represented by an 18-inch globe, the difference
+between the polar and equatorial diameter would only be about one-sixteenth
+of an inch, the highest mountains upon the earth's surface
+would be represented by about one-eightieth of an inch, and the average
+elevation of the continents would be hardly greater than that of
+a film of varnish. The earth is really relatively smoother and
+rounder than most of the balls in a bowling-alley.
+
+\includegraphicsouter{illo044}{\textsc{Fig.~44.}---Curvature of the Earth's Surface.}
+
+\begin{fineprint}
+\sloppy
+\nbarticle{134.} An approximate measure of the diameter is easily obtained. Erect
+upon a level plain three
+rods in line, a mile apart,
+and cut off their tops at
+the same level, carefully
+determined with a surveyor's
+levelling instrument.
+It will then be
+found that the line $AC$,
+Fig.~44, joining the extremities
+of the two terminal rods, passes about eight inches below $B$, the
+top of the middle rod.
+
+\fussy
+Suppose the circle $ABC$ completed, and that $E$ is the point on the circumference
+opposite $B$, so that $BE$ equals the diameter of the earth $(=2\,R)$.
+\begin{flalign*}
+&\text{\indent By geometry,}& &BD:BA = BA:BE, &&\phantom{\text{\indent By geometry,}}\\
+&\text{whence }& &BE = \frac{BA^{2}}{BD}, \text{ or }
+  R = \frac{BA^{2}}{2BD}.  &&
+\end{flalign*}
+
+Now $BA$ is one mile, and $BD = \frac{2}{3}$ of a foot, or $\frac{1}{7920}$ of a mile.
+
+Hence $2R = \dfrac{1^2}{\frac{1}{7920}}$, or 7920 miles: a very fair approximation.
+
+On account of refraction, however, the result cannot be made \textit{exact} by
+any care in observation. The line of sight, $AC$, is not \textit{strictly} straight,  but
+curves slightly towards the earth, and differently as the weather changes.
+\end{fineprint}
+
+\nbarticle{135.} The best method of ascertaining the size of the earth---in
+fact the only one of real value---is by measuring the arcs of the meridian
+in order to ascertain \textit{the number of miles or kilometers in one degree},
+from which we immediately get the circumference of the earth.
+%% -----File: 104.png---Folio 93-------
+This measure involves two distinct operations. One---the measure
+of the number of miles---is purely \textit{geodetic}; the other---the determination
+of the number of degrees, minutes, and seconds between
+the two stations---is purely \textit{astronomical}.
+
+We have to find by \textit{astronomical} observation the angle between two
+radii drawn from the centre of the earth to the two stations (regarding
+the earth as spherical); or, what is the same thing, the \textit{angular distance
+in the sky between their respective zeniths}. The two stations being
+on the same meridian, all that is necessary is to measure their \textit{latitudes}
+by any of the methods which have been given in \chapref{CHAPTERIV}{Chapter~IV.} and take
+the difference. This will be the angle wanted. If, for instance, the
+distance between the two stations was found by measurement to be
+120 miles, and the difference of latitude was found by astronomical
+observations to be $1°\,44'.2$, we should get 69.27 miles for one degree.
+Three hundred and sixty times this would be the circumference of
+the earth, a little less than 25,000 miles, and the diameter would be
+found by dividing this by $\pi$, which would give 7920 miles.
+
+\begin{fineprint}
+\nbarticle{136.}  Eratosthenes of Alexandria seems to have understood the matter
+as early as 250~\textsc{b.c.} His two stations were Alexandria and Syene in Upper
+Egypt. At Syene he observed that at noon of the longest day in summer
+there was no shadow at the bottom of a well, the sun being then vertically
+overhead. On the other hand, the gnomon at Alexandria, on the same day,
+by the length of the shadow, gave him $\frac{1}{50}$ of a circumference, or $7°\,12'$ as the
+distance of the sun from the zenith at that place, which, therefore, is the
+difference of latitude between Alexandria and Syene.
+
+The weak place in his work was in the measurement of the distance between
+the two places. He states it as 5000 stadia, thus making the circumference
+of the earth 250,000 stadia; but we do not know the length of his
+stadium, nor does he give any account of the means by which he measured
+the distance, if he measured it at all. There seem to have been as many
+different stadia among the ancient nations as there were kinds of ``feet'' in
+Europe at the beginning of this century.
+
+The first really valuable measure of the arc of a meridian was that made
+by Picard in Northern France in 1671---the measure which served Newton so
+well in his verification of the idea of gravitation.
+\end{fineprint}
+
+\section*{II.}
+
+\nbarticle{137.}  \nbparatext{The Rotation of the Earth.}---At the time of Copernicus the
+only argument in favor of the earth's rotation\footnote
+  {The word \textit{rotate} denotes a spinning motion like that of a wheel on its axis.
+  The word \textit{revolve} is more general in its application, and may be applied either to
+  describe such a spinning motion, or (and this is the more usual use in astronomy)
+  to describe the motion of one body around another, as that of the earth around
+  the sun.}
+was that the hypothesis
+%% -----File: 105.png---Folio 94-------
+was \textit{more probable} than that the heavens themselves revolved.
+All phenomena \textit{then known} would be sensibly the same on either
+supposition. A little later, analogy could be adduced, for when the
+telescope was invented, we could \textit{see} that the sun, moon, and several
+of the planets are rotating globes.
+
+At present we are able to adduce experimental proofs which absolutely
+demonstrate the earth's rotation, and some of them even make
+it visible.
+
+\includegraphicsouter{illo045}{\textsc{Fig.~45.}\\
+Eastward Deviation
+of a Falling
+Body.}
+
+\nbarticle{138.} 1.~\textit{The Eastward Deviation of Bodies falling from a Great
+Height}.---The idea that such a deviation ought to occur was first
+suggested by Newton. Evidently, since the top of a tower, situated
+anywhere but at the pole of the earth, describes every
+day a larger circle than its base, it must move faster.
+A body which is dropped from the top, retaining its excess
+of eastward motion as it descends, must therefore
+strike \textit{to the east} of the point which is vertically under
+its starting-point, provided it is not deflected in its fall
+by the resistance of the air or by air-currents. \figref{illo045}{Fig.~45}
+illustrates the principle. A body starting from $A$, the
+top of the tower, reaches the earth at $D$ ($BD$ being
+equal very approximately to $AA'$), while during
+its fall the \textit{bottom} of the tower has only moved from
+$B$ to $B'$. The experiments are delicate, since the deviation
+is very small, and it is not easy to avoid the
+effect of air-currents. It is also extremely difficult to
+get balls so perfectly spherical that they will not sheer
+off to one side or the other in falling.
+
+\begin{fineprint}
+The best experiments of this kind so far have been these of Benzenberg,
+performed at Hamburg in 1802, and those of Reich, performed in 1831, in
+an abandoned mine shaft near Freiberg, in Saxony. The latter obtained a
+free fall of 520 feet, and from the mean of 160 trials, the eastern deviation
+observed was 1.12 inches, while theory would make it 1.08. The experiment
+also gave a southern deviation of 0.17 of an inch, unexplained by theory.
+It seems to indicate the probable error of observation. The balls in falling
+sometimes deviated two or three inches one side or the other from the
+average.
+%% -----File: 106.png---Folio 95-------
+
+\includegraphicsouter{illo046}{\textsc{Fig.~46.}---Foucault's Pendulum Experiment.}
+
+The formula given by Worms in his treatise on ``The Earth and its
+Mechanism,'' is
+\[
+  x = \frac{4\pi t(H-\frac{1}{2}\Delta)\cos{\phi}}{3T} ;
+\]
+where $x$ is the deviation, $t$ is the number of seconds occupied in falling, $T$
+the number of seconds in a sidereal day, $H$ the height fallen through, and
+$\Delta$ the difference between $H$ and the height through which a body would fall
+in $t$ seconds if there were no resistance (so that $\Delta=\frac{1}{2}gt^2-H$). Finally, $\phi$ is
+the latitude of the place of observation. In latitude $45°$ a fall of 576 feet
+should give, neglecting the resistance of the air, a deviation of 1.47 inches.
+The resistance would increase it a little.
+
+It will be noted that \textit{at the pole}, where the cosine of the latitude equals
+zero, \textit{the experiment fails}. The largest deviation is obtained at the equator.
+\end{fineprint}
+
+\nbarticle{139.} 2.~\textit{Foucault's Pendulum Experiment}.---In 1851 Foucault,
+that most ingenious of French
+physicists, devised and first executed
+an experiment which actually
+shows the earth's rotation to the
+eye. From the dome of the Pantheon
+in Paris he suspended a heavy
+iron ball about a foot in diameter
+by a wire more than 200 feet long
+(\figref{illo046}{Fig.~46}). A circular rail some
+twelve feet across, with a little
+ridge of sand built upon it, was
+placed under the pendulum in such
+a way that a pin attached to the
+swinging ball would just scrape
+the sand and leave a mark at each
+vibration. The ball was drawn
+aside by a cotton cord and allowed
+to come absolutely to rest; then
+the cord was burned, and the pendulum
+set to swinging in a true
+plane; but this plane seemed to
+\textit{deviate slowly towards the right}, cutting the sand in a new place at
+each swing and shifting at a rate which would carry it completely
+around in about thirty-two hours if the pendulum did not first come
+to rest. In fact, the floor of the Pantheon was soon turning under
+the plane of the pendulum's vibration. The experiment created
+%% -----File: 107.png---Folio 96-------
+great enthusiasm at the time, and has since been very frequently
+performed and always with substantially the same results.
+
+\nbarticle{140.} The approximate theory of the experiment is very simple.
+Such a pendulum, consisting of a round ball hung by a \textit{round} wire or
+else suspended on \textit{point, so as to be equally free to swing in any plane}
+(unlike the common clock pendulum in this freedom), being set up
+at the pole of the earth, would appear to shift around in twenty-four
+hours. Really, the plane of vibration remains invariable and the
+earth turns under it, the plane of vibration in this case being unaffected
+by the motion of the earth. This can be easily shown
+by setting up a similar apparatus, consisting of a ball hung by a
+thread, upon a table, and then turning the table around with as little
+jar as possible. The plane of the swing will remain unchanged by
+the motion of the table.
+
+\includegraphicsouter{illo047}{\textsc{Fig.~47.}\\
+Explanation of the Foucault
+Pendulum Experiment.}
+
+It is easy to see, further, that at the equator there would be no such
+tendency to shift. In any other latitude the effect will be intermediate,
+and the time required for the pendulum to complete the revolution
+of its plane will be \textit{twenty-four hours
+divided by the sine of the latitude}. The northern
+edge of the floor of a room (in the northern
+hemisphere) is nearer the axis of the earth
+than its southern edge, and therefore is carried
+more slowly eastward by the earth's rotation.
+Hence it must \textit{skew around} continually,
+like a postage stamp gummed upon a whirling
+globe anywhere except at the globe's equator.
+The southern extremity of every north and
+south line on the floor continually works toward
+the east faster than the northern extremity,
+causing the line itself to shift its direction
+accordingly, compared with the direction
+it had a few minutes before. A free pendulum,
+set at first to swing along such a line, must therefore apparently
+deviate continually at the same rate in the opposite direction. In
+the \textit{northern} hemisphere its plane moves \textit{dextrorsum}; \textit{i.e.}, with the
+hands of a watch: in the \textit{southern}, its motion is \textit{sinistrorsum}.
+
+\includegraphicsouter{illo048}{\textsc{Fig.~48.}---Developed Cone.}
+
+\begin{fineprint}
+\nbarticle{141.} Suppose a parallel of latitude drawn through the place in question,
+and a series of tangent lines drawn toward the north at points an inch or so
+apart on this parallel. All these tangents would meet at some point, $V$, \figref{illo047}{Fig.~47},
+which is on the earth's axis produced; and taken together these tangents
+%% -----File: 108.png---Folio 97-------
+would form a cone with its point at $V$. Now if we suppose this cone cut
+down upon one side and opened up (technically, ``\textit{developed}''), it would give
+us a sector of a circle, as in \figref{illo048}{Fig.~48}, and the angle of the sector would be the
+sum total of the angles between all the adjacent
+meridians tangent to the earth on that parallel.
+Now it is easy to prove that the angle of this sector
+equals $360°× \sin{\phi}$ ($\phi$ being the latitude).
+(1)~The circumference of the parallel $AB$ (\figref{illo047}{Fig.~47})
+$= 2\pi× R\cos{\phi}$, since $R×\cos{\phi}=AD$, which is
+the radius of the parallel, $R$ being $AC$, the radius
+of the globe, and the angle $ACD$ equal to $90°-\phi$.
+(2)~The side $AV$ of the cone (which will be the
+radius of the sector when the cone is developed)
+$= R\cot{\phi}$; so that the circumference of the circle
+which has $VA$ for its radius would be $2\pi× R\cot{\phi}$.
+
+Hence ($ABA'$ being the circumference of the
+parallel forming the edge of the developed sector $ABA'V$ in \figref{illo048}{Fig.~48}), the
+angle of the sector $AVA'$ (greater than $180°$ in the \figref{illo048}{figure}): $360°=arc\
+ABA'$: whole circumference $ABA'm$; or angle $V : 360°=R\cos{\phi} : R\cot{\phi}$;
+\begin{flalign*}
+&\text{whence }&&
+  V = 360°\frac{\cos{\phi}}{\cot{\phi}}
+    = 360°\sin{\phi}.  &&\phantom{whence }
+\end{flalign*}
+
+$V$ is the total angle described by the plane of the pendulum in a day.
+
+At the pole the cone produced by the tangent lines becomes a little
+``button,'' a complete circle. At the equator it becomes a \textit{cylinder},
+and the angle is zero.
+
+In order to make the experiment successfully, many precautions must be
+taken. It is specially important that the pendulum should vibrate in a true
+plane, without any lateral motion. To secure this end, it must be carefully
+guarded against all jarring motion and air-currents. To diminish the effect
+of all such disturbances, which will always occur to a certain extent, the
+pendulum should be very heavy and very long, and of course the suspended
+ball must be truly round and smooth. Ordinary clock-work cannot be used
+to keep the pendulum in vibration, since it must be free to swing in every
+plane. Usually, the apparatus once started is left to itself until the vibrations
+cease of their own accord; but Foucault contrived a most ingenious
+electrical apparatus, which we have not space to describe, by means of which
+the vibration could be kept up for days at a time without producing any
+hurtful disturbance whatever.
+
+It will be noticed that this experiment is most effective precisely where
+the experiment of the falling bodies fails. This is best near the pole, the
+other at the equator.
+\end{fineprint}
+
+\includegraphicsouter{illo049}{\textsc{Fig.~49.}---Foucault's Gyroscope.}
+
+\nbarticle{142.} 3.~\textit{By the Gyroscope}, an experiment also due to Foucault
+and proposed and executed soon after the pendulum experiment.
+%% -----File: 109.png---Folio 98-------
+
+The instrument shown in \figref{illo049}{Fig.~49} consists of a wheel so mounted in
+gimbals that it is free to turn in every direction, and so delicately
+balanced that it will stay in any position if undisturbed. If the
+wheel be set to rotating rapidly, \textit{it will maintain the direction of its
+axis invariable, unless acted upon by extraneous force}. If, then,
+we set the axis horizontal and arrange a microscope to watch a
+mark upon one of the gimbals,
+it will appear slowly to
+shift its position as the earth
+revolves, in the same way as
+the plane of the pendulum
+behaves.
+
+\begin{fineprint}
+\nbarticle{143.} 4.~There are many other
+phenomena which depend upon
+and really demonstrate the earth's
+rotation. We merely mention
+them:---
+
+\begin{asparaenum}[\itshape a.\normalfont]
+\item \textit{The Deviation of Projectiles.}
+In the northern hemisphere a
+projectile always deviates towards
+the right; in the southern hemisphere
+toward the left.
+
+\item \textit{The Trade Winds.}
+
+\item \textit{The Vorticose Revolution of
+the Wind in Cyclones.} In the
+northern hemisphere the wind in
+a cyclone moves spirally towards
+the centre of the storm, whirling
+\textit{counter clock-wise}, while in the
+southern, the spiral motion is \textit{with the hands of a watch}. The motion is
+explained in either case by the fact that currents of air, setting out for the
+centre of disturbance where the cyclone is formed, deviate like projectiles,
+to the right in the northern hemisphere, and towards the left in the southern
+hemisphere, so that they do not meet squarely in the centre of disturbance.
+
+\item \textit{The Ordinary Law of Wind-change;} that is, in the northern hemisphere
+the north wind, under ordinary circumstances, changes to a northeast, a
+northeast wind to an east, east to southeast, etc. When the wind changes
+in the opposite direction, it is said to ``\textit{back}'' around. In the southern
+hemisphere it of course \textit{usually} backs around, much to the disconcertment
+of the early Australian settlers.
+\end{asparaenum}
+\end{fineprint}
+
+It might seem at first time the rotation of the earth, which occupies
+twenty-four hours, is not a very rapid motion. A point on the equator,
+%% -----File: 110.png---Folio 99-------
+however, has to move nearly one thousand miles an hour, which
+is about fifteen hundred feet per second, and very nearly the speed
+of a cannon-ball.
+
+\nbarticle{144.} \nbparatext{Invariability of the Earth's Rotation.}---It is a question of
+great importance whether the day changes its length. Theoretically it
+must almost necessarily do so. The friction of the tides, and the
+deposits of meteoric matter upon the earth both tend to lengthen it;
+while on the other hand, the earth's loss of heat by radiation and
+consequent shrinkage must tend to shorten it. Then geological
+changes, the elevation and subsidence of continents, and the transportation
+of matter by rivers, act, some one way, some the other. At
+present it can only be said that the change, if any has occurred since
+astronomy became accurate, has been too small to be detected. The
+day is certainly not longer or shorter by $\frac{1}{100}$ of a second than in the
+days of Ptolemy, and \textit{probably} has not changed by $\frac{1}{1000}$ of a second.
+The criterion is found in comparing the \textit{times} at which celestial
+phenomena, such as eclipses, transits of Mercury, etc., occur.
+
+\section*{III.}
+
+\nbarticle{145.} \nbparatext{The Earth's Form}, more accurately stated, \textit{is that of a
+spheroid of revolution, having an equatorial radius of} 6,377,377 \textit{metres,
+and a polar radius of} 6,355,270 \textit{metres, according to Listing $(1873);$
+or of} 6,378,206.4 \textit{and} 6,356,583.8 respectively, \textit{according to Clarke}.\footnote
+  {This is Clarke's spheroid of 1866, and is adopted by the United States Coast
+  and Geodetic Survey. See \hyperref[app:spheroid]{Appendix} for his spheroid of 1878.}
+It must be understood, also, that this statement is only a second
+\textit{approximation} (the first being that the earth is a globe). Owing
+to mountains and valleys, etc., the earth's surface does not strictly
+correspond to that of any geometrical solid whatever.
+
+The flattening at the poles is the necessary consequence of the
+earth's rotation, and might have been cited in the preceding section
+as proving it.
+
+\includegraphicsouter{illo050}{\textsc{Fig.~50.}---A Triangulation.}
+
+\nbarticle{146.} There are two ways of determining the form of the earth:
+one, by \textit{measurement of distances upon its surface in connection with
+the latitudes and longitudes of the points of observation}. This gives
+not only the \textit{form}, but the \textit{dimensions}. The other method is by the
+observation of the \textit{varying force of gravity at various points},---observations
+which are made by means of a pendulum apparatus of some
+kind, and determine \textit{only the form}, but not the size of the earth.
+%% -----File: 111.png---Folio 100-------
+
+\nbarticle{147.} 1.~\textit{Measurements of Arcs of Meridian in Different Latitudes}.---To
+determine the size of the earth regarded as a sphere, a \textit{single}
+arc of meridian in any latitude is sufficient. Assuming, however,
+that the earth is not a sphere, but a spheroid with elliptical meridians,
+we must measure at least \textit{two} such arcs, one of which should be near
+the equator, the other near the pole.
+
+The \textit{astronomical work} consists simply in finding with the greatest
+possible accuracy the \textit{difference of latitude} between the terminal stations
+of the meridian arc. The \textit{geodetic work} consists in measuring
+their \textit{distance} from each other in miles, feet, or metres, and it is this
+part of the work which consumes the most time and labor. The
+process is generally that known as triangulation.
+
+\begin{fineprint}
+
+Two stations are selected for the extremities of a
+\textit{base line} six or seven miles long, and the ground
+between them is levelled as if for a railroad. The
+distance between these stations ($A$ and $B$ in \figref{illo050}{Fig.~50})
+is then carefully measured by an apparatus especially
+designed for the purpose and with an error not to
+exceed half an inch or so in the whole distance. A
+third station, $1$, is then chosen, so situated that it will
+be visible from both $A$ and $B$, and all the angles of
+the triangle $AB\,1$ are measured with great care by a
+theodolite. A fourth station, $2$, is then selected, such
+that it will be visible from $A$ and $1$ (and if possible
+from $B$ also), and the angles of the triangle $A\,1\,2$ are
+measured in the same way. In this manner the whole
+ground between the two terminal stations is covered
+with a network of triangulation, the two terminal stations themselves being
+made two of the triangulation points. Knowing \textit{one distance and all the
+angles} in this system, it is possible to compute with great accuracy the exact
+length of the line $1\,5$ and its direction.
+\end{fineprint}
+
+The sides of the triangles are usually from twenty-five to thirty
+miles in length, though in a mountainous country not infrequently
+much longer ones are available. Generally speaking, the fewer the
+stations necessary to connect the extremities of the arc, and the
+longer the lines, the greater will be the ultimate accuracy. In this
+way it is possible to measure distances of 200 or 300 miles with a
+probable error not exceeding two or three feet.
+
+\begin{fineprint}
+Many arcs of meridians have been measured in this way,---not less than
+twenty or thirty in different parts of the earth, the most extensive being the
+%% -----File: 112.png---Folio 101-------
+so-called Anglo-French arc, extending more than twelve degrees in length;
+the Indian arc, nearly eighteen degrees long; and the great Russo-Scandinavian
+arc, more than twenty-five degrees in length, and reaching from
+Hammerfest to the mouth of the Danube. One short arc has been measured
+in South America and one in South Africa.
+\end{fineprint}
+
+In a general way, it appears that the higher the latitude the longer
+the arc. Thus, near the equator the length of a degree has been
+found to be 362,800 feet in round numbers, while in northern Sweden,
+in latitude $66°$, it is 365,800 feet; in other words, the earth's surface
+is \textit{flatter near the poles}. It is necessary to travel 3000 feet further in
+Sweden than in India to increase the latitude one degree, as measured
+by the elevation of the celestial pole.
+
+\nbarticle{148.}
+The deduction of the exact form of the earth from such
+measurements is an abstruse problem. Owing to errors of observation
+and local deviations in the direction of gravity, the different arcs do
+not give strictly accordant results, and the best that can be done is to
+find the result \textit{which most nearly satisfies all the observations}.
+
+If we assume that the form is that of an \textit{exact spheroid of revolution},
+with all the meridians true ellipses and all exactly alike, the problem
+is simplified somewhat, though still too complicated for discussion
+here. Theory indicates that the form of a revolving mass, fluid
+enough to yield to the forces acting in such a case, \textit{might, and probably
+would, be such a spheroid}; but other forms are also theoretically
+possible, and some of the measurements rather indicate that
+the equator of the earth is not a true circle, but an oval flattened by
+nearly half a mile. On the whole, however, astronomers are disposed
+to take the ground that since no regular geometrical solid
+whatsoever can \textit{absolutely} represent the form of the earth, we may as
+well assume a regular spheroid for the standard surface, and consider
+all variations from it as local phenomena, like hills and valleys.
+
+\begin{fineprint}
+\includegraphicsouter{illo051}{\textsc{Fig.~51.}\\
+Radii of Curvature of the Meridian.}
+
+\nbarticle{149.}  Each measurement of a degree of latitude gives the ``\textit{radius of curvature},''
+as it is called, of the meridian at the degree measured. The length
+of a degree from $44°\,30'$ to $45°\,30'$, multiplied by 57.29 (the number of degrees
+in a radian), gives the radius of the ``\textit{osculatory circle},'' which would
+just fit the curve of the meridian at that point, Having a table giving
+the actual length of each degree of latitude, we could construct the earth's
+meridian graphically as follows:---
+
+Draw the line $AX$, \figref{illo051}{Fig.~51}. On it lay off $Aa$, equal to the radius of curvature
+of the first measured degree (that is, 57.3 times the length of the degree),
+%% -----File: 113.png---Folio 102-------
+and with $a$ as centre, describe an arc $AB$, making the angle $AaB$ just one
+degree. Next produce the line $Ba$ to $b$,
+making $Bb$ the radius of curvature of
+the second degree, and draw this second
+degree-arc; and so proceed until the
+whole ninety have been drawn. This
+will give one quarter of the meridian, and
+of course the three other quarters are all
+just like it. $a$, $b$, $c$, etc., are called the
+``centres of curvature'' of the different
+degrees.
+
+If we assume the curve to be an
+ellipse, then the equatorial \DPtypo{semidiameter}{semi-diameter}
+$AO$, and the polar, $PO$, are given respectively
+by the two formulas, $AO = \sqrt[3]{qp^{2}}$
+and $PO = \sqrt[3]{q^{2}p}$, $q$ and $p$ being the radii
+of curvature ($Aa$ and $Pe$ in the \figref{illo051}{figure}) at the equator and pole.
+\end{fineprint}
+
+\nbarticle{150.} The ``\textit{ellipticity}'' or ``\textit{oblateness}'' of an ellipse is the fraction
+found by dividing the difference of the polar and equatorial diameters
+by the equatorial, and is expressed by the equation
+\[
+d = \frac{A - B}{A}.
+\]
+In the case of the earth this is $\frac{1}{295}$, according to Clarke's spheroid,
+of 1866. Until within the last few years Bessel's smaller value,
+viz., $\frac{1}{299}$, was generally adopted. Listing's larger value, $\frac{1}{288}$, is now
+preferred by some.
+
+The \textit{ellipticity} of an ellipse must not be confounded with its
+\textit{eccentricity}. The latter is
+\[
+e = \frac{\sqrt{A^{2} - B^{2}}}{A},
+\]
+and is always a much larger numerical quantity than the ellipticity.
+In the case of the earth's meridian, it is $\frac{1}{12.1}$ as against $\frac{1}{295}$. Its
+symbol is usually $e$.
+
+\begin{fineprint}
+\nbarticle{151.} \textit{Arcs of longitude} are also available for determining the earth's form
+and size. On a spherical earth a degree of longitude measured along any
+parallel of latitude would be equal to one degree of the equator multiplied by
+the cosine of the latitude. On an oblate or orange-shaped spheroid (the surface
+of which lies wholly within the sphere having the same equator) the degrees
+%% -----File: 114.png---Folio 103-------
+of longitude are evidently everywhere shorter than on the sphere,
+the difference being greatest at a latitude of $45°$.
+
+In fact, \textit{arcs in any direction between stations of which both the latitude and
+longitude are known} can be utilized for the purpose; and thus the extensive
+surveys that have been made in different countries have given us a pretty
+accurate knowledge of the earth's dimensions. It is very desirable, that in
+some way the chain of actual measurements should be extended from the
+eastern continent to the western, but the immense difficulties of so doing
+are obvious.
+
+At present the distance from a point on the earth's surface (say the observatory
+at Washington) to any other point in the opposite hemisphere (say
+the observatory at the Cape of Good Hope) is uncertain to perhaps the
+extent of a quarter of a mile.
+\end{fineprint}
+
+\nbarticle{152.} 2.~\textit{Pendulum Experiments}.---Since
+\[
+  t = \pi\sqrt{\frac{l}{g}}\ \text{(Physics, p.~72)},\
+  g = \frac{\pi^{2}l}{t^{2}};
+\]
+we can therefore measure the variations of the force of gravity, $g$,
+at different parts of the earth, either by taking a pendulum of invariable
+length and determining $t$, the time of its vibration; or by
+measuring the length, $l$, of a pendulum which will vibrate seconds.
+Extensive surveys of this sort have been made, and are still in progress,
+and it is found that the \textit{force of gravity at the pole exceeds that at
+the equator by about $\frac{1}{190}$ part}. In other words, a person who weighs
+190 pounds at the equator (\textit{by a spring balance}) would, if carried to
+the pole, show 191 pounds by the same balance.
+
+\begin{fineprint}
+The apparatus most used at present for the purpose of measuring the
+force of gravity is a modification of the so-called Kater's pendulum. The
+pendulum itself usually consists of a brass tube about an inch in diameter
+and about four feet long, carrying a ball three or four inches in diameter at
+each end, \textit{both balls being exactly of the same size, but one solid while the other
+is hollow}. Two knife edges are inserted through the rod at right angles,
+one near the heavy ball and the other at just the same distance from the
+lighter one, and the weights and dimensions of the apparatus are so adjusted
+that the \textit{time of vibration will be very approximately the same whether the pendulum
+is swung heavy end up or light end up, and will be not far from one second}.
+The distance between the knife edges will then, according to the theory of
+the pendulum, be very nearly equal to the length of a simple pendulum
+vibrating in the same time; and the small difference can be accurately calculated
+when we know the exact time of vibration, each end up. The
+knife edges swing on agate planes which are fastened upon a firm support;
+%% -----File: 115.png---Folio 104-------
+and great pains must be taken to have the support really firm. Professor
+Peirce of our Coast Survey a few years ago detected important errors in a
+majority of the earlier pendulum observations, due to insufficient care in
+this respect.
+
+\nbarticle{153.} The observations consist in comparing the pendulum with a clock,
+either by noting the ``\textit{coincidences},'' or by an electrical record automatically
+made on a chronograph. A pin attached to the end of the pendulum
+touches a globule of mercury (which is momentarily raised for the purpose
+once in eight or ten minutes), and so records the swing upon the chronograph
+sheet. The observations need to be carefully corrected for \textit{temperature}
+(which, of course, affects the distance between the knife edges), for the
+\textit{length of arc} through which the pendulum is swinging, and for the \textit{resistance
+of the air}. The observations determine the ``\textit{force of gravity}'' (French
+``\textit{pesanteur}'') at the station. This ``force of gravity,'' however, thus determined,
+is not simply the earth's \textit{attraction}, but includes also the effects of the
+centrifugal force, due to the earth's rotation, which we must consider and
+allow for.
+\end{fineprint}
+
+\nbarticle{154.}
+At the equator the centrifugal force acts vertically in direct
+opposition to gravity, and is given by the well-known formula
+\[
+C=\frac{V^2}{R}
+\]
+(see Physics, p.~62), in which $V$ is the velocity of the earth's surface
+at the equator, and $R$ the earth's radius. Since $V$ is equal to
+the earth's circumference divided by the number of seconds in a
+sidereal day, we have
+\[
+V=\frac{2\pi R}{t}, \text{ and } C = \frac{4\pi^2 R}{t^2}.
+\]
+\includegraphicsouter{illo052}{\textsc{Fig.~52.}\\
+The Earth's Centrifugal Force.}
+Now $R$, the radius of the earth, equals 20,926,000 feet; and $t$ equals
+86,164 mean-time seconds. $C$, therefore, comes out to 0.111 feet, which
+is $\frac{1}{289}$ of $g$, $g$ being $32\frac{1}{6}$ feet.
+
+\begin{fineprint}
+We may remark in passing that if the rate of rotation were seventeen
+times as great, $C$ would be $17^2$, or 289 times greater than now, and would
+equal gravity; so that on that supposition bodies at the equator would weigh
+absolutely nothing, and any greater velocity of rotation would send them
+flying.
+\end{fineprint}
+
+At any other latitude, since $MN = OQ\cos{MOQ}$,\footnote
+  {This is not \textit{exact}, since $MN$ in an oblate spheroid is less than $OQ×\cos{MOQ}$;
+  but the difference is unimportant in the case of the earth.}
+the centrifugal
+%% -----File: 116.png---Folio 105-------
+force, $c$, equals $C \cos {\phi}$, acting at right angles to the axis of the earth
+and parallel to the plane of the equator. Now, this centrifugal
+force $c$ is not \textit{wholly} effective in diminishing
+the weight of a body, but only
+that portion of it ($MR$ in \figref{illo052}{Fig.~52}) which
+is directed vertically. $c$ is $MT$ in the
+\figref{illo052}{figure}, and $MR$ is equal to $c$ multiplied
+by the cosine of $\phi$, which finally gives
+us $C×\cos^2\phi$ for the amount by which
+the centrifugal force diminishes gravity
+at a station whose latitude is $\phi$.
+
+Every observation, therefore, of the
+``force of gravity,'' obtained by the pendulum, needs to be increased
+by the quantity
+\[
+\frac{g}{289}×\cos^2{\phi},
+\]
+in order to get the real value of the earth's \textit{gravitational attraction}
+at the point of observation.
+
+\begin{fineprint}
+The other component of $c$ (viz.~$MS$) acts at right angles to gravity and
+parallel to the earth's surface, and is given by the formula
+\[
+C\cos{\phi} \sin{\phi} = \tfrac{1}{2}C\sin{2\phi}.
+\]
+The direction of still water is determined by the resultant of the earth's
+attraction combined with this deflecting force acting towards the equator;
+so that this surface is not perpendicular to a line drawn towards the centre
+of the earth anywhere excepting at the equator and the poles.
+\end{fineprint}
+
+\nbarticle{155.} Having a series of pendulum observations, we can then form
+a table showing the force of gravity at each station; and correcting
+this by adding the amount of the centrifugal force at each place, we
+shall have the force of the earth's attraction. This is greater the
+nearer each station is to the centre of the earth; but unfortunately
+there is no simple relation connecting the force with the distance. The
+attraction depends not only on the distance from the centre of the earth,
+but also upon the form of the earth and the constitution of its interior,
+and the arrangement of its strata of different density. We may safely
+assume, however, that the earth is made up \textit{concentrically}, so to speak;
+the strata of equal density being arranged like the coats of an onion.
+On this hypothesis Clairaut, in 1742, demonstrated the relation given
+below, which is always referred to as Clairaut's equation.
+%% -----File: 117.png---Folio 106-------
+
+Let $w$ be the loss of weight between the equator and the pole, and
+$C$ the centrifugal force at the planet's equator, both being expressed
+as fractions of the equatorial force of gravity, and let $d$ be the
+ellipticity of the planet.
+
+Then, as Clairaut proved,
+\begin{flalign*}
+&&&                d + w = 2\tfrac{1}{2}× C;  &&\\
+&\text{whence }&&  d = 2\tfrac{1}{2}C-w.  &&\phantom{whence }
+\end{flalign*}
+
+In the case of the earth we have
+\begin{flalign*}
+&&&   d = 2\tfrac{1}{2}×\frac{1}{289}-\frac{1}{190},  &&\\
+&\text{which gives }&&  d = \frac{1}{292.8}.  &&\phantom{which gives }
+\end{flalign*}
+
+Considering all the data, the most that can safely be said as to $d$ is
+that it lies between the fractions $\frac{1}{290}$ and $\frac{1}{295}$. (Clarke's later values
+for $d$ are larger than that adopted by the Coast Survey.)
+
+\nbarticle{156.}
+\nbparatext{Astronomical, Geographical, and Geocentric Lati\-tudes.}---The
+astronomical latitude of a place has been defined as \textit{the elevation of
+the pole}, or, what comes to the same thing, it is \textit{the angle between the
+plane of the equator and the direction of gravity} at that place, however
+that direction may be affected by local causes.
+
+The \textit{geocentric} latitude, on the other hand, is the angle made at
+the centre of the earth (as the word implies)
+between the plane of the equator and a line
+drawn from the observer to the centre of
+the earth, which line of course does not
+coincide with the direction of gravity, since
+the earth is not spherical.
+
+The \textit{geographical} or \textit{geodetic} latitude of
+a station is the angle formed with the plane
+of the equator by a line drawn from the
+station \textit{perpendicular to the surface of the
+standard spheroid}.
+
+\begin{fineprint}
+
+\includegraphicsouter{illo053}{\textsc{Fig.~53.}\\
+Astronomical and Geocentric
+Latitude.}
+
+If the earth's surface were \textit{strictly spher\-oidal, and there were no local variations
+of gravity}, the astronomical latitude and the geographical latitude
+would coincide---and they never differ greatly; but the geocentric latitude
+differs from them by a very considerable quantity---as much as $11'$ in latitude
+$45°$. The geocentric latitude is but little used except in certain astronomical
+calculations where parallax is involved.
+%% -----File: 118.png---Folio 107-------
+
+In \figref{illo053}{Fig.~53}, the angle $MOQ$ is the geocentric latitude of $M$, while $MNQ$ is
+the geographical latitude. $MNQ$ is also the astronomical latitude, unless
+there is some local disturbance of the direction of gravity. The angle $OMN$,
+which is the difference between the geocentric and astronomical latitudes, is
+called ``\textit{the angle of the vertical}.''
+\end{fineprint}
+
+\nbarticle{157.} It will be noticed that the astronomical latitude of a place is
+the only one of these three latitudes which is \textit{determined directly by
+observation}. In order to know the \textit{geocentric} and \textit{geographical} latitudes
+of a place, we must know the form and dimensions of the earth,
+which are ascertained only by the help of observations made elsewhere.
+
+The geocentric degrees are longer near the equator than near the
+poles, and it is worth noticing that if we form a table giving the length
+of each degree of \textit{geographical} latitude from the equator to the pole,
+the same table, read \textit{backwards}, gives the length of \textit{geocentric} degrees.
+
+Since the earth is ellipsoidal instead of spherical, it is evident that
+lines of ``level'' on the earth's surface are affected by the earth's rotation.
+If this rotation were to cease, the direction of gravity would
+be so much changed that the Gulf of Mexico would run up the Mississippi
+River, because the distance from the centre of the earth to
+the head of the river is less by some thousands of feet than the
+distance from the mouth of the river to the centre of the earth.
+
+\begin{fineprint}
+\nbarticle{158.} \nbparatext{Station Errors.}---The irregularities in the direction of gravity
+are by no means insensible as compared with the accuracy of modern astronomical
+observation, and the difference between the astronomical latitude
+and longitude of a place and the geographical latitude and longitude of the
+same place constitute what is called the ``\textit{station error}.'' In the eastern part
+of the United States these station errors, according to the Coast Survey
+observations, average about $1\frac{1}{2}''$. Errors of from $4''$ to $6''$ are not uncommon,
+and in mountainous countries, as for instance in the Caucasus and in
+Northern India, these errors occasionally amount to $30''$ or $40''$. They are
+not ``errors'' in the sense that the astronomical latitude of the place has not
+been determined correctly, but are merely the effects of the irregular distribution
+of matter in the crust of the earth in altering the direction of gravity.
+Pendulum observations show local variations in the \textit{force} of gravity quite
+proportional to the deviations which the station-errors show in its \textit{direction}.
+\end{fineprint}
+
+\section*{IV.}
+
+\nbarticle{159.} \nbparatext{The Earth's Mass and Density.}---The `\textit{mass}' of a body is the
+\textit{quantity of matter} that it contains, the unit of mass being the quantity
+of matter contained in a certain arbitrary body which is taken as a
+standard. For instance, a ``kilogram'' is the quantity of matter
+%% -----File: 119.png---Folio 108-------
+contained in the block of platinum preserved at Paris as the standard
+of mass.\footnote
+  {This was meant to be just equal to the mass contained in a cubic decimeter
+  of water at its maximum density, and is so very nearly indeed.}
+A pound is similarly defined by reference to the prototypes
+at Washington and London.
+
+Two masses of matter are defined as equal which \textit{require the same
+expenditure of energy to give them the same velocity}; or \textit{vice versa,
+those are equal which, when they have the same velocity, possess the
+same energy, and, in giving up their motion and coming to rest, do
+the same amount of work}.
+
+\begin{fineprint}
+Masses can therefore be compared \textit{by placing them in the same field of
+force and comparing the energies developed in them when they have moved equal
+distances under the action of the force}. This method, however, is seldom
+convenient.
+\end{fineprint}
+
+\nbarticle{160.} \nbparatext{Proportionality of Mass to Weight.}---Newton showed by his
+experiments with pendulums of different substances, that at any
+given point the attraction of the earth for a body of any kind of
+matter is proportional to the mass of that body; the attraction being
+measured as a pull or ``stress'' in this case, and called ``\textit{the weight}''
+of the body. In other and more common language, \textit{the mass of a
+body is proportional to its weight} (we must not say it \textit{is} its weight),
+provided the weighing of the bodies thus compared is done, in cases
+where scientific accuracy is essential, at the same place on the earth's
+surface. Practically, therefore, we \textit{usually measure the masses of
+bodies by simply weighing them}. It is to be carefully observed, however,
+that the words ``kilogram,'' ``pound,'' ``ton,'' etc., have also
+a secondary meaning, as denoting units of pull and push,---of
+``\textit{stress}'' speaking strictly and technically,---or of ``force,'' as that
+much abused word is very generally used.
+
+\begin{fineprint}
+It is, from a literary point of view, just as proper to speak of a \textit{stress or a
+pull} of a hundred pounds\footnote
+  {Of course the student will remember that we have a unit of stress,---the
+  \textit{dyne},---which is wholly free from this objection of ambiguity.}
+as of a \textit{mass} of a hundred pounds, but the word
+``pound'' means an entirely different thing in the two cases. At the surface
+of the earth the relation between the ideas, however, is so close that
+the way in which the ambiguity came about is perfectly obvious, and it is
+hardly probable that language will ever change so as to remove it. To a
+certain extent it is admittedly unfortunate, and the student must always be
+on his guard against it. At the earth's surface a \textit{mass} of 100 pounds always
+%% -----File: 120.png---Folio 109-------
+``\textit{weighs}'' very nearly 100 pounds; but, to anticipate slightly, at an elevation
+of 4000 miles above the surface, the same mass would ``weigh'' only 25
+pounds; at the distance of the moon about half an ounce; while on the
+surface of the sun it would ``weigh'' nearly 2800 pounds.
+\end{fineprint}
+
+\nbarticle{161.} \nbparatext{Gravity.}---The law of gravitation discovered by Newton declares
+that \textit{any particle of matter attracts any other particle with a force}
+(``\textit{stress},'' if the bodies are prevented from moving) \textit{proportional
+inversely to the square of the distance between them, and directly to the
+product of their masses;} or, as a formula, we may write,
+\[
+F = k\frac{M_{1}× M_{2}}{d^{2}},
+\]
+in which $M_{1}$ and $M_{2}$ are the two masses, and $d$ the distance between
+them, while $k$ is a constant numerical factor depending upon the units
+employed.\footnote
+  {It will not do to write the formula
+\[
+F = \frac{M_{1}× M_{2}}{d^{2}}
+\]
+(omitting the $k$), unless the units are so chosen that the unit of force shall be
+equal to the attraction between two masses each of one unit, at a distance of one
+\textit{unit}. It is not true that the attraction between two particles, each having a
+mass of one \textit{pound}, at a distance of \textit{one foot}, is equal to a stress of \textit{one pound} (of
+force), as would rather naturally be inferred if we should write the equation
+without the constant factor.}
+
+\begin{fineprint}
+We must not imagine the word ``\textit{attract}'' to mean too much. It merely
+states the fact that there is a \textit{tendency} for the bodies to move toward each
+other, without including or implying any explanation of the fact. So far,
+no explanation has appeared which is less difficult to comprehend than
+the fact itself. Whether bodies are \textit{drawn} together by some outside action,
+or \textit{pushed} together; or whether they themselves can act across space with
+mathematical intelligence,---in what way it is that ``attraction'' comes
+about, is still unknown,---apparently as inscrutable as the very nature and
+constitution of an atom of matter itself; \textit{it is simply a fundamental fact}.
+\end{fineprint}
+
+\nbarticle{162.} When the distance between attracting bodies is large as compared
+with their own magnitude, then reckoning the distance between
+their centres of mass as their true distance, the formula is sensibly
+true for them as it would be for mere particles. When, however, the
+distance is not thus great, the calculation of the attraction becomes a
+very serious problem, involving what is known as a ``double integration.''
+%% -----File: 121.png---Folio 110-------
+We must find the attraction of each particle of the first body
+upon each particle of the other body, and take the sum of all these
+infinitesimal stresses. Newton, however, showed that if the \textit{bodies are
+spheres, either homogeneous or of concentric structure, then they attract
+and are attracted precisely as if the matter in them were wholly collected
+at their centres}. The earth, for instance, attracts a body at its surface
+very nearly as if it were all collected at its own centre, 4000
+miles distant; not exactly so, because the earth is not strictly
+spherical; but in what follows we shall neglect this slight inaccuracy.
+
+\nbarticle{163.}
+In order, then, to find the mass of the earth in kilograms,
+pounds, or tons, we must find some means of accurately comparing its
+attraction for some object on its own surface with the attraction of the
+same object by some body of known mass, at a measured distance.
+The difficulty lies in the fact that the attraction produced by any body,
+not too large to be handled conveniently, is so excessively small that
+only the most delicate operations serve to detect and measure it.
+
+The first successful attack upon the problem was made in 1774 by
+Maskelyne, the Astronomer Royal, by means of what is now usually
+referred to as,---
+
+\nbarticle{164.} 1.~``\textsc{The Mountain Method},'' because, in fact, the earth
+in this operation is weighed against a mountain.
+
+\includegraphicsouter{illo054}{\textsc{Fig.~54.}\\
+The Mountain Method of Determining the
+Earth's Density.}
+
+Two stations were chosen on the same meridian, one north and one
+south of the mountain Schehallien, in Scotland. In the first place, a
+careful topographical survey was
+made of the whole region, giving the
+precise distance between the stations,
+as well as the exact dimensions
+of the mountain, which is a
+``hog-back'' of very regular contour.
+From the known dimensions of the
+earth and the measured distance,
+the difference of the \textit{geographical} latitudes of the two places $M$ and $N$
+(\figref{illo054}{Fig.~54})
+can be accurately computed; \textit{i.e.}, the angle which the plumb
+lines at $M$ and $N$ would have made if there were no mountain there.
+
+\includegraphicsouter[8]{illo055}{\textsc{Fig.~55.}}
+
+In this case it was $41''$. The next operation was to observe the
+\textit{astronomical} latitude at each station. This astronomical difference of
+latitude, \textit{i.e.}, the angle which the plumb lines actually do make, was
+found to be $53''$, the plumb lines at $M$ and $N$ being drawn inward
+out of their normal position by the attraction of the mountain to the
+%% -----File: 122.png---Folio 111-------
+extent of $6''$ on each side; so that the astronomical difference of
+latitude was increased by $12''$ over the geographical.
+
+\begin{fineprint}
+Now, in such a case the ratio of gravity to the deflecting force,
+according to the laws of the composition of forces, is that of
+$aM$ to $aA'$ in the figure (\figref{illo055}{Fig.~55}), or the ratio of 1 to the tangent
+of the deflection, $\delta$; that is, calling the deflecting force $f$,
+we have $\dfrac{g}{f}=\cot{\delta},=\cot{6}''$ in this case.
+
+By the law of gravitation, the earth's attracting force at its
+surface is given by the formula
+\begin{align*}
+g&=k\frac{E}{R^{2}},\\
+\intertext{where $E$ is the mass of the earth (the unknown quantity of our problem),
+and $R$ its radius, 4000 miles. Similarly, if $C$ in the \figref{illo055}{figure} is the centre
+of attraction of the mountain, we have}
+f&=k\frac{m}{d^{2}},\\
+\intertext{$m$ being the mass of the mountain, and $d$ the distance from $C$ to the station.
+Combining this with the preceding, we get}
+\frac{E}{m} &= \left(\frac{g}{f}\right) \left(\frac{R}{d}\right)^{2},\\
+\intertext{or, in this case,}
+\qquad\qquad %compensate for illustration on right hand side
+\frac{E}{m} &= \cot{6''} \left(\frac{R}{d}\right)^{2}.
+\end{align*}
+\end{fineprint}
+
+We thus get the \textit{ratio of the earth's mass to that of the mountain};
+and provided we can find the mass of the mountain in tons
+or any other known unit of mass, the problem will be completely
+solved. By a careful geological survey of the mountain, with deep
+borings into its strata, the mass of the mountain was determined as
+accurately as it could be (though here is the weakest point of the
+method), and thus the \textit{mass} of the earth was finally computed.
+
+Now, knowing the diameter of the earth, its volume in cubic feet is
+easily found, and from the volume and the known number of mass-pounds,
+($62\frac{1}{2}$ nearly) in a cubic foot of water, the weight the earth
+would have, if composed of water, follows. Comparing this with the
+mass actually found, we get the density, which in this experiment came
+out 4.71.
+
+A repetition of the work in 1832 at Arthur's Seat, near Edinburgh,
+gave 5.32.
+%% -----File: 123.png---Folio 112-------
+
+\includegraphicsouter{illo056}{\textsc{Fig.~56.}---Plan of the Torsion Balance.}
+
+\nbarticle{165.} 2.~Much more trustworthy results, however, are obtained by
+the method of the \textsc{torsion balance}, first devised by Michell, but first
+employed by Cavendish in 1798. A light rod, carrying two small
+balls at its extremities, is suspended horizontally at its centre by a
+long fine metallic wire. If it be allowed
+to come to rest, and then a very
+slight deflecting force be applied, the
+rod will be pulled out of position by
+an amount depending on the stiffness
+and length of the wire, as well as
+the force itself. When the deflecting
+force is removed, the rod will vibrate
+back and forth until brought to rest
+by the resistance of the air. The
+``\textit{torsional coefficient},'' as it is called
+(\textit{i.e.}, the stress corresponding to a torsion
+of one revolution), can be accurately
+determined by observing the time
+of vibration when the dimensions and
+weight of the rod and balls are known.
+If, now, two large balls $A$ and $B$ are
+brought near the smaller ones, as in
+Fig.~56, a deflection will be produced by their attraction, and the
+small balls will move from $a$ and $b$ to $a'$ and $b'$. By shifting the
+large balls to the other side at $A'$ and $B'$, we get an equal deflection
+in the opposite direction, \textit{i.e.}, to $a''$ and $b''$, and the difference between
+the two positions assumed by the small balls, \textit{i.e.}, $a'a''$ and
+$b'b''$, will be twice the deflection.
+
+\begin{fineprint}
+It is not necessary, nor even best, to wait for the balls to come to rest.
+We note the extremities of their swing. The middle point of the swing
+gives the point of rest, and the time occupied by the swing is the time of
+vibration, which we need in determining the coefficient of torsion. We
+must also measure accurately the distance, $Aa'$ and $Bb'$ between the centre
+of each of the large balls and the point of rest of the small ball when
+deflected.
+\end{fineprint}
+
+The \textit{earth's attraction} on each of the small balls of course equals
+\textit{the ball's weight}. The \textit{attractive force of the large ball} on the small one
+near it is found directly from the experiment. If the deflection, for
+instance, is $1°$ and the coefficient of torsion is such that it takes
+\textit{one grain} to twist the wire around one whole revolution, then the
+%% -----File: 124.png---Folio 113-------
+deflecting force, which we will call $f$ as before, will be $\frac{1}{360}$ of a grain.
+Call the mass of the large ball $B$, and let $d$ be the measured distance
+from its centre to that of the deflected ball. We shall then have
+\begin{align*}
+f &= k\frac{B}{d^{2}},\\
+\intertext{also, $w$ being the weight of the small ball,}
+w &= k\frac{E}{R^{2}},\\
+\intertext{whence we get, very much as in the preceding case,}
+\frac{E}{B} &= \frac{w}{f}\left(\frac{R}{d}\right)^{2} \!\! ,
+\end{align*}
+which gives the mass of the earth in terms of $B$.
+
+The method differs from the preceding in that we use a large ball
+of metal instead of a mountain, and measure its deflecting force by a
+laboratory experiment instead of comparing astronomical observations
+with geodetic measurements.
+
+\begin{fineprint}
+\nbarticle{166.} In the earlier experiments by this method the small balls were
+of lead, about two inches in diameter, at the extremities of a light wooden
+rod, five or six feet long, enclosed in a case with glass ends, and their position
+and vibration was observed by a telescope looking directly at them from
+a distance of several feet. The attracting masses, $B$, were balls also of lead,
+about one foot in diameter, mounted on a frame pivoted in such a way
+that they could be easily brought to the required positions.
+
+Great difficulty was caused by air currents in the case, and it was necessary
+to enclose the whole apparatus in a small room of its own which was
+covered with tin-foil on the outside, and to avoid going near the room or
+allowing any radiant heat to strike it for hours before the observations.
+Baily, in England, and Reich, in Germany, between 1838 and 1842, made
+very extensive series of observations of this kind. Baily obtained 5.66
+for the earth's density, and Reich 5.48.
+
+The experiment was repeated in 1872 by Cornu, in Paris, with a modified
+apparatus.
+
+The horizontal bar was in this case only half a metre long, of aluminium,
+with small platinum balls at the end. For the large balls, glass globes were
+used, which could be pumped full of mercury or emptied at pleasure. The
+whole was enclosed in an air-tight case, and the air exhausted by an air-pump.
+The deflections and vibrations were observed by means of a telescope
+watching the image of a scale reflected in a small mirror attached to
+the aluminium beam near its centre, according to the method first devised
+%% -----File: 125.png---Folio 114-------
+by Gauss and now so generally used in galvanometers and similar apparatus.
+Cornu obtained 5.56 as the result, and showed that Baily's figure required a
+correction which, when applied, would reduce it to 5.55.
+\end{fineprint}
+
+\nbarticle{167.} 3.~\textsc{Potsdam Observations}.---During 1886 and 1887 another
+series of observations was made by Wilsing, at Potsdam, with apparatus
+similar in principle to the torsion balance, except that the bar carrying
+the balls to be attracted was \textit{vertical}, and turned on knife edges
+very near its centre of gravity. The knife edges, like those of an
+ordinary balance, rested upon agate planes, and the centre of gravity
+of the apparatus was so adjusted that one vibration of the pendulum,
+under the influence of gravity alone, would occupy from two to four
+minutes. The deflecting weights in this case were large cylinders of
+cast iron, suspended in such a way that they could be brought opposite
+the small balls, first on one side and then on the other. The
+whole was set up in a basement, and carefully and very effectually
+guarded against all changes of temperature, the arrangements being
+such that all manipulations and observations could be effected from
+the outside without entering the room. The deflections and vibrations
+were observed by a reflected scale, as in Cornu's observations.
+The result obtained was 5.59.
+
+Several other methods have been used; of less scientific value,
+however.
+
+\begin{fineprint}
+\nbarticle{168.}  \textit{a}. The mass of the earth can he deduced by ascertaining \textit{the force
+of gravity at the top of a mountain} and \textit{at its base, by means of pendulum experiments}.
+The mass of the mountain must be determined by a survey, just as
+in the Schehallien method, which makes the method unsatisfactory. At the
+top of a mountain the height of which is $h$, and the distance of its centre of
+attraction from the top is $d$, gravity will be made up of two parts, one the
+attraction of the earth at a distance from its centre equal to $R + h$, and the
+other the attraction of the mountain alone considered. Calling the mass of
+the mountain $m$, and gravity at its summit $g'$ ($g$ being the force of gravity at
+the earth's surface), we shall have the proportion
+\[
+g:g' = \frac{E}{R^{2}}:\left[\frac{E}{(R+h)^{2}}+\frac{m}{d^{2}}\right],
+\]
+the second fraction in the last term of the proportion being the attraction
+of the mountain. When $g$ and $g'$ are ascertained by the pendulum experiments,
+$E$ remains as the only unknown quantity, and can be readily found. Observations
+of this kind were made by Carlini, in 1821, at Mt.~Cenis, and
+the result was 4.95.
+%% -----File: 126.png---Folio 115-------
+
+\nbarticle{169.} \textit{b. By means of pendulum observations at the earth's surface compared
+with those at the bottom of a mine of known depth}. This method was employed
+by Airy in 1843, at Harton Colliery, 1200 feet deep; result, 6.56. In this case
+the principle involved is somewhat different. \textit{At any point within a hollow,
+homogeneous, spherical shell, gravity is zero}, as Newton has shown. The
+attraction balances in all directions. If, then, we go down into a mine,
+the effect on gravity is the same as if a shell composed of all that part of the
+earth above our level had been removed. At the same time our distance
+from the earth's centre has been decreased by $d$, the depth of the mine.
+\begin{flalign*}
+&\text{At the surface }&
+  g  &= k\frac{E}{R^{2}},\text{ as before.}  &&
+\\[1ex]
+&\text{At the bottom of the mine }&
+  g' &= k\frac{E - \text{``shell''}}{(R - d)^{2}}.
+&&\phantom{\text{At the bottom of t}}
+\end{flalign*}
+
+Comparing the two equations, we find $E$ in the terms of the shell, since
+the ratio of $g$ to $g'$ is given by pendulum observations. Obviously, however,
+the mass of the ``shell'' is difficult to determine with accuracy. And it is by
+no means homogeneous, so that there is no great reason for surprise at the
+discordant result.  $g'$ was found to be actually greater than $g$, showing that
+although at the centre of the earth the attraction necessarily becomes zero,
+yet as \textit{we descend below the surface, gravity increases for a time} down to some
+unknown but probably not very great depth, where it becomes a maximum.
+
+\nbarticle{170.}  \textit{c}. By \textit{experiments with a common balance}. If a body be hung from
+one of the scale-pans of a balance, its apparent weight will obviously be
+increased when a large body is brought very near it underneath; and this
+increase can be measured. Poynting in England and Jolly in Germany
+have recently used this method, and have obtained results agreeing very
+fairly with these got from the torsion balance. The experiment, with some
+modifications, is soon to be tried again on a very large scale in Germany.
+\end{fineprint}
+
+\nbarticle{171.} \nbparatext{Constitution of the Earth's Interior.}---Since the average density
+of the earth's crust does not exceed three times that of water,
+while the mean density of the whole earth is about 5.58 (taking the
+average of all the most trustworthy results), it is obvious that at the
+centre the density must be very much greater than at the surface,---very
+likely as high as eight or ten times that of water, and equal to
+the density of the heavier metals. There is nothing in this that might
+not have been expected. If the earth were ever fluid, it is natural to
+suppose that in the solidification the densest materials would settle
+towards the interior.
+
+\begin{fineprint}
+Whether the interior of the earth is solid or fluid it is difficult to say with
+certainty. Certain tidal phenomena, to be discussed hereafter, have led Sir
+%% -----File: 127.png---Folio 116-------
+William Thomson and the younger Darwin to conclude that the earth as a
+whole is solid throughout, and ``more rigid than glass,'' volcanic centres
+being mere pustules in the general mass. To this many geologists demur.
+
+As regards the temperature at the earth's centre, it is hardly an astronomical
+question, though it has very important astronomical relations. We can only
+take space to say that the temperature appears to increase from the surface
+downward at the rate of about one degree Fahrenheit for every fifty or sixty
+feet, so that at the depth of a few miles the temperature must be very
+high.
+\end{fineprint}
+\chelabel{CHAPTERV}
+%% -----File: 128.png---Folio 117-------
+
+\Chapter{VI}{The Earth's Orbital Motion}
+\nbchapterhang{THE APPARENT MOTION OF THE SUN AMONG THE STARS,
+AND THE EARTH'S ORBITAL MOTION.---THE EQUATION
+OF TIME, PRECESSION, NUTATION, AND ABERRATION.---VARIOUS
+KINDS OF ``YEAR.''---THE CALENDAR.}
+
+\nbarticle{172.} \nbparatext{The Annual Motion of the Sun.}---The apparent \textit{annual motion
+of the sun} must have been one of the earliest noticed of all astronomical
+phenomena. Its discovery antedates history.
+
+As seen by the people in Europe and Asia, the sun, starting in
+the spring, mounts higher in the sky each day at noon for three
+months, appears to stand still for a few days at the summer solstice,
+and then descends towards the south, reaching in the autumn
+the same noonday elevation it had in the spring. It keeps on its
+southward course to a winter solstice in December, and then returns
+to its original height at the end of the year, marking and causing the
+seasons by its course. A year, the interval between the successive
+returns of the sun to the same position, was very early found to
+consist of a little more than three hundred and sixty days.
+
+Nor is this all. The sun's motion is \textit{not merely a north-and-south
+motion}, but it also moves \textit{eastward among the stars}; for in the
+spring the stars which are rising in the eastern horizon at sunset are
+different from these which are found there in the summer or winter.
+In the spring, the most conspicuous of the eastern constellations at
+sunset are Leo and Boötes; a little later, Virgo appears; in the summer,
+Ophiuchus and Libra; still later, Scorpio; and in mid-winter,
+Orion and Taurus are in the eastern sky.
+
+\nbarticle{173.} So far as mere appearances go, everything would be explained
+by assuming that the earth is at rest and the sun moving
+around it; but equally by the converse supposition,---for if the earth
+as seen from the sun appears at any point in the heavens, the sun as
+seen from the earth must appear in exactly the opposite point, and
+must keep opposite, moving through the same path in the sky (but
+%% -----File: 129.png---Folio 118-------
+six months behind), and always in the same ``angular direction,'' if
+we may use the expression. (Just as two opposite teeth on a gear-wheel
+move in the same \textit{angular direction}, though at any moment
+they are moving in opposite \textit{linear} directions.)
+
+\nbarticle{174.} That it is really the earth which moves, and not the sun, is
+absolutely demonstrated by two phenomena, too minute and delicate
+for pre-telescopic observations, but accessible enough to modern
+methods. We can only mention them here, leaving their fuller discussion
+for the present. One of them is \textit{the aberration of light}, the
+other \textit{the annual parallax of the fixed stars}. These can be explained
+only by the actual motion of the earth.
+
+\nbarticle{175.} \nbparatext{The Ecliptic.}---By observing with a meridian circle daily
+the declination of the sun, and the difference between its right
+ascension and that of some star (Flamsteed used $\alpha$~Aquilæ for the
+purpose), we shall obtain a series of positions of the sun's centre
+which can be plotted on a celestial globe; and we can thus make out
+the path of the sun among the stars, and find the place where it
+cuts the celestial equator, and the angle it makes. This path turns
+out to be a \textit{great circle}, as is shown by its cutting the equator at two
+points just $180°$ apart (the so-called equinoctial points or equinoxes),
+and makes an angle with it of approximately $23\frac{1}{2}°$. This great circle
+is called the \textsc{Ecliptic}, because, as was early discovered, eclipses
+happen only when the moon is crossing it. It may be defined as \textit{the
+trace of the plane of the earth's orbit upon the celestial sphere}, just as
+the celestial equator is the trace of the plane of the terrestrial equator
+on the same sphere.
+
+\nbarticle{176.} \nbparatext{Definitions.}---The angle which the ecliptic makes with the
+equator is called the \textit{Obliquity of the ecliptic}, and the points midway
+between the equinoxes are called the \textit{Solstices} (\textit{sol-stitium}), because at
+these points the sun ``\textit{stands}'' or stops moving in declination for a
+short time.
+
+Two circles parallel to the equator, drawn through the solstices, are
+called the \textit{Tropics} (Greek $\tau\rho\acute{\epsilon}\pi\omega$), or ``\textit{turning-lines},'' because there
+the sun turns from its northward motion to a southward, or \textit{vice versa}.
+The obliquity is, of course, simply equal to the \textit{sun's maximum declination},
+or greatest distance from the equator, which is reached in
+June and December.
+%% -----File: 130.png---Folio 119-------
+
+\begin{fineprint}
+The ancients were accustomed to determine it by means of the gno\-mon\footnote
+  {The Chinese claim to have made an observation of this kind about 4000~\textsc{b.c.},
+and the result given is very nearly what it should have been at that time. (The
+obliquity changes slightly in centuries.) If their observation is genuine, it is
+  by far the oldest of all astronomical records.}
+(\artref{Art.}{107}). The length of the shadow at noon on the solstitial days determines
+the zenith distance of the sun on these days, and the difference of the
+zenith distances at the two solstices is twice the angle desired. The gnomon
+also determined for the ancients the length of the year, it being only necessary
+to observe the interval between days in the spring or autumn, when the
+shadow had the same length at noon.
+\end{fineprint}
+
+\nbarticle{177.} \nbparatext{The Zodiac and its Signs.}---A belt $16°$ wide, $8°$ on each side
+of the ecliptic, is called the \textit{Zodiac}. The name is said to be derived
+from $\zeta\acute{\omega}o\nu$, a living creature, because the constellations in it (except
+Libra) are all figures of animals. It was taken of that particular
+width by the ancients simply because the moon and the then known
+planets never go further than $8°$ from the ecliptic.
+
+\begin{fineprint}
+This belt is divided into the so-called \textsc{signs}, each $30°$ in length, having
+the following names and symbols:---
+\[
+\begin{array}{l@{\;}ll@{\;}l}
+\text{Spring} &\left\{
+ \begin{array}{ll}
+    \text{Aries,}  & \aries  \\
+    \text{Taurus,} & \taurus \\
+    \text{Gemini,} & \gemini \\
+ \end{array}
+\right.
+%
+&\text{Autumn} &\left\{
+ \begin{array}{ll}
+    \text{Libra,}                                    & \libra \\
+    \text{Scorpio,}                                  & \scorpio  \\
+    \rlap{Sagittarius,}\phantom{\text{Capricornus,}} & \sagittarius \\ %phantom text to ensure width matches lower half
+ \end{array}
+\right.
+\\
+\text{Summer} &\left\{
+ \begin{array}{ll}
+    \rlap{Cancer,}\phantom{\text{Gemini,}} & \cancer \\ %phantom text to ensure width matches upper half
+    \text{Leo,}                            & \leo    \\
+    \text{Virgo,}                          & \virgo  \\
+ \end{array}
+\right.
+%
+&\text{Winter} &\left\{
+ \begin{array}{ll}
+    \text{Capricornus,} & \capricornus \\
+    \text{Aquarius},    & \aquarius    \\
+    \text{Pisces},      & \pisces      \\
+ \end{array}
+\right.
+\end{array}
+\]
+
+The symbols are for the most part conventionalized pictures of the objects.
+The symbol for Aquarius is the Egyptian character for water. The
+origin of the signs for Leo, Virgo, and Capricornus is not quite clear. It
+has been suggested that $\leo$ is simply a ``\textit{cursive}'' form for $\Lambda$, the initial of
+$\Lambda\acute{\epsilon}\omega\nu$; $\virgo$ for $\Pi\alpha\rho$ ($\Pi\alpha\rho\Theta\acute{\epsilon}\nu o\varsigma$), and $\capricornus$ for T$\rho$ (T$\rho\acute{\alpha}\gamma o\varsigma$).
+\end{fineprint}
+
+\section*{CELESTIAL LATITUDE AND LONGITUDE.}
+
+\nbarticle{178.} Since the moon and all the principal planets always keep
+within the zodiac, the ecliptic is a very convenient circle of reference,
+and was used us such by the ancients. Indeed, until the invention
+of pendulum clocks, it was on the whole more convenient than the
+equator, and more used.
+
+The two points in the heavens $90°$ distant from the ecliptic are called
+the \textit{Poles of the ecliptic}. The northern one is in the constellation
+%% -----File: 131.png---Folio 120-------
+of Draco, about half-way between the stars $\delta$ and $\zeta$~Draconis. Now,
+suppose a set of great circles drawn, like meridians, through these
+poles of the ecliptic, and hence perpendicular to that circle; these
+are \textit{Circles of latitude} or \textit{secondaries to the ecliptic}. The \textsc{Longitude}
+of a star or any other heavenly body is, then, \textit{the angle made at the
+pole of the ecliptic, between the circle of latitude, which passes through
+the vernal equinox, and the circle of latitude passing through the body};
+or, what comes to the same thing, it is \textit{the arc of the ecliptic included
+between the vernal equinox and the foot of the circle of latitude passing
+through the body}. Celestial longitude is always reckoned \textit{eastward}
+from the vernal equinox, completely around the ecliptic, so that
+the longitude of the sun when $10°$ \textit{west} of the vernal equinox would be
+written as $350°$, and not as $-10°$.
+
+The \textsc{Latitude} of a star is simply \textit{its distance north or south of
+the ecliptic measured on the star's circle of latitude}.
+
+\nbarticle{179.} It will be seen that \textit{longitude differs from right ascension in
+being reckoned on the ecliptic instead of
+on the equator}, nor can it be reckoned
+in \textit{time}, but only in degrees, minutes,
+and seconds. \textit{Latitude differs from declination
+in that it is reckoned from the
+ecliptic instead of from the equator}.
+
+\includegraphicsouter{illo057}{\textsc{Fig.~57.}\\
+Relation between Celestial Latitude and
+Longitude, and Right Ascension and
+Declination.}
+
+The relation between right ascension
+and declination on the one hand, and
+longitude and latitude on the other,
+may be made clearer by the accompanying
+diagram (\figref{illo057}{Fig.~57}), in which
+$EC$ is the ecliptic and $EQ$ the equator,
+$E$ being the vernal equinox. $S$
+being a star, its right ascension ($\alpha$) is
+$ER$ and its declination ($\delta$) is $SR$; its longitude ($\lambda$) is $EL$, and its latitude
+($\beta$) is $SL$. $P$ and $K$ are the poles of the equator and ecliptic
+respectively, and the circle $KPCQ$ is the \textit{Solstitial Colure}, so called.
+
+\begin{fineprint}
+The student can hardly take too great care to avoid confusion of celestial
+latitude and longitude with right ascension and declination or with \textit{terrestrial}
+latitude and longitude. It is, of course, unfortunate that latitude in
+the sky should not be analogous to latitude upon the earth, or celestial longitude
+to terrestrial. The terms right ascension and declination are, however,
+of comparatively recent introduction, and found the ground preoccupied,
+celestial latitude and longitude being much older.
+%% -----File: 132.png---Folio 121-------
+
+\nbarticle{180.} \nbparatext{Conversion of $\lambda$ and $\beta$ into $\alpha$ and $\delta$, or Vice Versa.}---Right
+ascension and declination can, of course, always be converted into longitude
+and latitude by a trigonometrical calculation. We proceed as follows: In
+the triangle $ERS$, right-angled at $R$, we have given $ER$ and $RS$  ($\alpha$ and $\delta$),
+from which we find the hypothenuse $ES$ and the angle $RES$. Next in the
+triangle $ELS$, right-angled at $L$, we have the hypothenuse $ES$ and the angle
+$LES$, which is equal to $RES-LEQ$ ($LEQ$ being {$\omega$}, the obliquity of the
+ecliptic). Hence we easily find $EL$ and $LS$.
+\end{fineprint}
+
+\nbarticle{181.} \nbparatext{The Earth's Orbit in Space.}---The \textit{ecliptic is not the earth's
+orbit}, and must not be confounded with it. It is a \textit{great circle} of the
+infinite celestial sphere, the \textit{trace} made upon the sphere by the plane
+of the earth's orbit, as was stated in its definition. The fact that
+it is a great circle gives us no information about the earth's orbit,
+except that \textit{the orbit all lies in one plane passing through the sun}. It
+tells us nothing as to its real form and size.
+
+By reducing the observations of the sun's right ascension and
+declination through the year to longitude and latitude (the latitude
+will always be zero, of course, except for some slight perturbations)
+and combining them with observations of the sun's apparent diameter,
+we can, however, ascertain the real form of the earth's orbit and the
+law of its motion in this orbit. But the \textit{size} of the orbit---the scale
+of miles---cannot be fixed until we can find the sun's distance.
+
+\includegraphicsouter{illo058}{\textsc{Fig.~58.}\\
+Determination of the Form of the Earth's Orbit.}
+
+\nbarticle{182.} \nbparatext{To find the Form of the Orbit}, we may proceed thus: Take
+a point $S$ for the sun and draw
+from it a line $SO$, \figref{illo058}{Fig.~58},
+directed towards the vernal
+equinox as the origin of longitudes.
+Lay off from $S$ indefinite
+lines, making angles with
+$SO$ equal to the earth's longitude
+on each of the days observed
+through the year; \textit{i.e.},
+the angle $OS$\:10, is the longitude
+at the time of the 10th
+observation; and so on. We
+shall thus get a sort of ``spider,'' showing the \textit{directions}
+as seen from the earth on these days.
+
+Next, as to \textit{relative distances}. While the apparent diameter of the
+sun does not tell us its real distance from the earth, unless we first
+%% -----File: 133.png---Folio 122-------
+know the sun's real diameter in miles, the changes in the apparent
+diameter do inform us as to the \textit{relative} distance of the earth at
+different times, since the nearer we are, the larger the sun appears,---the
+distance being inversely proportional to the apparent diameter
+(\artref{Art.}{6}). If, then, we lay off on the arms of our ``spider'' distances
+inversely proportional to the number of seconds of arc in the
+sun's measured diameter at each date, these distances will be \textit{proportional}
+to the true distance of the earth from the sun, and the curve
+joining the points thus obtained will be a true map of the earth's
+orbit, though without any scale of
+miles upon it.
+
+\includegraphicsouter{illo059}{\textsc{Fig.~59.}---The Ellipse.}
+
+When the operation is performed,
+we find that the orbit is an ellipse
+of small eccentricity (about one-sixtieth),
+with the sun, not in the
+centre, but at one focus.
+
+\begin{fineprint}
+\nbarticle{183.} For the benefit of any who
+may not have studied conic sections
+we define the ellipse. It is a curve such
+that \textit{the sum of the two distances from any point on its circumference to
+two points within, called the foci, is always constant}, and equal to what is
+called the major-axis of the ellipse, $SP + PF = AA'$, in \figref{illo059}{Fig.~59}. $AC$ is
+called the semi-major-axis, and is usually denoted by $A$ or $a$. $BC$ is the
+semi-minor-axis, denoted by $B$ or $b$. The eccentricity, denoted by $e$, is the
+fraction $\dfrac{SC}{AC}$.
+\begin{flalign*}
+&\text{Since $BS$ is equal to $A$,}&&
+  SC = \sqrt{A^{2} - B^{2}};  &&\phantom{\text{Since $BS$ is equal to $A$,}}
+\\[1ex]
+&\text{and }&&
+  e = \frac{\sqrt{A^{2} - B^{2}}}{A}.  &&
+\end{flalign*}
+\end{fineprint}
+
+The points where the earth is nearest to and most remote from the
+sun are called respectively \textit{perihelion} and \textit{aphelion} and the line
+that joins them is, of course, the major axis of the orbit. This line, considered
+as indefinitely produced in both directions, is called the \textit{line
+of apsides},---the major-axis being a limited piece or ``sect'' of the
+line of apsides.
+
+\begin{fineprint}
+\nbarticle{184.} The variations of the sun's diameter are too small to be detected
+without a telescope (amounting to only about three per cent), so that
+the ancients were unable to perceive them. Hipparchus, however, about
+%% -----File: 134.png---Folio 123-------
+150~\textsc{b.c.}, discovered that the earth is not in the centre of the circular orbit
+which he supposed the sun to describe around it. Everybody assumed, on
+\textit{à priori} grounds, never disputed until the time of Kepler, that the sun's orbit
+must be a circle and described with a uniform motion, because a circle is
+the only ``perfect'' curve, and uniform motion the only perfect motion.
+Obviously, however, the sun's \textit{apparent} motion is not uniform, because it
+takes 186 days for the sun to pass from the vernal equinox to the autumnal
+through the summer months, and only 179 days to return during the winter.
+Hipparchus explained this difference by the hypothesis that the earth
+is out of the centre of the sun's path.
+\end{fineprint}
+
+\nbarticle{185.} \nbparatext{To find the Eccentricity of the Orbit.}---Having the greatest
+and least apparent diameters of the sun, the eccentricity, $e$, is easily
+found. In \figref{illo059}{Fig.~59}, since, by definition, $e = CS÷ CA$, we have $CS =
+CA× e$, or $Ae$. The perihelion distance $AS$ is therefore equal to
+$A× (1-e)$, and the aphelion distance $SA'$ to $A (1+e)$. Suppose
+now that the greatest and least measured diameters of the sun are $p$
+and $q$. This gives us the proportion $p:q = A(1+e):A(1-e)$,
+since the diameters are \textit{inversely} proportional to the distances. From
+this we get
+\[
+e = \frac{p-q}{p+q}.
+\]
+The actual values of $p$ and $q$ are $32'\,36''.4$ and $31'\,31''.8$, which give
+$e = 0.01678:$ this is about $\frac{1}{60}$, as has been stated.
+
+\includegraphicsouter{illo060}{\textsc{Fig.~60.}\\
+Equable Description of Areas.}
+
+\nbarticle{186.} \nbparatext{To find the Law of the Earth's Motion.}---By comparing the
+measured apparent diameter with the differences of longitude from day
+to day, we can also deduce the \textit{law} of the earth's motion. On making
+a table of daily motions and apparent diameters, we find that these
+\textit{daily motions} vary \textit{directly as the squares of the diameters}; from which
+it directly follows that the earth moves
+in such a way that its \textit{radius-vector
+describes areas proportional to the times}
+(a law which Kepler first brought to
+light in 1609). The radius-vector is
+the line which joins the earth to the
+sun at any moment.
+
+\begin{fineprint}
+\nbarticle{187.}  Consider a small elliptical sector,
+$dSc$ (\figref{illo060}{Fig.~60}), described by the earth
+in a unit of time. Regarding it as a triangle,
+its area is given by the formula $\frac{1}{2}SC× SD\sin{cSd}$; and calling this
+angle $\theta$ (which will be very small), and considering that in so short a time
+%% -----File: 135.png---Folio 124-------
+$Sd$ and $Sc$ would remain sensibly equal, each being equal to $R$ (the radius-vector
+at the middle point of the arc), this formula becomes,
+\begin{flalign*}
+&&  &\text{Area of sector } = \tfrac{1}{2}R^{2}\theta.  &&\phantom{whence}\\
+\rlap{\text{\indent Now, calling the sun's apparent diameter $D$, we have:}}\\
+&&  &R = \frac{k}{D},  &&\\
+\rlap{\text{($k$ being a constant, and depending on the sun's diameter in miles);}}\\
+&\text{whence }&  &R^{2} = \frac{k^{2}}{D^{2}}.  &&\\
+\rlap{\text{But our measurements show that $\theta = k_{1}D^{2}$, $k_{1}$ being another constant.}}\\
+\rlap{\text{Substitute these values of $R^{2}$ and $\theta$ in the formula for the area, and we have}}\\
+&&  &\text{Area of sector } = \tfrac{1}{2}\frac{k^{2}}{D^{2}}× k_{1}D^{2} = \tfrac{1}{2}k^{2}k_{1},
+\end{flalign*}
+a constant; that is, the area described by the radius-vector in a unit of time
+is always the same. The planet near perihelion moves so much faster, that
+the areas $aSb, cSd$, and $eSf$ are all equal to each other, if the arcs are described
+in the same time.
+\end{fineprint}
+
+\includegraphicsouter{illo061}{\textsc{Fig.~61.}---Kepler's Problem}
+
+\nbarticle{188.} \nbparatext{\stretchyspace Kepler's Problem.}---{\stretchyspace As the case} stands so far, this is a mere
+fact of observation; but as we shall see hereafter, and as was demonstrated
+by Newton, the fact shows
+that the earth moves under the action
+of a force \textit{always directed in line
+with the sun}. In such a case the
+``equable description of areas'' is a
+necessary mechanical consequence.
+It is true in every case of elliptical
+motion, and enables us to find the
+position of the earth or any planet
+in its orbit at any time, when we
+once know the time of its orbital revolution (technically the period),
+and the time when it was at perihelion. Thus, the angle $ASP$ (\figref{illo061}{Fig.~61}),
+which is called the \textit{Anomaly} of the planet, must be such that the area of
+the elliptical sector $ASP$ will be that portion of the whole ellipse which
+is represented by the fraction $\dfrac{t}{T}$, $t$ being the number of days since the
+planet last passed the perihelion, and $T$ the number of days in the
+whole period. For instance, if the earth last passed perihelion on
+Dec.~31 (which it did), its place on May~1 must be such that the
+%% -----File: 136.png---Folio 125-------
+sector $ASP$ will be $\frac{121}{365\frac{1}{4}}$ of the whole of the earth's orbit; since from
+Dec.~31 to May~1 is 121 days. The solution of this problem, known
+as ``\textit{Kepler's problem},'' leads to transcendental equations, and lies
+beyond our scope.
+
+\begin{fineprint}
+See Watson's ``Theoretical Astronomy,'' pp.~53 and 54, or any other similar
+work.
+\end{fineprint}
+
+\nbarticle{189.} \nbparatext{Anomaly and Equation of the Centre.}---The angle $ASP$,
+which has been termed simply the ``\textit{Anomaly},'' is strictly the \textit{true}
+Anomaly, as distinguished from the \textit{mean} Anomaly. The \textit{former}
+may be defined as \textit{the angle actually made at any time by the radius-vector
+of a planet with the line of apsides}, the angle being reckoned
+from the perihelion point completely around in the direction of the
+planet's motion. The \textit{mean} Anomaly is what the Anomaly \textit{would be
+at the given moment if the planet had moved with uniform} angular
+\textit{velocity, completing the orbit in the same period, and passing perihelion
+at the same time}, as it actually does. The difference between the
+two anomalies is called the \textit{Equation of the Centre}. This is zero at
+perihelion and aphelion, and a maximum midway between them. In
+the case of the sun, its greatest value is nearly $2°$, the sun getting
+alternately that amount ahead of, and behind, the position it would
+occupy if its apparent daily motion were uniform.
+
+\nbarticle{190.} \nbparatext{The Seasons.}---The earth in its motion around the sun
+always keeps its axis parallel to itself, for the mechanical reason that
+a revolving body necessarily maintains the direction of its axis invariable,
+unless disturbed by extraneous force, as is very prettily illustrated
+by the gyroscope. About March 20 the earth is so situated
+that the plane of its equator passes through the sun, the sun's declination
+being zero on that day.
+
+At that time, the line which separates the illuminated portions of
+the earth passes through the two poles, and day and night are everywhere
+equal. The same is again true of the 22d of September, when
+the sun is at the autumnal equinox on the opposite side of the orbit.
+
+About the 21st of June the earth is so situated that its \textit{north} pole
+is inclined towards the sun by about $23\frac{1}{2}°$, which is the sun's northern
+declination on that date. The south pole is then in the obscure half
+of the earth's globe, while the north pole receives sunlight all day
+long; and in all portions of the northern hemisphere the day is longer
+than the night, the difference between the day and night depending
+upon the latitude of the place, while in the southern hemisphere the
+%% -----File: 137.png---Folio 126-------
+days are shorter than the nights. At the time of the winter solstice
+these conditions are reversed. At the equator (of the earth) the day
+and night are equal at all times of the year. The sun when in northern
+declination of course always rises at a point on the horizon \textit{north
+of east}, and sets at a point north of west, so that for a portion of
+the time each day it shines on the north side of a house.
+
+\begin{fineprint}
+\nbarticle{191.} \nbparatext{Diurnal Phenomena near the Pole.}---At the north pole, where
+the celestial pole is in the zenith, and the diurnal circles are parallel with
+the horizon, the sun will maintain the same elevation all day long, except
+for the slight change caused by the variation of its declination in twenty-four
+hours. The sun will appear on the horizon at the date of the vernal
+equinox (in fact, about three days before, on account of refraction), and
+slowly wind upward in the sky until it reaches its maximum elevation of
+$23\frac{1}{2}°$ on June 21. Then it will retrace its course until a day or two after
+the autumnal equinox, when it sinks out of sight.
+
+At points between the north pole and the polar circle the sun will appear
+above the horizon earlier in the year than March 20, and will rise and set
+daily until its declination becomes equal to the observer's distance from the
+pole, when it will make a complete circuit of the heavens, touching the horizon
+at midnight at the northern point; and after that never setting again
+until it reaches the same declination in its southward course after passing
+the solstice. From that time it will again rise and set daily until it reaches
+a southern declination just equal to the observer's polar distance, when the
+long night begins; to continue until the sun, having passed the southern
+solstice, returns again to the same declination at which it made its appearance
+in the spring. At the polar circle itself (or, more strictly speaking,
+owing to refraction, about one-half a degree south of it) the ``\textit{midnight sun}''
+will be seen on just one day in the year, the day of the summer solstice;
+and there will also be one absolutely sunless day, viz., the day of the winter
+solstice. The same remarks apply in the southern hemisphere, by making
+the obvious changes.
+\end{fineprint}
+
+\nbarticle{192.} \nbparatext{Effects on Temperature.}---The changes in the duration of
+``\textit{insolation}'' (exposure to sunshine) at any place involve changes
+of temperature and other climatic conditions, thus producing the seasons.
+Taking as a standard the amount of heat received in twenty-four
+hours on the day of the equinox, it is clear that the surface of the
+soil at any place in the northern hemisphere will receive more than
+this average amount of heat whenever the sun is north of the celestial
+equator, for two reasons.
+
+\includegraphicsouter{illo062}{\textsc{Fig.~62.}\\
+Effect of Sun's Elevation on Amount
+of Heat Imparted to the Soil.}
+
+1. Sunshine lasts more than half the day.
+
+2. The \textit{mean elevation} of the sun during the day is greater than
+%% -----File: 138.png---Folio 127-------
+when it is at the equinoxes, since it is higher at noon, and in any
+case reaches the horizon at rising and setting. Now, the more obliquely
+the rays strike, the less heat they bring to each square inch of surface,
+as is obvious from \figref{illo062}{Fig.~62}. A beam of sunshine having the cross-section
+$ABCD$, when it strikes the
+surface at an angle $h$ (equal to the
+sun's altitude) is spread over a much
+larger surface, $Ac$, and of course
+the amount of heat per square inch
+is proportionately reduced. If $Q$
+is the amount of heat per square
+inch brought by the ray when falling
+perpendicularly, as on the surface
+$AC$, then on $Ac$ the amount
+per square inch will be $Q×\sin{h}$,
+since $AB = Ab×\sin{h}$. This difference in favor of the more nearly
+vertical rays is exaggerated by the absorption of heat in the atmosphere,
+because rays that are nearly horizontal have to traverse a
+much greater thickness of air before reaching the ground.
+
+For these two reasons, at a place in the northern hemisphere, the
+temperature rises rapidly as the sun comes north of the equator, thus
+giving us our summer.
+
+\nbarticle{193.} \nbparatext{Time of Highest Temperature.}---We, of course, receive the
+most heat \textit{per diem} at the time of the summer solstice; but this is
+not the hottest time of the summer, for the obvious reason that the
+weather is then all the time \textit{getting hotter}, and the maximum will not
+be reached \textit{until the increase ceases}; that is, not until the \textit{amount of
+heat lost in the night} equals that \textit{stored up by day}.
+
+\begin{fineprint}
+If the earth's surface threw off the same amount of heat hourly whether
+it were hot or cold, then this maximum would not come until \textit{the autumnal
+equinox}. This, however, is not the case. The soil loses heat faster when
+warm than it does when cold, the loss being nearly proportional to the difference
+between the temperature of the soil and that of surrounding space;
+(Newton's law of cooling); and so the time of the maximum is made to
+come not far from the end of July, or the first of August, in our latitude.
+For similar reasons the minimum temperature of winter occurs about Feb.~1,
+about half-way between the solstice and the vernal equinox. Since, however,
+our weather is not entirely ``made on the spot where it is used,'' but
+is affected by winds and currents that come from great distances, the actual
+time of the maximum temperature cannot be determined by any mere astronomical
+considerations, but varies considerably from year to year.
+\end{fineprint}
+%% -----File: 139.png---Folio 128-------
+
+\nbarticle{194.} \nbparatext{\stretchyspace Difference between Seasons in Northern and Southern Hemispheres.}---Since
+in December the distance of the earth from the sun
+is about three per cent less than it is in June, the earth (as a whole)
+receives hourly about six per cent more heat in December than in
+June, the heat varying inversely as the \textit{square} of the distance. For
+this reason the southern summer, which occurs in December and
+January, is \textit{hotter} than the northern. It is, however, seven days
+\textit{shorter}, because the earth moves more rapidly in that part of its
+orbit. The total amount of heat per acre, therefore, received during
+the summer is sensibly the same in each hemisphere, the shortness
+of the southern summer making up for its increased warmth.
+
+\begin{fineprint}
+\nbarticle{195.} The southern \textit{winter}, however, is both longer and colder than the
+northern; and it is maintained by certain geologists, Mr.~Croll especially,
+that, on the whole, the mean annual temperature of the hemisphere which
+has its winter at the time when the earth is in aphelion is lower than that
+of the opposite one. It has been attempted to account for the glacial epochs
+in this way. It is certain that at present, at any place in the southern hemisphere,
+the difference between the maximum temperature of summer and
+the minimum of winter must be greater than in the case of a station in the
+northern hemisphere, similarly situated as to elevation, etc. We say ``at
+present'' because, on account of certain slow changes in the earth's orbit, to
+be spoken of immediately, the state of things will be reversed in about ten
+thousand years, the northern summer being then the hotter and shorter one.
+\end{fineprint}
+
+\nbarticle{196.} \nbparatext{Secular Changes in the Orbit of the Earth.}---The orbit of
+the earth is not absolutely unchangeable in \textit{form} or \textit{position}, though
+it is so in the long run as regards the \textit{length of its major axis} and the
+\textit{duration of the year}.
+
+\sloppy
+\nbarticle{197.} 1.~\textit{Change in Obliquity of the Ecliptic}.---The ecliptic slightly
+and very slowly shifts its position among the stars, thus altering
+the latitudes of the stars and the angle between the ecliptic and
+equator, \textit{i.e.}, the obliquity of the ecliptic.  This obliquity is at
+present about $24'$ less than it was 2000 years ago, and is still
+\textit{decreasing} about half a second a year. It is computed that this
+diminution will continue for about 15,000 years, reducing the obliquity
+to $22\frac{1}{4}°$, when it will begin to increase. The whole change,
+according to J.~Herschel, can never exceed about $1°\, 20'$ on each side
+of the mean.
+
+\fussy
+\nbarticle{198.} 2.~\textit{Change of Eccentricity.}---At present the eccentricity of
+the earth's orbit (which is now 0.0168) is also slowly diminishing.
+%% -----File: 140.png---Folio 129-------
+According to Leverrier, it will continue to decrease for about 24,000
+years, until it becomes 0.003, and the orbit will be almost circular.
+Then it will increase again for 40,000 years, until it becomes
+0.02.
+
+\begin{fineprint}
+In this way the eccentricity will oscillate backwards and forwards, always,
+however, remaining between zero and 0.07; but the oscillations are not
+equal either in amount or time, and so cannot properly be compared to the
+``vibrations of a mighty pendulum,'' which is rather a favorite figure of
+speech.
+\end{fineprint}
+
+\nbarticle{199.} 3.~\textit{Revolution of the Apsides of the Earth's Orbit.}---The line
+of apsides of the orbit, which now stretches in both directions towards
+the constellations of Sagittarius and Gemini, is also slowly and
+steadily moving eastward, and at a rate which will carry it around
+the circle in about 108,000 years.
+
+\nbarticle{200.} These so-called ``\textit{secular}'' changes are due to the action of
+the other planets upon the earth. Were it not for their attraction, the
+earth would keep her orbit with reference to the sun strictly unaltered
+from age to age, except that possibly in the course of millions of
+years the effects of falling meteoric matter and the attraction of the
+nearer fixed stars might make themselves felt.
+
+\begin{fineprint}
+Besides these secular perturbations of the earth's orbit, the earth itself is
+continually being slightly disturbed in its orbit. On account of its connection
+with the moon, it oscillates each month a few hundred miles above and
+below the true plane of the ecliptic, and by the action of the other planets it is
+sometimes set forwards or backwards to the extent of a few thousand miles.
+Of course every such change produces a corresponding slight change in the
+apparent position of the sun.
+\end{fineprint}
+
+\nbarticle{201.} \nbparatext{Equation of Time.}---We have stated a few pages back
+(\artref{Art.}{111}), that the interval between the successive passages of the
+sun across the meridian is somewhat variable, and that for this
+reason apparent solar, or sun-dial, days are unequal. On this account
+mean time has been adopted, which is kept by a ``\textit{fictitious}''
+or ``\textit{mean}'' sun moving uniformly in the equator at the same
+average rate as that of the real sun in the ecliptic. The hour-angle
+of this mean sun is, as has been already explained, \textit{the
+local mean time} (or clock time); while the hour-angle of the
+real sun is the \textit{apparent} or \textit{sun-dial time}. The \textit{Equation of Time}
+is the difference between these two times, reckoned as \textit{plus} when
+%% -----File: 141.png---Folio 130-------
+the sun-dial is \textit{slower} than the clock, and \textit{minus} when it is faster. It
+is the \textit{correction which must be added} (algebraically) \textit{to apparent time
+in order to get mean time}. As it is the difference between the two
+hour-angles, it may also be defined as \textit{the difference between the
+right ascensions of the mean sun and the true sun}; or as a formula
+we may write: $E = \alpha_{t} - \alpha_{m}$, in which $\alpha_{m}$ is the right ascension of the
+mean sun, and $\alpha_{t}$, of the true sun.
+
+The principal causes of this difference are two:---
+
+\includegraphicsouter{illo063}{\textsc{Fig.~63.}
+Effect of Obliquity of Ecliptic in producing
+Equation of Time.}
+
+\nbarticle{202.} 1.~\textit{The Variable Motion of the Sun in the Ecliptic, due to the
+Eccentricity of the Earth's Orbit.}---Near perihelion, which occurs
+about Dec.~31, the sun's motion in longitude is most rapid. Accordingly,
+at this time the apparent solar days exceed the sidereal
+by more than the average amount, making the sun-dial days longer
+than the mean. (The average solar day, it will be remembered, is $3^{\text{m}}\, 56^{\text{s}}$
+longer than the sidereal.) The sun-dial will therefore lose time at
+this season, and will continue to do so for about three months, until
+the angular motion of the sun falls to its mean value. Then it will
+gain until aphelion, when, if the clock and the sun were started
+together at perihelion, they will once more be together. During
+the next half of the year the action will be reversed. Thus, twice
+a year, so far as the eccentricity of the earth's orbit is concerned,
+the clock and sun would be together at perihelion and aphelion,
+while half-way between they would differ by about \textit{eight minutes};
+the equation of time (so far as due to this cause only) being $+8$
+minutes in the spring, and $-8$ minutes in the autumn.
+
+\nbarticle{203.} 2.~\textit{The Inclination of the
+Ecliptic to the Equator.}---Even if
+the \textit{sun's} (\textit{apparent}) \textit{motion in longitude}
+(\textit{i.e.}, along the ecliptic) were
+uniform, its motion in \textit{right ascension}
+would be variable. If the true
+and fictitious suns started together
+at the equinox, they would indeed
+be together at the solstices and at
+the other equinox, because it is just
+$180°$ from equinox to equinox, and
+the solstices are exactly half-way
+between them. But at \textit{intermediate points}, between the equinoxes
+and solstices, they would not be together on the same hour-circle.
+%% -----File: 142.png---Folio 131-------
+This is best seen by taking a celestial globe and marking \textit{on the ecliptic}
+a point, $m$, half-way between the vernal equinox and the solstice,
+and also marking a point $n$ \textit{on the equator}, $45°$ from the equinox. It
+will at once be seen that the former point, $m$ in \figref{illo063}{Fig.~63},\footnote
+  {Fig.~63 represents a celestial globe viewed from the \textit{west} side, the axis being
+vertical, and $K$, the pole of the ecliptic, on the meridian, while $E$ is the vernal
+  equinox.}
+is \textit{west} of $n$,
+so that $m$ in the daily westward motion of the sky will come to the
+meridian first; in other words, when the sun is half-way between
+the vernal equinox and the summer solstice, the \textit{sun-dial} is \textit{faster}
+than the clock, and the equation of time is \textit{minus}. The difference,
+measured by the arc $m'n$, amounts to nearly ten minutes; and of
+course the same thing holds, \textit{mutatis mutandis}, for the other quadrants.
+
+\includegraphicsmid{illo064}{\textsc{Fig.~64.}---The Equation of Time.}
+
+\nbarticle{204.} \nbparatext{Combination of the Effects of the Two Causes.}---We can represent
+graphically these two components of the equation of time and
+the result of their combination as follows (\figref{illo064}{Fig.~64}):---
+
+The central horizontal line is a scale of dates one year long, the
+letters denoting the beginning of each month. The dotted curve
+shows the equation of time due to the \textit{eccentricity} of the earth's orbit,
+above considered. Starting at perihelion on Dec.~31, this component
+is then zero, rising from there to a value of about $+8^{\text{m}}$ on
+April~2, falling to zero on June~30, and reaching a second maximum
+of $-8^{\text{m}}$ on Oct.~1. In the same way the broken-line curve denotes
+the effect of the \textit{obliquity of the ecliptic}, which, by itself alone
+considered, would produce an equation of time having \textit{four} maxima
+of, approximately, $10^{\text{m}}$ each, about the 6th of February, May,
+%% -----File: 143.png---Folio 132-------
+August, and November (alternately $+$ and $-$), and reducing to zero
+at the equinoxes and solstices.
+
+The full-lined curve represents their combined effect, and is constructed
+by making its ordinate at each point equal to the sum
+(algebraic) of the ordinates of the two other curves. At the 1st of
+February, for instance, the distance $F$,~3, in the \figref{illo064}{figure} $=F,\,1 + F,\,2$.
+So, also, $M,\, 6 = M,\, 4 + M,\, 5$; the components, however, in this case
+have opposite signs, so that the \textit{difference} is actually taken.
+
+The equation of time is zero \textit{four} times a year, viz.: on April~15,
+June~14, Sept.~1, and Dec.~24. The maxima are February 11, $+14^{\text{m}}\,32^{\text{s}}$;
+May 14, $-3^{\text{m}}\,55^{\text{s}}$; July 26, $+6^{\text{m}}\,12^{\text{s}}$, and Nov.~2, $-16^{\text{m}}\,18^{\text{s}}$.
+But the dates and amounts vary slightly from year to year.
+
+\begin{fineprint}
+The two causes above discussed are only the \textit{principal} ones effective in
+producing the equation of time. Every perturbation suffered by the earth
+comes in with its own effect; but all other causes combined never alter the
+equation by more than a few seconds.
+\end{fineprint}
+
+\nbarticle{205.} \nbparatext{Precession of the Equinoxes.}---The length of year was
+found in two ways by the ancients:---
+
+1.~By the gnomon, which gives the time of the equinox and solstice;
+and
+
+2.~By observing the position of the sun with reference to the
+stars,---their rising and setting at sunrise or sunset.
+
+Comparing the results of observations made by these two methods
+at long intervals, Hipparchus (120~\textsc{b.c.}) found that the two do not
+agree; the former year (from equinox to equinox) being $20^{\text{m}}\,23^{\text{s}}$
+shorter than the other (according to modern data). The equinox
+is plainly moving \textit{westward} on the ecliptic, as if it \textit{advanced to
+meet the sun} on each annual return. He therefore called the motion
+the ``\textit{precession}'' of the equinoxes.
+
+On comparing the \textit{latitudes} of the stars in the time of the ancient
+astronomers with the present latitudes, we find that they have
+changed very slightly indeed; and we know therefore that the ecliptic
+and the plane of the earth's orbit maintains its position sensibly
+unaltered. On the other hand, the \textit{longitudes} of the stars have been
+found to increase regularly at the rate of about $50''.2$ annually,---fully
+$30°$ in the last 2000 years. Since longitudes are reckoned
+from the equinox (the intersection between the ecliptic and equator),
+and since the ecliptic does not move, it is evident that the
+motion must be in the \textit{celestial equator}; and accordingly we find that
+both the \textit{right ascension and the declination} of the stars are constantly
+changing.
+%% -----File: 144.png---Folio 133-------
+
+\nbarticle{206.} \nbparatext{Motion of the Pole of the Equator around the Pole of the
+Ecliptic.}---The obliquity of the ecliptic, which equals the distance in
+the sky between the pole of the equator and the pole of the ecliptic
+(\artref{Art.}{178}), has remained nearly constant. Hence the pole of the
+equator must be describing a circle around the pole of the ecliptic in a
+period of about 25,800 years ($360°$ divided by $50''.2$). The pole of
+the ecliptic has remained practically fixed among the stars, but the
+other pole has changed its position materially. At present the pole
+star is about $1\frac{1}{4}°$ from the pole. At the time of the star catalogue of
+Hipparchus it was $12°$ distant from it, and during the next century
+it will approach to within about $30'$, after which it will recede.
+
+\begin{fineprint}
+
+\nbarticle{207.} If upon a celestial globe we take the pole of the ecliptic as a centre,
+and describe about it a circle with a radius of $23\frac{1}{2}°$, we shall get the
+track of the celestial pole among the stars, and shall find that the circle
+passes very near the star $\alpha$~Lyræ, on the opposite side of the pole of the
+ecliptic from the present pole star. About 12,000 years hence $\alpha$~Lyræ
+will be the pole star. Reckoning backwards, we find that some 3000
+years ago $\gamma$~Draconis was the pole star; and it is a curious circumstance
+that certain of the tunnels in the pyramids of Egypt face exactly to the
+north, and slope at such an inclination that this star at its lower culmination
+would have been visible from their lower end at the date when the pyramids
+are supposed to have been built. It is probable that these passages were
+arranged to be used for the purpose of observing the transits of their then
+pole star.
+
+\nbarticle{208.} \nbparatext{Effect of Precession upon the Signs of the Zodiac.}---Another
+effect of precession is that the \textit{signs} of the zodiac do not now agree with the
+\textit{constellations} which bear the same name. The sign of Aries is now in the
+constellation of Pisces; and so on, each sign having ``backed,'' so to speak,
+into the constellation west of it.
+\end{fineprint}
+
+\includegraphicsouter{illo065}{\textsc{Fig.~65.}\\
+Effect of Attraction
+on a Spheroid.}
+
+\nbarticle{209.} \nbparatext{Physical Cause of Precession.}---The \textit{physical cause} of this
+slow conical rotation of the earth's axis around the pole of the ecliptic
+lies in the two facts that \textit{the earth is not exactly spherical}, and that
+\textit{the attractions of the sun and moon\footnote
+  {The \textit{planets}, by their action upon the plane of the earth's orbit (\artref{Art.}{197}),
+slightly disturb the equinox \textit{in the opposite direction}. This effect amounts to about
+  $0''.16$ annually.}
+act upon the equatorial ring of matter
+which projects above the true sphere, tending to draw the plane of the
+equator into coincidence with the plane of the ecliptic} by their greater
+attraction on the nearer portions of the ring. The action is just what
+it would be if a spheroidal ball of iron of the shape of the earth had
+%% -----File: 145.png---Folio 134-------
+a magnet brought near it. The magnet, as illustrated in \figref{illo065}{Fig.~65},
+would tend to draw the plane of the equator into the line $CM$ joining its
+pole with the centre of the globe, because it attracts the nearer portion
+of the equatorial protuberance at $E$ more strongly than the remoter at
+$Q$. If it were not for the earth's rotation, this attraction would bring the
+two planes of the ecliptic and equator together; but since the earth is
+spinning on its axis, we get the same result that we do with the whirling
+wheel of a gyroscope by hanging a weight at one end of the axis.
+We then have the result of the combination of two
+rotations at right angles with each other, one the
+whirl of the wheel, the other the ``tip'' which the
+weight tends to give the axis. (See Brackett's Physics,
+pp.~53--56.)
+
+\nbarticle{210.} In this case, if the wheel of the gyroscope is turning swiftly
+\textit{clock-wise}, as seen from above (\figref{illo066}{Fig.~66}), the weight at the (lower)
+end of the axis will make the axis move slowly around, \textit{counter-clockwise},
+without at all changing its inclination. If we regard the horizontal
+plane passing through the gyroscope as representing the ecliptic,
+and the point in the ceiling vertically above the gyroscope as the
+pole of the ecliptic, the line of the axis of the wheel produced upward
+would describe on the ceiling a circle around this imaginary
+ecliptic pole, acting precisely as does the pole of the earth's axis in
+the sky. The swifter the wheel's rotation, the slower would be this
+``precessional'' motion of its axis; and of course, the rate of motion
+also depends upon the magnitude of the suspended weight.
+
+\includegraphicsmid{illo066}{\textsc{Fig.~66.}\\
+Precession Illustrated by the Gyroscope.}
+
+\begin{fineprint}
+\nbarticle{211.} A full treatment of the subject would be too complicated for our
+pages. An elementary notion of the way the action takes place, correct as
+%% -----File: 146.png---Folio 135-------
+far as it goes, is easily obtained by reference to \figref{illo067}{Fig.~67}. Let $XY$ be the axis of
+the gyroscope, the wheel being seen in section edge-wise, and the eye being
+on the same level as the centre of the wheel; the wheel turning so that the
+point $B$ is coming towards the observer. Now, suppose a weight hung on
+the lower end of the axis. If the wheel
+were not turning, the point $B$ would
+come to some point $F$ in the same time
+it now takes to reach $C$ (that is, after a
+quarter of a revolution). By combination
+of the two motions it will come to a
+point $K$ at the end of the same time,
+having crossed the horizontal plane $AD$
+at $L$; and this can be effected only by
+a backward ``skewing around'' of the
+whole wheel, axis and all. This does not,
+of course, explain why the inclination of
+the axis does not change under the action
+of the weight, but is only a very partial illustration, showing merely why
+the plane of the wheel regresses. A complete discussion would require the
+consideration of the motion of every point on the wheel by a thorough and
+difficult analytical treatment, in order to give the complete explanation of
+the reason why the depressing weight, however heavy, does not cause the end
+of the axis to fall perceptibly. (See article, ``Gyroscope,'' in Johnson's
+Cyclopædia.)
+\end{fineprint}
+
+\nbarticle{212.} \nbparatext{Why Precession is so Slow.}---The slowness of the precession
+depends on three things: (\textit{a})~the enormous ``moment of rotation''
+of the earth---a point on the equator moves with the speed of a
+cannon ball; (\textit{b})~the smallness of the mass (compared with that of
+the whole earth) of the protuberant ring to which precession is due;
+and (\textit{c})~the minuteness of the force which tends to bring this ring into
+coincidence with the ecliptic, a force which is not constant and persistent,
+like the weight hung on the gyroscope axis, but very variable.
+
+\begin{fineprint}
+\nbarticle{213.} \nbparatext{The Equation of the Equinox.}---Whenever the sun is in the
+plane of the equator (which is twice a year, at the time of the equinoxes), the
+\textit{sun's} precessional force disappears entirely, its attraction then having no
+tendency to draw the equator out of its position. The moon's action, on account
+of her proximity, is still more powerful than that of the sun; on the
+average two and a half times as great. Now, the moon crosses the celestial
+equator twice every month, and at these times \textit{her} action ceases.
+
+\includegraphicsouter{illo067}{\textsc{Fig.~67.}\\
+Regression of the Gyroscope Wheel.}
+
+There is still another cause for variation in the effectiveness of the \textit{moon's}
+attraction. As we shall see hereafter, she does not move in the ecliptic, but
+in a path which cuts the ecliptic at an angle of about $5°$, at two points called
+the \textit{Nodes}; the \textit{ascending} node being the point where she crosses the ecliptic
+%% -----File: 147.png---Folio 136-------
+from south to north. These nodes move westward on the ecliptic (\artref{Art.}{455}),
+making the circuit once in about nineteen years. Now, when the ascending
+node of the moon's orbit is at $B$ (\figref{illo068}{Fig.~68}), near the autumnal equinox $F$, its
+inclination to the equator will
+be, as the \figref{illo068}{figure} shows, \textit{less}
+than the obliquity of the ecliptic
+by about $5°$; \textit{i.e.}, it will
+be only about $18°$. On the
+other hand, nine and a half
+years later, when the node has
+backed around to a point $A$, near the vernal equinox, the inclination of the
+moon's orbit to the equator will be nearly $28°$. When the node is in this
+position, the moon will produce nearly twice as much precessional movement
+each month as when the node was at $B$.
+
+\includegraphicsouter{illo068}{\textsc{Fig.~68.}\\
+Variation in the Inclination of Moon's Orbit to Equator.}
+
+The precession, therefore, is not uniform, but variable, almost ceasing at
+some times and at others becoming rapid. The average amount, as has
+been stated, is $50''.2$ a year; and the variation is taken account of in what
+is called the \textit{equation of the equinox}, which is the difference between the
+actual position of the equinox at any time and the position it would have at
+that moment if the precession had been all the time going on \textit{uniformly}.
+\end{fineprint}
+
+\nbarticle{214.} \nbparatext{Nutation.}---Not only does the precessional force vary in
+amount at different times, but in most positions of the disturbing
+body with respect to the earth's equator there is a \textit{slight thwartwise
+component of the force, tending directly to accelerate or retard} the precessional
+movement of the pole---just as if one should gently draw
+the weight $W$ (\figref{illo063}{Fig.~63}) horizontally.  The consequence is what
+is called \textit{Nutation} or ``nodding.'' The axis of the earth, instead of
+moving smoothly in a circle, nods in and out a little with respect to
+the pole of the ecliptic, describing a wavy curve resembling that
+shown in \figref{illo069}{Fig.~69}, but with nearly 1400 indentations in the entire circumference
+traversed in 26,000 years.
+
+\begin{fineprint}
+
+\includegraphicsouter{illo069}{\textsc{Fig.~69.}---Nutation.}
+
+\nbarticle{215.}  We distinguish three of these nutations, (\textit{a})~The \textit{Lunar Nutation},
+depending upon the motion of the moon's nodes.
+This has a period of a little less than nineteen
+years, and amounts to $9''.2$, (\textit{b})~The \textit{Solar Nutation},
+due to the changing declination of the sun.
+Its period is a year, and its amount $1''.2$. (\textit{c})~The
+\textit{Monthly Nutation}, precisely like the solar nutation,
+except that it is due to the \textit{moon's} changes of declination
+during the month. It is, however, too
+small to be certainly measured, not exceeding one-tenth
+of a second.
+%% -----File: 148.png---Folio 137-------
+
+Nutation was detected by Bradley in 1728, but not fully explained until
+1748.
+
+\textit{Neither} precession \textit{nor} nutation \textit{affects the} latitudes \textit{of the stars, since they
+are not due to any change in the position of the ecliptic, but only to displacements
+of the earth's axis}. \textit{The} longitudes \textit{alone are changed by them}.
+
+The \textit{right ascension} and \textit{declination} of a star are both affected.
+\end{fineprint}
+
+\nbarticle{216.} \nbparatext{The Three Kinds of Year.}---In consequence of the motion
+of the equinoxes caused by precession, the \textit{sidereal} year and the
+equinoctial or ``\textit{tropical}'' year do not agree in length. Although the
+sidereal year is the one which represents the earth's \textit{true} orbital revolution
+around the sun, it is not used as the year of chronology and
+the calendar, because the \textit{seasons} depend on the sun's place in relation
+to the equinoxes. The tropical year is the year usually employed,
+unless it is expressly stated to the contrary. The length of the
+Sidereal year is $365^{\text{d}}\ 6^{\text{h}}\ 9^{\text{m}}\ 9^{\text{s}}$; that of the Tropical year is about
+$20^{\text{m}}$ less, $365^{\text{d}}\ 5^{\text{h}}\ 48^{\text{m}}\ 46^{\text{s}}$.
+
+The third kind of year is the \textit{anomalistic} year, which is the time
+from perihelion to perihelion again. As the line of apsides of the
+earth's orbit moves always slowly towards the east, this year is a little
+longer than the sidereal. Its length is $365^{\text{d}}\ 6^{\text{h}}\ 13^{\text{m}}\ 48^{\text{s}}$.
+
+\nbarticle{217.} \nbparatext{The Calendar.}---The natural units of time are \textit{the day, the
+month}, and \textit{the year}. The day, however, is too short for convenient
+use in designating extended periods of time, as for instance in
+expressing the age of a man. The month meets with the same
+objection, and for all chronological purposes, therefore, the year is
+the unit practically employed. In ancient times, however, so much
+regard was paid to the month, and so many of the religious beliefs
+and observances connected themselves with the times of the new and
+full moon, that the early history of the calendar is largely made up
+of attempts to fit the month to the year in some convenient way.
+Since the two are incommensurable, the problem is a very difficult,
+and indeed strictly speaking, an impossible, one.
+
+In the earliest times matters seem to have been wholly in the hands
+of the priesthood, and the calendar then was predominantly \textit{lunar},
+with months and days intercalated from time to time to keep the
+seasons in place. The Mohammedans still use a purely lunar calendar,
+having a year of twelve lunar months, and containing alternately
+354 and 355 days. In their reckoning the seasons fall, of course,
+continually in different months, and their calendar gains about one
+year in thirty-three upon the reckoning of Christian nations.
+%% -----File: 149.png---Folio 138-------
+
+\nbarticle{218.} \nbparatext{The Metonic Cycle.}---Among the Greeks the discovery of
+the so-called lunar or Metonic cycle by Meton, about 433~\textsc{b.c.}, considerably
+simplified matters. This cycle consists of 235 \textit{synodic
+months} (from new moon to new again), which is very approximately
+equal to 19 common years of $365\frac{1}{4}$ days.
+
+\begin{fineprint}
+\sloppy
+235 months equal $6939^{\text{d}}\ 16^{\text{h}}\ 31^{\text{m}}$; 19 tropical years equal $6939^{\text{d}}\ 14^{\text{h}}\ 27^{\text{m}}$;
+so that at the end of the 19 years, the new and full moon recur again on the
+same days of the year, and at the same time of day within about two
+hours. The calendar of the phases of the moon, for instance, for 1889 is the
+same as for 1870 and 1908 (except that intervening leap-years may change
+the dates by one day).
+
+\fussy
+The ``\textit{Golden number}'' of a year is its number in this Metonic cycle, and
+is found by adding 1 to the ``date-number'' of the year and dividing by 19.
+The remainder, unless zero, is the ``golden number'' (if it comes out zero, 19
+is taken instead). Thus the golden number for 1888 is found by dividing
+1889 by 19, and the remainder 8, is the golden number of the year.
+
+This cycle is still employed in the ecclesiastical calendar in finding the
+time of Easter.
+
+{\footnotesize
+For further information on the subject, consult Johnson's Encyclopædia, or Sir
+Edmund Becket's ``Astronomy without Mathematics.''
+}
+\end{fineprint}
+
+\nbarticle{219.} \nbparatext{Julian Calendar.}---Until the time of Julius Cæsar the Roman
+calendar seems to have been based upon the lunar year of twelve
+months, or 355~days, and was substantially like the modern Mohammedan
+calendar, with arbitrary intercalations of months and days made by
+the priesthood and magistrates from time to time in order to bring it
+into accordance with the seasons. In the later days of the Republic,
+the confusion had become intolerable. Cæsar, with the help of the astronomer
+Sosigenes, whom he called from Alexandria for the purpose,
+reformed the system in the year 45~\textsc{b.c.}, introducing the so-called
+``Julian calendar,'' which is still used either in its original shape or
+with a very slight modification. He gave up entirely the attempt to coordinate
+the month with the year, and adopting $365\frac{1}{4}$ days as the true
+length of the tropical year, he ordained that every fourth year should
+contain an extra day, the \textit{sixth day before the Kalends of March on that
+year being counted twice}, whence the year was called ``\textit{bissextile}''
+Before his time the year had begun in March (as indicated by the
+Roman names of the months,---September, seventh month; October,
+eighth month, etc.), but he ordered it to begin on the 1st of January,
+which in that year (45~\textsc{b.c.}) was on the day of the new moon next following
+the winter solstice. In introducing the change it was necessary
+to make the preceding year 445~days long, and it is still known in
+%% -----File: 150.png---Folio 139-------
+the annals as ``the year of confusion.'' He also altered the name of
+the month Quintilis, calling it ``July'' after himself.
+
+\begin{fineprint}
+There was some irregularity in the bissextile years for a few years after
+Cæsar's death, from a misunderstanding of his rule for the intercalary day;
+but his successor Augustus remedied that, and to put himself on the same
+level with his predecessor, he took possession of the month Sextilis, calling
+it ``August''; and to make its length as great as that of July, he robbed
+February of a day.
+
+From that time on, the Julian calendar continued unbrokenly in use until
+1582; and it is still the calendar of Russia and of the Greek Church.
+\end{fineprint}
+
+\nbarticle{220.} \nbparatext{The Gregorian Calendar.}---The Julian calendar is not quite
+correct. The true length of the tropical year is 365~days 5~hours
+48~minutes and 46~seconds, and this leaves a difference of 11~minutes
+and 14~seconds by which the Julian calendar year is the longer, being
+exactly $365\frac{1}{4}$~days. As a consequence, the date of the equinox comes
+gradually earlier and earlier by about three days in 400~years.
+($400× 11\frac{1}{6}^{\text{m}} = 4467\ \text{minutes} = 3^{\text{d}}\ 2^{\text{h}}\ 27^{\text{m}}$.) In the year 1582, the
+date of the vernal equinox had fallen back 10~days to the 11th of
+March, instead of occurring on the 21st of March, as at the time of
+the Council at Nice, 325~\textsc{a.d.}
+Pope Gregory, therefore, acting under
+the advice of the Jesuit astronomer, Clavius, ordered that the day
+following Oct.~4 in the year 1582 should be called not the 5th, but
+the 15th, and that the rule for leap-year should be slightly changed
+so as to prevent any such future displacement of the equinox. The
+rule now stands: \textit{All years whose date-number is divisible by four
+without a remainder are leap-years, unless they are century years} (1700,
+1800, etc.). \textit{The century years are not leap-years unless their date-number
+is divisible by} 400, \textit{in which case they are}: that is, 1700, 1800,
+and 1900 are not leap years; but 1600, 2000, and 2400 are.
+
+\nbarticle{221.} \nbparatext{Adoption of the New Calendar.}---The change was immediately
+adopted by all Catholic nations; but the Greek Church and
+most of the Protestant nations, rejecting the Pope's authority, declined
+to accept the correction. In England it was at last adopted
+in the year 1752, at which time there was a difference of eleven days
+between the two calendars. (The year 1600 was a leap-year according
+to the Gregorian system as well as the Julian, but 1700 was not.)
+Parliament in 1751 enacted that the day following the 2d of September,
+in the year 1752, should be called the 14th instead of the 3d;
+and also that this year (1752), and all subsequent years, should begin
+on the first of January.
+%% -----File: 151.png---Folio 140-------
+
+\begin{fineprint}
+The change was made under very great opposition, and there were violent
+riots in consequence in different parts of the country, especially at Bristol,
+where several persons were killed. The cry of the populace was, ``Give us
+back our fortnight,'' for they supposed they had been robbed of eleven days,
+although the act of Parliament was carefully framed to prevent any injustice
+in the collection of interest, payment of rents, etc.
+
+At present, since the year 1800 was not a leap-year according to the
+Gregorian calendar, while it was so according to the Julian, the difference
+between the two calendars amounts to twelve days; thus in Russia the 19th
+of August would be reckoned as the 7th. In Russia, however, for scientific
+and commercial purposes \textit{both} dates are very generally used, so that the date
+mentioned would be written Aug.~$\frac{7}{19}$. When Alaska was annexed to the
+United States, its calendar had to be altered by \textit{eleven} days. (See \artref{Art.}{123}.)
+
+\nbarticle{222.} \nbparatext{The Beginning of the Year.}---The beginning of the year has
+been at several different dates in the different countries of Europe. Some
+have regarded it as beginning at Christmas, the 25th of December; others,
+on the 1st of January; others still, on the 1st of March; others, on the
+25th; and others still, at Easter, which may fall on any day between the
+22d of March and the 25th of April.
+
+In England previous to the year 1752 the legal year commenced on the
+25th of March, so that when the change was made, the year 1751 necessarily
+lost its months of January and February, and the first twenty-four days of
+March. Many were slow to adopt this change, and it becomes necessary,
+therefore, to use considerable care with respect to English dates which occur
+in the months of January, February, or March about that period. The
+month of February, 1755, for instance, would by some writers be reckoned
+as occurring in 1754. Confusion is best avoided by writing, Feb.~$\frac{1754}{1755}$.
+
+\nbarticle{223.} \nbparatext{First and Last Days of the Year.}---Since the ordinary civil
+year consists of 365 days, which is 52 weeks and one day, the last day of
+each common year falls on the same day as the first; so that any given date
+will fall one day later in the week than it did on the preceding year, unless
+a 29th of February has intervened, in which case it will be \textit{two} days later;
+that is, if the 3d of January, 1889, falls on Thursday, the same date in 1890
+will fall on Friday.
+\end{fineprint}
+
+\nbarticle{224.} \nbparatext{Aberration.}---{\footnotesize Although
+in strictness the discussion of aberration does
+not belong to a chapter describing the earth and its motions, yet since it is a phenomenon
+due to the earth's motion, and affects the right ascension and declination of
+the stars in much the same ways as do precession and nutation, it may properly enough
+be considered here.}
+
+\textit{Aberration is the apparent displacement of a star, due to the combination
+of the motion of light with the motion of the observer.}
+
+The direction in which we have to point our telescope in observing
+a star is not the same that it would be if the earth were at rest. It
+%% -----File: 152.png---Folio 141-------
+lies beyond our scope to show that according to the wave theory of
+light the apparent direction of a ray will be affected by the observer's
+motion precisely in the same way (within very narrow limits) as it
+would be if light consisted of corpuscles shot off from a luminous
+body, as Newton supposed. This
+is the case, however, as Doppler
+and others have shown; and assuming
+it, the explanation of
+aberration is easy:---
+
+\includegraphicsouter{illo070}{\textsc{Fig.~70.}---Aberration of a Raindrop.}
+
+Suppose an observer standing
+at rest with a tube in his hand in
+a shower of rain where the drops
+are falling vertically. If he wishes
+to have the drops descend axially
+through the tube without touching
+the sides, he must of course keep
+it vertical; but if he advances in
+any direction, he must draw back the bottom of the tube by an
+amount which equals the advance he makes in the time while the
+drop is falling through the tube, so that when the drop falling from
+$B$ reaches $A'$, the bottom of the tube will be there also; \textit{i.e., he must
+incline the tube forward by an angle $\alpha$, such that} $\tan{\alpha} = u÷ V$,
+where $V$ is the velocity of the raindrop and $u$ that of his own motion.
+In \figref{illo070}{Fig.~70} $BA'= V$ and $AA'=u$.
+
+\includegraphicsouter{illo071}{\textsc{Fig.~71.}---Aberration of Light.}
+
+\nbarticle{225.} Now take the more general case.
+Suppose a star sending us light with a
+velocity $V$ in the direction $SP$, \figref{illo071}{Fig.~71},
+which makes the angle $\theta$ with the line of
+the observer's motion. He himself is
+carried by the earth's orbital velocity in
+the direction $QP$. In pointing the telescope
+so that the light may pass exactly
+along its optical axis, he will have to
+draw back the eye-end by an amount
+$QP$ which just equals the distance he is
+carried, by the earth's motion during the
+time that the light moves from $O$ to $P$. The star will thus apparently
+be displaced towards the point towards which he is moving,
+the angle of displacement $POQ$, or $\alpha$, being determined by the relative
+length and direction of the two sides $OP$ and $QP$ of the triangle
+%% -----File: 153.png---Folio 142-------
+$OPQ$. These sides are respectively proportional to the velocity of
+light, $V$, and the orbital velocity of the earth, $u$.
+
+\begin{fineprint}
+The angle at $P$ being $\theta$, the angle $OQP$ will be ($\theta - \alpha$), and we shall
+have from trigonometry the proportion $\sin{\alpha}:\sin{(\theta - \alpha)} = u : V$.
+
+To find $\alpha$ from this, develop the second term of the proportion and divide
+the first two terms by $\sin{\alpha}$, which gives us
+\begin{flalign*}
+&&&1 : \sin{\theta}\cot{\alpha} - \cos{\theta} = u : V,&&\\
+&\text{whence } &&u\sin{\theta}\cot{\alpha} = V + u\cos{\theta},&&\\
+&\text{and } &&\cot{\alpha} = \frac{V + u\cos{\theta}}{u\sin{\theta}}&&\phantom{whence }
+\intertext{\indent Taking the reciprocal of this we have }
+&&  &\tan{\alpha} = \frac{u}{V + u\cos{\theta}}\sin{\theta}. &&
+\intertext{The second term in the denominator is insensible, since $u$ is only about one
+ten-thousandth of $V$, so that we may neglect it.\footnotemark\
+This gives the formula in
+the shape in which it ordinarily appears, viz.,}
+&& &\tan{\alpha} = \frac{u}{V}\sin{\theta}.  &&
+\end{flalign*}
+  \footnotetext{The velocity of light, according to the latest determinations of Newcomb and
+  Michelson, is 299860 kilometers${}\pm 30$ kilometers (which equals 186,330 miles${}\pm 20$
+  miles). The mean velocity of the earth in its orbit, if we assume the solar
+  parallax to be $8''.8$, is 29.77 kilometers, or 18.50 miles; this makes the constant
+  of aberration $20''.478$, a little smaller than that given in the text.}
+\end{fineprint}
+\vspace*{-2ex} % end{flalign*} followed be end{fineprint} gives rather large space
+
+The value of $\alpha$ (denoted by $\alpha_0$) which obtains when $\theta = 90°$ and
+$\sin{\theta} =$ unity, is called the \textit{Constant of Aberration}.
+
+\includegraphicsouter{illo072}{\textsc{Fig.~72.}\\
+Aberrational Orbit of a Star.}
+
+The latest, and probably the most accurate, determination of this
+constant (derived from the Pulkowa Observations
+by Nyrén in 1882) is $20''.492$.
+Aberration was discovered and explained
+by Bradley, the English Astronomer Royal,
+in 1726.
+
+\nbarticle{226.} \nbparatext{The Effect of Aberration upon the
+Apparent Places of the Stars.}---As the earth
+moves in an orbit nearly circular, and with
+a velocity so nearly uniform that we may
+for our present purpose disregard its variations,
+it is clear that a star at the pole of
+the ecliptic will be always displaced by the
+same amount of $20''.5$, but in a direction
+%% -----File: 154.png---Folio 143-------
+continually changing. It must, therefore, appear to describe a
+little circle $41''$ in diameter during the year, as shown in \figref{illo072}{Fig.~72}.
+Now the direction of the earth's orbital motion is always in the plane
+of the ecliptic, and towards the right hand as we stand facing the
+sun. At the vernal equinox, therefore, we are moving toward the
+point of the ecliptic, which is $90°$ \textit{west of the sun, i.e.}, towards
+the winter solstitial point, and the star is then displaced in that
+direction. Three months later the star will be displaced in a line
+directed towards the vernal equinox, and so on. The earth, therefore,
+so to speak, \textit{drives the star before it} in the aberrational orbit,
+keeping it just a quarter of a revolution ahead of itself.
+
+A star on the ecliptic simply appears to oscillate back and forth in
+a straight line $41''$ long.
+
+Generally, in any latitude whatever, the aberrational orbit is an
+ellipse, having its major axis parallel to the ecliptic, and always
+$41''$ long, while its minor axis is $41'' × \sin{\beta}$, $\beta$ being the star's latitude,
+or distance from the ecliptic.
+
+\begin{fineprint}
+\nbarticle{226*.}
+\nbparatext{Diurnal Aberration.}---The motion of an observer due to the
+earth's \textit{rotation} also produces a slight effect known as the \textit{diurnal aberration}.
+Its ``constant'' is only $0''.31$ for an observer situated at the equator; anywhere
+else it is $0''.31\cos{\phi}$, $\phi$ being the latitude of the observer.
+
+For any given star it is a maximum when the star is crossing the meridian,
+and then its whole effect is slightly to \textit{increase the right ascension} by an
+amount given by the formula
+\[
+\Delta\alpha = 0''.31\cos{\phi}\sec{\delta},
+\]
+$\delta$ being the star's declination.
+
+\smallskip
+See Chauvenet, ``Practical Astronomy,'' I. p.~638.
+
+\end{fineprint}
+\chelabel{CHAPTERVI}
+%% -----File: 155.png---Folio 144-------
+
+\Chapter{VII}{The Moon}
+\nbchapterhang{\stretchyspace THE MOON: HER ORBITAL MOTION AND VARIOUS KINDS OF
+MONTH.---DISTANCE AND DIMENSIONS, MASS, DENSITY AND
+GRAVITY.---ROTATION AND LIBRATIONS.---PHASES.---LIGHT
+AND HEAT.---PHYSICAL CONDITION AND INFLUENCES EXERTED
+ON THE EARTH.---TELE\-SCOPIC ASPECT.---SURFACE
+AND POSSIBLE CHANGES UPON IT.}
+
+\nbarticle{227.}  \textsc{We} pass next to a consideration of our nearest neighbor in
+the celestial spaces, the moon, which is a satellite of the earth and
+accompanies us in our annual motion around the sun. She is much
+smaller than the earth, and compared with most of the other heavenly
+bodies, a very insignificant affair; but her proximity makes her
+far more important to us than any of them except the sun. The
+very beginnings of Astronomy seem to have originated in the study
+of her motions and in the different phenomena which she causes,
+such as the eclipses and tides; and in the development of modern
+theoretical astronomy the lunar theory with the problems it raises
+has been perhaps the most fertile field of invention and discovery.
+
+\nbarticle{228.} \nbparatext{Apparent Motion of the Moon.}---Even superficial observation
+shows that the moon moves eastward among the stars every
+night, completing her revolution from \textit{star to star} again in about $27\frac{1}{4}$ days.
+In other words, she revolves around the earth in that time; or,
+more strictly speaking, they both revolve about their common centre
+of gravity. But the moon is so much smaller than the earth that
+this centre of gravity is situated within the ball of the earth on the
+line joining the centres of the two bodies at a point about 1100
+miles below its surface.
+
+As the moon moves eastward so much faster than the sun, which
+takes a year to complete its circuit, she every now and then, at the
+time of the new moon, overtakes and passes the sun; and as the
+phases of the moon depend upon her position with reference to the
+sun, this interval from new moon to new moon is what we ordinarily
+understand as the month.
+%% -----File: 156.png---Folio 145-------
+
+\sloppy
+\nbarticle{229.} \nbparatext{Sidereal and Synodic Revolutions.}---\textit{The} \textsc{Sidereal} \textit{revolution
+of the moon is the time occupied in passing from a star
+to the same star again}, as the name implies. It is equal to
+$27^\text{d}\ 7^\text{h}\ 43^\text{m}\ 11^\text{s}.545 \pm 0^\text{s}.01$, or $27^\text{d}.32166$. The moon's mean daily
+motion among the stars equals $360°$ divided by this, which is
+$13°\ 11'$ (nearly).
+
+\fussy
+\textit{The} \textsc{Synodic} \textit{revolution is the interval from new moon to new
+moon again}, or from full to full. It varies somewhat on account
+of the eccentricity of the moon's orbit and of that of the earth
+around the sun, but its \textit{mean} value is $29^\text{d}\ 12^\text{h}\ 44^\text{m}\ 2^\text{s}.684\pm 0^\text{s}.01$,
+or $29^\text{d}.53059$; and \textit{this is the ordinary month}. (The word synodic
+is derived from the Greek \mytextgreek{s'un} and \mytextgreek{<od'os}, and has nothing to
+do with the \textit{nodes} of the moon's orbit. The word is \textit{syn-odic}, not
+\textit{sy-nodic}).
+
+A synodical revolution is longer than the sidereal, because during
+each sidereal month of 27.3 days the sun has advanced among the
+stars, and must be caught up with.
+
+\nbarticle{230.} \nbparatext{Elongation, Syzygy, etc.}---The angular distance of the moon
+from the sun is called its \textit{Elongation}. At new moon it is zero, and
+the moon is then said to be in ``\textit{Conjunction}.'' At full moon it is
+$180°$, and the moon is then in ``\textit{Opposition}.'' In either case the
+moon is said to be in ``\textit{Syzygy}'' (\mytextgreek{s'un zug'on}). When the elongation
+is $90°$, as at the half-moon, the moon is in ``\textit{Quadrature}.''
+
+\nbarticle{231.} \nbparatext{Determination of the Moon's Sidereal Period.}---This is
+effected directly by observations of the moon's right ascension and
+declination (with the meridian circle), kept up systematically for a
+sufficient time.
+
+If it were not for the so-called ``secular acceleration'' of the
+moon's motion (Arts.\ \arnref{459}--\arnref{461}), an exceedingly accurate determination
+of the moon's \textit{synodic} period could be obtained by comparing
+ancient eclipses with modern.
+
+The earliest authentically recorded eclipse is one that was observed
+at Nineveh in the year 763~\textsc{b.c.}\ between 9 and 10~o'clock on the
+morning of June 15th.
+
+By comparing this eclipse with (say) the eclipse of August, 1887,
+we have an interval of more than 35,000 months, and so an error of
+ten hours even, in the observed time of the Nineveh eclipse, would
+make only about one second in the length of the month. But the
+month is a little shorter now than it was 2000 years ago.
+%% -----File: 157.png---Folio 146-------
+
+\sloppy
+\nbarticle{232.} \nbparatext{Relation of Sidereal and Synodic Periods.}---The fraction of
+a revolution described by the moon in one day equals $\frac{1}{M}$, $M$ being the
+length of the \textit{sidereal} month. In the same way $\frac{1}{E}$ represents
+the earth's daily motion in its orbit, $E$ being the length of the year.
+The difference of these two equals the fraction of a revolution which
+the moon \textit{gains} on the sun during one day. In a synodic month, $S$, it
+gains one whole revolution, and therefore must gain each day $\frac{1}{S}$ of a
+revolution; so that we have the equation
+\[
+\frac{1}{M} - \frac{1}{E} = \frac{1}{S};
+\]
+or, substituting the numerical values of $E$ and $S$,
+\[
+\frac{1}{M} - \frac{1}{365.25635} = \frac{1}{29.53059},
+\]
+whence we derive the value of $M$.
+
+\fussy
+\begin{fineprint}
+Another way of looking at it is this: In a year there must be \textit{exactly one
+more} sidereal revolution than there are synodic revolutions, because the sun
+completes one entire circuit in that time. Now the number of synodic revolutions
+in a year is given by the fraction
+\[
+  \cfrac{365\frac{1}{4}}{S} = 12.369\!+\!.
+\]
+There will therefore be 13.369 sidereal revolutions in the year, and the
+length of one sidereal revolution equals $365\frac{1}{4}$ days divided by this number
+13.369, which will be found to give the length of the sidereal revolution as
+before.
+\end{fineprint}
+
+\nbarticle{233.} \nbparatext{Moon's Path among the Stars.}---By observing with the meridian
+circle the right ascension and declination of the moon daily
+during the month, just as in the case of the sun, we obtain the position
+of the moon for each day, and joining the points thus found, we
+can draw the path of the moon in the sky. It is found to be a great
+circle inclined at a mean angle of $5°\, 8'$ to the ecliptic, which it cuts
+in two points called the \textit{nodes} (from \textit{nodus}, a ``knot'').
+
+We say the path is found to be a great circle. This must be taken
+with some reservation, since at the end of the month the moon never
+returns \textit{precisely} to the position it occupied at the beginning, owing
+%% -----File: 158.png---Folio 147-------
+to the regression of the nodes and other so-called ``perturbations,''
+which will be discussed hereafter.
+
+\begin{fineprint}
+\nbarticle{234.} \nbparatext{Moon's Meridian Altitude.}---Since the moon's orbit is inclined
+to the ecliptic $5°\, 8'$, its inclination to the equator varies from $28°\, 36'$ ($23°\, 28'
++ 5°\, 8'$), when the moon's ascending node is the vernal equinox, to $18°\, 20'$,
+when, $9\frac{1}{2}$ years later, the same node is at the autumnal equinox. In the first
+case the moon's declination will change during the month by $57°\, 12'$,
+from $- 28°\, 36'$ to $+ 28°\, 36'$. In the other case it will change only by $36°\, 40'$,
+so that at different times the difference in the behavior of the moon in
+this respect is very striking.
+\end{fineprint}
+
+\nbarticle{235.} \nbparatext{Interval between Moon's Transits.}---On the average the moon
+gains $12°\, 11'.4$ on the sun daily, so that it comes to the meridian about
+51 minutes of solar time later each day.
+
+To find the mean interval between the successive transits of the
+moon we may use the proportion
+\[
+(360° - 12°\, 11'.4) : 360° = 24^\text{h} : x;\quad \text{whence } x = 24^\text{h}\, 50^\text{m}.6.
+\]
+The variations of the moon's motion in right ascension, which are
+very considerable (much greater than in the case of the sun), cause
+this interval to vary from $24^\text{h}\ 38^\text{m}$ to $25^\text{h}\ 06^\text{m}$.
+
+\nbarticle{236.} \nbparatext{The Daily Retardation of the Moon's Rising and Setting.}---The
+\textit{average} daily retardation of the moon's rising and setting is,
+of course, the same as that of her passage across the meridian,
+viz., $51^\text{m}$; but the actual retardation of rising is subject to very
+much greater variations than these of the meridian passage, being
+affected by the moon's changes in declination as well as by
+the inequalities of her motion in right ascension. When the moon
+is very far north, having her maximum declination of $28°\, 36'$, she
+will rise in our latitudes much earlier than when she is farther
+south.
+
+In the latitude of New York the least possible daily retardation
+of moon-rise is 23
+minutes, and the greatest is 1~hour and 17~minutes.
+In higher latitudes the variation is greater yet.
+
+\begin{fineprint}
+\nbarticle{237.} \nbparatext{Harvest and Hunter's Moons.}---The variations in the retardation
+of the moon's rising attract most attention when they occur at the time
+of the full moon. When the retardation is at its minimum, the moon rises
+soon after sunset at nearly the same time for several successive evenings;
+whereas, when the retardation is greatest, the moon appears to plunge nearly
+%% -----File: 159.png---Folio 148-------
+vertically below the horizon by her daily motion. When the full moon
+occurs at the time of the autumnal equinox, the moon itself will be near the
+first of Aries.
+
+Now, as will be seen by reference to \figref{illo073}{Fig.~73}, the portion of the ecliptic
+near the first of Aries makes a much smaller angle with the eastern horizon
+than the equator.
+
+[The line $HN$ is the horizon, $E$ being the east point---the \figref{illo073}{figure} being
+drawn to represent a celestial globe, as if the observer were looking at the
+eastern side of the celestial sphere \textit{from the outside}.]
+
+$EQ$ is the equator. Now, when the autumnal equinoctial point or first of
+Libra is on the horizon at $E$, the position of the ecliptic will be that represented
+by $ED$; more steeply inclined to the horizon than $EQ$ is, by the
+angle $QED$, $23\frac{1}{2}°$. But when the first of Aries is at $E$, the ecliptic will be in
+the position $JJ'$. And if the ascending node of the moon's orbit happens
+then to be near the first of Aries, the moon's path will be $MM'$.
+
+\includegraphicsmid{illo073}{\textsc{Fig.~73.}---Explanation of the Harvest Moon.}
+
+Accordingly, when the moon is in Aries, it, so to speak, coasts along the
+eastern horizon from night to night, its time of rising not varying very
+much; and this, when it occurs near the full of the moon, gives rise to the
+phenomenon known as the harvest moon, the \textit{harvest moon being the full moon
+nearest to the autumnal equinox}. The full moon next following is called the
+\textit{hunter's} moon.
+
+In Norway and Sweden, under these circumstances, the moon's orbit may
+actually coincide with the horizon, so that she will rise at absolutely the same
+time for a considerable number of successive evenings.
+\end{fineprint}
+
+\nbarticle{238.} \nbparatext{The Moon's Orbit.}---As in the case of the sun, the observation
+of the moon's path in the sky gives no information as to the real
+size of its orbit; but its \textit{form} may be found by measuring the apparent
+diameter of the moon, which ranges from $33'\ 30''$ to $29'\ 21''$ at
+different points. The orbit turns out to be an ellipse like the orbit
+of the earth, but with an eccentricity more than three times as great---about
+%% -----File: 160.png---Folio 149-------
+$\frac{1}{18}$ on the average, but varying from $\frac{1}{14}$ to $\frac{1}{22}$ on account of
+perturbations.
+
+The extremities of the major axis of the moon's orbit are called
+the \textit{perigee} and \textit{apogee} (from \mytextgreek{per\`i g\newtie h} and \mytextgreek{>ap\`o g\newtie h}).
+
+The line of apsides, which passes through these two points, moves
+around towards the east once in about nine years, also on account
+of perturbations.
+
+\nbarticle{239.} \nbparatext{Parallax and Distance of the Moon.}---These can be found in
+several ways, of which the simplest is the following: At two observatories
+$B$ and $C$ (\figref{illo074}{Fig.~74}) on, or
+very nearly on, the same meridian
+and very far apart (in the
+northern and southern hemispheres
+if possible; Greenwich
+and the Cape of Good Hope, for
+instance) let the moon's zenith
+distance $ZBM$ and $Z'CM$ be observed
+simultaneously with the
+meridian circle. This gives in
+the quadrilateral $BOCM$ the two
+angles at $B$ and $C$, each of
+which is the supplement of the geocentric zenith distance. The
+angle at the centre of the earth, $BOC$, is the difference of the geocentric
+latitudes and is known from the geographical positions of
+the two observatories. Knowing the three angles in the quadrilateral,
+the fourth at $M$ is of course known, since the sum of the four
+must he four right angles. The sides $BO$ and $CO$ are known, being
+radii of the earth; so that we can solve the whole quadrilateral by
+a simple trigonometrical process.
+
+\includegraphicsouter{illo074}{\textsc{Fig.~74.}---Determination of the Moon's Parallax.}
+
+\begin{fineprint}
+First find from the triangle $BOC$ the partial angles $OCB$ and $OBC$, and
+the side $BC$. Then in the triangle $BCM$ we have $BC$ and the two angles
+$CBM$ and $MCB$, from which we can find the two sides $BM$ and $CM$.
+Finally, in the triangle $OBM$, we now know the sides $OB$ and $BM$ and the
+included angle $OBM$, so that the side $OM$ can be computed, which is the
+distance of the moon from the earth's centre. Knowing this, the horizontal
+parallax $KMO$, or the semi-diameter of the earth as seen from the moon,
+follows at once.
+
+The moon's parallax can also be deduced from observations at a single
+station on the earth, but not so simply. If she did not move among the
+stars, it would be very easy, as all we should have to do would be to compare
+her apparent right ascension and declination at different points in her diurnal
+%% -----File: 161.png---Folio 150-------
+circle. Near the eastern horizon the parallax (always depressing an object)
+increases her right ascension; at setting, \textit{vice versa}. On the meridian
+the declination only is affected. But the motion of the moon must be allowed
+for, as the observations to be compared are necessarily separated by
+considerable intervals of time, and this complicates the calculation.
+
+A third, and a very accurate, method is by means of occultations of
+stars, observed at widely separated points on the earth. These occultations
+furnish the moon's place with great accuracy, and so determine the parallax
+very precisely; but the calculation is not very simple, as the moon's
+motion in this case also enters into it, since the observations cannot be
+simultaneous.
+\end{fineprint}
+
+\nbarticle{240.} \nbparatext{The Distance of the Moon is continually changing} on account
+of the eccentricity of its orbit, varying all the way, according to Neison,
+between 252,972 and 221,614 miles; the \textit{mean} distance being
+238,840 miles, or 60.27 times the equatorial radius of the earth.
+The mean parallax of the moon is $57'\ 2''$, subject to a similar percentage
+of change. This value of the parallax, it will be noted,
+indicates that the earth, as seen from the moon, has a diameter of
+nearly $2°$.
+
+Knowing the size of the moon's orbit and the length of the month,
+the velocity of her motion around the earth is easily calculated. It
+comes out 2288 miles per hour, or about 3350 feet a second.
+
+\includegraphicsmid{illo075}{\textsc{Fig.~75.}---Moon's Path with Reference to the Sun.}
+
+\nbarticle{241.} \nbparatext{Form of the Moon's Orbit with Reference to the Sun.}---While
+the moon moves in a small elliptical orbit around the earth, it
+also moves around the sun in company with the earth. This \textit{common}
+%% -----File: 162.png---Folio 151-------
+motion of the moon and earth, of course, does not affect their \textit{relative}
+motion; but to an observer outside the system the moon's motion
+around the earth would only be a very small component of the moon's
+movement as seen by \textit{him}.
+
+The distance of the moon from the earth, 239,000 miles, is very
+small compared with that of the earth from the sun, 93,000,000 miles---being
+only about $\frac{1}{400}$ part. The speed of the earth in its orbit
+around the sun is also more than thirty times faster than that of the
+moon in its orbit around the earth, so that for the moon the resulting
+path in space is one which is always \textit{concave towards the sun},
+as shown in \figref{illo075}{Fig.~75}. It is \textit{not} like Figs.~76 and 77, as often represented.
+If we represent the orbit of the earth by a circle with a
+radius of 100~inches (8~feet 4~inches), the moon would only move
+out and in a quarter of an inch, crossing the circumference twenty-five
+times in going once around it.
+
+\begin{center}
+\begin{tabular}{cc}
+  \includegfx{illo076}&
+  \includegfx{illo077}\\
+  \footnotesize\textsc{Fig.~76.} & \footnotesize\textsc{Fig.~77.}
+\end{tabular}
+\captionof*{figure}{False Representations of Moon's Motions.}
+\end{center}
+
+\nbarticle{242.} \nbparatext{Diameter of the Moon.}---The mean apparent diameter of the
+moon is $31'\ 7''$. This gives it a real diameter of 2163 miles (plus or
+minus one mile), which equals 0.273 of the earth's diameter. Since
+the surfaces of globes are as the squares of their diameters, and their
+volumes as their cubes, this makes the surface of the moon 0.0747 of
+the earth's (between $\frac{1}{13}$ and $\frac{1}{14}$); and the volume 0.0204 of the earth's
+volume (almost exactly $\frac{1}{49}$); that is, it would take 49 balls each as
+large as the moon in bulk to make a ball of the size of the earth.
+
+\nbarticle{243.} \nbparatext{Mass of the Moon.}---This is about $\frac{1}{80}$ of the earth's mass,
+different authorities giving the value from $\frac{1}{75}$ to $\frac{1}{85}$. It is not easy
+to determine it with accuracy. In fact, though the moon is the
+nearest of all the heavenly bodies, it is more difficult to ``weigh'' her
+than to weigh Neptune, although he is the most remote of the planets.
+
+There are four ways of approaching the problem: (1)~(perhaps
+easiest to understand) \textit{by finding the position of the common centre of
+gravity of the earth and moon with reference to the centre of the earth}.
+Since it is this \textit{common centre of gravity} of the two bodies which
+describes around the sun the ellipse which we have called the
+earth's orbit, and since the earth and moon revolve around this
+common centre of gravity once a month, it follows that this monthly
+motion of the earth causes an alternate eastward and westward
+displacement of the sun in the sky, which can be measured.
+At the time of the new and full moon this displacement is zero, the
+centre of gravity being on the line which joins the earth and sun;
+%% -----File: 163.png---Folio 152-------
+but when the moon is at \textit{quadrature} (that is, $90°$ from the sun, as at
+the time of half-moon), the sun
+is apparently displaced in the sky
+\textit{towards the moon}, as is evident
+from \figref{illo078}{Fig.~78}. It will be about
+$6''.3$ east of its mean place at the
+first quarter of the moon, \figref{illo078}{Fig.~78}
+($B$), and as much west at the time
+of the last quarter, \figref{illo078}{Fig.~78} ($A$);
+(\textit{i.e.}, when the angle $MGS$ is $90°$,
+the angle $MCS$ is always \textit{less} than
+$90°$ by $6''.3$, which is therefore the
+value of the angle $CSG$). Now
+since the parallax of the sun
+(which is the earth's semi-diameter
+seen from the sun---the
+angle $CSK$) is about $8''.8$, it follows
+that the distance of the centre
+of gravity of the earth and
+moon from the centre of the
+earth is the fraction $\frac{6.3}{8.8}$ of the
+earth radius, or, about 2830 miles.
+This is just about $\frac{1}{80}$ of the distance
+from the earth to the moon, whence we conclude that the mass
+of the earth is 80 times that of the moon.
+
+\includegraphicsouter[21]{illo078}{\textsc{Fig.~78.}\\
+Apparent Displacement of Sun at First and
+Third Quarters of the Month.}
+\begin{fineprint}
+\nbarticle{244.} (2) A second method is by comparing the moon's \textit{actual period} with
+the \textit{computed period to which a single particle at the moon's distance from the earth
+ought to have}, according to the known force of gravity of the earth, as determined
+by pendulum experiments. The explanation of this method cannot
+be given until we have further studied the motion of bodies under the law
+of gravitation. It is not susceptible of great accuracy.
+
+(3) Still another method is by comparing the \textit{tides produced by the moon
+with these produced by the sun}. This gives us the mass of the moon as compared
+with that of the sun; and the mass of the sun compared with that of
+the earth being known, it gives us ultimately the mass of the moon compared
+with that of the earth.
+
+(4) The ratio of the moon's mass to the sun's can also be computed from
+the \textit{nutation} of the earth's axis, (See \chapref{CHAPTERXIII}{Chap.~XIII.})
+
+\nbarticle{245.}  No other satellite is nearly as large as the moon, in comparison
+with its primary planet. The earth and moon together, as seen from a
+%% -----File: 164.png---Folio 153-------
+distant star, are really in many respects more like a \textit{double planet} than like
+a planet and satellite, as ordinarily proportioned to each other. At a time,
+for instance, when Venus happens to be near the earth, at a distance of
+about twenty-five millions of miles, the earth to her would appear just about
+as bright as Venus at her best does to us; and the moon would be about as
+bright as Sirius, at a distance of about half a degree from the earth.
+\end{fineprint}
+
+\nbarticle{246.} \nbparatext{Density and Superficial Gravity of the Moon.}---Since the
+density of a body is equal to $\dfrac{\text{mass}}{\text{volume}}$, the density of the moon as
+compared with the earth is found from the fraction
+\[
+\frac{\frac{1}{80}}{\frac{1}{49}}, \text{ or }\frac{0.125}{0.0204}.
+\]
+
+This makes the moon's density 0.613 of the earth's density, or
+about $3\frac{4}{10}$ the density of water---somewhat above the average density
+of the rocks which compose the crust of the earth.
+
+\begin{fineprint}
+This small density of the moon is not surprising, nor at all inconsistent
+with the belief that it once formed part of the same mass with the earth,
+since if such were the case, the moon was probably formed by the separation
+of the \textit{outer portions} of that mass, which would be likely to have a smaller
+specific gravity than the rest.
+\end{fineprint}
+
+\nbarticle{247.} \nbparatext{The superficial gravity}, or the attraction of the moon for bodies
+at its own surface, may be found by the equation
+\[
+g' = g × \frac{m}{r^{2}},
+\]
+in which $g'$ signifies the superficial gravity of the moon, $g$ is the force
+of gravity on the earth, while $m$ and $r$ are the mass and radius of the
+moon as compared with those of the earth. This gives us
+\[
+g' = g × \frac{0.0125}{0.0747},
+\]
+or (very approximately) $g'$ equals \textit{one-sixth} of $g$; that is, a body which
+weighs six pounds on the earth's surface would at the surface of the
+moon weigh only one (in a spring balance). A man on the moon
+could jump six times as high as he could on earth and could throw
+%% -----File: 165.png---Folio 154-------
+a stone six times as far. This is a fact to be remembered in connection
+with the enormous scale of the surface-structure of the moon.
+Volcanic forces, for instance, upon the moon would throw the rejected
+materials to a vastly greater distance there than on the earth.
+
+\includegraphicsouter[9]{illo079}{\textsc{Fig.~79.}}
+
+\nbarticle{248.} \nbparatext{Rotation of the Moon.}---The moon rotates on its axis once a
+month, in precisely the same time as that occupied
+by its revolution around the earth. In the
+long run it therefore keeps the same side always
+towards the earth: we see to-day precisely the
+same face and aspect of the moon as Galileo did
+when he first looked at it with his telescope, and
+the same will continue to be the case for thousands
+of years more, if not forever.
+
+\begin{fineprint}
+It is difficult for some to see why a motion of this
+sort should be considered a \textit{rotation} of the moon, since
+it is essentially like the motion of a ball carried on a
+revolving crank. See \figref{illo079}{Fig.~79}. Such a ball, they say, ``\textit{revolves} around the
+shaft, but does not \textit{rotate} on its own axis.'' It does rotate, however. The
+shaft being vertical and the crank horizontal, suppose that a compass
+needle be substituted for the ball, as in \figref{illo080}{Fig.~80}. The pivot turns underneath
+it as the crank whirls, but the compass
+needle does not rotate, maintaining always
+its own direction with the marked end north.
+On the other hand, if we mark one side of
+the ball (in the preceding \figref{illo079}{figure}), we shall
+find the marked side presented successively to
+every point of the compass as the crank revolves,
+so that the ball as really turns on its
+own axis as if it were whirling upon a pin
+fastened to a table. The ball has \textit{two} distinct
+motions by virtue of its connection with the crank: \textit{first}, the motion
+of translation, which carries its centre of gravity, like that of the compass
+needle, in a circle around the axis of the shaft; \textit{secondly}, an additional
+motion of rotation around a line drawn through its centre of gravity parallel
+to the shaft.
+
+\includegraphicsouter{illo080}{\textsc{Fig.~80.}}
+
+\sloppy
+\nbarticle{248*.}
+\nbparatext{Definition of Rotation.}---A body ``\textit{rotates}'' whenever \textit{a line
+drawn from its centre of gravity outward, through any point selected at random
+in its mass, describes a circle in the heavens}. In every rotating body, one such
+line can be so drawn that the circle described by it in the sky becomes infinitely
+small, This is the \textit{axis} of the body. Another set of points can be
+found such that lines drawn from the centre of gravity outward through
+%% -----File: 166.png---Folio 155-------
+\textit{them} describe a great circle in the sky $90°$ distant from the point pierced
+by the axis, and these points constitute the \textit{equator} of the body.
+\end{fineprint}
+
+\fussy
+\nbarticle{249.} \nbparatext{Librations of the Moon.}---1.~\textit{Libration in Latitude}. The axis
+of revolution of the moon is not perpendicular to its orbit. It makes
+a constant angle of about $88\frac{1}{2}°$ with the ecliptic, and the moon's equator
+is so placed that it is always edge-wise to the earth when the moon is
+at her node, being maintained in that position by an action of the earth,
+which produces a precessional motion of the moon's axis. The angle
+between the moon's equator and the plane of her orbit, therefore, is $1\frac{1}{2}°+$
+the inclination of the moon's orbit, which together make up an angle
+of a little more than $6\frac{1}{2}°$; but, as the inclination of the moon's orbit
+to the ecliptic is constantly varying slightly, this inclination of the
+moon's axis to her orbit also changes correspondingly. This inclination
+of the moon's axis produces changes in the aspect of the moon
+towards the earth similar to these produced by the inclination of the
+earth's axis towards the ecliptic. At one time, just as the north pole
+of the earth is turned towards the sun, so also the north pole of the
+moon is tipped towards the earth at an angle of $6\frac{1}{2}°$, and in the opposite
+half of the moon's orbit the south pole is similarly presented to
+us. In consequence we alternately look over the northern and southern
+portions of the moon's disc.
+
+\begin{fineprint}
+The period of this libration is the time of the moon's revolution from
+node to node, called a \textit{nodical
+revolution}. This is 27.21 days---about
+2~hours and 38~minutes
+shorter than the sidereal revolution
+of the moon, since the
+nodes always move westward,
+completing the circuit in about
+19~years.
+\end{fineprint}
+
+\nbarticle{250.} 2.~\textit{Libration in Longitude}.
+The moon's orbit
+being eccentric, she moves
+faster when near perigee,
+and slower when near apogee;
+half-way between perigee
+and apogee she is more
+than $6°$ ahead of the position
+she would have if she had moved with the \textit{mean} angular velocity.
+Now the rotation is \textit{uniform}. A point, therefore, on the moon's
+%% -----File: 167.png---Folio 156-------
+surface which is directed toward the earth at perigee will not
+have revolved far enough to keep it directed toward the earth when
+she is half-way (\textit{in time}) between perigee and apogee, as is evident
+from \figref{illo081}{Fig.~81}. For in the quarter-month next following the perigee,
+the moon will travel to a point $M$, considerably more than half-way to
+apogee. But the point $a$ will have made only one quarter-turn, which
+is not enough to bring it to the line $ME$. We shall therefore see a little
+around the \textit{western} edge. Similarly on the other side of the orbit,
+half-way between apogee and perigee, we shall look around the \textit{eastern}
+edge to the same extent. At perigee and apogee both, the libration
+is, of course, zero. The amount of this libration is evidently at any
+moment just the same as that of the so-called ``equation of the centre,''
+which, it will be remembered, is the difference between the \textit{mean} and
+\textit{true anomalies} of the moon at any moment. Its maximum possible
+value is $7°\ 45'$.
+
+\includegraphicsouter{illo081}{\textsc{Fig.~81.}---The Libration in Longitude.}
+
+\begin{fineprint}
+The period of this libration is the time it takes the moon to go around
+from perigee to perigee---the so-called \textit{anomalistic revolution}, which is 27.555
+days, about 5~hours and 36~minutes longer than the sidereal month, and 8
+hours 14~minutes longer than the moon's \textit{nodical} revolution, which determines
+the libration in latitude.
+
+The cause of the increased length of the anomalistic revolution is of
+course the fact that the line of apsides continually advances \textit{eastward}, making
+one revolution every nine years.
+\end{fineprint}
+
+\nbarticle{251.} 3.~\textit{Diurnal Libration}. This is strictly a libration not of the
+moon, but of the observer; still, as far as the aspect of the moon
+goes, the effect is precisely the same as if it were a true lunar libration.
+The moon's motions have reference to the earth's centre. We,
+on the surface of the earth, look down over the western edge of the
+moon when it is rising, and over the eastern when it is setting, by
+an amount which is equal to the semi-diameter of the earth as seen
+from the moon; that is, about one degree (the moon's parallax).
+
+On the whole, taking all three librations into account, we see considerably
+more than half the moon, the portion which never disappears
+being about \textit{forty-one per cent} of the moon's surface, that never visible
+also \textit{forty-one} per cent, while that which is alternately visible and
+invisible is \textit{eighteen} per cent.
+
+\nbarticle{252.} The agreement between the moon's time of rotation and of
+her orbital revolution cannot be accidental. It is probably due to the
+action of the earth on some slight protuberance on the moon's surface,
+%% -----File: 168.png---Folio 157-------
+analogous to a tidal wave. If the moon were ever plastic, such a
+bulge must have been formed on the side of the moon next the earth,
+and would serve as the handle by which the earth always keeps the
+same face of the moon towards herself. This subject will be resumed
+later.
+
+\includegraphicsmid{illo082}{\textsc{Fig.~82.}---Explanation of the Phases of the Moon.}
+
+\sloppy
+\nbarticle{253.} \nbparatext{The Phases of the Moon.}---Since the moon is an opaque
+globe, shining entirely by reflected light, we can see only that hemisphere
+of her surface which happens to be illuminated, and of course
+only that part of the illuminated hemisphere which is at the time turned
+towards the earth. At new moon, when the moon is between the
+earth and the sun, the dark side is towards us. A week later, at the
+end of the first quarter, half of the illuminated hemisphere is seen,
+and we have the half moon, just as we do a week after the full. Between
+the new moon and the half moon, during the first and last
+quarters of the lunation, we see less than half of the illuminated portion,
+and then have the ``crescent'' phase. See \figref{illo082}{Fig.~82} (in which the
+%% -----File: 169.png---Folio 158-------
+light is supposed to come from a point far above the moon's orbit).
+Between the half moon and the full, during the second and third
+quarters of the lunation, we see more than half of the moon's illuminated
+side, and have what is called the ``gibbous'' phase.
+
+\fussy
+Since the terminator or line which separates the dark portion of the
+disc from the bright is always a \textit{semi-ellipse} (being a semi-circle viewed
+obliquely), the illuminated surface is always a figure made up of a
+\textit{semi-circle} plus or minus a \textit{semi-ellipse}, as shown in \figref{illo083}{Fig.~83}, $A$.
+
+\includegraphicsouter{illo083}{\textsc{Fig.~83.}}
+
+\begin{fineprint}
+It is sometimes incorrectly attempted to represent the crescent form
+by a construction like \figref{illo083}{Fig.~83}, $B$ (where a
+smaller circle is cut by a larger one). It is
+to be noticed that $ab$, the line which \DPtypo{joints}{joins}
+the cusps, is always perpendicular to the line
+directed to the sun, and \textit{the horns are always
+turned away from the sun}; so that the precise
+position in which they will stand at any time is
+always predictable, and has nothing whatsoever
+to do with the weather. Artists are sometimes careless in the manner in
+which they introduce the moon into landscapes. One occasionally sees the
+moon near the horizon with the \textit{horns turned downwards}, a piece of drawing
+fit to go with Hogarth's barrel which shows both its heads at once.
+
+\nbarticle{254.} \nbparatext{Earth-Shine on the Moon.}---Near the time of new moon the whole
+disc of the satellite is easily visible, the portion on which sunlight does not fall
+being illuminated by a pale ruddy light. This light is \textit{earth-shine}, the earth
+as seen from the moon being then nearly full; for seen from the moon the earth
+shows all the phases that the moon does, the earth's phase in every case being
+exactly supplementary to that of the moon as seen by us.
+
+As the earth has a diameter nearly four times that of the moon, the earth-shine
+at any phase would be about thirteen times as strong as moonlight, if
+the reflective power of the earth's surface were the same. Probably, taking
+the clouds and snow into account, the earth's surface on the whole is rather
+more brilliant than the moon's, so that near new moon the earth-shine, by
+which the dark side of the moon is then illuminated, is from fifteen to
+twenty times as strong as full moonlight. The ruddy color is due to the
+fact that light sent to the moon from the earth has twice penetrated our
+atmosphere and so has acquired the sunset tinge.
+\end{fineprint}
+
+\nbarticle{255.} \nbparatext{Physical Characteristics of the Moon.}---1.~\textit{Its Atmosphere}.
+The moon's atmosphere, if it has any at all, is extremely rare, probably
+not producing a barometric pressure to exceed $\frac{1}{25}$ of an inch
+of mercury, or $\frac{1}{750}$ of the pressure at the earth's surface. The
+evidence on this point is twofold.
+%% -----File: 170.png---Folio 159-------
+
+(\textit{a}) \textit{The telescopic appearance}. The parts of the moon near the
+edge of the disc, which, if there were any atmosphere, would be
+seen through its greatest possible depth, are seen without the least
+distortion: there is no haze, and all shadows are perfectly black.
+There is no sensible twilight at the cusps of the moon; no evidences
+of clouds or storms, or anything like atmospheric phenomena.
+
+(\textit{b}) \textit{The absence of refraction} when the moon intervenes between
+us and any more distant object. For instance, at an eclipse of the
+sun there is no distortion of the sun's limb where the moon cuts it,
+nor any ring of light running out on the edge of the moon like that
+which encircles the disc of Venus at the time of a transit. The most
+striking evidence of this sort comes, however, from occultations of
+the stars. When the moon hides a star from sight, the phenomenon,
+if it occurs at the moon's dark edge, is an exceedingly striking
+one. The star retains its full brightness in the field of the telescope
+until all at once, without the least warning, it simply is not there,
+the disappearance generally being absolutely instantaneous. Its reappearance
+is of the same sort, and still more startling. Now if the
+moon had any perceptible atmosphere (or the star any sensible diameter)
+the disappearance would be gradual. The star would change
+color, become distorted, and fade away more or less gradually.
+
+The spectroscope adds its evidence in the same direction. There is
+no modification of the spectrum of the star in any respect at the time
+of its disappearance; and we may add that the spectrum of moonlight
+is identical with that of sunlight pure and simple, there being no
+traces of any effect whatever produced upon the sunlight by its reflection
+from the moon, nor any signs of its having passed through
+an atmosphere.
+
+\nbarticle{256.} The time during which a star would be hidden behind the
+moon would also be decreased by the refraction of any sensible
+atmosphere, making the observed duration of an occultation less
+than that computed from the known diameter of the moon and its rate
+of motion. Certain Greenwich observations \textit{apparently show a difference},
+amounting to about \textit{two seconds} of time. This may possibly be
+due in some part to the action of a \textit{real}, but exceedingly rare, lunar
+atmosphere; for if the whole phenomenon were due simply to atmospheric
+action, it would indicate an atmosphere having a density about
+$\frac{1}{2000}$ of our own,---far within the limits which were stated above.
+But the difference may be, and very probably is, attributable, in part
+at least, to a slight error in the measured diameter of the moon, due to
+%% -----File: 171.png---Folio 160-------
+\textit{irradiation}: the diameter of a bright object always appears a little
+larger than it really is. An error of about $2''$ of this sort would
+explain the whole discrepancy, without any need of help from an
+atmosphere.
+
+\begin{fineprint}
+\nbarticle{257.} \nbparatext{What has become of the Moon's Atmosphere.}---if the moon
+ever formed a part of the same mass as the earth, she must once have had
+an atmosphere. There are a number of possible and more or less probable
+hypotheses to account for its disappearance. It has been surmised (1)~that
+there may be great cavities left within the moon's mass by volcanic eruptions,
+and that the rocks themselves have been transformed into a sort of
+pumice-stone structure, and that the air has retired into these internal
+cavities.
+
+(2) That the air has been absorbed by the inner lunar rocks in cooling. A
+heated rock expels any gases that it may have absorbed; but if it afterwards
+cools slowly, it reabsorbs them, and can take up a very great quantity. The
+earth's core is supposed to be now too intensely heated to absorb much gas;
+but if it goes on cooling, it will absorb more and more, and in time it may
+rob the surface of the earth of all its air. There are still other hypotheses,
+which we can not take space even to mention.
+\end{fineprint}
+
+\nbarticle{258.} \nbparatext{Water on the Moon's Surface.}---Of course without an atmosphere
+there can be no water, since the water would immediately evaporate
+and form an atmosphere of water vapor if there were no air
+present. It is not impossible, however, or even improbable, that
+\textit{solid} water, that is, ice and snow, may exist on the moon's surface
+at a temperature too low for any sensible evaporation. There are
+many things in the moon's appearance that seem to indicate the former
+existence of seas and oceans on her surface, and the same
+hypotheses have been suggested to account for their disappearance
+that were suggested in the case of the moon's atmosphere. It may
+be added also that many kinds of molten rock in crystallizing would
+take up large quantities of water of crystallization, not merely absorbed
+as a sponge absorbs water, but chemically united with the other
+constituents of the rock. In whatever way, however, it may have
+come about, it is certain that \textit{now} no substances that are gaseous, or
+that can be evaporated at low temperatures, exist in any quantity on
+the moon's surface---at least, not \textit{on our side} of the moon.
+
+\begin{fineprint}
+There have been speculations that on the other side---that celestial country
+so near us and so absolutely concealed from us---there may be air and
+water and abundant life; the idea being that our side of the moon is a great
+table land many miles in elevation, while the other side is a corresponding
+%% -----File: 172.png---Folio 161-------
+depression, like the valley of the Caspian Sea, only vastly deeper. An insufficiently
+grounded conclusion of Hansen's, that the centre of gravity of
+the moon is some thirty miles farther from us than its centre of figure, for a
+time gave color to the idea, but it is now practically abandoned, Hansen's
+conclusion having been shown to be unwarranted by the facts.
+\end{fineprint}
+
+\nbarticle{259.} \nbparatext{The Moon's Light.}---As to \textit{quality} it is simple sunlight, showing
+a spectrum which, as has been said, is identical in every detail
+with that of light coming directly from the sun. Its \textit{brightness} as
+compared with that of sunlight is difficult to measure accurately, and
+different experimenters have found results for the ratio between full
+moonlight and sunlight ranging all the way from $\frac{1}{300000}$ (Bouguer)
+to $\frac{1}{800000}$ (Wollaston). The value now usually accepted is that
+determined by Zöllner, viz., $\frac{1}{618000}$. According to this, if the whole
+visible hemisphere were packed with full moons, we should receive
+from it about one-eighth part of the light of the sun.
+
+\begin{fineprint}
+It is found, also, that the half moon does not give even nearly half as
+much light as the full moon. The law which connects the phase of the
+moon with the amount of light given at the time, is rather complicated, but
+the gist of the matter is that at any time, except at the full, the visible surface
+is more or less darkened by the shadows cast by the irregularities of the
+surface. Zöllner has calculated that an average angle of $52°$ for these elevations
+and depressions would account for the law of illuminations actually
+observed.
+\end{fineprint}
+
+The average ``\textit{albedo},'' or reflecting power of the moon's surface,
+Zöllner states as 0.174; that is, the moon's surface \textit{reflects a little
+more than one-sixth} part of the light that falls upon it. This is about
+the albedo of a rather light-colored sandstone, and agrees well with
+the estimate of Sir John Herschel, who found the moon to be very
+exactly of the same brightness as the rock of Table Mountain when
+it was setting behind it, illuminated as were the rocks themselves by
+the light of the rising sun. There are, however, great variations in
+the brightness of different portions of the moon's surface. Some
+spots are nearly as white as snow or salt, and others as dark as
+slate.
+
+\nbarticle{260.} \nbparatext{Heat of the Moon.}---For a long time it was impossible to
+detect the moon's heat. It is too feeble to be detected by the most
+delicate mercurial thermometer even when concentrated by a large
+lens. The first sensible effect was obtained by Melloni, in 1846,
+%% -----File: 173.png---Folio 162-------
+with the then newly invented thermopile, by a series of observations
+from the summit of Vesuvius. Since then several physicists have
+worked upon the subject with more or less success, but the most recent
+and reliable investigations are those of Lord Rosse and Professor
+Langley. With modern apparatus there is no difficulty in detecting
+the heat in the lunar radiations, but \textit{measurements} are extremely
+difficult and liable to error. A considerable percentage of the lunar
+heat seems to be heat simply \textit{reflected} (like light), while the rest,
+perhaps three-fourths of the whole, is ``\textit{obscure heat}''; that is, heat
+which has been first absorbed by the moon's surface and then \textit{radiated},
+like the heat from a brick surface that has been warmed by sunshine.
+This is shown by the fact that a comparatively thin plate of glass
+cuts off some 86 per cent of the heat received from the moon in the
+same way that it does the heat of a stove, while the heat of direct
+sunlight, or of an electric arc, would pass through the same plate
+with very little diminution. The same thing appears also from direct
+measurements upon the \textit{heat-spectrum} of the moon made by Langley
+with his bolometer, described further on. (\artref{Art.}{343}.)
+
+\nbarticle{261.} As to the \textit{temperature of the moon's surface}, it is difficult to
+affirm much with certainty. On one hand, the lunar rocks are exposed
+to the sun's rays in a cloudless sky for fourteen days at a time, so
+that if they were blanketed by air like our own rocks they would certainly
+become intensely heated. A few years ago, Lord Rosse inferred
+from his observations that the temperature of the lunar surface
+rose at its maximum (about three days after full moon) far above
+that of boiling water.
+
+But his own later investigations and these of Langley throw great
+doubt on this conclusion. There is no air-blanket at the moon's
+surface to prevent it from losing heat as fast as it receives it; and
+it now seems rather more probable that the temperature never rises
+above the freezing-point of water, as is the case on the highest of
+our mountains, where there is perpetual ice, and the temperature is
+always low even at noon. So far as we can judge, the condition of
+things on the moon's surface must correspond to an elevation many
+times higher than any mountain on the earth; for no terrestrial mountain
+is so high that the density of the air at its summit is even nearly
+as low as that of the densest supposable lunar atmosphere.
+
+This idea, that the temperature is low, is borne out, also, by the
+fact that the bolometer shows the presence, in the lunar radiations,
+of a considerable quantity of heat having a wave-length greater than
+that of the heat radiated from a block of ice.
+%% -----File: 174.png---Folio 163-------
+
+At the end of the long lunar night of fourteen days the temperature
+must fall appallingly low, \textit{certainly} $200°$ below zero.
+
+The whole amount of heat sent by the full moon to the earth is
+estimated by Rosse as \textit{about one eighty-thousandth part of that sent by
+the sun}.
+
+\nbarticle{262.} \nbparatext{Lunar Influences on the Earth.}---The moon's attraction cooperates
+with that of the sun in producing tides, of which we shall
+speak hereafter. There are also certain distinctly ascertained disturbances
+of \DPtypo{terrestial}{terrestrial}
+magnetism connected with the approach and
+recession of the moon at perigee and apogee; and this ends the
+chapter of \textit{ascertained} lunar influences.
+
+\begin{fineprint}
+The multitude of current beliefs as to the controlling influence of the
+moon's phases and changes over the weather and the various conditions of
+life are mostly unfounded, and in the strict sense of the word ``superstitions,''---mere
+survivors from a past credulity.
+
+It is quite certain that if there is any influence at all of the sort it is extremely
+slight---so slight that it cannot be demonstrated with certainty,
+although numerous investigations have been made expressly for the purpose
+of detecting it. We have never been able to ascertain, for instance, with
+certainty, whether it is \textit{warmer or not}, or \textit{less cloudy or not}, at the time of the
+full moon. Different investigations have led to contradictory results.
+
+The frequency of the moon's changes is so great that it is always easy to
+find instances by which to verify a belief that changes of the moon control
+conditions on the earth. A change of the moon necessarily occurs about
+once a week, the interval from quarter to quarter being between seven and
+eight days. \textit{All} changes, of the weather for instance, must therefore occur
+within three and three-fourths days of a change of the moon, and fifty per
+cent of them ought to occur within forty-six hours of a change, even if there
+were no causal connection whatever.
+
+Now it requires only a very slight prepossession in favor of a belief in
+the effectiveness of the moon's changes to make one forget a few of the
+weather changes that occur too far from the proper time. Coincidences
+enough can easily be found to justify a preëxisting belief.
+\end{fineprint}
+
+\section*{THE MOON'S SURFACE.}
+
+\nbarticle{263.} Even to the naked eye the moon is a beautiful object,
+diversified with darker and lighter markings which have given rise to
+numerous popular superstitions. With a powerful telescope these
+naked-eye markings mostly vanish, and are replaced by a countless
+multitude of smaller details, which are interesting in the highest
+degree. The moon on the whole, on account of this diversity of
+%% -----File: 175.png---Folio 164-------
+detail, is the finest of all telescopic objects; especially to moderate-sized
+instruments, say from six to ten inches in diameter, which
+generally give a more pleasing view of our satellite than instruments
+either larger or smaller.
+
+\includegraphicsouter{illo084}{\textsc{Fig.~84.}---A Normal Lunar Crater (Nasmyth.)}
+
+\nbarticle{264.} \nbparatext{How near the Telescope brings the Moon.}---An instrument
+of this size, with magnifying powers between 250 and 500, brings
+up the moon virtually to a distance ranging from 1000 miles to 500;
+and since an object a mile in diameter on the moon subtends an
+angle of about $0''.86$, with the higher powers of such an instrument
+objects less than a mile in diameter become visible under favorable
+atmospheric conditions. A long line or streak, even less than a
+quarter of a mile across, could probably be seen. With larger telescopes
+the power can now and then be carried at least twice as high,
+and correspondingly smaller details made out. When everything is
+at its best, the great Lick telescope of 30 inches aperture, with a
+power of 2500 or so, may possibly reduce the virtual distance of our
+satellite to about 100 miles for visual purposes. It is evident that
+while with our telescopes we should be able to see such objects as
+lakes, rivers, forests, and great cities, if they exist on the moon, it
+will be hopeless to expect to distinguish single buildings, or any of
+the ordinary operations and indications of life, if such there are.
+
+\begin{fineprint}
+There are a few mountains on the earth from which a range of 100 miles
+is obtained in the landscape. Those who have seen such a landscape know
+how little is to be made out with the naked eye at that distance. Still, the
+comparison is not quite fair, because in looking at a terrestrial object a hundred
+miles away the line of vision passes through a dense atmosphere, while
+in looking upward towards the moon it penetrates a much less thickness
+of air.
+\end{fineprint}
+
+\nbarticle{265.} \nbparatext{The Moon's Surface Structure.}---The moon's surface for the
+most part is extremely uneven and broken, far more so than that of
+the earth. The structure, however, is not like that of the earth's
+surface. On the earth the mountains are mostly in long ranges, such
+as the Alps, the Andes, and Himalayas. On the moon such mountain
+ranges are few in number, though they exist; but the surface is
+pitted all over with great craters, resembling very closely the volcanic
+craters on the earth's surface, though on an immensely greater
+scale. One of the largest craters upon the earth, if not the largest,
+is the Aso San in Japan, about seven miles across. Many of those
+on the moon are fifty and sixty miles in diameter, and some are
+%% -----File: 176.png---Folio 165-------
+over 100 miles across, while smaller ones from a half-mile to eight
+or ten miles in diameter are counted by the thousand.
+
+\includegraphicsmid{illo085}{\textsc{Fig.~85.}---Map of the Moon.}
+
+The normal lunar crater is nearly circular, surrounded by an elevated
+ring of mountains
+which rise anywhere from
+1000 to 20,000 feet above
+the surrounding country.
+Within the floor of the
+crater the surface may be
+either above or below the
+outside level. Some craters
+are deep, some filled
+nearly to the brim. In
+some cases the surrounding
+mountain ring is entirely
+absent, and the crater is a mere hole in the plain. In the
+centre of the crater there usually rises a group of peaks, which attain
+about the same elevation as the encircling ring, and these central
+peaks often show little holes or craterlets in their summits.
+
+\begin{fineprint}
+In most cases the resemblance of these formations to terrestrial volcanic
+structures, like those exemplified by Vesuvius and others in the surrounding
+region, makes it natural to assume that they had a similar origin.
+This, however, is not absolutely certain, for there are considerable difficulties
+in the way, especially in the case of the great ``Bulwark Plains,'' so called,
+which are so extensive that a person standing in the centre could not see
+the summit of the surrounding ring at any point; and yet no line of demarcation
+can be drawn between them and the smaller craters. The series
+is continuous. Moreover, on the earth, volcanoes necessarily require the
+action of air and water, which do not now exist on the moon. It is obvious,
+therefore, that if these lunar craters are the result of true volcanic eruptions,
+they must be fossil formations; for it is quite certain that no evidence of
+existing volcanic activity has ever been found. The moon's surface appears
+to be absolutely quiescent---still in death.
+\end{fineprint}
+
+On some portions of the moon these craters stand very thickly.
+Older craters have been encroached upon, or more or less completely
+obliterated by the newer, and the whole surface is a chaos, of which
+the counterpart is hardly to be found on the earth, even in the roughest
+portions of the Alps. This is especially the case near the moon's
+south pole. It is noticeable that, as on the earth the newest mountains
+are generally the highest, so on the moon the more newly
+formed craters are generally deeper and more precipitous than the
+older ones.
+%% -----File: 177.png---Folio 166-------
+
+{\stretchyspace \nbarticle{266.} \nbparatext{Lunar Nomenclature.}---The great plains were called by
+Galileo} oceans or seas (\textit{Maria}), and some of the smaller ones
+marshes (\textit{Paludes}) and lakes, for he supposed that the grayish surfaces
+visible to the naked eye, and conspicuous in a small telescope,
+were covered with water. Thus we have the ``Oceanus Procellarum,''
+the ``Mare Imbrium,'' and a number of other ``seas,'' of which
+``Mare Fecunditatis,'' ``Mare Serenitatis,'' and ``Mare Tranquilitatis,''
+are the most conspicuous. There are twelve of them in all, and
+eight or nine Paludes, Lacus, and Sinus.
+
+\sloppy
+The ten mountain ranges on the moon are mostly named after
+terrestrial mountains, as Caucasus, Alps, Apennines, though two or
+three bear {\stretchyspace the names of astronomers, like Leibnitz,} Dörfel, etc.
+%% -----File: 178.png---Folio 167-------
+The conspicuous craters bear the names of the more eminent ancient
+and mediæval astronomers and philosophers, as Plato, Archimedes,
+Tycho, Copernicus, Kepler, and Gassendi; while hundreds of smaller
+and less conspicuous formations bear the names of more modern or
+less noted astronomers.
+
+\fussy
+\begin{fineprint}
+The \textit{system} seems to have originated with Riccioli in 1650, but most of
+the names have been more recently assigned by the later map-makers, the
+most eminent of whom have been the German astronomers Beer and Maedler
+(who published their map in 1837), and Schmidt of Athens, whose great
+map of the moon, on a scale seven feet in diameter, was published by the
+Prussian government a few years ago. It is not at all too much to say that
+our maps of the earth's surface do not, on the whole, compare in fulness and
+accuracy with our maps of the moon. Of course this is not true of such
+countries as France and England, or others that have been trigonometrically
+surveyed; but there are no such \textit{lucunæ} in our maps of the moon as exist in
+our maps of Asia and Africa, for instance.
+\end{fineprint}
+
+\includegraphicsouter[16]{illo086}{\textsc{Fig.~86.}---Archimedes and the Apennines (Nasmyth).}
+
+\sloppy
+\nbarticle{267.} \nbparatext{Other Lunar Formations.}---The craters and mountains are
+not the only interesting formations
+on the moon's surface.
+There are many deep,
+narrow, crooked valleys that
+go by the name of ``rills''
+(German \textit{Rillen}), some of
+which may once have been
+watercourses. Then there
+are numerous ``clefts,'' half
+a mile or so wide and of unknown
+depth, running in
+some cases several hundred
+miles, straight through mountain
+and valley, without any
+apparent regard for the accidents
+of the surface.
+They seem to be deep
+cracks in the crust of our
+satellite. Several of them
+are shown in \figref{illo086}{Fig.~86}. Most
+curious and interesting of all are the light-colored \textit{streaks} or ``\textit{rays}''
+which radiate from certain of the craters, extending in some cases
+a distance of several hundred miles. They are usually from five to
+ten miles wide, and neither elevated nor depressed to any extent with
+%% -----File: 179.png---Folio 168-------
+reference to the general surface. They pass across mountain and
+valley, and sometimes through craters without any change in width
+or color. We do not know whether they are like the so-called ``trap-dykes''
+on the earth,---fissures which have been filled up from below
+with some light-colored material,---or whether they are mere surface
+markings. No satisfactory explanation has ever been given.
+
+\fussy
+The most remarkable system of ``rays'' of this kind is the one
+connected with the great crater Tycho, not very far from the moon's
+south pole. They are not very conspicuous until within a few days
+of full moon, but at that time they, and the crater from which they
+radiate, constitute by far the most striking feature of the whole lunar
+landscape.
+
+\includegraphicsouter{illo087}{\textsc{Fig.~87.}---Gassendi (Nasmyth).}
+
+\nbarticle{268.} \hspace{-2.1pt}\nbparatext{Changes on the Moon.}\\---It is certain that there are no \textit{conspicuous}
+changes. The observer
+has before him no
+such ever-varying vision
+as he would have in looking
+toward the earth,---no
+flying clouds, no alternations
+of seasons with the
+transformation of the snowy
+wastes to green fields, nor
+any considerable apparent
+movement of objects on the
+disc. The sun rises on them
+slowly as they come one
+after the other to the terminator,
+and sets as slowly.
+At the same time it is confidently
+maintained by many
+observers that here and there
+changes are still going on in
+the details of the surface.
+Others as stoutly dispute it.
+
+\begin{fineprint}
+\nbarticle{269.} Probably the most notable and best advocated instance of such a
+change is that of the little crater Linné, in the Mare Serenitatis. It was observed
+by Schroeter very early in the century, and is figured and described by
+Beer and Maedler as being about five and a half or six miles in diameter, quite
+deep and very bright. In 1866 Schmidt, who had several times observed it
+before, announced that it had disappeared. A few months later it was
+%% -----File: 180.png---Folio 169-------
+visible again, and there were many reported changes in its appearance
+during the next year or two. There is no question that it does not now at
+all agree in conspicuousness and size with the representation of Beer and
+Maedler, for it is at present, and has been for several years, only a minute
+dark spot, with a whitish spot surrounding it. Astronomers would feel
+more confident that this was a case of real change were it not that Schroeter's
+earlier picture much more resembles the present appearance than does
+that of Beer and Maedler. As the latter observers worked with rather a
+small telescope, and had no reason for taking any special pains in the delineation
+of this particular object, the evidence
+is less conclusive than it might
+seem at first. The change, however, if
+real, was certainly as great as in the
+instance of \DPtypo{Krakatoa}{Krakatão}, the great volcano
+whose eruption in 1883 filled the earth's
+atmosphere with smoke and vapor for
+more than two years, and caused the
+``twilight conflagrations'' of the sky.
+The phenomenon in the case of Linné,
+if real, was probably a falling in of the
+walls of the crater, exposing fresh unweathered
+surfaces.
+
+The reason why it is so difficult to
+be sure of changes lies in the great
+variations in the appearance of a lunar object under the varying illumination.
+To insure certainty in such delicate observations, comparisons
+must be made between the appearance seen at precisely the same phase of
+the moon, with telescopes (and eyes too) of equal power; and under substantially
+the same conditions otherwise, such as the height of the moon
+above the horizon, the clearness and steadiness of the air, etc. It is of
+course very difficult to secure such identity of conditions.
+\end{fineprint}
+
+\includegraphicsouter{illo088}{\textsc{Fig.~88.}}
+
+\nbarticle{270.} \nbparatext{Measurements of Heights of Lunar Mountains.}---When the
+terminator approaches a lunar mountain, the top of the mountain
+catches the sunlight first, and appears as a star entirely detached
+from the rest of the illuminated portion, like the little bright spots
+opposite $a$ and $b$ in \figref{illo088}{Fig.~88}. As time passes, the bright spot becomes
+larger as the light extends lower down the mountain side, until
+the terminator reaches and passes it.
+
+If now we measure the apparent distance, $AD$ or $a$, \figref{illo089}{Fig.~89}, from the
+peak to the terminator at the moment when it first appears like a star,
+it is easy to compute $AB$, and from this, the height of the mountain.
+
+\begin{fineprint}
+
+In \figref{illo089}{Fig.~89} the angle $S'CN$ very approximately the moon's ``elongation''
+at the time of observation, since the line from the \textit{earth} to the sun is nearly
+%% -----File: 181.png---Folio 170-------
+parallel to $S'C$ (the moon's distance being only about $\frac{1}{400}$ of the sun's). Now
+the angle $BAD = GBE'' = 90° - S'CN$, so that $AB$, or $b$, $= AD$ (or $a$)$÷\sin{S'CN}$.
+Knowing $b$, and the radius of the moon $r$, we get
+\[
+(r + h)^{2} = r^{2} + b^{2},
+\]
+in which $h$ is the height of the mountain, and $b$ is the distance $AB$. From
+this equation we find
+\begin{flalign*}
+&&  h &= \sqrt{r^{2} + b^{2}} - r,    &&\phantom{\text{or (very nearly) }}\\
+&\text{or (very nearly) }&
+    h &= \tfrac{1}{2}\frac{b^{2}}{r}. &&
+\end{flalign*}
+
+If, for instance, $b$ were 60 miles,
+\[
+h\text{(in miles)} = \frac{3600}{2× 1081} \text{ or } 1.664 \text{ miles } = 8845 \text{ feet. }
+\]
+\includegraphicsouter{illo089}{\textsc{Fig.~89.}\\
+Measurement of the Height of a Lunar Mountain.}
+
+Since the terminator is very ragged, it is sometimes best to measure from
+the mountain top clear across
+to the edge of the moon, as
+indicated by the little arrows
+in \figref{illo088}{Fig.~88}. The position of
+the theoretical terminator
+(the terminator as it would
+be if the moon were a smooth
+sphere) is known from the
+moon's age, so that $AB$ can
+be deduced by measuring
+from the limb as well as
+from the terminator.
+\end{fineprint}
+
+The height of a mountain
+can also be ascertained
+by measuring the
+length of its shadow in
+cases where the shadow
+falls on a reasonably level surface. A few of the lunar mountains
+reach the height of 22,000 or 23,000 feet, but there are none which
+attain the elevation of the very highest terrestrial mountains. Heights
+ranging from 10,000 to 20,000 feet are common.
+
+\begin{fineprint}
+\nbarticle{271.} \nbparatext{The Best Time to look at the Moon with a Telescope.}---The
+moon when full is not so satisfactory an object as when near the half, because
+at the full moon there are no shadows, so that at that time the ``relief''
+of the surface structure is entirely lost. Certain features, however, as has
+been before mentioned, are then best seen, as, for instance, the streaks or
+%% -----File: 182.png---Folio 171-------
+rays. Generally, any particular mountain, crater, rill, or cleft, is best studied
+when it is just on or very near the terminator, that is, at the time when the
+sun is rising or setting near it, because then the shadows are longest. The
+best general view of the moon is that obtained a few days after the half
+moon, when Copernicus and Tycho are both near the terminator, and Plato
+is still near enough to it to show very well.
+
+\nbarticle{272.} \nbparatext{Photographs of the Moon.}---A great deal of attention has been
+paid to this subject, and some fine results have been reached. The earliest
+success was that of Bond in 1850, with the old daguerreotype process; then
+followed the work of De la Rue in England, and of Dr.~Henry Draper, and
+especially of Mr.~Rutherfurd in this country. Rutherfurd's pictures have
+remained absolutely unrivalled until very recently.
+
+To these older experimenters the moon's motion offered a great difficulty,
+but now that with the sensitive plates at present used, a fraction of a second
+is a sufficient exposure, that difficulty has disappeared, and the plates which
+have recently been taken at Cambridge, Mass., are far in advance even of
+Rutherfurd's, showing such craters as Copernicus or Ptolemy with a diameter
+of two inches, on a scale larger than that of Schmidt's map. The Lick
+Observatory has also taken up the work, and is making admirable pictures.
+\end{fineprint}
+\chelabel{CHAPTERVII}
+%% -----File: 183.png---Folio 172-------
+
+\Chapter{VIII}{The Sun}
+\nbchapterhang{\stretchyspace
+THE SUN: DISTANCE AND DIMENSIONS.---MASS AND DEN\-SITY.---ROTATION.---STUDY
+OF THE SURFACE: GENERAL VIEWS AS
+TO THE SUN'S CONSTITUTION.---SUN SPOTS: THEIR APPEARANCE,
+NATURE, DISTRIBUTION, AND PERIODICITY.---THE
+SPECTROSCOPE AND THE SOLAR SPECTRUM.---CHEMICAL
+ELEMENTS RECOGNIZED IN THE SUN.---THE CHROMOSPHERE
+AND PROMINENCES.---THE CORONA.}
+
+\nbarticle{273.} \textsc{The} SUN is simply a \textit{star}; a hot, self-luminous globe of
+enormous magnitude as compared with the earth and the moon,
+though probably only of medium size among its stellar compeers.
+But to the earth and the other planets which circle around it, it is
+the grandest of all physical objects. Its attraction confines its
+planets to their orbits and controls their motions, and its rays supply
+the energy which maintains every form of activity upon their
+surfaces and makes them habitable.
+
+\nbarticle{274.} \nbparatext{Its Distance and Dimensions.}---Its distance is determined by
+finding its horizontal parallax; that is, the semi-diameter of the
+earth as seen from the sun. The mean value of this parallax is
+probably very near $8''.8$,\footnote
+  {In the American Ephemeris the value deduced by Newcomb in 1867 is used, viz.,
+$8''.85$. The British ``Nautical Almanack'' uses the same value, and the French the
+value deduced by Leverrier a little earlier, $8''.86$; but more recent observations
+seem to show that this value is a little too large, and that the number stated,
+$8''.8$, is more probably correct. The difference is of no importance for \textit{almanac
+  purposes}.}
+plus or minus $0''.03$.
+
+We reserve to a separate \chapref{CHAPTERXVI}{chapter} the discussion of the methods by
+which this most fundamental and important of all astronomical data
+has been ascertained, merely remarking here that the problem is one
+of extreme \textit{practical} difficulty, though the principles involved are
+simple enough.
+
+Assuming the parallax at $8''.8$, the mean distance of the sun (putting
+$r$ for the earth's radius) equals
+\[
+r÷\sin{8''.8} = 23,\!439× r.
+\]
+
+With Clarke's value of $r$ (\artref{Art.}{145}), this gives 149,500000 kilometers,
+%% -----File: 184.png---Folio 173-------
+or 92,897,000 miles; which, however, is uncertain by at least 200,000
+miles, and is \textit{variable}, also, to the extent of about three million
+miles on account of the eccentricity of the earth's orbit, the earth
+being nearer the sun in December than in June.
+
+\nbarticle{275.} This distance is so much greater than any with which we
+have to do on the earth that it is possible to reach a conception of it
+only by illustrations of some sort. Perhaps the simplest is that
+drawn from the motion of a railway train. Such a train going 1000
+miles a day (nearly forty-two miles an hour, and faster than the
+Chicago Limited on the Pennsylvania Railroad) would take $254\frac{1}{3}$
+years to make the journey.
+
+If sound were transmitted through interplanetary space, and at the
+same rate as through our own atmosphere, it would make the passage
+in about fourteen years; \textit{i.e.}, an explosion on the sun would be
+heard by us fourteen years after it actually occurred. A cannon-ball
+moving unretarded, at the rate of 1700 feet per second, would
+travel the distance in nine years. Light does it in 499 seconds.
+
+\sloppy
+\nbarticle{276.} \nbparatext{Diameter.}---The sun's mean apparent diameter is $32'\ 04''\pm2''$.
+Since at the sun one second equals 450.36 miles, its diameter
+equals 866,500 miles, or $109\frac{1}{2}$ times the diameter of the earth. It
+is quite possible that this diameter is variable to the extent of a few
+%% -----File: 185.png---Folio 174-------
+hundred miles, since, as will appear hereafter, the sun (at least the
+surface which we see) is not solid.
+
+\fussy
+Representing the sun by a globe two feet in diameter, the earth
+would be $\frac{22}{100}$ of an inch in diameter,---the size of a very small pea,
+or a ``22-calibre'' round pellet. Its distance from the sun on that
+scale would be just about 220 feet, and the \textit{nearest star} (still on the
+same scale) \textit{would be eight thousand miles away, at the antipodes}.
+
+If we were to place the earth in the centre of the sun, supposing
+it to be hollowed out, the sun's surface would be 433,000 miles away
+from us. Since the distance of the moon is only about 239,000
+miles, it would be only a little more than half-way out from the
+earth to the inner surface of the hollow globe, which would thus
+form a very good background for the study of the lunar motions.
+
+\begin{fineprint}
+It is perhaps worth noticing, as a help to memory, that the sun's diameter
+exceeds the earth's just about as many times as it is itself exceeded by the
+radius of the earth's orbit; or, in other words, the sun's diameter is \textit{nearly} a
+mean proportional between the earth's distance from the sun and the earth's
+diameter, 110 being the common ratio.
+\end{fineprint}
+
+\includegraphicsmid{illo090}{\textsc{Fig.~90.}---Dimensions of the Sun compared with the Moon's Orbit.}
+
+\nbarticle{277.} \nbparatext{Surface and Volume.}---Since the \textit{surfaces} of globes are proportional
+to the \textit{squares} of their radii, the surface of the sun exceeds
+that of the earth in the ratio of $(109.5)^2$ to 1; that is, its surface is
+about 12,000 times the surface of the earth.
+
+The \textit{volumes} of spheres are proportional to the \textit{cubes} of their radii;
+hence the \textit{sun's volume} is $(109.5)^3$, or 1,300000 times that of the earth.
+
+\nbarticle{278.} \nbparatext{The Sun's Mass.}---The mass of the sun is \textit{very nearly three
+hundred and thirty-two thousand times that of the earth}, subject to a
+probable error of at least one per cent. There are various ways of
+getting at this result. For our purpose here, perhaps the most convenient
+is by comparing the earth's attraction for bodies at her surface
+(as determined by pendulum experiments) with the attraction of the
+sun for the earth,---the central force which keeps her in her orbit.
+Put $f$ for this force (measured, like gravity, by the velocity it generates
+in one second), $g$ for the force of gravity (32~feet 2~inches
+per second), $r$ the earth's radius, $R$ the sun's distance, and let $E$
+and $S$ be the masses of the earth and sun respectively. Then, by
+the law of gravitation, we have the proportion
+\begin{flalign*}
+            &&&f : g :: \frac{S}{R^2} : \frac{E}{r^2}, \text{ or } S = E\left(\frac{f}{g}\right)\left(\frac{R}{r}\right)^2. &&&(a)&\\
+&\text{Now, }&&\frac{R}{r} = 23,440 \text{ (nearly)}.&&&\phantom{Now, }&
+\end{flalign*}
+%% -----File: 186.png---Folio 175-------
+Its square equals 549,433,600. $g = 386$ inches. To find $f$ we have
+from Mechanics (Physics, p.~62),
+\[
+  f = \frac{V^2}{R}, \tag{\textit{b}}
+\]
+this being the expression for the ``central force'' in the case of a
+body revolving in a circle. (We may neglect the eccentricity of the
+earth's orbit in a merely approximate treatment of the problem.)
+$V$ is the orbital velocity of the earth, which is found by dividing the
+circumference of the orbit, $2\pi R$, by $T$, the number of seconds in a
+sidereal year. This velocity comes out 18.495 miles per second.
+Putting this into formula (\textit{b}), we get $f = 0.2333$ inches,
+\begin{flalign*}
+&\text{so that }&  &\frac{f}{g} = 0.0006044 = \frac{1}{1654} \text{ (nearly)};&&\\
+&\text{whence }&  &S = E× \frac{1}{1654}× 519,\!433,\!600; \text{ or }S \text{ equals 332,000.}&&
+\end{flalign*}
+We may note in passing that half of $f$ expresses the distance by
+which the earth \textit{falls towards the sun} every second, just as half $g$ is
+the distance a body at the earth's surface falls in a second. This
+quantity (0.116 inch), a trifle more than a ninth of an inch, is the
+amount by which the earth's orbit deviates from a straight line in a
+second. In travelling \textit{eighteen and one-half miles} the deflection is
+only \textit{one-ninth of an inch}.
+
+\begin{fineprint}
+\nbarticle{278*.} By substituting $\dfrac{2\pi R}{T}$ for $V$ in equation (\textit{b}), we get
+\begin{flalign*}
+&&&f=\frac{4{\pi}^2 R}{T^2};&&
+\intertext{and putting this value $f$ into equation (\textit{a}) and reducing, we obtain}
+&&&S = E\left[ \left( \frac{4{\pi}^2}{T^2} \right)
+               \left( \frac{r}{g}\right)
+               \left( \frac{R}{r}\right)^3 \right],  &(\textit{c})&\\
+&\text{or, since }&  &\frac{R}{r}=\frac{1}{\sin p}  &\phantom{or\ since }&
+\intertext{($p$ being the sun's horizontal parallax), we have finally}
+&&&S = E\left[ \left( \frac{4{\pi}^2}{T^2} \right)
+               \left( \frac{r}{g} \right) \frac{1}{\sin^3p} \right].   &(\textit{d})&
+\end{flalign*}
+
+It will be noticed that in this expression the \textit{cube} of the parallax appears,
+and this is the reason why an uncertainty of one per cent in $p$ involves an
+uncertainty of three per cent in $S$.
+%% -----File: 187.png---Folio 176-------
+
+In obtaining the mass of the sun it will be seen that we require as
+data, $T$, the length of the sidereal year in seconds; the value of
+gravity, $g$ (which is derived from pendulum experiments); the radius
+of the earth, $r$ (deduced from geodetic surveys); and finally (and
+most difficult to get), the sun's parallax, $p$, or else, what comes to
+the same thing, the ratio $\dfrac{r}{R}$.
+\end{fineprint}
+
+\nbarticle{279.} \nbparatext{The Sun's Density.}---This density\footnote
+  {The determination of the sun's density does not \textit{necessarily} involve its parallax.
+Put $\rho$ for the sun's radius, and $Ds$ for its density: also let $De$ be the earth's mean
+density. Substitute in equation ($c$), and we have $\frac{4}{3}\pi\rho^{3}Ds = \frac{4}{3}\pi r^{3} De \left(\dfrac{r}{g}\right) \left(\dfrac{R}{r}\right)^{3}$,
+whence $Ds = De \left(\dfrac{r}{g}\right)  \left(\dfrac{R}{\rho}\right)^{3}$. But $\left(\dfrac{\rho}{R}\right) = \sin{\Sigma}$, $\Sigma$ being the sun's (angular) semi-diameter.
+   Hence, finally, $Ds = De \left(\dfrac{r}{g}\right) \dfrac{1}{\sin^{3}{\Sigma}}$.}
+as compared with that of
+the earth is found by simply dividing its mass by its volume (both as
+compared with the earth); that is, it equals the fraction
+\[
+\frac{332\, 000}{1300\, 000} = 0.255,
+\]
+a little more than \textit{a quarter} of the earth's density. To get its
+``\textit{specific gravity}'' (\textit{i.e.}, density as compared with \textit{water}), we must
+multiply this by 5.58, the earth's mean specific gravity. This gives
+1.41; that is, \textit{the sun's mean density is not $1\frac{1}{2}$ times that of water},---a
+most significant result as bearing on its physical condition.
+
+\nbarticle{280.} \nbparatext{Superficial Gravity.}---This is found by dividing its mass by
+the square of its radius; that is,
+\[
+\frac{332\, 000}{(109\frac{1}{2})^{2}},
+\]
+which equals 27.6. A body weighing one pound on the earth's surface
+would there weigh 27.6 lbs. A body would fall 444 feet in a
+second, instead of sixteen feet, as here.
+
+\nbarticle{281.} \nbparatext{The Sun's Rotation.}---The sun's surface often shows spots
+upon it, which pass across the disc from east to west. These are
+evidently attached to its surface, and not bodies circling around the
+sun at a distance above it, as was imagined by some early astronomers,
+because, as Galileo early demonstrated, they continue in sight
+just as long as the time during which they are invisible; which would
+not be the case if they were at any considerable elevation.
+%% -----File: 188.png---Folio 177-------
+
+\textbf{Period of Rotation.}---The average time occupied by a spot in passing
+around the sun and returning to the same position again is 27.25
+days,---\textit{average} because different spots show considerable differences
+in this respect. This interval, however, is not the \textit{true time} of solar
+rotation, but the \textit{synodic}, since the earth advances in the interval of
+a revolution so that the sun has to turn on its axis a little farther each
+time to bring the spot again into conjunction with the earth. The
+equation by which the true period is deduced from the synodic is the
+same as in the case of the moon (\artref{Art.}{232}), viz.:
+\begin{flalign*}
+&&  &\frac{1}{T} - \frac{1}{E} = \frac{1}{S},  &&
+\intertext{$T$ being the true period of the sun's rotation, $E$ the length of the
+year, and $S$ the observed \textit{synodic} rotation;}
+&\text{whence }&  &\frac{1}{T} = \frac{1}{27.25} + \frac{1}{365.25},
+&&\phantom{whence }
+\end{flalign*}
+which gives $T = 25^{\text{d}}.35$. Different observers get slightly different
+results. Carrington finds $25^{\text{d}}.38$; Spoerer, $25^{\text{d}}.23$.
+
+\nbarticle{282.} \nbparatext{Position of the Sun's Axis.}---On watching the spots with
+care as they cross the disc, it appears that they usually describe paths
+more or less oval, showing that the sun's axis is inclined to the
+ecliptic. Twice a year, however, the paths become straight, at the
+times when the earth is in the plane of the sun's rotation. These
+dates are about June~3 and Dec.~5.
+
+\begin{fineprint}
+The ascending node of the sun's equator is in celestial longitude $73°\, 40'$
+(Carrington), and the inclination of its equator to the plane of the ecliptic
+is $7°\, 15'$. Its inclination to the plane of the terrestrial equator is $26°\, 25'$.
+The position of the point in the sky towards which the sun's pole is directed
+is in right ascension $18^{\text{h}}\ 44^{\text{m}}$, declination $+64°$, almost exactly half-way
+between the bright star $\alpha$~Lyræ and the Pole Star.
+\end{fineprint}
+
+\nbarticle{283.} \nbparatext{Peculiar Law of the Sun's Rotation.}---\textit{Equatorial Acceleration}.
+The earth rotates \textit{as a whole}, every point on its surface making
+its diurnal revolution in the same time; so also with the moon and
+with the planet Mars. Of course it is necessarily so with any \textit{solid}
+globe. But this is not the case with the sun. It was noticed quite
+early that the different spots give different results for the rotation
+period, but the researches of Carrington about thirty years ago first
+%% -----File: 189.png---Folio 178-------
+brought out the fact that the differences follow a regular law, showing
+that at the solar equator the time of rotation is less than on either
+side of it. Thus spots near the sun's equator give $T = 25$ days; at
+solar latitude $20°$, $T= 25.75$ days; at solar latitude $30°$, $T = 26.5$
+days; at solar latitude $40°$, $T = 27$ days. The time of rotation in
+latitude $40°$ is fully two days longer than at the solar equator; but we
+are unable to follow the law further towards the poles, because the
+spots are rarely found beyond the parallels of $45°$ on each side of the
+equator, and there are no well-defined markings between this point
+and the poles by which we can accurately determine the motion.
+
+\begin{fineprint}
+\nbarticle{284.} Various formulæ have been proposed to represent this law of rotation.
+Carrington gives for the daily motion of a spot $X = 865' - 165'× \sin^{\frac{7}{4}}{l}$,
+$l$ being the solar latitude of the spot. Faye, from the same observations, considering
+that the exponent $\frac{7}{4}$ could have no physical justification, deduced
+$X = 862' - 186'× \sin^{2}{l}$, which agrees almost as well with the observations.
+Still other formulæ have been deduced by Spoerer, Zöllner, and Tisserand,
+all giving substantially the same results.
+\end{fineprint}
+
+The law, in any case, is simply \textit{empirical}; that is, it is deduced from
+the observations, without being based upon any satisfactory physical
+explanation, for no such explanation of this strange equatorial acceleration
+has yet been found. Probably it has its origin somehow in
+the effects produced by the outpour of heat from the sun's surface;
+still, just how such a result should follow in the case of a cooling
+globe, of which the particles are free to move among each other, is
+not yet evident.
+
+\begin{fineprint}
+\nbarticle{285.} It is possible that the spots move on the surface of the sun, changing
+their places just as do clouds or railroad trains upon the surface of the
+earth, so that their motion does not represent the sun's true rotation. This
+however, as we shall see later, is hardly probable, and if it were the case, it
+would still be no less difficult to account for this systematic difference in
+their behavior at the solar equator and in the higher latitudes.
+
+\includegraphicsouter{illo091}{\textsc{Fig.~91.}---Telescope and Screen.}
+
+It has been suggested that the spots may be due to the fall of matter from
+a considerable elevation above the sun's surface, matter which has remained
+at that elevation for some time, and acquired a corresponding velocity of
+rotation due to that elevation. It can be shown that if the matter forming
+the spots had thus fallen from an elevation of about 20,000 miles, it would
+account for their apparent acceleration. Matter so falling would have an
+apparent eastward motion, just as do bodies on the earth when falling from
+the summit of a tower (\artref{Art.}{138}). From this point of view it is very interesting
+to inquire whether the minuter markings upon the sun's surface, such
+as the ``granules,'' to be spoken of very soon, do, or do not, possess the same
+%% -----File: 190.png---Folio 179-------
+rate of motion as the spots. There is no decisive observational\footnote
+  {Mr.~Crew has recently made at Baltimore, under the direction of Professor
+Rowland, an extensive series of observations upon the displacement of the lines
+of the spectrum at the eastern and western limbs of the sun. This displacement,
+which is very slight, is due, according to Doppler's principle (\artref{Art.}{321}), to the
+rotation of the sun; and Mr.~Crew's results, so far as they can be considered
+decisive, go to show that the absorbing layer of gases by which the Fraunhofer
+lines are formed \textit{does not} behave like the sun spots, but is slightly \textit{retarded} at the
+sun's equator. The observations are so delicate, however, that the conclusion,
+though made very probable, can hardly be considered to be absolutely proved
+  beyond question.}
+evidence at
+present that they do not. But the subject is an extremely difficult one; and
+yet important, because the solution of the problem of the sun's equatorial
+acceleration will probably throw much light upon its real constitution.
+\end{fineprint}
+
+\nbarticle{286.} \nbparatext{The Phenomena of the Sun's Surface.}---In order to study
+the sun with the telescope it is necessary to be provided with some
+special forms of apparatus. Its heat
+and light are so intense that it is
+impossible to look directly at it, as
+we do at the moon. A very convenient
+method of exhibiting the sun
+to a number of persons at once is simply
+to attach to the telescope a frame
+carrying a screen of white paper at a
+distance of a foot or more from the
+eye-piece, as shown in \figref{illo091}{Fig.~91}.  On
+pointing the instrument to the sun and
+properly adjusting the focus, a distinct image is formed on the
+screen, which shows the main features very fairly. It is, however,
+much more satisfactory to look at it directly, with a proper eye-piece.
+With a small telescope, not more than two and a half or
+three inches in diameter, a mere dark glass between the eye-piece and
+the eye can be used, but this dark glass soon becomes very hot, and
+is apt to crack. With larger instruments, it is necessary to use
+eye-pieces especially designed for the purpose and known as \textit{solar
+eye-pieces} or \textit{helioscopes}.
+
+\begin{fineprint}
+\includegraphicsouter{illo092}{\textsc{Fig.~92.}---Herschel Eye-piece.}
+
+The simplest of them, and a very good one for ordinary purposes, is one
+known as Herschel's, in which the sun's rays are reflected at right angles by a
+plane of unsilvered glass (\figref{illo092}{Fig.~92}). This reflector is made either of a prismatic
+form or concave, in order that the reflection from the back surface may not
+%% -----File: 191.png---Folio 180-------
+interfere with that from the front. About nine-tenths of the light passes
+through this reflector, and is allowed to pass
+out uselessly through the open end of the
+tube. The remaining tenth is sent through
+the eye-piece, and though still too intense
+for the eye to endure, it requires only a
+comparatively thin shade of neutral-tinted
+glass to reduce it sufficiently, and in this
+case the shade does not become uncomfortably
+heated. It is well to have the shade-glass
+made wedge-shaped,---thinner at one
+end than at the other---so that one can
+choose the particular thickness which is
+best adapted to the magnifying power
+employed.
+
+\nbarticle{287.} The polarizing eye-pieces are still better when well made. In
+these the light is reflected twice at plane surfaces of glass at the ``angle of
+polarization'' (Physics, p.~480), and is then received on a second pair
+of reflectors of black glass. When
+the upper pair of reflectors is in either
+of the two positions shown in \figref{illo093}{Fig.~93},
+a strong beam of light is received at
+$C$,---too strong for the eye to bear,
+although more than ninety per cent
+of it has already been rejected; but
+by simply turning the box which
+carries the upper reflectors one-quarter
+of a revolution around the line
+$BB'$ as an axis, the light may be
+wholly extinguished; and any desired
+gradation may be obtained by setting
+it at the proper angle, without the
+use of a shade-glass.
+\end{fineprint}
+
+\nbarticle{288.} It may be asked why it
+will not answer merely to ``cap''
+the object-glass, and so cut off
+part of the light, instead of rejecting
+it after it has once been
+allowed to enter the telescope. It
+is because of the fact, mentioned in \artref{Article}{43}, that the smaller the
+object-lens of the telescope, the larger the image it makes of a luminous
+point, or the wider its image of a sharp line. To cut down the
+%% -----File: 192.png---Folio 181-------
+aperture, therefore, is to sacrifice the definition of delicate details.
+With a low power there is no objection to reducing the amount of
+heat admitted into the telescope tube in that way, but with the higher
+powers the whole aperture should always be used.
+
+\includegraphicsouter{illo093}{\textsc{Fig.~93.}---Polarizing Helioscope.}
+
+\nbarticle{289.} \hspace{-3.2pt}\nbparatext{Photography.}---In the study of the sun's surface photography
+is for some purposes very advantageous and much used. The
+instrument must have a special object-glass (\artref{Article}{42}), with an apparatus
+for the quick exposure of plates. Such instruments are called
+photo-heliographs, and with them photographs of the sun are made daily
+at numerous observatories. The necessary exposure varies from $\frac{1}{500}$
+to $\frac{1}{10}$ of a second, in different cases. The pictures made by these
+instruments are usually from two inches up to eight or ten inches in
+diameter, and some of Janssen's, made at Meudon, bear enlarging
+up to forty inches in diameter. Photographs have the advantage of
+freedom from prejudice and prepossession on the part of the observer;
+but they take no advantage of the instants of fine seeing.
+They represent the surface as it happened to be at the moment
+when the plate was uncovered.
+
+\begin{fineprint}
+\nbarticle{290.} The study of the sun has become so important from a scientific
+point of view that several observatories have recently been established
+mainly for that purpose, though most of them connect with it that of other
+topics in astronomical physics. The two most important of these solar or
+astro-physical observatories, are the observatory at Meudon and the so-called
+``Sonnenwarte'' at Potsdam. There ought to be one in this country.
+\end{fineprint}
+
+\nbarticle{291.} \nbparatext{General Views.}---Before passing to a discussion of the
+details of the different solar phenomena, it will be well to give a
+very brief summary of the objects and topics to be considered.
+
+1. \textit{The photosphere; i.e.}, the luminous surface of the sun directly
+visible to our telescopes. It is probably a sheet of \textit{luminous clouds}
+formed by condensation into little drops and crystals (like the water-drops
+and ice-crystals in our terrestrial clouds) of certain substances
+which within the central mass of the sun exist in a gaseous form,
+but are cooled at its surface below the temperature necessary for
+their condensation; perhaps such substances as carbon, boron, and
+silicon. The granules, faculæ, and spots are all phenomena in this
+photosphere.
+
+2. The so-called ``\textit{reversing layer}'' is a stratum of unknown thickness,
+but probably shallow, just above the photosphere, containing
+the vapors of many of the familiar terrestrial elements; of which
+%% -----File: 193.png---Folio 182-------
+the presence, and to some extent their physical condition, can be investigated
+by means of the spectroscope.
+
+\includegraphicsouter{illo094}{\textsc{Fig.~94.}\\
+Constitution of the Sun. From ``The Sun,'' by permission
+of the Publishers.}
+
+3. Above the photosphere, interpenetrating the atmosphere of
+vapors just spoken of,
+and perhaps indistinguishable
+from it, is an envelope
+of \textit{permanent} gases; that
+is, gases which, under the
+solar conditions, cannot
+be condensed into clouds
+of solid or liquid particles.
+Among them hydrogen
+is most conspicuous.
+This envelope is the so-called
+\textit{Chromosphere}; and
+from it the \textit{prominences}
+of various kinds rise,
+sometimes to the height
+of hundreds of thousands
+of miles. These beautiful
+objects are best seen
+at total eclipses of the
+sun, but to a certain extent
+they can also be studied
+at any time by the
+help of a spectroscope.
+
+4. Higher yet rises the
+mysterious \textit{Corona}, of material
+still less dense, and so
+far observable only during
+total eclipses of the sun.
+
+Fig.~94 shows the relative positions of these different elements of
+the solar constitution.
+
+5. A fifth subject deals with the \textit{measurement of the sun's light}
+and the relative brightness of different parts of the solar surface.
+
+6. Another most interesting and important topic relates to the
+amount of \textit{heat} radiated by the sun,---the sun's \textit{probable temperature}
+and the mechanism by which its heat-supply is maintained.
+
+\includegraphicsmid{illo095}{\textsc{Fig.~95.}\\
+The Great Sun Spot of September, 1870, and the Structure of the Photosphere. From a Drawing
+by Professor Langley. From ``The New Astronomy,'' by permission of the Publishers.}
+
+\nbarticle{292.} \nbparatext{The Photosphere.}---The sun's visible surface is called the
+\textit{photosphere}, and when studied under favorable atmospheric conditions,
+%% -----File: 194.png---Folio 183-------
+with rather a low magnifying power, it looks like rough drawing-paper.
+With higher powers it is seen to be, as shown in
+Fig.~95, made up of a comparatively darkish background sprinkled
+over with grains, or ``nodules,'' as Herschel called them, of something
+much more brilliant,---like \DPtypo{snowflakes}{snow-flakes} on gray cloth, according
+to Langley. These are from 400 to 600 miles across, and in the
+finest seeing are themselves resolved into more minute ``granules.''
+For the most part, these nodules are about as broad as they are long,
+though of irregular form; but here and there, especially in the
+neighborhood of the spots, they are drawn out into long streaks.
+Nasmyth seems first to have observed this structure, and called the
+filaments ``willow leaves.'' Secchi called them ``rice grains.''
+According to Huggins they were ``dots''; and there was for a long
+time a pretty lively controversy as to their true form. Their shape,
+however, unquestionably varies very much in different parts of the
+%% -----File: 195.png---Folio 184-------
+surface and under different circumstances. They are probably luminous
+clouds floating in a less luminous atmosphere.
+
+Near the edge the photosphere appears generally much less brilliant;
+but certain bright streaks called ``faculæ'' (from \textit{fax}, a
+torch), which though visible are not very obvious at points further
+from the limb, become there conspicuous. These faculæ are elevations,---masses
+of the same material as the rest of the photosphere
+but elevated above the general level and intensified in brightness.
+When one of them passes off the edge of the sun, it is sometimes
+seen as a little projection. They are most abundant near the sun-spots,
+and they are more conspicuous near the edge of the disc, as
+shown in \figref{illo096}{Fig.~96}, because the sun's surface is overlaid by a gaseous
+atmosphere which absorbs more of the light there than it does near
+the centre, and these faculæ push up through it like mountains.
+
+\includegraphicsmid{illo096}{\textsc{Fig.~96.}---Faculæ at Edge of the Sun. (De La Rue.)}
+
+\nbarticle{293.} \nbparatext{The Sun Spots.}---Whenever these are present upon the sun's
+surface, they are the most conspicuous objects to be seen upon it.
+The appearance of a normal sun spot, \figref{illo097}{Fig.~97}, fully formed and not
+yet beginning to break up, is that of a dark central ``\textit{umbra},'' more
+or less nearly circular, with a fringing ``\textit{penumbra},'' composed of
+filaments directed radially. The umbra itself is not uniformly dark
+throughout, but is overlaid with filmy clouds which require a good
+telescope and helioscope to make them visible. Usually, also, in the
+%% -----File: 196.png---Folio 185-------
+umbra there are several round and very black spots, which are sometimes
+called ``\textit{nucleoli},'' but are often referred to us ``Dawes'
+holes,'' after the name of their first discoverer. But while this is the
+appearance of what may be taken as a normal spot, very few are
+strictly normal. Most of them are more or less irregular in form.
+They are often gathered in groups with a common penumbra, and
+partly covered by brilliant ``\textit{bridges}'' extending across from the outside
+photosphere. Often the umbra is out of the centre of the penumbra,
+or has a penumbra only on one side, and the penumbral
+filaments, instead of being strictly radial, are frequently distorted in
+every conceivable way. In fact, the normal spots form a very small
+proportion of the whole number.
+
+\includegraphicsmid{illo097}{\textsc{Fig.~97.}---A Normal Sun Spot. (Secchi; modified.)}
+
+The darkest portions of the umbra are dark only by contrast.
+Photometric observations (by Langley) show that even the nucleus
+gives at least one per cent as much light as a corresponding area
+of the photosphere; that is to say, as we shall see hereafter, the
+darkest portion of a sun spot is brighter than a calcium light.
+
+\nbarticle{294.} The spots are unquestionably \textit{cavities} or \textit{depressions} in the
+photosphere, filled with gases and vapors which are cooler than the
+surrounding portions, and therefore absorb a considerable proportion
+of light. The fact that they are \textit{cavities} is shown by the change
+%% -----File: 197.png---Folio 186-------
+in the appearance of a spot as it approaches the edge of the disc.
+When a normal spot is near the centre of the disc, the nucleus is
+nearly central. As it approaches the edge, the penumbra becomes
+wider on the outer edge and narrower on the inner, and just before
+the spot disappears around the limb of the sun, the penumbra on the
+inner edge entirely disappears,---the appearance being precisely
+such as would be shown by a saucer-shaped cavity in the surface
+of a globe, the bottom of the cavity being painted black to represent
+the umbra, and the sloping sides gray for the penumbra. \figref{illo098}{Fig.~98}
+represents the phenomena in a schematic way. Observations
+upon a single spot would hardly be sufficient to substantiate this,
+because the spots are so irregular in their form; but by observing
+the behavior of several hundred of them the truth comes out quite
+clearly. Occasionally, when a very large spot passes off the sun's
+limb, the depression can be seen with the telescope.
+
+\includegraphicsmid{illo098}{\textsc{Fig.~98.}---Sun Spots as Cavities.}
+
+The fact was first discovered by Wilson of Glasgow something
+more than a hundred years ago. Previously it had very commonly
+been supposed that the spots were \textit{elevated} above the general surface
+of the sun, and the idea still survives in certain quarters, though certainly
+incorrect.
+
+\nbarticle{295.} The \textit{penumbra} is usually composed of ``thatch-straws,'' or
+long drawn-out granules of photospheric matter, which, as has been
+said, converge in a general way towards the centre of the spot. At
+the inner edge the penumbra, from the convergence of these filaments,
+is usually brighter than the outer. The inner ends of the filaments
+are generally club-formed; but sometimes they are drawn out into
+fine points, which seem to curve downward into the umbra like the
+rushes over a pool of water. The outer edge of the penumbra is
+usually pretty definite, and the penumbra there is darker. Around
+%% -----File: 198.png---Folio 187-------
+the spot the photosphere is much disturbed and elevated into faculæ,
+which sometimes radiate outward from the spot like streams of
+lava from a crater, though, of course, they are really nothing of the
+sort.
+
+\nbarticle{296.} \nbparatext{Dimensions of Sun Spots.}---The diameter of the umbra of
+a sun spot ranges all the way from 500 to 1000 miles in the case of
+a very small one, to 50,000 or 60,000 miles in the case of the larger
+ones. The penumbra surrounding a group of spots is sometimes
+150,000 miles across, though that would be rather an exceptional
+size. Not infrequently sun spots are large enough to be seen by the
+naked eye, and they have been often so seen at sunset or through a
+fog. The depth by which the umbra is depressed below the general
+surface of the photosphere is very difficult to determine, but according
+to Faye, Carrington, and others, it seldom exceeds 2500 miles,
+and more often is between 500 and 1500.
+
+\nbarticle{297.} \nbparatext{Development and Changes of Form.}---Generally the origin
+of a sun spot fails to be observed. It begins from an insensible
+point, and rapidly grows larger, the penumbra usually appearing
+only \textit{after the nucleus is fairly developed}.
+
+If the disturbance which causes the spot is violent, the spot usually
+breaks up into several fragments, and these again into others which
+tend to separate from each other. At each new disturbance the forward
+portions of the group show a tendency to advance eastward
+on the sun's surface, leaving behind them a trail of smaller spots.
+
+\begin{fineprint}
+\nbarticle{298.} The ``segmentation'' of a spot, as Faye calls it, is usually effected
+by the formation of a ``bridge,'' or streak of brilliant light, which projects
+itself across the penumbra and umbra from the outside photosphere. These
+bridges are mere extensions of the surrounding faculæ, and are often intensely
+bright.
+
+Occasionally a spot shows a distinct cyclonic motion, the filaments being
+drawn inward spirally; and in different members of the same group of spots
+the cyclonic motions are not seldom in opposite directions.
+
+When a spot at last vanishes it is usually by the rapid encroachment of
+the photospheric matter, which, as Secchi expresses it, appears to ``fall pell-mell
+into the cavity,'' completely burying it and leaving its place covered
+by a group of faculæ. Figs.~99--104 (see page~\figpage{99-104}) show the changes which
+took place in the great spot of September, 1870. They are from photographs
+by Mr.~Rutherfurd of New York, and are borrowed from ``The New Astronomy''
+of Professor Langley, through the courtesy of his publishers.
+\end{fineprint}
+%% -----File: 199.png---Folio 188-------
+
+\nbenlargepage
+\nbarticle{299.} Spots within $15°$ or $20°$ of the sun's equator generally, on
+the whole, drift a little \textit{towards} it, while these in higher latitudes
+drift \textit{away from} it; but the motion is slight, and exceptions are frequent.
+
+In and around the spot itself the motion is usually \textit{inward towards
+the centre}, and \textit{downward at} the centre. Not infrequently the fragments
+at the inner end of the penumbral filaments appear to draw
+off, move towards the centre of the spot, and then descend. Occasionally,
+though seldom, the motion is vigorous enough to be detected
+by the displacement of lines in the spectrum.
+
+\begin{includegraphicspage}{99-104}
+  \begin{tabular}{c@{\qquad\qquad}c}
+    \includegfx{illo099}           & \includegfx{illo100}\\
+    \footnotesize\textsc{Fig.~99.}---Sept.~19.  & \footnotesize\textsc{Fig.~100.}---Sept.~20.\\[4ex]
+    \includegfx{illo101}           & \includegfx{illo102}\\
+    \footnotesize\textsc{Fig.~101.}---Sept.~21. & \footnotesize\textsc{Fig.~102.}---Sept.~22.\\[4ex]
+    \includegfx{illo103}           & \includegfx{illo104}\\
+    \footnotesize\textsc{Fig.~103.}---Sept.~23. & \footnotesize\textsc{Fig.~104.}---Sept.~26.\\[4ex]
+  \end{tabular}
+  \captionof*{figure}{The Great Sun Spot of 1870.}
+\end{includegraphicspage}
+
+\nbarticle{300.} \nbparatext{Duration.}---The duration of the spots is very various, but,
+astronomically speaking, they are always short-lived phenomena,
+sometimes lasting for only a few days, more frequently, perhaps, for
+a month or two. In a single instance, a spot has been observed
+through as many as eighteen successive revolutions of the sun.
+
+\nbarticle{301.} \nbparatext{Distribution.}---It is a significant fact that the spots are confined
+mostly to two zones of the sun's surface between $5°$ and $40°$ of
+latitude north and south. A few are found near the equator, none
+beyond the latitude of $45°$. \figref{illo105}{Fig.~105} shows the distribution of several
+thousand spots as observed by Carrington and Spörer.
+
+\includegraphicsmid{illo105}{\textsc{Fig.~105.}---Distribution of Sun Spots in Latitude.}
+
+Occasionally, what Trouvelot calls ``veiled spots'' are seen beyond
+the $45°$ limits---grayish patches surrounded by faculæ, which look as
+if a dark mass were submerged below the surface and dimly seen
+through a semi-transparent medium.
+%% -----File: 200.png---Folio 189-------
+
+\nbarticle{302.} \nbparatext{Theories as to the Nature of the Spots.}---We first mention
+(\textit{a})~the theory of Sir William Herschel, because it still finds place
+in certain text-books, though certainly incorrect. His belief was
+that the spots were openings through two luminous strata, which he
+supposed to surround the central globe of the sun. This globe he
+supposed to be \textit{dark} (\textit{and even habitable!}). The outer stratum, the
+photosphere, was the brighter of the two, and the opening in it the
+larger, while the inner shell between it and the solid globe was of less
+luminous substance, and formed the penumbra. He thought the opening
+through these might be caused by volcanoes on the globe beneath.
+
+\nbarticle{303.} (\textit{b}) Another theory, now abandoned, was proposed independently
+both by Secchi and Faye about 1868. They supposed that
+the spots were openings in the photosphere caused by the bursting
+outward of the imprisoned gases underneath it; the photosphere at
+that time being supposed to be \textit{liquid}.
+
+\begin{fineprint}
+They explained the darkness of the centre of the spot by the fact that a
+heated gas at a given temperature has a lower radiating power and sends
+out much less light than a \textit{liquid surface}, or than \textit{clouds} formed by the condensation
+of the same material at even a lower temperature. This is true
+of gases at low pressure, but not of gases under great compression, such as
+must be the case within the body of the sun. Besides, if the gases possessed
+the small radiating power necessary to this theory, they would also possess
+small \textit{absorbing} power, and therefore would be transparent; the inner side of
+the photosphere on the opposite side of the sun would therefore be visible
+through the opening, so that the centre of such an eruption would not be
+\textit{dark}, but, if anything, brighter than the general solar surface. Moreover, as
+we now know from the spectroscopic evidence, the motion at the centre of a
+spot is \textit{inward}, not \textit{outward}.
+\end{fineprint}
+
+\nbarticle{304.} (\textit{c}) Faye more recently has proposed and now maintains a
+theory which has numerous good points about it, and is accepted by
+many, viz.: that the spots are analogous to storms on the earth, being
+\textit{cyclones}, due to the fact that the portions of the sun's surface near
+the equator make their revolution in a shorter time than these in
+higher latitudes. This causes a relative drift in adjacent portions of
+the photosphere, and according to him gives rise to \textit{vortices} or \textit{whirlpools}
+like these in swiftly running water. The theory explains the
+distribution of the spots (which abound precisely in the regions
+where this relative drift is at the maximum) and many other facts,
+such as their ``segmentation.'' According to it, however, \textit{all} spots
+should be cyclonic, and the spiral motion of all the spots in the
+%% -----File: 201.png---Folio 190-------
+northern hemisphere should be \textit{clock-wise}, while in the southern hemisphere
+they should be \textit{counter-clock-wise}. Now, as a matter of fact,
+only a very few of the spots show such spiral motions, and there is
+no such agreement in the general direction of the motion as the
+theory requires.
+
+\begin{fineprint}
+Faye attempts to account for this by saying that we do not see the vortex
+itself, but only the cloud of cooler materials which is drawn together by
+the down-rushing vortex, itself hidden beneath this cloud. Still, it would
+seem that in such a case the cloud itself should gyrate. Moreover, the
+relative drift of the adjacent portions of the photosphere is too small to
+account for the phenomena satisfactorily. In the solar latitude of $20°$ two
+points separated by $1'$ of the sun's surface (123 miles) have a relative daily
+drift of only about four and one-sixth miles, insufficient to produce any sensible
+whirling.
+\end{fineprint}
+
+\nbarticle{305.} (\textit{d}) Secchi's later theory. He supposed the spots to be due
+to eruptions from the inner portions of the sun's surface, not \textit{in} the
+spot, however, but only \textit{near} it; the spot itself being formed by
+the settling down upon the photosphere of materials thrown out by
+the eruption and cooled by their expansion and their motion through
+the upper regions. We have, however, in fact, as a usual thing, not
+a single eruption, but a ring of eruptions all around every large spot,
+all of them converging their bombardment, so to speak, upon the
+same centre,---a fact very difficult to explain if the spot originates
+in the eruption, but not difficult to understand if the eruptions
+are the result of the spot.
+
+Perhaps the true explanation may be that when an eruption occurs
+at any spot, the \textit{photosphere somewhere in the neighborhood settles
+down in consequence of the diminution of the pressure beneath}, thus
+forming a ``\textit{sink},'' so to speak, which is of course covered by a
+greater depth of cooler vapors above, and so looks dark.
+
+\nbarticle{306.} (\textit{e}) Mr. Lockyer, in his recent work on the chemistry of the
+sun, revives an old theory, first suggested by Sir John Herschel and
+accepted by the late Professor Peirce, that the spots are not formed
+by any action from within, but by \textit{cool matter descending from above},---matter
+very likely of meteoric origin; but it is difficult to see how
+the distribution of the spots with reference to the sun's equator can
+be accounted for in this way.
+
+On the whole it is impossible to say that the problem of the origin
+of sun spots is yet satisfactorily solved. There is no question that
+%% -----File: 202.png---Folio 191-------
+sun spots are closely associated with eruptions from beneath; but
+which is cause and which effect, or whether both are due to some
+external action, remains undetermined.
+
+\nbarticle{307.} \nbparatext{Periodicity of Sun Spots.}---In 1851 Schwabe of Dessau, by
+the comparison of an extensive series of observations running over
+nearly thirty years, showed that the sun spots are \textit{periodic}, being at
+times vastly more numerous than at others, with a roughly regular
+recurrence every ten or eleven years. This had been surmised
+by Horrebow more than a century before, though not proved.
+
+\sloppy
+Subsequent study fully confirms this remarkable result of Schwabe.
+Wolf of Zurich has collected all the observations discoverable and
+finds a pretty complete record back to 1610. From these records is
+constructed the annexed diagram, \figref{illo106}{Fig.~106}. The ordinates of the
+curve represent what Wolf calls his ``relative numbers,''\footnote
+  {This ``relative number'' is formed in rather an arbitrary manner from the
+observations which Wolf hunted up as the basis of his investigation. The
+formula is, $r$ (the relative number)${}- k (10g + f)$, in which $g$ is the number
+of groups and isolated spots observed, $f$ the total number of spots which can
+be counted in these groups and singly, while $k$ is a coefficient which depends
+upon the observer and the size of his telescope: it is large for a small telescope
+  and not very persistent observer, and approaches unity the more likely the
+  observer may be supposed to have noted every sun spot that appeared during
+  the time covered by his observations.}
+which he
+has adopted as representing the spottedness.
+%% -----File: 203.png---Folio 192-------
+
+\fussy
+\includegraphicsmid{illo106}{\textsc{Fig.~106.}---Wolf's Sun-Spot Numbers.}
+
+The average period is eleven and one-tenth years, but, as the \figref{illo106}{figure}
+shows, the spot maxima are quite irregular, both in time and as to
+the extent of spottedness. The last spot maximum occurred in
+1883--84 (a year or two behind time), and we are now (1888) approaching
+a minimum. During a maximum the surface of the sun
+is never free from spots, from twenty-five to fifty being frequently
+visible at once. During a minimum, on the other hand, weeks often
+pass without the appearance of a single one.
+
+\begin{fineprint}
+\nbarticle{308.} \nbparatext{Possible Cause of the Periodicity.}---The cause of this periodicity
+is not known. It has been attempted to connect it with planetary action.
+Some things in the Kew statistics of the sun spots look as if Venus, Mercury,
+and the Earth had something to do with it, the sun's surface being
+more spotted when these planets approach nearer; but the evidence is
+insufficient, or at least needs to be supplemented by further comparisons.
+Jupiter also has been suspected. His period is 11.86 years, which is not
+very different from the mean sun-spot period; but an examination of the
+different spot maxima show that some of them have occurred when he was
+near perihelion, and others when he was near aphelion; and on the whole
+there is very little reason for supposing that he has any considerable influence
+in the matter.
+
+Sir John Herschel suggested that it might be due to streams of meteors
+moving in an oval orbit with a period of about eleven years, and approaching
+so near at perihelion that numerous members of the meteoric group actually
+fall into the sun; but, as has been said before, the distribution of the
+spots would seem to contradict the idea.
+\end{fineprint}
+
+\nbarticle{309.} \nbparatext{Terrestrial Influence of the Sun Spots.}---One influence of
+the sun spots upon the earth is perfectly demonstrated. When the
+spots are numerous, magnetic disturbances (the so-called magnetic
+storms) are most numerous and violent upon the earth, a fact not to
+be wondered at since violent disturbances upon the sun's surface
+have been in many individual cases immediately followed by magnetic
+storms, with a brilliant exhibition of the Aurora Borealis. The
+nature and mechanism of the connection is as yet unknown, but of
+the fact there can be no question. The dotted lines in the figure of
+the sun-spot periodicity (\figref{illo106}{Fig.~106}) represent the magnetic storminess
+of the earth at the indicated dates; and the correspondence between
+these curves and the curve of spottedness makes it impossible to
+doubt the connection.
+%% -----File: 204.png---Folio 193-------
+
+\nbarticle{310.} It has been attempted, also, to show that greater or less
+disturbance of the sun's surface, as indicated by the greater frequency
+of the sun's spots, is accompanied by \textit{effects upon the meteorology
+of the earth}, upon its temperature, barometric pressure, storminess,
+and the amount of rain-fall. The researches of Mr.~Meldrum
+of Mauritius with respect to the cyclones in the Indian Ocean appear
+to bear out the conclusion that there may be some such connection in
+that case, but the general results are by no means decisive. In some
+parts of the earth the rain-fall seems to be greater during a spot
+maximum; in others, less.
+
+As to the temperature, it is still uncertain whether it is higher or
+lower at the time of a spot maximum. The \textit{spots themselves are
+cooler} (as Henry, Secchi, and Langley have shown) than the general
+surface of the photosphere; but their extent is never sufficient to
+reduce the amount of heat radiated from the sun by us much as $\frac{1}{1000}$
+part. On the other hand, when the spots are most numerous, the
+generally disturbed condition of the photosphere would, as Langley
+has shown, necessarily be accompanied by an increased radiation.
+
+Dr.~Gould considers that the meteorological records in the Argentine
+Republic between 1875 and 1885 show an indubitable connection
+between the \textit{wind currents} and the number of sun spots. But the
+\textit{demonstration} of such a relation really requires observations running
+through several spot periods. On the whole, it is now quite certain
+that whatever influence the sun spots exert upon terrestrial meteorology
+is very slight, if it exists at all.
+
+\section*{THE SOLAR SPECTRUM AND ITS REVELATIONS.}
+
+\nbarticle{311.}  About 1860 the spectroscope appeared in the field as a new
+and powerful instrument of astronomical research, at once resolving
+many problems as to the nature and constitution of the heavenly bodies
+which before had not seemed to be even open to investigation.
+
+The essential part of the apparatus is either a prism or train of
+prisms, or else a diffraction grating,\footnote
+  {The grating is merely a piece of glass or speculum metal, ruled with many
+thousand straight, equidistant lines, from 5000 to 20,000 in the inch. Usually the
+surface before ruling is accurately plane, but for some purposes the \textit{concave} gratings,
+  originated by Professor Rowland, are preferable.}
+which is capable of performing
+the same office of dispersing---that is, of spreading and sending in
+different directions---the light rays of different wave-lengths. If,
+with such a ``\textit{dispersion piece},'' as it may be called (either prism or
+grating), one looks at a distant point of light, as a star, he will see
+%% -----File: 205.png---Folio 194-------
+instead of a point a long streak of light, red at one end and violet
+at the other. If the object looked at be not a point, but a \textit{line of
+light} parallel to the edge of the prism or to the lines of the grating,
+then, instead of a mere colored streak without width, one gets a
+\textit{spectrum}, a colored band of light, which may show markings that
+will give the observer most valuable information. (Physics, pp.~458--460.)
+For convenience' sake it is usual to form this line
+of light by admitting the light through a narrow ``\textit{slit},'' which is
+at one end of a tube having at the other end an achromatic object-glass
+at such a distance that the slit is in its principal focus.
+This tube with slit and lens constitutes the ``\textit{collimator},'' so called because
+it is precisely the same as the instrument used in connection with
+the transit instrument to adjust its line of collimation (\artref{Article}{60}).
+
+Instead of looking at the spectrum with the naked eye, however,
+it is better in most cases to use a small telescope; called the ``\textit{view-telescope},''
+to distinguish it from the large telescope, to which the
+spectroscope is often attached.
+
+\nbarticle{312.} \nbparatext{Construction of the Spectroscope.}---The instrument, therefore,
+as usually constructed, and shown in \figref{illo107}{Fig.~107}, consists of three
+parts,---collimator, dispersion-piece, and view-telescope; but in the
+direct-vision spectroscope, shown in the \figref{illo107}{figure}, the view-telescope is
+omitted. If the slit, $S$, be illuminated by strictly homogeneous light
+%% -----File: 206.png---Folio 195-------
+all of one wave-length, say yellow, the ``real image'' of the slit
+will be found at $Y$. If at the same time light of a different
+wave-length be also admitted, say red, a second image will be
+formed at $R$, and the observer will see a spectrum with two ``bright
+lines,'' the lines being really nothing more than \textit{images of the slit}. If
+light from a candle be admitted, there will be an infinite number of
+these slit-images, close packed, like the pickets in a fence, without
+interval or break, and we then get a continuous spectrum; but if we
+look at sunlight or moonlight, we shall find a spectrum continuous in
+the main, but crossed by numerous dark lines, as if some of the
+``pickets'' had been knocked off, leaving gaps.
+
+\includegraphicsmid{illo107}{\textsc{Fig.~107.}---Different Forms of Spectroscope.}
+
+\sloppy
+\nbarticle{313.} \nbparatext{Integrating and Analyzing Spectroscope.}---If we simply
+direct the collimator of a spectroscope towards a distant luminous
+object, every part of the slit receives light from every part of the
+object, so that in this case every elementary streak of the spectrum
+%% -----File: 207.png---Folio 196-------
+is a spectrum of the entire body, without distinction of parts. A
+spectroscope used in this way is said to be an \textit{integrating} instrument.
+
+\fussy
+If, however, we interpose a lens (the object-glass of a telescope)
+between the luminous object and the slit, so as to have in the plane
+of the slit a distinct, real image of the object, then the top of the
+slit, for instance, will be illuminated wholly by light from one part
+of the object, the middle of it by light from another point, and the
+bottom by light from still a third. The spectrum formed by the top
+of the slit belongs, then, to the light from that particular point of the
+object whose image falls upon that part of the slit; and so of the
+rest. We thus separate the spectra of the different parts of the
+object, and so \textit{optically analyze} it. An instrument thus used is
+spoken of as an ``\textit{analyzing spectroscope}.'' The combined instrument
+formed by attaching a spectroscope to a large telescope for
+the spectroscopic observation of the heavenly bodies has been called
+by Mr.~Lockyer a ``telespectroscope.'' \figref{illo108}{Fig.~108} shows the apparatus
+used by the writer for some years at Dartmouth College.
+
+For solar purposes a grating spectroscope is generally better than a
+prismatic, being less complicated and more compact for a given power.
+
+\nbenlargepage
+\includegraphicsmid{illo108}{\textsc{Fig.~108.}---The Telespectroscope.}
+
+\nbarticle{314.} \hspace{-1.7pt}\nbparatext{Principles upon which Spectrum Analysis depends.}---These,
+substantially as announced by Kirchoff in 1858, are the three
+following:---
+
+1st, A \textit{continuous spectrum} is given by every incandescent body,
+the molecules of which so interfere with each other as to prevent
+their free, independent, luminous vibration; that is, by bodies which
+are either \textit{solid} or \textit{liquid}, or, if gaseous, are \textit{under high pressure}.
+
+2d, The spectrum of a gaseous element, \textit{under low pressure}, is
+discontinuous, made up of \textit{bright lines}, and these lines are characteristic;
+that is, the same substance under similar conditions always
+gives the same set of lines, and generally does so even under widely
+different conditions.
+
+\includegraphicsmid{illo109}{\textsc{Fig.~109.}---Reversal of the Spectrum.}
+
+\sloppy
+3d, A gaseous substance \textit{absorbs} from white light passing through
+it \textit{precisely these rays of which its own spectrum consists}. The spectrum
+of white light which has been transmitted through it then exhibits
+a ``\textit{reversed}'' spectrum of the gas; that is, one which shows dark
+lines instead of the characteristic bright lines.
+
+\fussy
+Fig.~109 illustrates this principle. Suppose that in front of the slit
+of the spectroscope we place a spirit lamp with a little carbonate of
+soda and some salt of thallium upon the wick. We shall then get a
+spectrum showing the two yellow lines of sodium and the green line
+%% -----File: 208.png---Folio 197-------
+of thallium, \textit{bright}. If now the lime-light be started right behind the
+lamp flame, we shall at once get the effect shown in the lower \figref{illo109}{figure},---a
+continuous spectrum crossed by \textit{black} lines just where the bright
+lines were before. Insert a screen between the lamp flame and the
+lime, and the dark lines instantly show bright again.
+
+\includegraphicsouter{illo110}{\textsc{Fig.~110.}---The Comparison Prism.}
+
+\nbarticle{315.} \nbparatext{Chemical Constituents of the Sun.}---By taking advantage
+of these principles we can detect the presence of a large number of
+well-known terrestrial elements in the sun. The solar spectrum is
+crossed by dark lines, which, with
+an instrument of high dispersion,
+number several thousand, and by
+proper arrangements it is possible
+to identify among these lines many
+which are due to the presence in
+the sun's lower atmosphere of
+known terrestrial elements in the
+state of vapor.  To effect the comparison
+necessary for this purpose,
+the spectroscope must be so arranged that the observer can have before
+him, side by side, the spectrum of sunlight and that of the substance
+to be tested. In order to do this, half of the slit is fitted with a little
+``\textit{comparison prism},'' so-called (\figref{illo110}{Fig.~110}), which reflects into it the
+light from the sun, while the other half of the slit receives directly
+%% -----File: 209.png---Folio 198-------
+the light of some flame or electric spark. On looking into the eye-piece
+of the spectroscope, the observer will then see a spectrum, the
+\textit{lower} half of which, for instance, is made by sunlight, while the
+\textit{upper} half is made by light coming from an electric spark between
+two metal points, say of iron.
+
+\includegraphicsmid{illo111}{\textsc{Fig.~111.}\\
+Comparison of the Solar Spectrum with that of Iron. From a Negative by Prof.~Trowbridge.}
+
+Photography may also be most effectively used in these comparisons
+instead of the eye. \figref{illo111}{Fig.~111} is a rather unsatisfactory reproduction,
+on a reduced scale, of a negative recently made by Professor
+Trowbridge at Cambridge. The lower half is the violet portion
+of the spectrum of the sun, and the upper half that of the vapor
+of iron in an electric arc. The reader can see for himself with what
+absolute certainty such a photograph indicates the presence of iron
+in the solar atmosphere. A few of the lines in the photograph
+which do not show corresponding lines in the solar spectrum are due
+to impurities in the carbon, and not to iron.
+
+\nbarticle{316.} As the result of such comparisons we have the following list
+of twelve elements, which are \textit{certainly} known to exist in the sun,
+viz.:---\\
+\begin{tabular}{llll}
+\hspace{.2\textwidth}
+&hydrogen,      &&cobalt,\\
+&iron,          &&chromium,\\
+&titanium,      &&barium,\\
+&calcium,       &&sodium,\\
+&manganese,     &&magnesium,\\
+&nickel,        &&platinum.\\[1ex]
+\multicolumn{4}{@{}p{\textwidth}@{}}{There are evidences, perhaps not quite conclusive, of the presence of
+nearly as many more, viz.:---}\\
+&copper,        &&uranium,\\
+&palladium,     &&aluminium,\\
+&vanadium,      &&cadmium,\\
+&molybdenum,    &&carbon,\\
+&&lead.
+\end{tabular}
+%% -----File: 210.png---Folio 199-------
+
+As to carbon, however, the spectrum is so peculiar, consisting of
+bands rather than lines, that it is very difficult to be sure, but the
+tendency of the latest investigations (of Rowland and Hutchins) is to
+establish its right to a place on the list.
+
+\begin{fineprint}
+The more recent researches have thrown much doubt on the presence of
+several substances which, a few years ago, were usually included in the list,
+as, for instance, strontium and cerium. It has been generally admitted also
+that the photographs of Dr.~Henry Draper had demonstrated the presence
+of oxygen in the sun, represented in the solar spectrum, not by dark lines
+like other elements, but by certain \textit{wide, bright bands}. The latest work,
+while it does not absolutely refute Dr.~Draper's conclusion, appears however
+to turn the balance of evidence the other way.
+\end{fineprint}
+
+\nbarticle{317.} It will be noticed that all the bodies named in the list, carbon
+alone excepted, are \textit{metals} (chemically hydrogen is a metal), and that
+a multitude of the most important terrestrial elements fail to appear;
+oxygen~(?), nitrogen, chlorine, bromine, iodine, sulphur, phosphorus,
+silicon, and boron are all missing. We must be cautious, however,
+as to \textit{negative} conclusions. It is quite conceivable that the spectra of
+these bodies under solar conditions may be so different from their spectra
+as presented in our laboratories that we cannot recognize them; for
+it is now quite certain that some substances, nitrogen, for instance,
+under different conditions, give two or more widely different spectra.
+
+Among the many thousand lines of the solar spectrum only a few
+hundred are so far identified.
+
+\sloppy
+\nbarticle{318.} \nbparatext{Mr.~Lockyer's Views.}---Mr.~Lockyer thinks it more probable
+that the missing substances are not truly elementary, but are
+decomposed or ``dissociated'' on the sun by the intense heat, and so
+do not exist there, but are replaced by their components; he believes,
+in fact, that none of our so-called elements are really elementary, but
+that all are decomposable, and, to some extent actually decomposed
+in the sun and stars, and some of them by the electric spark in our
+own laboratories. Granting this, a crowd of interesting and remarkable
+spectroscopic facts find easy explanation. At the same time the
+hypothesis is encumbered with great difficulties and has not yet been
+finally accepted by physicists and chemists. For a full statement
+of his views the reader is referred to his ``Chemistry of the Sun.''
+
+\fussy
+\nbarticle{319.} \nbparatext{The Reversing Layer.}---According to Kirchoff's theory the
+dark lines are formed by the passing of light from the minute solid
+and liquid particles of which the photospheric clouds are supposed to
+%% -----File: 211.png---Folio 200-------
+be formed, through vapors containing the substances which we recognize
+in the solar spectrum. If this be so, the spectrum of the gaseous
+envelope, which by its absorption forms the \textit{dark} lines, should by itself
+show a spectrum of corresponding bright lines. The opportunities are
+of course rare when it is possible to obtain the spectrum of this gas-stratum
+alone by itself; but at the time of a total eclipse, at the
+moment when the sun's disc has just been obscured by the moon,
+and the sun's atmosphere is still visible beyond the moon's limb,
+if the slit of the spectroscope be carefully adjusted to the proper
+point, the observer ought to see this bright-line spectrum. The
+author succeeded in making this very observation at the Spanish
+eclipse of 1870. The lines of the solar spectrum, which up to the
+final obscuration of the sun had remained dark as usual (with the
+exception of a few belonging to the spectrum of the chromosphere),
+were suddenly ``reversed,'' and the whole length of the spectrum was
+filled with brilliant-colored lines, which flashed out quickly and then
+gradually faded away, disappearing in about two seconds,---a most
+beautiful thing to see. Substantially the same thing has since then
+been several times observed.
+
+\begin{fineprint}
+\nbarticle{320.}  The natural interpretation of this phenomenon is, that \textit{the formation
+of the dark lines in the solar spectrum is mainly, at least, produced by a very
+thin layer close down to the photosphere}, since the moon's motion in two seconds
+would cover a thickness of only about 500 miles. It was not possible, however,
+to be certain that \textit{all} the dark lines were reversed, and in this uncertainty
+lies the possibility of a different interpretation. Mr.~Lockyer doubts
+the existence of any such \textit{thin stratum}. According to his views the solar
+atmosphere is very extensive, and these lines of iron, which correspond to
+the more complex combinations of its constituents, are formed only in the
+regions of lower temperature, \textit{high up} in the sun's atmosphere. They should
+appear \textit{early} at the time of an eclipse and \textit{last long}, but not be very bright.
+Those due to the constituents of iron which are found only close down to
+the solar surface should be short and bright; and he thinks that the
+numerous bright lines observed under the conditions stated are due to such
+substances only. Observation needs to be directed to the special point to
+determine whether \textit{all} of the dark lines are reversed at the edge of the sun,
+or \textit{only a few};
+and if so, what ones.
+\end{fineprint}
+
+\includegraphicsouter{illo112}{\textsc{Fig.~112.}\\
+The C line in the Spectrum of a Sun Spot,
+Sept.~22, 1870.}
+
+\sloppy
+\nbarticle{321.} \hspace{0.8em}\nbparatext{Sun-Spot Spectrum.}---\hspace{0pt}
+This is like the general solar spectrum,
+except that certain lines are much widened, while certain others are
+thinned, and sometimes the lines of hydrogen become bright. It is
+to be noticed that by far the larger proportion of the dark lines of any
+%% -----File: 212.png---Folio 201-------
+%% -----File: 213.png---Folio 202-------
+given substance are not affected at all in the spot spectrum, but only
+a certain few of them, a point
+which Mr.~Lockyer considers very
+important. Not infrequently it
+happens that certain lines of the
+spectrum are crooked and broken
+in connection with sun spots, as
+shown by \figref{illo112}{Fig.~112}. Such phenomena
+are caused, according to
+\textit{Doppler's principle},\footnote
+  {Doppler's principle is this: that when we are approaching, or approached by,
+a body which is emitting regular vibrations, then the \textit{number} of waves received
+by us in a second is \textit{increased}, and their wave-length correspondingly \textit{diminished};
+and \textit{vice versa} when the distance of the vibrating body is increasing. Thus the
+\textit{pitch} of a musical tone rises while we are approaching the sounding body, and
+falls as we recede; in just the same way the ``refrangibility'' of the rays, say of
+hydrogen, emanating from the sun is increased (the wave-length being shortened)
+whenever we are approaching it with a speed which bears a sensible ratio to the
+velocity of light. Calling $\lambda$ the wave-length of the ray when the observer and
+the luminous object are relatively at rest, and $\lambda'$ the wave-length as affected by
+their relative motion; putting $V$ also for the velocity of light (about 186,330 miles
+per second), and $s$ for the speed with which the observer and source of light
+\textit{approach} each other, we have
+\[
+\lambda' = \lambda \left(\frac{V}{V + s}\right).
+\]
+[If the distance is increasing instead of diminishing, the denominator will be
+$(V - s)$.] With the most powerful spectroscopes motions of from one to two miles
+  per second along the line of sight can thus be detected.}
+by the swift
+motion of matter towards or from
+the observer. In the particular case shown in the \figref{illo112}{figure}, hydrogen is
+the substance, and the greatest motion indicated was towards the
+observer at the rate of about 300~miles a second---an unusual velocity.
+These effects are most noticeable, not \textit{in} the spots, but near them,
+usually just at the outer edge of the penumbra.
+
+\fussy
+\begin{fineprint}
+The dark and apparently continuous spectrum which is due to the nucleus
+of a sun spot is not truly continuous, but under high dispersion is resolved
+into a range of extremely fine, close-packed, dark lines, separated by narrow
+spaces. At least this is so in the green and blue portions of the spectrum; it
+is more difficult to make out this structure in the yellow and red. It appears
+to indicate that the absorbing medium which fills the hollow of a sun spot
+is \textit{gaseous}, and not composed of precipitated particles like smoke, as has
+been suggested.
+\end{fineprint}
+%% -----File: 214.png---Folio 203-------
+
+\nbarticle{322.} \nbparatext{The Chromosphere.}---The \textit{chromosphere} is a region of the
+sun's gaseous envelope which lies close above the photosphere, the
+``\textit{reversing layer},'' if it exists at all, being only the most dense and
+hottest part of it. The chromosphere is so called, because as seen
+for an instant, during a total solar eclipse, it is of a bright scarlet
+color, the color being due to the hydrogen which is its main constituent.
+It is from 5000 to 10,000 miles in thickness, and in structure
+is very like a sheet of scarlet flame, not being composed of horizontal
+sheets, but of (approximately) upright filaments. Its appearance
+has been compared very accurately to that of ``a prairie on fire'';
+but the student must carefully guard against the idea that there is
+any real ``burning'' in the case; \textit{i.e.}, any \textit{process of combination}
+between hydrogen and some other substance. The temperature is
+altogether too high for any formation of hydrogen compounds at the
+sun's surface.
+
+\nbarticle{323.} \nbparatext{The Prominences.}---At a total eclipse, after the totality has
+fairly set in, there are usually to be seen at the edge of the moon's
+disc a number of scarlet, star-like objects, which in the telescope
+appear as beautiful, fiery clouds of various form and size. These are
+the so-called ``\textit{prominences},'' which very non-committal name was
+given while it was still doubtful whether they were solar or lunar.
+Photography, in 1860, proved that they really belong to the sun,
+for the photographs taken during the totality showed that the moon
+obviously moves over them, covering those upon the eastern limb,
+and uncovering those upon the western.
+
+In 1868, during the observation of the eclipse which occurred
+that year in India, the spectroscope showed them to be gaseous,
+and that their main constituent is hydrogen. Their spectrum contains
+also a conspicuous yellow line known as the $D_{3}$ line, because
+it is very near the two ``$D$'' lines of sodium. This line
+is not due to hydrogen, and is not identified with any known element;
+but an element has been assumed for it and called provisionally
+``\textit{helium}.''
+
+In connection with this eclipse, Janssen, who observed it in India,
+found that the lines of the prominence spectrum were so bright
+that he was able to observe them the next day after the eclipse in
+full sunlight; and he also found that by a proper management of
+his instrument he could study the form and behavior of the prominences
+nearly as well without an eclipse as during one. Lockyer,
+in England, some time earlier had come to similar conclusions from
+%% -----File: 215.png---Folio 204-------
+theoretical grounds, and he practically perfected his discovery a few
+weeks later than Janssen, although without knowledge of what he
+had done. By a remarkable but accidental coincidence their discoveries
+were communicated to the French Academy on the same day;
+and in their honor the French have struck a medal bearing their united
+effigies.
+
+\sloppy
+\nbarticle{324.} \nbparatext{How the Spectroscope makes the Prominences Visible.}---The
+only reason we cannot see the prominences at any time is on
+account of the bright illumination of our own atmosphere. We can
+screen off the direct light of the sun; but we cannot screen off the
+reflected sunlight coming from the air which is directly between us
+and the prominences themselves; a light
+so brilliant that the prominences cannot
+be seen through it without some kind of
+aid.
+
+\fussy
+\includegraphicsouter{illo113}{\textsc{Fig.~113.}\\
+Spectroscope Slit adjusted for
+Observation of the Prominences.}
+
+The spectrum of this air-light is, of course,
+just the same as that of the sun---a continuous
+spectrum with the same dark lines
+upon it. When, therefore, we arrange the
+apparatus as indicated in \figref{illo113}{Fig.~113}, pointing
+the telescope so that the image of the sun's
+limb just touches the slit of the spectroscope,
+then, if there is a prominence at that point, we shall have in
+our spectroscope two spectra superposed upon each other; namely,
+the spectrum of the air-illumination and that of the prominence.
+The latter is a spectrum of \textit{bright lines}, or, if the slit is opened
+a little, of bright \textit{images} of whatever part of the prominence may
+fall within the edges of the slit. Now, the brightness of these
+images is not affected by any increase of dispersion in the spectroscope.
+Increase\footnote
+  {Too high dispersion injures the definition, however, because the lines in the
+  spectrum of hydrogen are rather broad and hazy.}
+of dispersion merely sets these images farther
+apart, without making them fainter. The spectrum of the aerial illumination,
+on the other hand, is made very faint by its extension; and,
+moreover, it \textit{presents dark lines} (or \textit{spaces} when the slit is opened)
+precisely at the points where the bright images of the prominences
+fall.
+
+\includegraphicsouter[14]{illo114}{\textsc{Fig.~114.}\\
+The Chromosphere and Prominences seen in the
+Spectroscope.}
+
+\hspace{-1.1pt}A spectroscope of dispersive power sufficient to divide the two $E$
+lines, attached to a telescope of four or five inches aperture, gives a
+%% -----File: 216.png---Folio 205-------
+very satisfactory view of these beautiful objects; the \textit{red} image corresponds
+to the $C$ line, and is by far the best for such observations,
+though the $D_{3}$ line or the $F$ line can also be used. When the instrument
+is properly adjusted, the slit opened a little, and the image of
+the sun's limb brought exactly to the edge of the slit, the observer
+at the eye-piece of the spectroscope will see things about as we have
+attempted to represent them in \figref{illo114}{Fig.~114}; as if he were looking
+at the clouds in an evening sky through a slightly opened window-blind.
+
+\sloppy
+\nbarticle{325.} \nbparatext{Different Kinds of Prominences; Their Forms and Motions.}---The
+prominences may be broadly divided into two classes,---the
+\textit{quiescent} or diffused, and the \textit{eruptive} or ``metallic,'' as Secchi
+calls them, because they show
+in their spectrum the lines of
+many metals besides hydrogen.
+The former, illustrated by \figref{illo115a}{Fig.~115}
+(see p.~\figpage{115}), are immense
+clouds, often 60,000 miles in
+height, and of corresponding
+horizontal dimensions, either
+resting upon the chromosphere
+or connected with it by slender
+stems like great banyan-trees.
+They are not very brilliant, and
+are composed almost entirely
+of hydrogen and ``helium.''
+They often remain nearly unchanged
+for days together as
+they pass over the sun's limb.
+They are found on all portions of the disc, at the poles and equator
+as well as in the spot zones. Some of them are clouds floating
+entirely detached from the sun's surface.
+
+\fussy
+Usually these clouds are simply the remnants of prominences which
+appear to have been thrown up from below, but in some cases they
+actually form and grow larger without any visible connection with
+the chromosphere---a fact of considerable importance, as showing in
+these regions the presence of hydrogen, invisible to our spectroscopes
+until somehow or other it is made to give out the rays of its familiar
+spectrum. All the forms and motions of the prominences, it
+may be said further, seem to indicate the same thing---that they
+%% -----File: 217.png---Folio 206-------
+%% -----File: 218.png---Folio 207-------
+%% -----File: 219.png---Folio 208-------
+exist and move, not \textit{in a vacuum}, but in a medium of density comparable
+with their own, as clouds do in our own atmosphere.
+
+\nbenlargepage
+\nbarticle{326.}  The \textit{eruptive} prominences, on the other hand, are brilliant
+and active, not \textit{usually} so large as the quiescent, but at times
+enormous, reaching elevations of 100,000, 200,000, or even 400,000
+miles. They are illustrated by \figref{illo116a}{Fig.~116}. Most frequently they are
+in the form of spikes or flames; but they present also a great variety
+of other fantastic shapes, and are sometimes so brilliant as to be
+visible with the spectroscope on the surface of the sun itself, and not
+merely at the limb. Generally prominences of this class are associated
+with active sun spots, while both classes appear to be connected
+with the faculæ. The \figref{illo115a}{figures} given are from drawings of
+individual prominences that have been observed by the author at
+different times.
+
+\begin{includegraphicspage}{115}
+  \begin{tabular}{c@{\qquad\qquad}c}
+    \includegfx{illo115a}           & \includegfx{illo115b}\\[1ex]
+    \footnotesize Clouds.      & \footnotesize Diffuse.\\[4ex]
+    \includegfx{illo115c}           & \includegfx{illo115d}\\[1ex]
+    \footnotesize Filamentary. & \footnotesize Stemmed.\\[4ex]
+    \includegfx{illo115e}           & \includegfx{illo115f}\\[1ex]
+    \footnotesize Plumes.      &  \footnotesize Horns.\\[4ex]
+  \end{tabular}
+  \captionof*{figure}{\textsc{Fig}.~115.\\
+  Quiescent Prominences. Scale 75,000 Miles to the Inch. From ``The Sun,''\\
+  by Permission of D. Appleton \& Co.}
+\end{includegraphicspage}
+
+\begin{includegraphicspage}{116}
+  \begin{tabular}{c@{\qquad\qquad}c}
+    \includegfx{illo116a}           & \includegfx{illo116b}\\[1ex]
+    \footnotesize Vertical Filaments.      & \footnotesize Prominences Sept. 7, 1871, 12.30 \textsc{p.m.}\\[4ex]
+    \includegfx{illo116c}           & \includegfx{illo116d}\\[1ex]
+    \footnotesize Cyclone. & \footnotesize Same at 1.15 \textsc{p.m.}\\[4ex]
+    \includegfx{illo116e}           & \includegfx{illo116f}\\[1ex]
+    \footnotesize Flame.      &  \footnotesize Jets near Sun's Limb, Oct. 5, 1871.\\[4ex]
+  \end{tabular}
+  \captionof*{figure}{\textsc{Fig}.~116.\\
+  Eruptive Prominences. From ``The Sun.'' By Permission of D. Appleton \& Co.}
+\end{includegraphicspage}
+
+These solar clouds are most fascinating objects to watch, on account
+of the beauty of their forms, and the rapidity of their changes.
+In the case of the eruptive prominences the swiftness of the changes
+is sometimes wonderful---portions can be actually seen to move, and
+this implies a real velocity of at least 250 miles a second, so that it
+is no exaggeration to speak of such phenomena as veritable ``explosions'':
+of course, in such cases the lines in the spectrum are greatly
+broken and distorted, and frequently a ``magnetic storm'' follows
+upon the earth, with a brilliant Aurora Borealis.
+
+The number visible at a single time is variable, but it is not very
+unusual to find as many as twenty on the sun's limb at once.
+
+\nbarticle{327.} \nbparatext{The Corona.}---This is a halo, or ``glory,'' of light which surrounds
+the sun at the time of the total eclipse. From the remotest
+times it has been well known, and described with enthusiasm, as being
+certainly one of the most beautiful of natural phenomena.
+
+The portion of the corona nearest the sun is almost dazzlingly bright,
+with a greenish, pearly tinge which contrasts finely with the scarlet
+blaze of the prominences. It is made up of streaks and filaments
+which on the whole radiate outwards from the sun's disc, though
+they are in many places strangely curved and intertwined. Usually
+these filaments are longest in the sun-spot zones, thus giving the
+corona a more or less quadrangular figure. At the very poles of
+the sun, however, there are often tufts of sharply defined threads.
+
+For the most part the streamers have a length not much exceeding
+the sun's radius, but some of them at almost every eclipse go
+%% -----File: 220.png---Folio 209-------
+far beyond this limit. In the clear air of Colorado during the eclipse
+of 1878, two of them could be traced for five or six degrees,---a
+distance of at least 9,000000 miles from the sun. A most striking
+feature of the corona usually consists of certain dark rifts which
+reach straight out from the moon's limb, clear to the extremest limit
+of the corona.
+
+The corona varies much in brightness at different eclipses, and of
+course the details are never twice the same. Its total light under
+ordinary circumstances is at least two or three times as great as that
+of the full moon.
+
+\nbarticle{328.} \nbparatext{Photographs of the Corona.}---While the eye can perhaps
+grasp some of its details more satisfactorily than the photographic
+plate can do, it is found that drawings of the corona are hardly to be
+trusted. At any rate, it seldom happens that the representations of
+two artists agree sufficiently to justify any confidence in their scientific
+accuracy. Photographs, on the other hand, may be trusted as far as
+they go, though they may fail to bring out some things which are
+conspicuous to the eye. \figref{illo117}{Fig.~117} is from the photograph of the Egyptian
+eclipse of 1882, when a little comet was found close to the sun.
+
+\includegraphicsmid{illo117}{\textsc{Fig.~117.}---Corona of the Egyptian Eclipse, 1882.}
+
+\begin{fineprint}
+Of course, as in the case of the prominences, the only reason we cannot
+see the corona without an eclipsed sun is the illumination of the earth's
+atmosphere. If we could ascend above our atmosphere, and manage to exist
+and to observe there, we could see it by simply screening off the sun's disc.
+%% -----File: 221.png---Folio 210-------
+So long, however, as the brightness of the illuminated air is more than about
+sixty times that of the corona, it must remain invisible to the eye. Dr.~Huggins
+has thought that it might be possible by means of photographs to
+detect differences of illumination less than $\frac{1}{60}$ (the limit of the eye's perception),
+and so to obtain pictures of the corona at any time; especially as it
+appears that the coronal light is far richer in ultra-violet rays (the photographic
+rays) than the general sunlight with which the air is illuminated.
+His attempts so far, however, have yielded only doubtful success.
+\end{fineprint}
+
+\nbarticle{329.} \nbparatext{Spectrum of the Corona.}---This was first definitely observed
+in 1869 during the eclipse which passed over the western part of the
+United States in that year. It was then found that its most remarkable
+characteristic is a bright line in the green, which the writer identified
+as coinciding with the dark line at 1474 on the scale of Kirchoff's
+map ($\lambda = 5316$). This line was also observed by Harkness.
+
+\begin{fineprint}
+This result was for a time very puzzling, since the dark line in question
+is given by \AA ngstrom and other authorities as due to the spectrum of \textit{iron}.
+The mystery has since been removed, however, by the discovery that under
+high dispersion the line is double, and that the corona line coincides with
+the more refrangible of the two components, while the other one is the line
+due to iron. We have as yet been unable to identify with any terrestrial
+element the substance to which this line is due, but the provisional name
+``\textit{coronium}'' has been proposed for it. The recent researches of Grünewald
+make it somewhat probable that both coronium and helium are components
+of hydrogen, which (in line with Mr.~Lockyer's speculations) is supposed to
+be partially decomposed under solar conditions.
+\end{fineprint}
+
+Besides this conspicuous green line, the hydrogen lines are also
+faintly visible in the spectrum of the corona; and by means of a photographic
+camera used during the Egyptian eclipse of 1882, it was
+found that the upper or violet portion of the spectrum is very rich in
+lines, among which $H$ and $K$ are specially conspicuous. There is
+also, through the whole spectrum, a faint continuous background,
+which, however, according to Mr.~Lockyer's statements, is not of
+uniform brightness, but ``\textit{banded}.'' In it some observers have reported
+the presence of a few of the more conspicuous dark lines of
+the ordinary solar spectrum, but the evidence on this point is rather
+conflicting.
+
+If during the totality we look at the eclipsed sun with a diffraction
+grating, or through a prism of high dispersive power, we see three
+rings which are really images of the corona. One of them, the brightest
+and the largest, is the green ring due to the 1474 line; the others
+%% -----File: 222.png---Folio 211-------
+are a red ring due to $C$, and a blue one due to the $F$ line of
+hydrogen.
+
+\nbarticle{330.} \nbparatext{Nature of the Corona.}---It is evident that the corona is a
+truly solar and not merely an optical or atmospheric phenomenon, from
+two facts: first, \textit{the identity of detail in photographs made at widely
+separate stations}. In 1871, for instance, photographs were obtained at
+the Indian station of Bekul, in Ceylon, and in Java, three stations separated
+by many hundreds of miles; but, excepting minute differences
+of detail, such as might be expected to have resulted from the changes
+that would naturally go on in the corona, during the half-hour while
+the moon's shadow was travelling from Bekul to Java, all the photographs
+agree exactly, which of course would not be the case if the
+corona depended in any way upon the atmospheric conditions at the
+observer's station.
+
+Second (but \textit{first} historically), \textit{the presence of bright lines in the
+spectrum of the corona} proves that it cannot be a terrestrial or lunar
+phenomenon, by demonstrating the presence in the corona of a \textit{self-luminous
+gas}, which observation fails to find either near to the moon
+or in our own atmosphere. It must, therefore, be at the sun.
+
+But while it is thus certain that the corona contains luminous gas, it
+also is very likely that finely divided solid or liquid matter may be present
+in the corona; that is, fog or dust of some kind.
+
+\nbarticle{331.}  The corona cannot be a true ``solar \textit{atmosphere}'' in any strict
+sense of the word. No gaseous envelope in any way analogous to the
+earth's atmosphere could possibly exist there in gravitational equilibrium
+under the solar conditions of pressure and temperature. The
+corona is probably a phenomenon due somehow to the intense activity
+of the forces there at work; meteoric matter, cometic matter, matter
+ejected from within the sun, are all concerned.
+
+That this matter is inconceivably rare is evident from the fact that
+in several cases comets have passed directly through the corona without
+experiencing the least perceptible disturbance of their motions. It is
+altogether probable that at a very few thousand miles above the sun's
+surface, its density becomes far less than that of the best vacuum we
+can make in an electric lamp.
+\chelabel{CHAPTERVIII}
+%% -----File: 223.png---Folio 212-------
+
+\Chapter{IX}{The Sun's Light and Heat}
+\nbchapterhang{\stretchyspace
+THE SUN'S LIGHT AND HEAT: COMPARISON OF SUNLIGHT WITH
+ARTIFICIAL LIGHTS.---MEASUREMENT OF THE SUN'S HEAT,
+AND DETERMINATION OF THE ``SOLAR CONSTANT.''---PYRHELIOMETER,
+ACTINOMETER, AND BOLOMETER.---THE SUN'S
+TEMPERATURE.---THEORIES AS TO THE MAINTENANCE OF
+THE SUN'S RADIATION, AND CONCLUSIONS AS TO THE SUN'S
+POSSIBLE AGE AND FUTURE DURATION.}
+
+\nbarticle{332.} \nbparatext{The Sun's Light.}---\textit{The Quantity of Sunlight}. It is very easy
+to compare (approximately) sunlight with the light of a standard\footnote
+  {A standard candle is a sperm candle weighing one-sixth of a pound and burning
+120 grains an hour. The French ``Carcel burner,'' used as a standard in their
+photometry, gives just ten times the quantity of light given by this standard
+candle. An ordinary gas-burner consuming five feet of gas hourly gives a light
+  equivalent to from twelve to fifteen standard candles.}
+candle; and the result is, that when the sun is in the zenith, it illuminates
+a white surface about 60,000 times as strongly as a standard
+candle at a distance of one metre. If we allow for the atmospheric absorption,
+the number would he fully 70,000. If we then multiply 70,000
+by the square of 150,000 million (roughly the number of metres in
+the sun's distance from the earth), we shall get what a gas engineer
+would call the sun's ``\textit{candle power}.'' The number comes out 1575
+billions of billions (English); \textit{i.e.}, 1575 with twenty-four ciphers
+following.
+
+\begin{fineprint}
+\nbarticle{333.} One way of making the comparison is the following: Arrange matters
+as in \figref{illo118}{Fig.~118}. The sunlight is brought into a darkened room by a
+mirror $M$, which reflects the rays through a lens $L$ of perhaps half an inch in
+diameter. After the rays pass the focus they diverge and form on the
+screen $S$ a disc of light, the size of which may be varied by changing the
+distance of the screen. Suppose it so placed that the illuminated circle is
+just ten feet in diameter; that is, 240 times the diameter of the lens. The
+illumination of the disc will then be less than that of direct sunlight in
+the ratio of $240^2$ (or 57,600) to 1 (neglecting the loss of light produced by
+%% -----File: 224.png---Folio 213-------
+the mirror and the lens, a loss which of course must he allowed for). Now
+place a little rod like a pencil near the screen, as at $P$, light a standard
+candle, and move the candle back and forth until the two shadows of the
+pencil, one formed by the candle, and the other by the light from the lens,
+are equally dark. It will be found that the candle has to be put at a distance
+of about one metre from the screen; though the results would vary a
+good deal from day to day with the clearness of the air.
+\end{fineprint}
+
+\includegraphicsmid{illo118}{\textsc{Fig.~118.}---Comparison of Sunlight with a Standard Candle.}
+
+\nbarticle{334.} When the sun's light is compared with that of the full moon
+and of various stars, we find, as stated (\artref{Art.}{259}), that it is about
+600,000 times that of the full moon. It is 7,000,000000 times as
+great as the light received from Sirius, and about 40,000,000000
+times that from Vega or Arcturus.
+
+\nbarticle{335.} \nbparatext{The Intensity of the Sun's Luminosity.}---This is a very
+different thing from the total quantity of its light, as expressed by
+its ``candle power'' (a surface of comparatively feeble luminosity
+can give a great quantity of light if large enough). It is the \textit{amount
+of light per square inch of luminous surface} which determines the
+intensity. Making the necessary computations from the best data
+obtainable (only roughish approximations being possible), it appears
+that the sun's surface is about 190,000 times as bright as that of a
+%% -----File: 225.png---Folio 214-------
+candle flame, and about 150 times as bright as the lime of the calcium
+light. \textit{Even the darkest part of a solar spot outshines the lime}. The
+intensely brilliant spot in the so-called ``crater'' of an electric arc
+comes nearer sunlight than anything else known, being from one-half
+to one-fourth as bright as the surface of the sun itself. But either
+the electric arc or the calcium light, when interposed between the
+eye and the sun looks like a dark spot on the disc.
+
+\nbarticle{336.}  \nbparatext{Comparative Brightness of Different Portions of the Sun's
+Surface.}---By forming a large image of the sun, say a foot in diameter,
+upon a screen, we can compare with each other the rays
+coming from different parts of the sun's disc. It thus appears that
+there is a great diminution of light at the edge, the light there, according
+to Professor Pickering's experiments, being just about one-third
+as strong as at the centre. There is also an obvious difference of
+color, the light from the edge of the disc being brownish red as compared
+with that from the centre. The reason is, that the red and
+yellow rays of the spectrum lose much less of their brightness at the
+limb than do the blue and violet. According to Vogel, the latter rays
+are affected nearly twice as much as the former. For this reason,
+photographs of the sun exhibit the darkening of the limb much more
+strongly than one usually sees it in the telescope.
+
+\includegraphicsouter{illo119}{\textsc{Fig.~119.}\\Cause of the Darkening of the Sun's Limb.}
+
+\nbarticle{337.} \nbparatext{Cause of the Darkening of the Limb.}---It is due unquestionably
+to the general absorption of the sun's rays by the lower portion
+of the overlying atmosphere.
+The reason is obvious from the
+figure (\figref{illo119}{Fig.~119}). The \textit{thinner} this
+atmosphere, other things  being
+equal, the \textit{greater the} ratio \textit{between
+the percentage of absorption at the
+centre and edge of the disc, and
+the more obvious the darkening of
+the limb}.
+
+Attempts have been made to
+determine from the observed differences
+between the brightness of centre and limb the total
+percentage of the sun's light thus absorbed. Unfortunately we have
+to supplement the observed data with some very uncertain assumptions
+in order to solve the problem; and it can only be said that
+it is \textit{probable} that the amount of light, absorbed by the sun's atmosphere
+%% -----File: 226.png---Folio 215-------
+lies between fifty and eighty per cent; \textit{i.e.}, the sun deprived of
+its gaseous envelope would probably shine from two to five times as
+brightly as now. It is noticeable also, as Langley long ago pointed
+out, that thus stripped, the ``complexion'' of the sun would be
+markedly changed from yellowish white to a good full \textit{blue}, since the
+blue and violet rays are much more powerfully absorbed than these
+at the lower end of the spectrum.
+
+\section*{THE SUN'S HEAT.}
+
+\nbarticle{338.} \nbparatext{Its Quantity; the ``Solar Constant.''} By the ``\textit{quantity of
+heat}'' received by the earth from the sun we mean the number of
+heat-units received in each unit of time by a square unit of surface
+when the sun is in the zenith. The heat-unit most employed by
+engineers is the \textit{calorie}, which is the quantity of heat required to
+raise the temperature of one kilogram of water one degree centigrade.
+It is found by observation that each square metre of surface exposed
+perpendicularly to the sun's rays receives from the sun each minute
+from twenty-five to thirty of these calories; or rather it \textit{would do so} if
+a considerable portion of the sun's heat were not stopped by the earth's
+atmosphere, which absorbs some thirty per cent of the whole, even
+when the sun is vertical, and a much larger proportion when the sun
+is near the horizon. This quantity, \textit{twenty-five calories\footnote
+  {For many scientific purposes the engineering calorie is inconveniently
+large, and a smaller one is employed, which replaces the kilogram of water
+by the \textit{gram} heated one degree---the smaller calorie being thus only $\frac{1}{1000}$ of
+the engineering unit. As stated by many writers (Langley, for instance), the
+solar constant is the number of these \textit{small} calories received per square \textit{centimetre}
+of surface in a minute. This would make the number 2.5 instead of 25. It
+would perhaps be better to bring the whole down to the ``c.g.s.\ system'' by substituting
+the \textit{second} for the minute; and this would give us for the solar constant,
+  on the ``c.g.s.\ system,'' 0.0417 (\textit{small}) \textit{calories per square centimetre per second}.}
+per square
+metre per minute} (using the smaller of the values mentioned, which
+\textit{certainly} is not too large), is known as the ``\textit{Solar Constant}.''
+
+\nbarticle{339.} \nbparatext{Method of determining the ``Solar Constant.''}---The method
+by which the solar constant is determined is simple enough in principle,
+though complicated with serious practical difficulties which
+affect its accuracy. It is done by allowing \textit{a beam of sunlight of
+known cross-section to shine upon a known weight of water} (\textit{or other
+substance of known specific heat}) \textit{for a known length of time, and}
+%% -----File: 227.png---Folio 216-------
+\textit{measuring the rise of temperature}. It is necessary, however, to determine
+and allow for the heat received from other sources during the
+experiment, and for that lost by radiation. Above all, the absorbing
+effect of our own atmosphere is to be taken into account, and
+this is the most difficult and uncertain part of the work, since the
+atmospheric absorption is continually changing with every change of
+the transparency of the air, or of the sun's altitude.
+
+\begin{fineprint}
+\includegraphicsouter{illo120}{\textsc{Fig.~120.}---Pouillet's Pyrheliometer.}
+
+\sloppy
+\nbarticle{340.} \nbparatext{Pyrheliometers and Actinometers.}---The instruments with
+which these measurements are made, are known as ``pyrheliometers'' and
+``actinometers.'' \figref{illo120}{Fig.~120} represents the pyrheliometer
+of Pouillet, with which in 1838 he
+made his determination of the solar constant,
+at the same time that Sir John Herschel was
+experimenting at the Cape of Good Hope in
+practically the same way. They were the
+first apparently to understand and attack the
+problem in a reasonable manner. The pyrheliometer
+consists essentially of a little cylindrical
+box $ab$, like a snuff-box, made of thin
+silver plate, with a diameter of one decimetre
+and such a thickness that it holds 100 grams
+of water. The upper surface is carefully
+blackened, while the rest is polished as brilliantly
+as possible. In the water is inserted
+the bulb of a delicate thermometer, and the
+whole is so mounted that it can be turned in
+any direction so as to point it directly towards
+the sun. It is used by first holding a screen
+between it and the sun for (say) five minutes,
+and watching the rise or fall of the mercury in
+the thermometer at $m$. There will usually be
+some slight change due to the radiation of
+surrounding bodies. The screen is then removed,
+and the sun is allowed to shine upon
+the blackened surface for five minutes, the
+instrument being continually turned upon
+the thermometer as an axis, in order to keep the water in the calorimeter
+box well stirred. At the end of the five minutes the screen is replaced
+and the rise of the temperature noted. The difference between this and
+the change of the thermometer during the first five minutes will give us the
+amount by which a beam of sunlight one decimetre in diameter has raised
+the temperature of 100 grams of water in five minutes, and were it not for
+the troublesome corrections which must be made, would furnish directly the
+value of the solar constant.
+%% -----File: 228.png---Folio 217-------
+
+\fussy
+\includegraphicsouter{illo121}{\textsc{Fig.~121.}---Violle's Actinometer.}
+
+\nbarticle{341.} The second apparatus, \figref{illo121}{Fig.~121}, is the actinometer of Violle, which
+consists of two concentric metal spheres, the inner of which is blackened on
+the inside, while the outer one is brightly polished, the space between the
+two being filled with water at a known temperature, kept circulating by a
+pump of some kind. The thermoscopic
+body in this case, instead of being a box
+filled with water, is the blackened bulb
+of the thermometer $T$; and the observations
+may be made either in the same
+way as with the pyrheliometer, or simply
+by noting the difference between the
+temperature finally attained by the thermometer
+$T$ after it has ceased to rise in
+the sun's rays, and the temperature of
+the water circulating in the shell.
+
+\nbarticle{342.} \nbparatext{Correction for Atmospheric
+Absorption.}---The correction for atmospheric
+absorption is determined by
+making observations at various altitudes
+of the sun between zenith and horizon.
+If the rays were \textit{homogeneous} (that is,
+all of one wave-length), it would be
+comparatively easy to deduce the true correction and the true value of
+the solar constant. In fact, however, the \textit{visible} solar spectrum is but a
+small portion of the whole spectrum of the sun's radiance, and, as Langley
+has shown, it is necessary to determine the coefficient of absorption separately
+for all the rays of different wave-length.
+
+\sloppy
+\nbarticle{343.} \nbparatext{The Bolometer.}---This he has done by means of his ``Bolometer,''
+an instrument which is capable of indicating exceedingly minute changes in
+the amount of radiation received by an extremely thin strip of metal. This
+strip is so arranged that the least change in its electrical resistance due to
+any change of temperature will disturb a delicate galvanometer. The
+instrument is far more sensitive than any thermometer or even thermopile,
+and has the especial advantage of being extremely quick in its response
+to any change of radiation. \figref{illo122}{Fig.~122} shows it so connected with
+a spectroscope that the observer can bring to the bolometer, $B$, rays of
+any wave-length he chooses. The rays enter through the collimator lens
+$L$, and are then refracted by the prism $P$ to the reflector $M$, whence they
+are sent back to $B$.
+
+\fussy
+Langley has shown that the corrections for atmospheric absorption deduced
+by earlier observers are all considerably too small, and has raised the received
+value of the solar constant, from 20 or 25, which was the value
+accepted a few years ago, to 30. We have, however, provisionally retained
+%% -----File: 229.png---Folio 218-------
+the 25, as his new results, though almost certainly correct, have not yet
+been universally accepted, and perhaps need verification.
+\end{fineprint}
+
+\includegraphicsouter{illo122}{\textsc{Fig.~122.}\\
+Langley's Spectro-Bolometer, as used for Mapping
+the Energy of the Prismatic Spectrum.}
+
+\nbarticle{344.} A less technical statement of the solar radiation may be
+made in terms of thickness of
+the quantity of \textit{ice} which would
+be melted by it in a given time.
+Since it requires about eighty
+calories of heat to melt a kilogram
+of ice, it follows that
+twenty-five calories per minute
+per square metre would liquefy
+in an \textit{hour} a sheet of ice \textit{one
+metre square and about nineteen
+millimetres thick}. According
+to this the sun's heat
+would melt about 174 feet of
+ice annually on the earth's
+equator; or $136\frac{1}{2}$ feet yearly all
+over the surface of the earth,
+if the heat annually received
+were equally distributed in all
+latitudes. (See note at end of
+the chapter, page~\pageref{pg:227}.)
+
+\nbenlargepage
+\nbarticle{345.} \nbparatext{Solar Heat expressed
+as Energy.}---Since according to
+the known value of the ``mechanical equivalent of heat'' (Physics, p.~159)
+a horse-power corresponds to about $10\frac{7}{10}$ calories per minute, it
+follows that \textit{each square metre of surface} (neglecting the air-absorption)
+\textit{would receive, when the sun is overhead}, about \textit{two and one-third
+horse-power continuously}. Atmospheric absorption cuts this
+down to about one and one-half horse-power, of which about one-eighth
+can be actually utilized by properly constructed machinery,
+as, for instance, the solar engines of Ericsson and Mouchot (see
+Langley's ``New Astronomy''). In Ericsson's apparatus the reflector,
+about 11~feet by 16~feet, collected heat enough to work a
+three-horse-power engine very well. Taking the earth's surface as a
+whole, the energy received during a year aggregates about sixty mile-tons
+for every square foot. That is to say, the \textit{heat annually received
+on each square foot of the earth's surface, if employed in a}
+%% -----File: 230.png---Folio 219-------
+\textit{perfect heat engine, would be able to hoist sixty tons to the height
+of a mile}.
+
+\nbarticle{346.} \nbparatext{Solar Radiation at the Sun's Surface.}---If, now, we estimate
+the amount of radiation at the sun's surface itself, we come to
+results which are simply amazing and beyond comprehension.  It is
+necessary to multiply the solar constant observed at the earth (which
+is at a distance of 93,000000 miles from the sun) by the square of
+the ratio between 93,000000 and 433,250, the radius of the sun. This
+square is about 46,000; in other words, the amount of heat emitted
+in a minute by a square metre of the sun's surface is about 46,000
+times as great as that received by a square metre at the earth. Carrying
+out the calculations, we find that this heat radiation at the surface
+of the sun amounts to \textit{over a million calories per square metre
+per minute}; that it is over 100,000 horse-power per square metre
+continuously acting; that \textit{if the sun were frozen over completely to
+a depth of fifty feet, the heat emitted is sufficient to melt this whole
+shell in one minute of time}; that if an ice bridge could be formed
+from the earth to the sun by a column of ice two and one-fourth
+miles square at the base and extending across the whole 93,000000
+of miles, and if by some means the whole of the solar radiation
+could be concentrated upon this column, it would be melted in one
+second of time, and in between seven and eight seconds more would
+be dissipated in vapor. To maintain such a development of heat \textit{by
+combustion} would require the \textit{hourly burning of a layer of the best
+anthracite coal from sixteen to twenty feet thick} over the sun's entire
+surface,---a ton for every square foot of surface,---at least nine
+times as much as the consumption of the most powerful blast furnace
+in existence. At that rate the sun, if made of solid coal, would
+not last 6000 years.
+
+\begin{fineprint}
+\nbarticle{347.} \nbparatext{Waste Of Solar Heat.}---Those estimates are of course based on
+the assumption that the sun radiates heat equally in all directions, and there
+is no assignable reason why it should not do so. On this assumption, however,
+\textit{so far as we can see}, only a minute fraction of the whole radiation ever
+reaches a resting-place. The earth receives about $\frac{1}{2200,000000}$ of the whole, and
+the other planets of the solar system, with the comets and the meteors, get
+also their shares; all of them together, perhaps ten or twenty times as much
+as the earth. Something like $\frac{1}{100,000000}$ of the whole seems to be utilized within
+the limits of the solar system. As for the rest, science cannot yet tell what
+becomes of it. A part, of course, reaches distant stars and other objects in
+interstellar space; but by far the larger portion seems to be ``wasted,'' according
+to our human ideas of waste.
+\end{fineprint}
+%% -----File: 231.png---Folio 220-------
+
+\sloppy
+\nbarticle{348.} Experiments with the thermopile, first conducted by Henry
+at Princeton in 1845, show that the heat from the edges of the sun's
+disc, like the light, is less than that from the centre---according
+to Langley's measurements about half as much. The explanation
+evidently lies in its absorption by the solar atmosphere.
+
+\fussy
+\includegraphicsouter{illo123}{\textsc{Fig.~123.}}
+
+\nbarticle{349.} \nbparatext{The Sun's Temperature.}---While we can measure with some
+accuracy the \textit{quantity} of heat sent us by the sun, it is different with
+its \textit{temperature} in respect to which we can only say that it must be
+very high---much higher than any temperature attainable by known
+methods on the surface of the earth.
+
+This is shown by a
+number of facts, for instance,
+by the great \textit{abundance
+of the violet and
+ultra-violet rays} in the
+sunlight.
+
+Again, by the \textit{penetrating}
+power of sunlight;
+a large percentage of the heat from a common fire, for instance,
+being stopped by a plate of glass, while nearly the whole of the solar
+radiation passes through.
+
+The most impressive demonstration, however, follows from this
+fact; viz., that at the focus of a powerful burning-lens all known
+substances melt and vaporize, as in an electric arc. Now at the
+focus of the lens the \textit{limit} of the temperature is that which would
+be produced by the sun's direct radiation at a point where the sun's
+angular diameter equals that of the burning-lens itself seen from the
+focus, as represented in \figref{illo123}{Fig.~123}. An object at $F$ would theoretically
+(that is, if there was no loss of heat conducted away by surrounding
+bodies and by the atmosphere) reach the same temperature
+as if carried to a point where the sun's angular diameter equals the
+angle $LFL'$. In the most powerful burning-lenses yet constructed
+a body at the focus is thus virtually carried up to within about
+240,000 miles of the sun's surface, where its apparent diameter
+would be about $80°$. Here, as has been said, the most refractory
+substances are immediately subdued. If the earth were to approach
+the sun as near as the moon is to us, she would melt and be
+vaporized.
+
+\begin{fineprint}
+\nbarticle{350.}  Ericsson in 1872 made an exceedingly ingenious and interesting
+experiment illustrating the intensity of the solar heat. He floated a calorimeter,
+%% -----File: 232.png---Folio 221-------
+containing about ten pounds of water, upon the surface of a large
+mass of molten iron by means of a raft of fire-brick, and found that the
+radiation of the metal was a trifle over 250 calories per minute for each
+square foot of surface; which is only $\frac{1}{400}$ part of the amount emitted by the
+same area of the sun's surface. He estimated the temperature of the metal
+at $3000°$~F. or $1649°$~C.
+\end{fineprint}
+
+\sloppy
+\nbarticle{351.} \nbparatext{Effective Temperature.}---The question of the sun's temperature
+is embarrassed by the fact that it has no \textit{one} temperature; the
+temperature at different parts of the solar photosphere and chromosphere
+must be very different. We evade this difficulty to some
+extent by substituting for the \textit{actual} temperature, as the object of
+inquiry, what has been called the sun's ``\textit{effective temperature}''; that
+is, the temperature which a sheet of \textit{lampblack} must have in order
+to radiate the amount of heat actually thrown off by the sun. (Physicists
+have taken the radiating power of lampblack as \textit{unity}.) If we
+could depend upon the laws\footnote
+  {A number of such laws have been formulated; for instance, the well-known
+law of Dulong and Petit (Physics, p.~470). The French physicists Pouillet and
+Vicaire, using this formula, have deduced values for the sun's effective temperature
+running from $1500°$ to $2500°$~C\@. Ericsson and Secchi, using Newton's law
+of radiation (which, however, is certainly inapplicable under the circumstances),
+put the figure among the millions. Zöllner, Spörer, and Lane give values ranging
+  from $25,000°$ to $50,000°$~C.}
+deduced from laboratory experiments,
+by which it has been sought to connect the temperature of the body
+with its rate of radiation, the matter would then be comparatively
+simple: from the known radiated \textit{quantity of heat} (in calories) we
+could compute the \textit{effective temperature} in degrees. But at present it
+is only by a very unsatisfactory process of extrapolation that we can
+reach conclusions. The sun's temperature is so much higher than
+any which we can manage in our laboratories, that there is not yet
+much certainty to be obtained in the matter. Rosetti, the most
+recent investigator, whose results seem to be on the whole the most
+probable, obtains $10,000°$~C. or $18,000°$~F. for the effective temperature.
+
+\fussy
+\nbarticle{352.} \nbparatext{Constancy of the Sun's Heat.}---It is an interesting and thus
+far unsolved problem, whether the total amount of the sun's radiation
+varies perceptibly at different times.  It is only certain that the
+variations, if real, are too small to be detected by our present means
+of observation.  Possibly, at some time in the future, observations
+on a mountain summit above the main body of our atmosphere may
+decide the question.
+%% -----File: 233.png---Folio 222-------
+
+It is not unlikely that changes in the earth's climate such as
+have given rise to glacial and carboniferous periods may ultimately
+be traced to the condition of the sun itself, especially to changes in
+the thickness of the absorbing atmosphere, which, as Langley has
+pointed out, must have a great influence in the matter. Since the
+Christian era, however, it is certain that the amount of heat annually
+received from the sun has remained practically unchanged. This is
+inferred from the distribution of plants and animals, which is still substantially
+the same as in the days of Pliny.
+
+\nbarticle{353.} \nbparatext{Maintenance of the Solar Heat.}---The question at once
+arises, if the sun is sending off such an enormous quantity of heat
+annually, how is it that it does not grow cold?
+
+(\textit{a}) The sun's heat cannot be kept up by \textit{combustion}. As has
+been said before, it would have burned out long ago, even if made
+of solid coal burning in oxygen.
+
+(\textit{b}) Nor can it be simply a \textit{heated body cooling down}. Huge as it
+is, an easy calculation shows that its temperature must have fallen
+greatly within the last 2000 years by such a loss of heat, even if it
+had a specific heat higher than that of any known substance.
+
+As matters stand at present, the available theories seem to be
+reduced to two,---that of Mayer, which ascribes the solar heat to
+the energy of meteoric matter falling on the sun; and that of Helmholtz,
+who finds the cause in a slow contraction of the sun's diameter.
+
+\nbarticle{354.} \nbparatext{Meteoric Theory of Sun's Heat.}---The first is based on the
+fact that when a moving body is stopped, its mass-energy becomes
+molecular energy, and appears mainly as heat. The amount of heat
+developed in such a case is given by the formula
+\[
+Q = \frac{MV^{2}}{8339},
+\]
+in which $Q$ is the number of calories of heat produced, $M$ the mass
+of the moving body in kilograms, and $V$ its velocity in metres
+per second; the denominator is the ``mechanical equivalent of heat''
+\DPtypo{}{(}Physics, p.~159) multiplied by $2g$ expressed in metres; \textit{i.e.}, $425 × 2 × 9.81$.
+
+Now, the velocity of a body coming from any considerable distance
+and falling into the sun can be shown to be about 380 miles per
+second, or more than 610 kilometres. A body weighing one kilogram
+%% -----File: 234.png---Folio 223-------
+would therefore, on striking the sun with this velocity, produce about
+45,000000 calories of heat,
+\[
+\left[\frac{(610000)^{2}}{8339}\right].
+\]
+This is 6000 times more than could be produced by \textit{burning} it, even
+if it were coal or solidified hydrogen burning in pure oxygen.
+
+Now, as meteoric matter is continually falling upon the earth, it
+must be also falling upon the sun, and in vastly greater quantities,
+and an easy calculation shows that a quantity of meteoric matter
+equal to $\frac{1}{100}$ of the earth's mass striking the sun's surface annually
+with the velocity of 600 kilometres per second would account for its
+whole radiation.
+
+\nbarticle{355.} \nbparatext{Objections to Meteoric Theory of Sun's Heat.}---There can be
+no question that a certain fraction of the sun's heat is obtained in this
+way, but it is very improbable that this fraction is a large one;
+indeed, it is hardly possible that it can be as much as \textit{one per cent} of
+the whole.
+
+\begin{fineprint}
+(1) The annual fall on the sun's surface of such a quantity of meteoric
+matter implies the presence \textit{near} the sun of a vastly greater mass; for, as we
+shall see hereafter, only a few of the meteors that approach the sun from
+outer space would strike the surface: most of them would act like the
+comets and swing around it without touching. Now, if there were any
+considerable quantity of such matter near the sun, there would result disturbances
+in the motions of the planets Mercury and Venus, such as observation
+does not reveal.
+
+(2) Professor Peirce has shown further that if the heat of the sun were
+produced in this way, the earth ought to receive from the meteors that strike
+her surface about half as much heat as she gets from the sun. Now the
+quantity of meteoric matter which would have to fall upon the earth to furnish
+us daily half as much heat as we receive from the sun, would amount to
+nearly fifty tons for each square mile. It is not likely that we actually get
+$\frac{1}{10,000000}$ of that amount. It is difficult to determine the amount of heat which
+the earth actually does receive from meteors, but all observations indicate
+that the quantity is extremely small. The writer has estimated it, from
+the best data attainable, as less in a \textit{year} than we get from the sun in
+a \textit{second}.
+\end{fineprint}
+
+\sloppy
+\nbarticle{356.} \nbparatext{Helmholtz's Theory of Solar Contraction.}---We seem to be
+shut up to the theory of Helmholtz, now almost universally accepted:
+namely, that the heat necessary to maintain the sun's radiation is
+principally supplied \textit{by the slow contraction of its bulk}, aided, however,
+%% -----File: 235.png---Folio 224-------
+by the accompanying liquefaction and solidification of portions of its
+gaseous mass. When a body falls through a certain distance, \textit{gradually},
+against resistance, and then comes to rest, the same total amount
+of heat is produced as if it had fallen \textit{freely, and been stopped instantly}.
+If, then, the sun does contract, heat is necessarily produced by the
+process, and that in enormous quantity, since the attracting force at
+the solar surface is more than twenty-seven times as great as terrestrial
+gravity, and the contracting mass is immense. In this process
+of contraction each particle at the surface moves inward by an
+amount equal to the diminution of the sun's radius: a particle below
+the surface moves less and under a diminished gravitating force; but
+every particle in the whole mass, excepting only that at the exact
+centre of the globe, contributes something to the evolution of heat.
+In order to calculate the precise amount of heat evolved by a given
+shrinkage it would be necessary to know the law of increase of the
+sun's density from the surface to the centre; but Helmholtz has
+shown that under the most unfavorable conditions \textit{a contraction in the
+sun's diameter of about two hundred and fifty feet a year} (125 feet in
+the sun's \textit{radius}) \textit{would account for the whole annual output of heat}.
+This contraction is so slow that it would be quite imperceptible to
+observation. It would require more than 9000 years to reduce the
+sun's diameter by a single second of arc; and nothing much less
+would be certainly detectible by our measurements. \textit{If the contraction
+is more rapid than this}, the mean temperature of the sun
+must be actually \textit{rising}, notwithstanding the amount of heat it is
+losing. Long observation alone can determine whether this is really
+the case or not.
+
+\fussy
+\begin{fineprint}
+\nbarticle{357.} \nbparatext{Lane's Law.}---It is a remarkable fact, first demonstrated by
+Lane of Washington, in 1870, that a gaseous sphere, losing heat by radiation
+and contracting under its own gravity, \textit{must rise in temperature and actually
+grow hotter}, until it ceases to be a ``perfect gas,'' either by beginning to
+liquefy, or by reaching a density at which the laws of perfect gases no longer
+hold. The kinetic energy developed by the shrinkage of a gaseous mass
+is more than sufficient to replace the loss of heat which caused the shrinkage.
+In the case of a \textit{solid or liquid} mass this is not so. The shrinkage
+of such a mass contracting under its own gravity on account of the loss
+of heat is never sufficient to make good the loss; but the temperature falls
+and the body cools. At present it appears that in the sun the relative
+proportions of true gases and liquids are such as to keep the temperature
+nearly stationary, the liquid portions of the sun being of course the little
+drops which are supposed to constitute the clouds of the photosphere.
+\end{fineprint}
+%% -----File: 236.png---Folio 225-------
+
+\nbarticle{358.} \nbparatext{Future Duration of the Sun.}---If this shrinkage theory of the
+solar heat is correct (and there is every reason to accept it), it follows
+that in time the sun's heat must come to an end, and, looking backwards,
+we see that there must have been a beginning.
+
+We have not sufficient data to enable us to calculate the future
+duration of the sun with exactness, though an approximate estimate
+can he made. According to Newcomb, if the sun maintains its
+present radiation, it will have shrunk to half its present diameter in
+about 5,000000 years at the longest. Since when reduced to this
+size it must be about eight times as dense as now, it can hardly
+then continue to be mainly gaseous, and its temperature must begin
+to fall. Newcomb's conclusion, therefore, is that it is not likely
+that the sun can continue to give sufficient heat to support such life
+on the earth as we are now acquainted with, for 10,000000 years
+from the present time.
+
+\nbarticle{359.} \nbparatext{Age of the Sun.}---As to the past of the solar history on this
+hypothesis, we can be a little more definite. It is only necessary to
+know the present amount of radiation, and the mass of the sun, to compute
+how long the solar fire can have been maintained at its present
+intensity by the processes of condensation. No conclusion of geometry
+is more certain than this,---that the contraction of the sun to its
+present size, from a diameter even many times greater than Neptune's
+orbit, would have furnished about 18,000000 times as much
+heat as the sun now supplies in a year, and therefore that the sun
+cannot have been emitting heat \textit{at the present rate} for more than
+18,000000 years, \textit{if its heat has really been generated in this manner}.
+
+\begin{fineprint}
+But of course this conclusion as to the possible past duration of the solar
+system rests upon the assumption that the sun has derived its heat \textit{solely in
+this way}; and moreover, that it radiates heat equally in all directions in
+space,---assumptions which possibly further investigations may not confirm.
+\end{fineprint}
+
+\nbarticle{360.} \nbparatext{Constitution of the Sun.}---(\textit{a}) As to the nature of the main
+body or nucleus of the sun, we cannot be said to have certain knowledge.
+It is probably \textit{gaseous}, this being indicated by its low mean
+density and its high temperature---enormously high even at the surface,
+where it is coolest. At the same time the gaseous matter at the
+nucleus must be in a very different state from gases as we commonly
+know them in our laboratories, on account of the intense heat and the
+extreme condensation by the enormous force of solar gravity. The
+central mass, while still strictly gaseous, because observing the three
+%% -----File: 237.png---Folio 226-------
+physical laws of Boyle, Dalton, and Gay Lussac, which characterize
+gases, would be denser than water, and viscous; probably something
+like tar or pitch in consistency.\footnote
+  {The law of Dalton (Physics, p.~181) is, that any number of different gases and
+vapors tend to \textit{distribute themselves throughout the space which they occupy in common,
+each as if the others were absent}. The law of Boyle or Mariotte (Physics, p.~110)
+is, \textit{that at any given temperature the volume of any given amount of gas varies inversely
+with the pressure: i.e., $pv = p'v'$.} The law of Gay Lussac (Physics, p.~185) is,
+that a gas \textit{under constant pressure expands in volume uniformly under uniform
+increment of temperature}, so that $V_{t} = V_{0} (1 + at)$. This is not true of \textit{vapors} in
+presence of the liquids from which they have been evaporated; for instance, of
+steam in a boiler.}
+
+While this doctrine of the gaseous constitution of the sun is generally
+assented to, there are still some who are disposed to consider
+the great mass of the sun as liquid.
+
+\nbarticle{361.} (\textit{b}) The \textit{photosphere} is probably a shell of \textit{incandescent
+clouds}, formed by the condensation of the vapors which are exposed
+to the cold of space.
+
+\nbarticle{362.} (\textit{c}) The photospheric clouds float in an atmosphere containing,
+still uncondensed, a considerable quantity \textit{of the same vapors
+out of which they themselves have been formed}, just as in our own
+atmosphere the air around a cloud is still saturated with water vapor.
+This vapor-laden atmosphere, probably comparatively shallow, constitutes
+the \textit{reversing layer}, and by its selective absorption produces
+the dark lines of the solar spectrum, while by its general absorption
+it probably produces the darkening at the limb of the sun.
+
+\begin{fineprint}
+But it will be remembered that Mr.~Lockyer and others are disposed to
+question the existence of any such shallow absorbing stratum, considering
+that the absorption takes place in all regions of the solar atmosphere even
+to a great elevation.
+\end{fineprint}
+
+\nbarticle{363.} (\textit{d}) The \textit{chromosphere and prominences} are composed of
+the \textit{permanent gases}, mainly hydrogen and helium, which are mingled
+with the vapors of the reversing stratum in the region near the
+photosphere, but usually rise to far greater elevations than do the
+vapors. The appearances are for the most part as if the chromosphere
+was formed of jets of heated hydrogen ascending through the
+interspaces between the photospheric clouds, like flames playing over
+a coal fire.
+%% -----File: 238.png---Folio 227-------
+
+\nbarticle{364.} (\textit{e}) The \textit{corona} also rests on the photosphere, and the peculiar
+green line of its spectrum (\artref{Art.}{329}) is brightest just at the surface
+of the photosphere, in the reversing stratum and in the chromosphere
+itself; but the corona extends to a far greater elevation than even
+the prominences ever reach, and seems to be not wholly gaseous,
+but to contain, besides the hydrogen and the mysterious ``coronium,''
+dust and fog of some sort, perhaps meteoric. Many of its phenomena
+are as yet unexplained, and since it can only be observed during
+the brief moments of total solar eclipses, progress in its study is
+necessarily slow.
+
+\begin{fineprint}
+\nbarticle{364*.} \nblabel{pg:227}\textit{Note to }\artref{\textit{\DPtypo{article}{Article}}}{344}. The total heat received by the earth from the
+sun in any given time is that intercepted by its diametrical cross-section,
+\textit{i.e.}, by the area of one of its great circles kept always perpendicular to the
+sun's rays. The quantity of ice which would be melted annually on this
+circular plane by the solar rays would be a sheet having a thickness of 166.5
+metres or 546 feet ($19^{\text{mm}} × 24 × 365\frac{1}{4} = 166.5^{\text{met}}$).
+
+The thickness of the ice which could be melted in a year on a narrow
+equatorial belt would be $\frac{546^{\text{ft}}}{\pi}$, or $174^{\text{ft}}$, since such a belt intercepts the rays
+that would otherwise fall on a diametrical strip of the same width upon the
+circular plane.
+
+If the sun's heat were \textit{uniformly} distributed over the earth's whole surface,
+which equals four great circles, ($4\pi R^{2}$), it could melt a shell having a thickness
+of $\frac{546^{\text{ft}}}{4}$, or $136\frac{1}{2}^{\text{ft}}$.
+
+It is true that at the sea-level the solar-constant is much diminished by
+atmospheric absorption; and probably does not exceed fifteen calories per
+minute \textit{directly} received from the sun's rays. But a large part of the solar
+heat absorbed by the atmosphere reaches the earth's surface \textit{indirectly}, so
+that it must not be considered as lost to the earth, because not directly
+measurable by the actinometer.
+\end{fineprint}
+\chelabel{CHAPTERIX}
+%% -----File: 239.png---Folio 228-------
+
+\Chapter{X}{Eclipses}
+\nbchapterhang{\stretchyspace
+ECLIPSES: FORM AND DIMENSIONS OF SHADOWS.---LUNAR
+ECLIPSES.---SOLAR ECLIPSES.---TOTAL, ANNULAR, AND PARTIAL.---ECLIPTIC
+LIMITS AND NUMBER OF ECLIPSES IN A
+YEAR.---THE SAROS.---OCCULTATIONS.}
+
+\nbarticle{365.} \textsc{The} word eclipse (Greek \mytextgreek{>'ekleiyic}) is strictly a medical term,
+meaning a \textit{faint} or \textit{swoon}. Astronomically it is applied to the darkening
+of a heavenly body, especially of the sun or moon, though
+some of the satellites of other planets besides the earth are also
+``eclipsed'' from time to time. An eclipse of the \textit{moon} is caused
+by its passage through the shadow of the earth; an eclipse of the
+\textit{sun}, by the interposition of the moon between the sun and the observer,
+or, what comes to the same thing, by the passage of the
+moon's shadow over the observer.
+
+\nbarticle{366.} \nbparatext{Shadows.}---If interplanetary space were slightly dusty, we
+should see, accompanying the earth and moon and each of the
+planets, a long black shadow projecting behind it and travelling
+with it. Geometrically speaking, this shadow of a body, the earth
+for instance, is a \textit{solid---not a surface}. It is the space from which
+sunlight is excluded. If we regard the sun and other heavenly
+bodies as truly spherical, these shadows are \textit{cones} with their axes
+in the line joining the centres of the sun and the shadow-casting
+body, the point being always directed away from the sun, because
+the sun is always the larger of the two.
+
+\nbarticle{367.} \nbparatext{Dimensions of the Earth's Shadow.}---The length of the
+shadow is easily found. In \figref{illo124}{Fig.~124} we have from the similar
+triangles $OED$ and $ECa$, $OD:Ea::OE:EC$ or $l$. $OD$ is the difference
+between the radii of the sun and the earth, $= R-r$.
+$Ea=r$, and $OE$ is the distance of the earth from the sun $= \Delta$.
+\begin{flalign*}
+&\indent\text{Hence }&&
+  l = \Delta× \left( \frac{r}{R - r}\right)
+    = \frac{1}{108.5}\Delta.  &&\phantom{\indent Hence }
+\end{flalign*}
+
+(The fractional factor is constant, since the radii of the sun and
+%% -----File: 240.png---Folio 229-------
+earth are fixed quantities. Substituting the values of the radii, we
+find it to be $\frac{1}{108.5}$.) This gives 857,200 miles for the length of the
+earth's shadow when $\Delta$ has its mean value of 93,000000 miles, regarding
+the earth as a perfect sphere and taking its mean radius.
+This length varies about 14,000 miles on each side of the mean as
+the earth changes its distance from the sun.
+
+\includegraphicsmid{illo124}{\textsc{Fig.~124.}---Dimensions of the Earth's Shadow.}
+
+\begin{fineprint}
+The semi-angle of the cone (the angle $ECb$, or $ECB$ in the \figref{illo124}{figure}) is
+found as follows. Since $OEB$ is exterior to the triangle $BEC$,
+\begin{flalign*}
+&&            OEB &= EBC + BCE,  &&\phantom{or }  \\
+&\text{or }&  BCE &= OEB - EBC.  &&
+\end{flalign*}
+Now, $OEB$ is the \textit{sun's apparent semi-diameter} as seen from the earth, and
+$EBC$ is the earth's semi-diameter as seen from the sun, which is the same
+thing as \textit{the sun's horizontal parallax} (\artref{Art.}{83}).
+\end{fineprint}
+
+Putting $S$ for the sun's semi-diameter, and $p$ for its parallax, we
+have---
+\[
+\text{Semi-angle at } C = S - p.\footnote
+  {Also, $l = \dfrac{r}{\sin{(S - p)}}$, an expression sometimes more convenient than the one
+  given above.}
+\]
+
+From the cone $aCb$ all sunlight is excluded, or would be were it
+not for the fact that the atmosphere of the earth by its refraction
+bends some of the rays into this shadow. The effect is to make the
+shadow a little larger in diameter, but less perfectly dark.
+
+\nbarticle{368.} \nbparatext{Penumbra.}---If we draw the lines $Ba$ and $Ab$, crossing at
+$C'$ between the earth and the sun, they will bound the \textit{penumbra}.
+Within this space a part, but not the whole, of the sunlight is cut off:
+an observer outside of the shadow, but within this cone-frustum,
+%% -----File: 241.png---Folio 230-------
+which tapers \textit{towards} the sun, would see the earth as a black body
+encroaching on the sun's disc. The semi-angle of the penumbra $EC'a$
+is easily shown to be $S + p$.
+
+\nbarticle{369.} Although \textit{geometrically} the boundaries of the shadow and
+penumbra are perfectly definite, they are not so optically. If a screen
+were placed at $M$ (\figref{illo124}{Fig.~124}) perpendicular to the axis of the shadow,
+no sharply defined lines would mark the boundaries of either shadow
+or penumbra; near the edge of the shadow, the penumbra would
+be very nearly as dark as the shadow itself, only a mere speck of the
+sun being visible there; and at the outer limit of the penumbra the
+shading would be still more gradual.
+
+\nbarticle{370.} \nbparatext{Eclipses of the Moon.}---The axis of the earth's shadow is
+always directed to a point exactly opposite the sun. If, then, at the
+time of full moon, the moon happens to be near the ecliptic (that is,
+\textit{not far from one of the nodes of her orbit}), she will pass into the
+shadow and be eclipsed. Since, however, the moon's orbit is inclined
+about five and one-fourth degrees to the plane of the ecliptic, this
+does not happen very often (seldom more than twice a year). Ordinarily
+the moon passes north or south of the shadow without touching
+it.
+
+Lunar eclipses are of two kinds,---partial and total: total when
+she passes into the shadow completely; partial when she only partly
+enters it, going so far to the north or south of the centre of the
+shadow that only a portion of her disc is obscured.
+
+\begin{fineprint}
+We may also have a ``penumbral eclipse'' when she passes merely through
+the penumbra, without touching the shadow. In this case, however, the loss
+of light is so gradual and so slight, unless she almost grazes the shadow,
+that an observer would notice nothing unusual.
+\end{fineprint}
+
+\nbarticle{371.} \nbparatext{Size of the Earth's Shadow at the Point where the Moon
+crosses it.}---Since $EC$ in \figref{illo125}{Fig.~125} is 857,000 miles, and the distance
+of the moon from the earth is on the average about 239,000 miles,
+$CM$ must be 618,000 miles, and $MN$, the semi-diameter of the shadow
+at this point, will be $\frac{618}{857}$ of the earth's radius. This gives $MN = 2854$
+miles, and makes the whole diameter of the shadow a little over 5700
+miles, about two and two-thirds times the diameter of the moon. But
+this quantity varies considerably. The shadow is sometimes more
+than three times as large as the moon, sometimes hardly more than
+twice its size.
+%% -----File: 242.png---Folio 231-------
+
+\begin{fineprint}
+\nbarticle{372.} We may reach the same result in another way. Considering the
+triangle $ECN$, \figref{illo125}{Fig.~125}, we have the angular semi-diameter of the cross-section
+of the shadow where the moon passes through it, as seen from the
+earth, represented by $MEN$.
+\begin{flalign*}
+&\indent\text{But }&&  ENa = MEN + ECN;  &&\phantom{whence }\\
+&\text{whence }    &&  MEN = ENa - ECN.
+\end{flalign*}
+
+Now $ENa$ is the semi-diameter of the earth as seen from the moon; that
+is; it is the moon's \textit{horizontal parallax}, for which we write $P$. Hence, substituting
+for $ECN$ its value $S - p$, we get
+\[
+MEN = P + p - S.
+\]
+$MEN$ is called ``the radius of the shadow.'' The mean value of $P$ is $57'\, 2''$;
+of $p$, $8''.8$; and of $S$, $16'\, 2''$, which makes the mean value of $MEN = 41'\, 9''$.
+The mean value of the moon's apparent semi-diameter is $15'\, 40''$, the ratio
+between the semi-diameter of the moon and the radius of the shadow being
+about $2\frac{2}{3}$, as before.
+
+\includegraphicsmid{illo125}{\textsc{Fig.~125.}---Diameter of Earth's Shadow where the Moon crosses it.}
+
+In computing a lunar eclipse, this angular value for the ``radius of the
+shadow,'' as it is called, is more convenient than its value in miles. It is
+customary to increase it by about $\frac{1}{60}$ part in order to allow for the effect of
+the earth's atmosphere, the value ordinarily used being $\frac{61}{60}(P + p - S)$.
+Some computers, however, use $\frac{51}{50}$, and others $\frac{76}{75}$. On account of the indistinctness
+of the edge of the shadow it is not easy to determine what precise
+value ought to be employed.
+\end{fineprint}
+
+\nbarticle{373.} \nbparatext{Duration of a Lunar Eclipse.}---When central, a total eclipse
+of the moon may, all things favoring, continue total for about two
+hours, the interval from the first contact to the last being about two
+hours more. This depends upon the fact that the moon's hourly
+motion is nearly equal to its own diameter. The whole interval from
+first contact to last is the time occupied by the moon in moving from
+%% -----File: 243.png---Folio 232-------
+$a$ to $d$ (\figref{illo126}{Fig.~126}). The totality lasts while it moves from $b$ to $c$.
+The duration of a non-central eclipse varies, of course, according to
+the part of the shadow through which the moon passes.
+
+\includegraphicsmid{illo126}{\textsc{Fig.~126.}---Duration of a Lunar Eclipse.}
+
+\nbarticle{374.} \nbparatext{Lunar Ecliptic Limit.}---The lunar \textit{ecliptic limit} is the greatest
+distance from the node of the moon's orbit at which the sun
+can be \DPtypo{consistently}{consistent} with having an eclipse. This limit depends upon
+the inclination of the moon's orbit, which varies a little, and also upon
+the radius of the shadow at the time of the eclipse and the moon's
+apparent semi-diameter, which quantities are still more variable.
+Hence we recognize two limits, the major and minor. If the distance
+of the sun from the node at the time of full moon exceeds the
+major limit, an eclipse is impossible; if it is less than the minor, an
+eclipse is inevitable. The major limit is found to be $12°\, 5'$; the minor,
+$9°\, 30'$.  Since the sun passes over an arc of $12°\, 5'$ in less than
+thirteen days, it follows that an eclipse of the moon cannot possibly
+take place more than thirteen days before or after the time when the
+sun crosses the node.
+
+\includegraphicsmid{illo127}{\textsc{Fig.~127.}---Lunar Ecliptic Limit.}
+
+\begin{fineprint}
+\nbarticle{375.} In \figref{illo127}{Fig.~127} let $NE$ be the ecliptic, and $NM$ the orbit of the moon,
+the point $N$ being the node, and the angle at $N$ the inclination of the moon's
+orbit. $E$ is the centre of the earth's shadow. The sun, of course, is directly
+opposite, and its distance from the opposite node is equal to $EN$. $M$ is the
+%% -----File: 244.png---Folio 233-------
+centre of the moon. Call the semi-diameter of the moon $S'$; then $EM$ (the
+greatest possible distance between $E$ and $M$ which permits an eclipse) equals
+the sum of the semi-diameters of the moon and shadow, or $S' + (P + p - S)$,
+and the corresponding ecliptic limit $EN$ is found by solving the spherical
+triangle $MNE$, having given $ME$ and the angle at $N$, which is about $5\frac{1}{4}°$.
+We must also know one other angle, and with sufficient approximation for
+such purposes we may regard the angle at $M$ as a right angle. The solution
+will give the value of the ecliptic limit by assigning the proper values to the
+quantities involved. The limit is always very nearly \textit{eleven} times $EM$, because
+the inclination of the moon's orbit is nearly $\frac{1}{11}$ of a ``radian.''
+\end{fineprint}
+
+\nbarticle{376.} \nbparatext{Phenomena of a Total Lunar Eclipse.}---Half an hour or so
+before the moon reaches the shadow its eastern limb begins to be
+sensibly darkened, and the edge of the shadow itself, when it is first
+reached, looks nearly black by contrast with the bright parts of the
+moon's surface. To the naked eye the outline of the shadow appears
+reasonably sharp; but with even a small telescope it is found to be
+indefinite and hazy, and with a large instrument and high magnifying
+power it becomes entirely indistinguishable. It is impossible to determine
+the exact moment when the edge of the shadow reaches any
+particular point on the moon within half a minute or so.
+
+\includegraphicsmid{illo128}{\textsc{Fig.~128.}---Light bent into Earth's Shadow by Refraction.}
+
+After the moon has wholly entered the shadow her disc is usually
+still distinctly visible, illuminated with a dull, copper-colored light,
+which is sunlight, deflected around the earth into the shadow by the
+refraction of our own atmosphere, or rather by that portion of our
+atmosphere which lies within ten or fifteen miles of the earth's surface.
+Since the ordinary horizontal refraction is $34'$,
+it follows that
+light which just grazes the earth's surface will be bent inwards by
+twice that amount, or $1°\, 8'$. Now, the maximum ``radius of the
+shadow'' is only $1°\, 2'$. In an extreme case, therefore, even when
+the moon is exactly central
+in the largest possible shadow, it receives
+some sunlight coming around the edge of the earth, as shown by
+Fig.~128. To an observer stationed on the moon, the disc of the
+earth would appear to be surrounded by a narrow ring of brilliant
+sunshine, colored with sunset hues by the same vapors which tinge
+%% -----File: 245.png---Folio 234-------
+terrestrial sunsets, but acting with double power because the light
+has traversed a double thickness of our air. If the weather happens
+to be clear at this portion of the earth (upon its \textit{rim} as seen
+from the moon), the quantity of light transmitted through the atmosphere
+is very considerable, and the moon is strongly illuminated.
+If, on the other hand, the weather happens to be stormy
+in this region, the clouds cut off nearly all the light. In the lunar
+eclipse of 1886 the moon was absolutely invisible to the naked eye,
+a very unusual circumstance on such an occasion. At the eclipse
+of Jan.~28, 1888, Pickering found that the \textit{photographic power} of
+the centrally eclipsed moon was about $\frac{1}{400\,000}$ that of the moon
+when uneclipsed.
+
+\begin{fineprint}
+\nbarticle{377.} \nbparatext{Uses made of Lunar Eclipses.}---In astronomical importance a
+lunar eclipse cannot be at all compared with a solar eclipse. It has its uses,
+however. \textit{a}.~Many dates in chronology are fixed by reference to certain
+lunar eclipses. For instance, the date of the Christian era is determined by
+a lunar eclipse which happened upon the night that Herod died. \textit{b}.~Before
+better methods were devised, lunar eclipses were made use of to some extent
+in determining the longitude. Unfortunately, as has been said (\artref{Art.}{119}),
+it is impossible to note the critical instants with any degree of accuracy, on
+account of the indefiniteness of the moon's shadow. \textit{c}.~The study of the
+spectrum of the eclipsed moon gives us some data as to the constitution of
+our own atmosphere. We are thus enabled to examine light which has
+passed through a greater thickness of air than is obtainable in any other
+way. \textit{d}.~The study of the heat radiated by the moon during the different
+phases of an eclipse gives us some important information as to the absorbing
+power and temperature of its surface. Observations have been made
+at Lord Rosse's observatory of all the recent lunar eclipses, with this end in
+view. \textit{e}.~Finally, at the time when the moon is eclipsed, it is possible to observe
+its passage over small stars which cannot be seen at all when near the
+moon except at such a time. Observations of these star occultations made
+at different parts of the earth, furnish the best possible data for computing
+the dimensions of the moon, its parallax, and for determining its precise
+position in its orbit at the time of observation. The eclipses of the last
+few years have been very carefully observed in this way by concert between
+the different leading observatories.
+\end{fineprint}
+
+\sloppy
+\nbarticle{378.} \nbparatext{Computation of a Lunar Eclipse.}---Since all the phases of
+a lunar eclipse are seen everywhere at the same absolute instant
+wherever the moon is above the horizon, it follows that a single computation
+giving the Greenwich times of the different phenomena is all
+that is needed, and can be made and published once for all.  Each
+observer has merely to correct the predicted time by simply adding
+%% -----File: 246.png---Folio 235-------
+or subtracting his longitude from Greenwich in order to get the
+true local time. The computation is very simple, but lies rather
+beyond the scope of this work.
+
+\fussy
+\section*{ECLIPSES OF THE SUN.}
+
+\nbarticle{379.} \nbparatext{Dimensions of the Moon's Shadow.}---By the same method
+as used for the shadow of the earth (merely substituting in the
+formulæ the radius of the \textit{moon} for that of the earth), we find that
+the length of the moon's shadow at any time is $\frac{1}{399.55}$ of its distance
+from the sun, and at new moon averages 232,150 miles. It varies
+not quite 4000 miles each way, and so ranges from 236,050 to
+228,300. The semi-angle of the moon's shadow is practically equal
+to the semi-diameter of the sun at the earth, or very nearly $16'$.
+
+\includegraphicsmid{illo129}{\textsc{Fig.~129.}---The Moon's Shadow on the Earth.}
+
+\nbarticle{380.} \nbparatext{The Moon's Shadow on the Earth's Surface.}---Since the
+mean length of the shadow is less than the mean distance of the
+moon from the earth (which is 238,800 miles), it is obvious that
+\textit{on the average} it will not reach the earth. On account of the
+eccentricity of the moon's orbit however, our satellite is much of
+the time considerably nearer than this mean distance, and may come
+within 221,600 miles from the earth's centre, or about 217,650
+miles from its surface. The shadow, also, under favorable circumstances,
+may have a length of 236,050 miles. Its point may therefore
+at times extend nearly 18,400 miles beyond the earth's surface.
+The cross-section of the shadow where the earth's surface cuts it
+(at $o$ in \figref{illo129}{Fig.~129}) will then be 167~miles. \textit{This is the largest value
+possible}.
+
+\begin{fineprint}
+Of course, if the shadow strikes obliquely on the surface of the earth, as
+it must except when the moon is in the zenith, the shadow spot will be \textit{oval}
+instead of circular, and the length of the oval along the earth's surface may
+much exceed the true cross-section of the shadow.
+\end{fineprint}
+
+\nbarticle{381.} \nbparatext{The ``Negative'' Shadow.}---Since the distance of the moon
+may be as great as 252,970 miles from the earth's centre, or nearly
+%% -----File: 247.png---Folio 236-------
+249,000 miles from its surface, while the shadow may be as short as
+228,300 miles, we may have the state of things indicated by placing
+the earth at $B$ in the \figref{illo129}{figure}. The vertex of the shadow, $V$, will then
+fall 21,700 miles short of the surface, and the cross-section of the
+``\textit{shadow produced}'' will have a diameter of 206 miles where the
+earth's surface cuts it. When the shadow falls near the edge of the
+earth, this cross-section may be as great as 230 miles. The shadow-spot
+which is formed by the intersection of the produced shadow-cone
+with the earth's surface is sometimes called the \textit{negative shadow},
+because in calculating an eclipse its radius comes out from the formulæ
+as a \textit{minus} quantity in case the shadow does not reach the earth.
+
+\nbarticle{382.} \nbparatext{Total and Annular Eclipses.}---To an observer within the
+true shadow cone, that is, between $V$ and the moon in \figref{illo129}{Fig.~129}, the
+sun will be \textit{totally} eclipsed; but an observer in the produced cone
+beyond $V$ will see the moon projected on the sun, leaving an uneclipsed
+ring around it. He will have what is called an \textit{annular}
+eclipse. These annular eclipses are considerably more frequent than
+total eclipses---nearly in the ratio of three to two.
+
+\includegraphicsmid{illo130}{\textsc{Fig.~130.}---Width of the Penumbra of the Moon's Shadow.}
+
+\nbarticle{383.} \nbparatext{The Penumbra and Partial Eclipses.}---The penumbra can
+easily be shown to have a diameter on the line $CD$ (\figref{illo129}{Fig.~129}) of very
+nearly twice the moon's diameter. We may take it as having an average
+diameter at this point of 4400 miles; but as the earth is often beyond
+$V$, its cross-section \textit{at the earth} is sometimes as much as 4800
+miles. An observer situated within the penumbra observes a partial
+eclipse: if he is near the shadow cone, the sun will be mostly covered
+by the moon; but if near the outer limit of the penumbra, the moon
+will only slightly encroach on the sun's disc. While, therefore, total
+and annular eclipses are visible as such only by an observer within
+the narrow path traversed by the shadow-spot, the same eclipse
+will be visible as a partial one everywhere within 2000 miles on
+%% -----File: 248.png---Folio 237-------
+either side of the shadow path; and the 2000 miles is to be reckoned
+\textit{perpendicularly} to the axis of the shadow. When, for instance,
+the penumbra falls, as shown in \figref{illo130}{Fig.~130}, the distance $BC$ \textit{measured
+along the earth's surface} will be over 3000 miles, although $BF$ is only
+2000.
+
+\nbarticle{384.} \nbparatext{Velocity of the Shadow and Duration of Eclipses.}---The
+moon advances along its orbit very nearly 2100 miles an hour, and
+were it not for the earth's rotation this is the rate at which the
+shadow would pass the observer. The earth, however, is rotating
+towards the east in the same general direction as that in which the
+shadow moves, and its surface moves at the rate of about 1040
+miles an hour at the equator. An observer, therefore, on the earth's
+equator, with the moon near the zenith, would be passed by the
+shadow with a speed of about 1060 miles per hour ($2100 - 1040$);
+and this is its slowest velocity, which is about equal to that of a
+cannon-ball.
+
+\includegraphicsmid{illo131}{\textsc{Fig.~131.}---Track of the Moon's Shadow, Eclipse of July, 1878.}
+
+In higher latitudes, where the velocity of the earth's rotation is less,
+the relative speed of the shadow is higher; and where the shadow falls
+very obliquely, as it does when an eclipse occurs near sunrise or sunset,
+%% -----File: 249.png---Folio 238-------
+the advance of the shadow along the earth's surface may become
+exceedingly swift,---as great as 4000 or 5000 miles per hour. \figref{illo131}{Fig.~131},
+which we owe to the courtesy of the publishers of Langley's
+``New Astronomy,'' shows the track of the moon's shadow during the
+eclipse of July 29, 1878.
+
+\nbarticle{385.} \nbparatext{Duration of an Eclipse.}---A \textit{total} eclipse of the sun observed
+at a station near the equator under the most favorable conditions possible
+(the shadow-spot having its maximum diameter of 167 miles),
+may continue total for \textit{seven minutes and fifty-eight seconds}. In latitude
+$40°$ the duration of totality can barely equal six and one-quarter
+minutes. The greatest possible excess of the radius of the moon
+over that of the sun is only $1'\, 19''$.
+
+An \textit{annular} eclipse may last for $12^{\text{m}}\, 24^{\text{s}}$ at the equator. The
+maximum width of the ring of the sun visible around the moon
+is $1'\, 37''$.
+
+In the observation of an eclipse four ``contacts'' are noted: the
+first when the edge of the moon first touches the edge of the sun; the
+second, when the eclipse becomes total or annular; the third, at
+the cessation of the total or annular phase; and the fourth, when
+the moon finally leaves the disc of the sun. From the first contact
+to the fourth the duration may be a little over two hours.
+
+\includegraphicsmid{illo132}{\textsc{Fig.~132.}---Solar Ecliptic Limits.}
+
+\nbarticle{386.} \nbparatext{The Solar Ecliptic Limits.}---It is necessary, in order to have
+an eclipse of the sun, that the moon should encroach on the cone
+$ACBD$ (\figref{illo132}{Fig.~132}), which envelops earth and sun. In this case the
+``true'' angular distance between the centres of the sun and moon---that
+is, their distance as seen from the centre of the earth---would
+be the angle $MES$ in the \figref{illo132}{figure}. This is made up of three angles:
+$MEF$, which equals the moon's semi-diameter, $S'$; $AES$, the sun's
+semi-diameter; $S$; and $FEA$. This latter angle is equal to the difference
+%% -----File: 250.png---Folio 239-------
+between $CFE$ and $FAE$. $CFE$ is the moon's horizontal parallax
+(the semi-diameter of the earth seen from the moon), and $FAE$
+or $CAE$ is the sun's parallax. $FEA$, therefore, equals $P - p$; and
+the whole angle $MES$ equals $S + S' + P - p$. This angle may range
+from $1°\, 34'\, 13''$ to $1°\, 24'\, 19''$, according to the changing distances\footnote
+  {We give herewith in a table the different quantities which determine the
+dimensions of the shadows of the earth and moon, as well as the ecliptic limits
+and the duration of eclipses.\\[1ex]
+\begin{tabular*}{\textwidth} {l@{\extracolsep{\fill}}c@{\extracolsep{1pt}} |c|c|c}
+\hline\hline
+&& Greatest. & \PadTo{Greatest.}{Least.} & \PadTo{Greatest.}{Mean.}\rule[-1.5ex]{0pt}{4.5ex}\\
+\cline{2-5}
+Apparent semi-diameter of sun
+&& $16'\, 18''$ & $15'\, 45''$ & $16'\, 02''$\rule{0pt}{3ex}
+\\
+Apparent semi-diameter of moon
+&& $16'\, 46''$ & $14'\, 44''$ & $15'\, 45''$
+\\
+Horizontal parallax of the sun
+&& $9''.05$ & $8''.55$ & $8''.80$
+\\
+Horizontal parallax of the moon
+&& $61'\, 18''$ & $53'\, 58''$ & $57'\, 38''$
+\\
+Inclination of moon's orbit
+&& $\phantom{0}5°\, 19'\,$ & $\phantom{0}4°\, 57'\,$
+& $5°\, 8'\, 40''$
+\\
+\multicolumn{5}{c}{Sun's radius, 433,200 miles; earth's (mean), 3956; moon's, 1081.5.\rule[-1.5ex]{0pt}{4.5ex}}\\
+\hline\hline
+\end{tabular*}}
+of
+the sun and moon from the earth.
+
+The corresponding distances of the sun from the node, calculated in
+the same way as the lunar ecliptic limits (taking the maximum inclination
+of the moon's orbit as $5°\, 19'$ and the minimum as $4°\, 57'$,
+according to Neison), give $18°\, 31'$ and $15°\, 21'$ for the major and
+minor ecliptic limits.
+
+In order that an eclipse may be \textit{central} (total or annular) at any
+part of the earth, it is necessary that the moon should lie wholly
+inside the cone $ACBD$, as at $M'$. In this case the angle $M'ES$ will
+be $S - S' + P - p$, and the corresponding major and minor \textit{central}
+ecliptic limits come out $11°\, 50'$ and $9°\, 55'$.
+
+\nbarticle{387.} \nbparatext{Phenomena of a Solar Eclipse.}---There is nothing of special
+interest until the sun is mostly covered, though before that time the
+shadows cast by foliage begin to look peculiar. The light shining
+through every small interstice among the leaves, instead of forming a
+little \textit{circle} on the earth, makes a little \textit{crescent}---an image of the
+partly covered sun.
+
+Some ten minutes before totality the darkness begins to be felt, and
+the remaining light, coming as it does from the edge of the sun only,
+is much altered in quality, producing an effect very like that of
+a calcium light rather than sunshine. Animals are perplexed, and
+birds go to roost. The temperature falls a few degrees, and sometimes
+dew appears.
+%% -----File: 251.png---Folio 240-------
+
+In a few moments, if the observer is so situated that his view commands
+a distant western horizon, the moon's shadow is seen coming
+much like a heavy thunder-storm. It advances with almost terrifying
+swiftness until it envelops him.
+
+For a moment the air appears to quiver, and on every white surface
+bands or ``\textit{fringes},'' alternately light and dark, appear. They
+are a few inches wide and from a foot to three feet apart, and on the
+whole seem to be parallel to the edge of the shadow. Probably they
+travel with the \textit{wind}; but observations on this point are as yet hardly
+decisive. The phenomenon is not fully explained, but is probably
+due to irregular atmospheric refraction of the light coming from the
+indefinitely narrow strip of the sun's limb on the point of disappearing.
+
+\nbarticle{388.} \nbparatext{Appearance of the Corona and Prominences.}---As soon as the
+shadow arrives, and sometimes a little before it, the corona and prominences
+become visible. The stars of the first three magnitudes make
+their appearance at the same time.
+
+The suddenness with which the darkness descends upon the observer
+is exceedingly striking; the sun is so brilliant that even the
+small portion which remains visible up to within a very few seconds
+of the time of totality so dazzles the eye that it is not prepared for
+the sudden transition. In a few moments, however, the vision becomes
+accustomed to the changed circumstances, and it is then found
+that the darkness is not really very intense. If the totality is of
+short duration,---that is, if the diameter of the moon exceeds that
+of the sun by less than a minute of arc,---the lower parts of the corona
+and chromosphere, which are very brilliant, give a light at least three
+or four times as great as that of the full moon. Since the shadow also
+in such a case is of small diameter, a large quantity of light is sent in
+from the surrounding air, where thirty or forty miles away the sun
+is still shining; and what may seem remarkable, this intrusion of
+outside light is greatest when the sky is clouded. In such an eclipse
+there is not much difficulty in reading an ordinary watch-face. In
+an eclipse of long duration (say five or six minutes) it is much
+darker, and lanterns are necessary.
+
+\begin{fineprint}
+\nbarticle{389.} \nbparatext{Observations of an Eclipse.}---A total solar eclipse offers an
+opportunity of making an immense number of observations of great importance
+which are possible at no other time, besides certain others which can
+also be made during a partial eclipse. We mention (\textit{a})~\textit{Times of the four
+contacts, and direction of the line joining the cusps during the partial phases}.
+%% -----File: 252.png---Folio 241-------
+These observations determine accurately the relative position of the sun and
+moon at the time, and so furnish the means for correcting the tables of their
+motion. (\textit{b})~\textit{The search for intra-mercurial planets}. It has been thought
+likely that there may be one or more planets between the orbit of Mercury
+and the sun, and during a total eclipse they would become visible, if ever.
+On the whole, however, the observations, so far made, negative the existence
+of any body of considerable size in this region, though in 1878, Professor
+Watson and Mr.~Swift, it was thought, had discovered one, if not two, such
+planets. (\textit{c})~\textit{Observations on the fringes}, which have been described as showing
+themselves at the commencement of totality. Probably the phenomenon
+is merely atmospheric and of little importance, but it is not yet sufficiently
+understood. (\textit{d})~\textit{Photometric measurements} of the intensity of the light at
+different stages of the eclipse and during totality. (\textit{e})~\textit{Telescopic observations
+of the details of the prominences and of the corona}. (\textit{f})~\textit{Spectroscopic observations},
+both visual and photographic, upon the spectra of the lower atmosphere
+of the sun, the prominences, and the corona. (\textit{g})~\textit{Observations with the
+polariscope} upon the polarization of the light of the corona, for the purpose
+of determining the relation between the reflected and intrinsic light, and
+perhaps the size of the reflecting particles which are distributed through the
+corona. (\textit{h})~\textit{Photography}, both of the partial phases and of the corona.
+
+\nbarticle{390.} \nbparatext{Calculation of a Solar Eclipse.}---The calculation of a solar
+eclipse cannot be dealt with in any such summary way as that of a lunar
+eclipse, owing to the moon's parallax, which makes the times of contact and
+other phenomena different at every different station. The moon's apparent
+path in the sky, \textit{relative to the centre of the sun}, is not even a portion of a
+great circle, nor is it described with a uniform velocity. Moreover, since
+the phenomena of a solar eclipse admit of very accurate observations, it is
+necessary to take account of numerous little details which are of no importance
+in a lunar eclipse.
+
+Certain data for each solar eclipse hold good wherever the observer may
+be. These are calculated beforehand and published in the nautical almanacs;
+and from them, with the knowledge of his geographical position, the
+observer can work out the results for his own station. But the calculations
+are somewhat complicated and lie beyond our scope. The reader is referred
+to any work on practical astronomy; Chauvenet and Loomis treat the matter
+very fully.
+\end{fineprint}
+
+\nbarticle{391.} \nbparatext{Number of Eclipses in a Year.}---The least possible number is
+\textit{two}, both central eclipses of the sun. The largest possible number is
+\textit{seven}, \textit{five} of the sun and \textit{two} of the moon. The eclipses each year
+happen at two seasons (which may be called the ``eclipse months''),
+half a year apart---about the times, of course, at which the sun in its
+annual path crosses the two nodes of the moon's orbit. If these nodes
+were stationary, the eclipse months would be always the same; but
+%% -----File: 253.png---Folio 242-------
+because the nodes retrograde around the ecliptic once in about nineteen
+years, the eclipse months are continually changing. The time
+required by the sun in passing around from a node to the same
+node again is 346.62 days, which is sometimes called the ``eclipse
+year.''
+
+\includegraphicsouter{illo133}{\textsc{Fig.~133.}---Number of Eclipses Annually.}
+
+\sloppy
+\nbarticle{392.} \nbparatext{Number of Lunar Eclipses.}---Representing the ecliptic by a
+circle (\figref{illo133}{Fig.~133}) with the two opposite nodes $A$ and $a$, it is easy to see
+\textit{first}, that there can he but two \textit{lunar} eclipses in a year (omitting for
+a moment one exceptional case). The major lunar ecliptic limit is
+$12°\, 15'$; hence there is only a space of twice that amount, or $24°\, 30'$,
+between $L$ and $L'$, at each ``node
+month,'' within which the occurrence
+of a full moon might give
+a lunar eclipse. Now, in a synodic
+month the sun moves along
+the ecliptic $29°\, 6'$, while the node
+moves in the opposite direction
+$1°\, 31'$, giving the \textit{relative} motion
+of the sun referred to the node
+equal to $30°\, 37'$; \textit{i.e., the full-moon
+points on the circle would fall
+at a distance of $30°\, 37'$ from each
+other}. Only \textit{one} full moon, therefore,
+can possibly occur within
+the lunar ecliptic limits each time that the sun passes the node.
+
+\fussy
+Since the \textit{minor} ecliptic limit for the moon is only $9°\, 30'$, it may
+easily happen that \textit{neither} of the full moons which occur nearest to
+the time when the sun is at the node will fall within the limit. There
+are accordingly many years which have no lunar eclipses.
+
+\textit{Three} lunar eclipses, however, may possibly happen in one calendar
+year in the following way. Suppose the first eclipse occurs about
+Jan.~1, the sun passing the node about that time; the second may then
+happen about June 25 at the other node, $a$. The first node, $A$, will
+run back during the year, so that the sun will encounter it again about
+Dec.~13 at $A'$, and thus a third eclipse may occur in December of
+the same year. This actually occurred in 1787, the dates of the three
+lunar eclipses being Jan.~3, June 30, and Dec.~24.
+
+\sloppy
+\nbarticle{393.} \nbparatext{Number of Solar Eclipses.}---Considering now \textit{solar} eclipses,
+we find that there must inevitably be \textit{two}. Twice the minor limit
+%% -----File: 254.png---Folio 243-------
+(\artref{Art.}{386}) of a solar eclipse ($15°\, 21'$) is $30°\, 42'$, which, is more than
+the sun's whole motion in a month. One new moon, at least, therefore,
+\textit{must} fall within the limiting distance of the node, and two
+\textit{may} do so, since in the \figref{illo133}{figure}, $SS'$ is always greater than the distance
+between the points occupied by two successive new moons.
+
+\fussy
+If the two \textit{new} moons in the two eclipse months happen to fall
+very near a node, the two \textit{full} moons, a fortnight earlier and later,
+will both be very likely to fall outside the lunar limit. In that case
+the year will have only two eclipses, both solar and both central; \textit{i.e.},
+either total or annular. This was the case in 1886.
+
+Again, if in any year two \textit{full moons} occur when the sun is very near
+the node, then since the \textit{major} solar limit is $18°\, 31'$, it may happen,
+and often does, that there will be two partial solar eclipses, one a
+fortnight before, the other a fortnight after, each of the lunar eclipses,
+and so the year will have three eclipses in each eclipse month---six
+eclipses in all, two lunar and four solar. A \textit{fifth} solar eclipse may
+also come in near the end of the year, if the node was passed about
+Jan.~15, in the same way that sometimes happens with a lunar eclipse:
+the year will then have \textit{seven} eclipses. This was the case in 1823.
+The most usual number of eclipses is four or five.
+
+\nbarticle{394.} \nbparatext{Relative Frequency of  Solar and Lunar Eclipses.}---Although,
+\textit{taking the whole  earth into account}, the  solar  eclipses are
+the most numerous, about in the proportion of four to three, it is
+not so with the eclipses \textit{visible at any given place}. A solar eclipse
+can be seen only from a limited portion of the globe, while a lunar
+eclipse is visible over considerably more than half the earth, either at
+the beginning or end, if not throughout its whole duration; and this
+more than reverses the proportion between lunar and solar eclipses
+for any given station.
+
+\nbarticle{395.} \nbparatext{Recurrence of Eclipses, and the Saros.}---It is not known how
+early it was discovered that eclipses recur at a regular interval of
+eighteen years and eleven and one-third days (\textit{ten} and one-third days,
+if there happen to be five leap years in the interval); but the Chaldeans
+knew the period very well, and called it the \textit{Saros} (which means
+``restitution'' or ``repetition''), and used it in predicting the recurrence
+of these phenomena. It is a period of 223 \textit{synodic months},
+which is almost exactly equal to nineteen eclipse years. The eclipse
+year is $346^{\text{d}}.6201$, and  nineteen of them equal $6585^{\text{d}}.78$, while 223
+months equal $6585^{\text{d}}.32$.
+%% -----File: 255.png---Folio 244-------
+
+The difference is only $\frac{46}{100}$ of a day (about 11~hours) in which time
+the sun moves $28'$. If, therefore, an eclipse should occur to-day at
+new moon, with the sun exactly at the node, then after 223 months
+(18~years 11~days) a new moon will occur again with the sun only $28'$
+west of the node; so that the circumstances of the first eclipse will
+be pretty nearly repeated. It would however occur about eight hours
+of longitude further west on the earth's surface, since the 223 months
+exceed the even 6585 days by $\frac{32}{100}$ of a day, or $7^{\text{h}}\ 42^{\text{m}}$.
+
+\begin{fineprint}
+\nbarticle{396.} \nbparatext{Number of Recurrences of a Given Eclipse.}---It is usual to
+speak of eclipses recurring at this regular interval as ``repetitions'' of one
+and the same eclipse. A lunar eclipse is usually thus ``repeated'' 48 or 49
+times. Beginning as a very small partial eclipse, with the sun about $12°$
+\textit{east} of the node, it will be a little larger at its next occurrence eighteen
+years later; and after 13 or 14 repetitions the sun will have come so near
+the node that the eclipse will have become total. It will then be repeated
+as a total eclipse 22 or 23 times, after which it will become partial again
+with the sun \textit{west} of the node, and after 13 more returns as a partial eclipse
+will finally dwindle away and disappear, having thus recurred regularly once
+in every 223 months during an interval of $865\frac{1}{2}$ years.
+
+The same thing happens with the solar eclipses, only since the solar
+ecliptic limit is larger than the lunar, a solar eclipse has from 65 to 70 returns,
+occupying some 1200 years. Of these about 25 are only partial
+eclipses, the sun being so near the ecliptic limit that the \textit{axis} of the shadow
+does not reach the earth at all. The 45 eclipses in the middle of the period
+are central somewhere or other on the earth, about 18 of them being total,
+and about 27 annular. These numbers vary somewhat, however, in different
+cases.
+\end{fineprint}
+
+\nbarticle{397.} It is to be noticed that the Saros exhibits not only a close
+commensurability of the \textit{synodic} months with the \textit{eclipse} years, but
+also with the \textit{nodical}\footnote
+  {The \textit{nodical} month is the time of the moon's revolution from one of its \textit{nodes}
+to the same node again, and is equal to $27^{\text{d}}.21222$; the \textit{anomalistic} month is the
+time of revolution from \textit{perigee} to perigee again, and equals $27^{\text{d}}.55460$. See
+  Arts.~\arnref{454}, \arnref{455}.}
+and \textit{anomalistic} months: 242 nodical months
+equal 6585.357 days; 239 anomalistic months equal 6585.549 days.
+This last coincidence is important. The moon at the end of the
+Saros of  223 months not only returns very closely to its original
+position \textit{with respect to the sun and the node}, but also with respect to
+\textit{the line of apsides} of its orbit. If it was at perigee originally, it will
+again be within five hours of perigee at the end of the Saros. If this
+were not so, the time of the eclipse might be displaced several hours
+%% -----File: 256.png---Folio 245-------
+by the perturbations of the  moon's motion, to be considered later,
+in \chapref{CHAPTERXII}{Chap.~XII.}
+
+\nbarticle{398.} \nbparatext{Number of Eclipses in a Single Saros.}---The total number is
+usually about seventy, varying two or three one way or the other, as
+new eclipses come in at the eastern limit and go out at the western.
+Of the 70, 29 are usually lunar and 41 solar; and of the solar, 27 are
+\textit{central}, 17 being annular and 10 total. (These numbers are necessarily
+only approximate.) It appears, therefore, that total solar eclipses,
+\textit{somewhere or other on the earth}, are not very rare, there being about
+ten in eighteen years. Since, however, the shadow track averages
+less than 100 miles in width, each total eclipse is visible, \textit{as total},
+over only a very small fraction of the earth's whole surface---about
+$\frac{1}{200}$ in the mean. This gives about one total eclipse in 360 years, in
+the long run, at any given station.
+
+\begin{fineprint}
+The total solar eclipses visible in the United States during the present
+century are the following:---
+
+June 10, 1806, in New York and New England, duration $4\frac{1}{2}$ minutes;
+Nov.~30, 1834, in Arkansas, Missouri, Alabama, and Georgia, duration 2
+minutes; July 18, 1860, in Washington Territory and Labrador, 3 minutes;
+Aug.~7, 1869, in Iowa, Illinois, Kentucky, North Carolina, $2\frac{3}{4}$ minutes; July 29,
+1878, in Wyoming, Colorado, Texas, $2\frac{1}{2}$ minutes; Jan.~11, 1880, in California,
+duration 32 seconds; Jan.~1, 1889, in California and Montana, $2\frac{1}{4}$ minutes;
+May 27, 1900, from Louisiana to Virginia, 2 minutes.
+\end{fineprint}
+
+\nbarticle{399.} \nbparatext{Occultations of Stars.}---In theory, and in the method of computation,
+the occultation of a star is precisely like a solar eclipse, except
+that the shadow of the moon projected by a star is a \textit{cylinder} instead
+of a \textit{cone}, since, compared with the distance of the sun, that of a
+star is infinite: moreover, the star is a mere point, so that there is
+no sensible penumbra. In other words, a star has neither parallax
+nor semi-diameter, and  these  circumstances somewhat simplify the
+formulæ.
+
+As the moon moves always towards the east, the disappearance of
+the star always takes place at the eastern limb, and the reappearance
+at the western. In the first half of the lunation the eastern limb
+is dark and invisible, and the star vanishes without warning. The
+suddenness with which it vanishes and reappears has already been
+referred to (\artref{Art.}{255}) as proof of the non-existence of a lunar atmosphere.
+Observations of this sort determine the moon's plane with
+great accuracy, and when corresponding observations are made at
+%% -----File: 257.png---Folio 246-------
+different places, they supply one of the best possible means of determining
+their difference of longitude.
+
+\begin{fineprint}
+In some cases observers have reported that a star, instead of disappearing
+instantaneously when struck by the moon's limb (faintly visible by earth-shine),
+has appeared to cling to the limb for a second or two before vanishing,
+and in a few instances they have even reported it as having reappeared
+and disappeared a second time, as if it had been for a moment visible through
+a rift in the moon's crust. Some of these anomalous phenomena have been
+explained by the subsequent discovery that the star was double, or had a
+faint companion.
+\end{fineprint}
+\chelabel{CHAPTERX}
+
+%% -----File: 258.png---Folio 247-------
+
+\Chapter{XI}{Central Forces}
+\nbchapterhang{\stretchyspace
+CENTRAL FORCES: EQUABLE DESCRIPTION OF AREAS.---AREAL,
+LINEAR, AND ANGULAR VELOCITIES.---KEPLER'S LAWS AND
+INFERENCES FROM THEM.---GRAVITATION DEMONSTRATED
+BY THE MOON'S MOTION.---CONIC SECTIONS AS ORBITS.---THE
+PROBLEM OF TWO BODIES.---THE ``VELOCITY FROM
+INFINITY,'' AND ITS RELATION TO THE SPECIES OF ORBIT
+DESCRIBED BY A BODY MOVING UNDER GRAVITATION.---THE
+INTENSITY OF GRAVITATION.}
+
+\nbarticle{400.} \textsc{A moving} body left to itself, according to Newton's first law
+of motion (Physics, p.~27), moves on forever in a straight line with a
+uniform velocity. If we find a body so moving, we may, therefore,
+infer that it is either acted on by \textit{no} force whatever, or, if forces are
+acting upon it, that they exactly balance each other.
+
+\begin{fineprint}
+It has been customary with some writers to speak of a body thus moving
+``uniformly in a straight line'' as actuated by a ``projectile \textit{force},'' a very
+unfortunate expression, which is a survival of the Aristotelian idea that rest
+is more ``natural'' to matter than motion, and that when a body moves,
+some force must operate to keep it moving. The mere uniform rectilinear
+motion of a material mass in empty space implies no physical cause, and
+demands no explanation. \textit{Change} of motion, either in speed or in direction---this
+alone implies \textit{force in operation}.
+\end{fineprint}
+
+\includegraphicsouter{illo134}{\textsc{Fig.~134.}---Curvature of an Orbit.}
+
+\nbarticle{401.} If a body moves in a straight line, with swiftness either
+increasing or decreasing, we infer a
+force acting exactly in the line of motion,
+and accelerating or retarding it.
+If it moves in a \textit{curve}, we know that
+some force is acting \textit{athwart} the line
+of motion. If the velocity in the curve
+increases, we know that the direction
+of the force that acts is \textit{forward}, like
+$ab$ (\figref{illo134}{Fig.~134}), making an angle of less
+than $90°$ with the ``line of motion'' ${a\,t}$
+(the tangent to the path of
+the body); and \textit{vice versa}, if the motion of the body is retarded.
+%% -----File: 259.png---Folio 248-------
+If the speed does not alter at all, we know that the force must act
+along the line $ac$, \textit{exactly perpendicular} to the line of motion.
+
+\begin{fineprint}
+Here, also, we find many writers, the older ones especially, bringing in
+the ``projectile force,'' and saying that when a body moves in a curve it does
+so under the action of \textit{two} forces, one a force that draws it sideways, the
+other the ``projectile force'' directed along its path. We repeat; this ``projectile
+force'' has no present existence or meaning in the problem. Such
+a force may have put the body in motion long ago, but its function has
+ceased, and \textit{now} we have only to do with the action of one single force,---\textit{the
+deflecting force}, which alters the direction of the body's motion. Of
+course it is not intended to deny that the deflecting force may itself be the
+resultant of any number of forces all acting together; but a single force acting
+athwart a body's line of motion is sufficient to cause it to describe a
+curvilinear orbit, and from such an orbit we can only infer the \textit{necessary}
+existence of \textit{one} such force.
+\end{fineprint}
+
+\includegraphicsouter{illo135}{\textsc{Fig.~135.}\\Description of Areas in Uniform Motion.}
+
+\nbarticle{402.} \nbparatext{Description of Areas.}---(\textit{a}) If a body is moving uniformly on
+a straight line, and if we connect the points $A$, $B$, $C$, etc., \figref{illo135}{Fig.~135}, which
+it occupies at the end of successive units of time with any point whatever,
+as $O$, we shall have a series
+of triangles, $AOB$, etc., \textit{which
+will all be equal}; since their bases
+$AB$, $BC$, etc., are equal and on
+the same straight line, and they
+have a common vertex at $O$.
+Calling the line from $A$ to $O$ its
+radius vector, and $O$ the ``centre,''
+we may say, therefore, that
+when a body is moving undisturbed
+by any force whatever,
+\textit{its radius vector, from any centre arbitrarily chosen, describes equal
+areas in equal times around that centre}. The area enclosed in the
+triangle described by the radius vector in a unit of time is called
+the body's ``\textit{areal} (or ``\textit{areolar}'') velocity,'' and in this case is
+constant.
+
+\nbarticle{403.} (\textit{b}) Moreover, \textit{any impulse in the line of the radius vector,
+either towards or from the centre, leaves unchanged both the plane of
+the body's motion and its areal velocity}.
+
+Suppose a body moving uniformly on the line $AC$ (\figref{illo136}{Fig.~136}) with
+such a velocity that it describes $AB$, $BC$, etc., in successive units of
+time; then, by the preceding section, the areal velocity will be constant,
+%% -----File: 260.png---Folio 249-------
+and measured by the area of any one of the equal triangles
+$AOB$, $BOC$, or $COL$. Suppose, now, that at $C$ the body receives
+an ``impulse'' directed along the radius vector towards $O$---a blow,
+for instance, which by itself would make the body describe $CK$ in a
+unit of time. The body will now take a new path, which will carry it
+to the point $D$, determined  by constructing the ``parallelogram  of
+motions'' $CKDL$, and thus combining the new motion $CK$ with the
+former motion $CL$, according to Newton's second law (Physics, p.~27).
+The new areal velocity, measured by the triangle $OCD$, will be the
+same as before, as is easily shown.
+
+\includegraphicsmid{illo136}{\textsc{Fig.~136.}---Description of Areas under an Impulse directed towards a Centre.}
+
+Triangle $BOC =$ triangle $COL$, because $BC = CL$, and $O$ is their
+common vertex.
+
+Also, triangle $COL =$ triangle $COD$, because they have the common
+base $OC$, and their summits $L$ and $D$ are on a line which was drawn
+parallel to this base in constructing the parallelogram of motions.
+Hence triangle $BOC =$ triangle $COD$, and the \textit{areal velocity remains
+unchanged}.
+
+Also, as may be seen by following out the same reasoning with
+$CK'$ and $CD'$, the same result would hold true if the impulse had
+been directed \textit{away from} $O$ instead of towards it.
+
+\begin{fineprint}
+\nbarticle{404.} This result depends entirely on the fact that the impulse $CK$ or
+$CK'$ was \textit{exactly along the radius vector} $CO$. If it has not been so, then in
+%% -----File: 261.png---Folio 250-------
+constructing the parallelogram of motions to find the points $D$ and $D'$, we
+should have had to draw $LD$ or $LD'$ \textit{not} parallel to $CO$, and the two triangles
+$BOC$ and $COD$ would necessarily have been unequal. $COD$ would be
+greater than $BOC$ if $CK$ were directed ahead of the radius vector, and less
+if behind it.
+\end{fineprint}
+
+As regards the plane of motion, the point $D$ is on the plane $OCL$,
+because $LD$ was drawn through $L$ parallel to $OC$. $OCL$ is a part
+of the plane which contains the triangles $BOC$ and $AOB$, and hence
+$OCD$ also lies in the same plane.
+
+\sloppy
+\nbarticle{405.} (\textit{c}) From this obviously follows the important general proposition
+that \textit{when a body is moving under the action of a force always
+directed towards or from a fixed centre, the radius vector will describe
+equal areas in equal times; and the path of the body will all lie in one
+plane}.
+
+\fussy
+Such a force constantly acting is simply equivalent to an indefinite
+number of separate impulses. Now if no single impulse directed
+along the radius vector can alter the areal velocity or plane of motion,
+neither can a succession of them. Hence the proposition follows.
+
+In case of a \textit{continuously} acting force the orbit, however, will
+become a curve instead of being a broken line.
+
+Observe that this proposition remains true whether the force is
+attractive or repulsive, and that it is independent of the \textit{law} of the
+force; that is, the force may vary directly \textit{with the distance}, or \textit{inversely
+as the square of the distance}, or as \textit{the logarithm} of it, or in
+any conceivable way; it may even be \textit{discontinuous}, acting only at
+intervals and ceasing between times: and still the law holds good.
+
+\nbarticle{406.} Conversely, \textit{if a body moves in this way, describing equal
+areas in equal times around a point, it is easily shown that all the
+forces acting upon the body must be directed toward that point}.
+
+We, however, leave the demonstration to the student.
+
+Since the earth moves very nearly in this way in its orbit around
+the sun, we conclude that the only force of any consequence acting
+upon the earth is directed towards the sun. We say, ``of any consequence,''
+because there are other small forces which do slightly
+modify the earth's motion, and prevent it from \textit{exactly} fulfilling the
+law of areas.
+
+{\centering\nbrule[.5\textwidth]\par}
+
+As a direct consequence of the law of equal areas we have certain
+laws with respect to the linear and angular velocities of a body moving
+under the action of a central force.
+%% -----File: 262.png---Folio 251-------
+
+\includegraphicsouter{illo137}{\textsc{Fig.~137.}\\Linear and Angular Velocities.}
+
+\nbarticle{407.} \nbparatext{Law of Linear Velocity.}---Suppose a body moving under
+the action of a force always directed towards $S$ (\figref{illo137}{Fig.~137}), and let
+$AB$ be a portion of its path which it describes
+in a second. Draw the tangent
+$Bb$. Regarding the sector $ASB$ as a
+triangle (which it will be, sensibly, since
+the curvature of the path in one second
+will be very small) the area of this triangle
+will be $\frac{1}{2}(AB× Sb)$. Now $AB$,
+the distance travelled in a second, is
+the \textit{linear velocity} of the body (called
+\textit{linear} because it is measured with the
+same units as any other \textit{line}; \textit{i.e.}, in
+\textit{miles} or in \textit{feet} per second), and $Sb$ is the
+distance from the centre of force to the ``line of motion,'' as the tangent
+$Bb$ is called. For $Sb$, $p$ is usually written; hence in every
+part of the same orbit, $V$ (the velocity in miles per second) $= \dfrac{2A}{p}$, and
+is inversely proportional to $p$. If $p$ were to become zero, $V$ would
+become infinite, unless $A$ were zero also.
+
+\nbarticle{408.} \nbparatext{Law of Angular Velocity.}---Referring again to the same
+\figref{illo137}{figure}, the area of $ASB$ is equal to $\frac{1}{2}(AS× BS×\sin{ASB})$, or
+$A = \frac{1}{2}r_{1}r_{2}\sin{\omega}$. If we draw $r$ to the middle point of $AB$, then $r_{1}r_{2} = r^{2}$,
+nearly, since in a second of time the distance would not
+change perceptibly as compared with its whole length. $\omega$ will also
+be a small angle, so that its sine will equal the angle itself expressed
+in radians;
+\begin{flalign*}
+&\text{hence }&& \tfrac{1}{2}r^{2}\omega = A, \text{ and }\omega = \frac{2A}{r^{2}}.  &&\phantom{hence }
+\end{flalign*}
+Now $\omega$ is the \textit{angular velocity} of the body; that is, the number of
+``radians'' which it describes in a second of time, as seen from $S$,
+while $r$ is the radius vector.
+
+\nbarticle{409.} In every case, therefore, of motion under a central force,
+I.~\textit{The Areal velocity $($acres per second$)$ is constant}; II.~\textit{The
+Linear velocity $($miles per second$)$ varies inversely as the distance from
+the centre of force to the body's line of motion at the moment}, which
+line of motion is the tangent to the orbit at the point where the body
+happens to be; III.~\textit{The Angular velocity $($radians, or degrees, per
+second$)$ varies inversely as the square of the distance of the body from
+the centre of force}.
+%% -----File: 263.png---Folio 252-------
+
+\nbarticle{410.} The student will remember that it was found by observation
+that the sun's angular velocity varies as the square of its apparent
+diameter, and from this (\artref{Art.}{186}) the law of equal areas was
+inferred as a fact with respect to the earth's motion. Newton was
+the first to point out that a body moving under the action of a
+central force must \textit{necessarily} observe this law of areas, and, conversely,
+that a body thus observing the law of areas must necessarily
+be under the control of a central force.
+
+\nbarticle{411.} \nbparatext{Circular Motion.}---In the case of a body moving in a
+\textit{circle} under the action of a central force, the force must be constant,
+and (Physics, p.~62) is given by the formula
+\[
+f = \frac{V^2}{r}, \eqno{(a)}
+\]
+in which $r$ is the radius of the circle and $V$ the velocity, while $f$ is
+the central force measured by the ``acceleration'' of the body
+towards the centre; that is, by the number of units of velocity
+which the force would generate in the body in a second of time; just
+as the force of gravity is expressed by writing, $g = 9.81$ metres.
+
+For many purposes it is desirable to have an expression which
+shall substitute for $V$ (a quantity not given directly by observation)
+the time of revolution, $t$, which is so given. Since $V$ equals the
+circumference of the circle divided by $t$, or $\dfrac{2\pi r}{t}$, we have at once, by
+substituting this value for $V$ in equation (\textit{a}),
+\[
+f = 4\pi^2\left(\frac{r}{t^2}\right). \eqno{(b)}
+\]
+
+This, of course, is merely the equivalent of equation (\textit{a}), but is
+often more convenient.
+
+\section*{KEPLER'S LAWS.}
+
+\nbarticle{412.} In 1607--1620 Kepler discovered as facts, without an explanation,
+three laws which govern the motions of the planets,---laws
+which still bear his name. He worked them out from a discussion
+of the observations which Tycho Brahe had made through many
+preceding years. The three laws are as follows:---
+
+\mbox{\phantom{II}}I\@. \textit{The orbit of each planet is an ellipse, with the sun in one of its foci}.
+
+\mbox{\phantom{I}}II\@. \textit{The radius vector of each planet describes equal areas in equal
+times}.
+%% -----File: 264.png---Folio 253-------
+
+III\@. The ``Harmonic law,'' so-called. \textit{The squares of the periods
+of the planets are proportional to the cubes of their mean distances
+from the sun;} i.e., $t_1^2:t_2^2 = a_1^3:a_2^3$.
+
+\begin{fineprint}
+\nbarticle{413.} To make sure that the student apprehends the meaning and scope
+of this third law, we append a few simple examples of its application.
+
+(1) What would be the \textit{period} of a planet having a mean distance from
+the sun of 100 astronomical units; \textit{i.e.}, a distance 100 times that of the earth?
+\[
+  (\textit{\footnotesize Earth's Dist.} )^3 : (\textit{\footnotesize Planet's Dist.} )^3
+= (\textit{\footnotesize Earth's Period})^2 : (\textit{\footnotesize Planet's Period})^2;
+\]
+\vspace{-1.5\baselineskip}
+\begin{flalign*}
+&\textit{i.e.,}  &&1^3:100^3 = 1^2(\text{year}):X^2(\text{years}),&&&&\\
+&\text{whence}   &&X = 100^\frac{3}{2} = 1000\text{ years}.&&&&\phantom{whence}
+\end{flalign*}
+
+(2) What would be the distance of a planet having a period of 125 years?
+\begin{flalign*}
+&&&(1)^2:125^2 = 1^3:X^3,&&&&\\
+&\text{whence}   &&X = 125^\frac{2}{3} = 25 \text{ (Astron.\ units)}.&&&&\phantom{whence}
+\end{flalign*}
+
+(3) How long would a planet require to fall to the sun?
+
+If the sun were collected in a single point at its centre, a body starting
+from a point on the planet's orbit with a slight \textit{side-motion, i.e.}, motion at right
+angles to the radius vector, would describe an extremely narrow ellipse
+around the sun, with its perihelion just \textit{at} the sun, and the aphelion at the
+starting-point. Practically it would ``\textit{fall to the sun},'' and return just as if it
+had \textit{rebounded} from a perfectly elastic surface: the time of ``falling'' would be
+just equal to that of returning---the two making up the whole period of the
+body in the narrow ellipse. Now the semi-major axis of this narrow ellipse
+is evidently \textit{one-half} the radius of the planet's orbit. Hence to find the period
+in this ellipse which is $2\tau$ ($\tau$ being taken as the time of ``falling''), we have
+\begin{flalign*}
+&&  & a^3 : (\tfrac{1}{2}a)^3 = t^2:(2\tau)^2, \text{ or }
+      1   : \tfrac{1}{8} = t^2:4\tau^2;  &\phantom{whence }&
+\\
+&\text{whence }&  & \tau = t\sqrt{\frac{1}{32}} = 0.1768t,
+&\llap{$t$ being the planet's period.}&
+\end{flalign*}
+
+In the case of the earth $\tau = 365\frac{1}{4}× 0.1768 = 64.56$ days.
+
+(4) What would be the period of a satellite revolving close to the earth's
+surface?
+\[
+(\textit{Moon's Dist.})^3:(\textit{Dist.\ of Satellite})^3 = (27.3\text{ days})^2:X^2,
+\]
+\vspace{-1.5\baselineskip}
+\begin{flalign*}
+&\text{or } &&60^3:1^3 = (27.3)^2:X^2,&&&\\
+&\text{whence } && X = \frac{27.3\text{ days}}{60^\frac{3}{2}} = 1^\text{h}\ 24^\text{m}.&&&
+\end{flalign*}
+
+(5) How much would an increase of 10 per cent in the earth's distance
+from the sun increase the length of the year? \hfil\penalty0\hfilneg\null\penalty5000\hfill\mbox{\quad  \textsc{Ans}. 56.13 days.}
+
+(6) What is the distance from the sun of an asteroid which has a period
+of $3\frac{1}{2}$ years? \hfil\penalty0\hfilneg\null\penalty5000\hfill\mbox{\quad  \textsc{Ans}. 2.305 Astron.\ units.}
+\end{fineprint}
+%% -----File: 265.png---Folio 254-------
+
+\nbarticle{414.} Many surmises were made as to the physical meaning of
+these laws. More than one astronomer \textit{guessed} that a force directed
+toward the sun, or emanating from it, might be the explanation.
+Newton proved it. He demonstrated the law of equal areas and its
+converse as necessary consequences of the laws of motion. He also
+proved (Physics, pp.~64--66) that if a body move in an ellipse having
+a centre of force at its focus, then the force at different points in the
+orbit must vary inversely as the square of the distance from that
+centre. And, finally, he showed that, granting the harmonic law,
+the force from planet to planet must also vary according to the same
+law of inverse squares.
+
+\begin{fineprint}
+\nbarticle{415.} The demonstration of this last proposition for circular orbits is so
+simple that we give it, merely adding (without proof) that the proposition
+is equally true for elliptical orbits, if for $r$ we put $a$, the semi-major axis of
+the orbit.
+
+In a circular orbit, from equation (\textit{b}), (\artref{Art.}{411}), we have
+\[
+f = 4\pi^{2}\left(\frac{r}{t^{2}}\right),
+\]
+where $r$ and $t$ are the distance and period of a planet. In the same way the
+force acting upon a second planet is found from the equation
+\begin{flalign*}
+&             && f_{1} = 4\pi^{2}\left(\frac{r_{1}}{{t_{1}}^{2}}\right),  &&\\
+&\text{whence}&& \frac{f}{f_{1}} = \frac{r}{t^{2}}× \left(\frac{{t_{1}}^{2}}{r_{1}}\right).  &&\\
+\intertext{\indent But by Kepler's third law}
+&             && t^{2}:{t_{1}}^{2}::r^{3}:{r_{1}}^{3},  &&\\
+&\text{whence}&& {t_{1}}^{2} = \frac{t^{2}{r_{1}}^{3}}{r^{3}}.&&\phantom{\text{whence}}
+\end{flalign*}
+
+Substitute this value of ${t_{1}}^{2}$ in the preceding equation; we have
+\begin{flalign*}
+&&  & \frac{f}{f_{1}}
+= \frac{r}{t^{2}} × \frac{t^{2}{r_{1}}^{3}}{r^{3}r_{1}}
+= \frac{{r_{1}}^{2}}{r^{2}};  &&\\
+&\textit{i.e.},& & f:f_{1} = {r_{1}}^{2}:r^{2}, &&\phantom{i.e. }
+\end{flalign*}
+which is the law of inverse squares.
+
+\nbarticle{416.} Conversely, the harmonic law is just as easily shown to be a
+necessary consequence of the law of gravitation in the case of circular orbits.
+
+From \artref{Art.}{411}, Eq.~(\textit{b}), we have
+\[
+f = 4\pi^{2} \frac{r}{t^{2}} ;
+\]
+%% -----File: 266.png---Folio 255-------
+also, from the law of gravitation,
+\[
+f = \frac{M}{r^{2}},\quad\rlap{\text{$M$ being the mass of the sun}.}
+\]
+
+Hence, equating the two values of $f$,
+\[
+\frac{M}{r^{2}} = 4\pi^{2}\frac{r}{t^{2}},\qquad\text{ and }\qquad t^{2} = \frac{4\pi^{2}}{M}r^{3}.
+\]
+
+Similarly for another planet,
+\[
+{t_{1}}^{2} =  \frac{4\pi^{2}}{M}{r_1}^{3}.
+\]
+\begin{flalign*}
+\text{\indent Whence,} &&t^{2}:{t_{1}}^{2} = r^{3}:{r_{1}}^{3}.&&\phantom{\text{Whence, }}
+\end{flalign*}
+
+The demonstration for elliptical orbits is a little more complicated,
+involving the ``law of areas.'' It is given in all works on Theoretical
+Astronomy, and may be found in Loomis's ``Elements of Astronomy,'' p.~134.
+
+\nbarticle{417.} \nbparatext{Correction of Kepler's Third Law.}---The ``harmonic law''
+as it stands is not \textit{exactly} true, though the difference is too small to appear
+in the observations which Kepler made use of in its discovery. It would be
+exactly true if the planets were mere particles of matter; but as a planet's
+mass is a sensible, though a very small fraction of the sun's mass, it comes
+into account. The planet Jupiter, for instance, attracts the sun as well as is
+attracted by it. If at the distance $r$ Jupiter is drawn towards the sun by a
+force which would give it in a second an acceleration expressed by $\frac{M}{r^{2}}$ (the
+sun's mass being $M$), then the sun in the same time is accelerated towards
+Jupiter by the quantity $\frac{m}{r^{2}}$ ($m$ being the mass of Jupiter). The rate at which
+the two tend to \textit{approach each other} is therefore expressed by $\frac{M + m}{r^{2}}$. Hence,
+in discussing the motions of the planet Jupiter around the \textit{centre of the sun},
+instead of writing
+\begin{flalign*}
+&&f &= \frac{M}{r^{2}}\text{ simply},&&\\
+&\text{we must put}& f &= \frac{M + m}{r^{2}};&&\phantom{\text{we must put}}\\
+\intertext{but (in the case of entirely circular motion)}
+&&f &= \frac{4\pi^{2}r}{t^{2}}.&&\\
+\intertext{\indent Hence, we find}
+&& t^{2}(M + m) &= 4\pi^{2}r^{3};&&\\
+\intertext{or, as a proportion,}
+&& t^{2}(M + m)&:{t_{1}}^{2}(M + m_{1}) = r^{3}:{r_{1}}^{3},&&\\
+\intertext{which is \textit{strictly} true as long as the planet's motions are undisturbed.}
+\end{flalign*}
+\end{fineprint}
+\vspace{-2\baselineskip}
+%% -----File: 267.png---Folio 256-------
+
+\nbarticle{418.} \nbparatext{Inferences from Kepler's Laws.}---From Kepler's laws we
+are entitled to infer---
+
+\textit{First} (from the second law), that the \textit{force} which retains the planets
+in their orbits \textit{is directed towards the sun}.
+
+\textit{Second} (from the first law), that on any given planet the \textit{force
+which acts varies inversely as the square of its distance from the
+sun}.
+
+\textit{Third} (from the harmonic law), that the force is the same for one
+planet as it would be for another in the same place; or, in other
+words, the attracting force \textit{depends only on the mass and distance of
+the bodies concerned, and is wholly independent of their physical conditions},
+such as their temperature, chemical constitution, etc. It
+makes no difference in the motion of a planet around the sun whether
+it be made of hydrogen or iron, whether it be hot or cold.
+
+\includegraphicsouter{illo138}{\textsc{Fig.~138.}\\
+Verification of the Hypothesis of Gravitation
+by Means of the Motion of the Moon.}
+
+\nbarticle{419.} \nbparatext{Verification of ``Gravitation'' by Means of the Moon's Motion.}---When
+the idea of gravitation
+first occurred to Newton (the
+apple story is probably apocryphal),
+he endeavored to verify it
+by comparing the force which keeps
+the moon in her orbit with the force
+of gravity at the earth's surface,
+reduced in the proper proportion.
+For lack, however, of an accurate
+knowledge of the earth's dimensions,
+he failed at first, there being
+a discrepancy of about sixteen per
+cent. He had assumed a degree
+to be exactly sixty miles in length.
+Some years afterward, when Picard's
+measure of the arc of a meridian
+in Northern France had been
+made and reported to the Royal Society, making a degree about
+sixty-nine miles long, he saw at once that the new value would
+reconcile the discrepancy; and he resumed his unfinished work and
+completed it.
+
+\nbarticle{420.} At the earth's surface a body falls about 193 inches in a
+second. The distance of the moon being very nearly sixty times the
+earth's radius, if gravity really varies inversely as the square of the
+%% -----File: 268.png---Folio 257-------
+distance, a stone at that distance from the earth should fall $\dfrac{1}{60^{2}}$ or
+$\dfrac{1}{3600}$ as far; that is, it ought to fall $\dfrac{193\ \text{inches}}{3600} = 0.0535$ inches,---a
+little more than one-twentieth of an inch. Now the distance which
+the moon actually \textit{does} fall towards the earth in a second, \textit{i.e.}, the
+\textit{deflection of its orbit from a straight line in a second of time}, is easily
+found; and if the force which keeps the moon in its orbit is really the
+same as that which makes bodies fall towards the centre of the earth,
+this deflection ought to come out equal to $0^{\text{in}}.0535$. Let $AE$ (\figref{illo138}{Fig.~138})
+be the distance the moon travels in a second $= \dfrac{2\pi r}{t}$, where $r$ is the
+radius of the moon's orbit, and $t$ the number of seconds in a month.
+Then, since $AEF$ is a right-angled triangle, we have,
+\[
+AB:AE::AE:AF\ (\text{or }2r);
+\]
+\[
+\text{whence }\qquad AB = \frac{AE^{2}}{2r}.
+\]
+The calculation is easy enough, though the numbers are rather large.
+As a result it gives us $AB = 0.0534$ inches, which is practically equal
+to the thirty-six hundredth part of $193$ inches.
+
+If the quantities did not agree in amount, the discrepancy would
+disprove the theory, and, as we have said, Newton loyally gave it up
+until he was able to show that the apparent discordance was the result
+of a mistake in the original data, and disappeared when the data were
+corrected. The agreement, however, does not \textit{establish} the theory,
+but only renders it probable. It does not establish it completely,
+because it is conceivable that the agreement might be a case of accidental
+coincidence, while the forces might really differ as much in
+their nature as an electrical attraction and a magnetic.
+
+\nbarticle{421.} Newton was not satisfied with merely showing that the principal
+motions of the planets and the moon could be explained by the
+law of gravitation; but he went on to investigate the converse problem,
+and to determine what must be the motions \textit{necessary} under that
+law. He found that the orbit of a body moving around a central mass
+is not of necessity a circle, or even a nearly circular ellipse like the
+planetary orbits, but that it may be a \textit{conic section} of any eccentricity
+whatever---a circle, ellipse, parabola, or even an hyperbola; but \textit{it
+must be a conic}.
+%% -----File: 269.png---Folio 258-------
+
+\nbarticle{422.} For the benefit of those of our readers who are not acquainted
+with conic sections we give the following brief account of them
+(\figref{illo139}{Fig.~139}):---
+
+\textit{a}. If a cone of any angle be cut \textit{perpendicularly to the axis}, the
+section will be a \textit{circle}---$MN$ in the
+\figref{illo139}{figure}.
+
+\textit{b}. If it be cut by a plane which
+makes with the axis an angle \textit{greater}
+than the semi-angle of the cone, \textit{so
+that the plane of section cuts completely
+across the cone} (as $EF$), the
+section, is an \textit{ellipse}; the circle being
+merely a special case of the ellipse.
+Ellipses, of course, differ greatly in
+form, from these which are very
+narrow to the perfect circle.
+
+\includegraphicsouter{illo139}{\textsc{Fig.~139.}---The Conics.}
+
+\textit{c}. \textit{The parabola} is formed by
+cutting the cone with a plane \textit{parallel
+to its side}; \textit{i.e.}, making with the axis
+an angle \textit{equal to the semi-angle of
+the cone}. $RPO$ is such a plane. As
+all circles are alike in form, so are
+all parabolas, \textit{whatever the angle
+of the cone at $V$ and wherever the
+point $P$ is taken}. If the cutting
+plane is thus situated, then, no
+matter what is the angle of the
+cone or the place where the cut is
+made, the curve will always be the
+same \textit{in shape}, though of course \textit{its
+size} will depend upon a variety of
+circumstances. We do not stop to
+prove the statement, at first a little surprising; but it is true.
+
+\textit{d}. If the cutting plane makes an angle with the axis of the cone
+\textit{less than the semi-angle at $V$}, so that the cutting plane gets continually
+deeper and deeper into the cone, then the curve is an \textit{hyperbola}; so
+called, because the plane in this case ``shoots over'' (\mytextgreek{<up'er b'allein})
+and intersects the ``cone produced,'' cutting out of this second cone
+a curve precisely like the curve cut from the original, as at $H'G'K'$ in
+the \figref{illo139}{figure}. The axis of the hyperbola lies outside of the curve itself,
+being the line $HH'$ in the \figref{illo139}{figure}, and the ``\textit{centre}'' of the curve is
+also outside of the curve at the middle point of this axis.
+%% -----File: 270.png---Folio 259-------
+
+\sloppy
+\nbarticle{423.} Philosophically speaking there are therefore but \textit{two} species
+of conic sections,---the ellipse and the hyperbola, with the parabola
+for a partition between them. (The circle, as has been said before,
+is merely a special case of the ellipse.) \figref{illo140}{Fig.~140} will give the reader
+perhaps a better idea of the nature of the curves as drawn on a plane.
+In the ellipse the sum of the distances from the two foci, $FN + F'N$,
+equals the major axis of the curve; in the hyperbola it is the difference
+of these two lines ($F''N' - FN'$) that equals the major axis;
+in the ellipse the \textit{eccentricity} is \textit{less than unity} (zero in the circle); in
+the hyperbola it is \textit{greater than unity}; in the parabola \textit{exactly unity}.
+
+\fussy
+\includegraphicsmid{illo140}{\textsc{Fig.~140.}---The Relation of the Conics to Each Other.}
+
+\begin{fineprint}
+The general equation of a conic in polar co-ordinates, applying alike to
+both the species, is
+\[
+r = \frac{p}{1 + e \cos{V}},
+\]
+in which $r$ is the distance $Fn$, or $Fn'$, $e$ is the fraction $\dfrac{FC}{PC}$ or $\dfrac{FC'}{PC'}$, the angle
+$V$ is the angle $PFn$, $PFn'$, or $PFn''$, and $p$ is the line $FY$, $FY'$ or $FY''$,
+called the ``semi-parameter.'' The word ``\textit{para}meter'' means the \textit{cross}-measure
+of a curve, just, as ``\textit{dia}meter'' means the \textit{through}-measure of a curve. If $e$
+is zero, the curve is a circle, and $r = p$. If $e < 1$, the curve is an ellipse; if
+$e > 1$, the curve is an hyperbola; if $e = 1$, it is a parabola.
+\end{fineprint}
+%% -----File: 271.png---Folio 260-------
+
+\nbarticle{424.} \nbparatext{Problem of Two Bodies.}---This problem, proposed and
+solved by Newton, is the following:---
+
+\textit{Given the masses of two spheres and their positions and motions at
+any moment; given, also, the law of gravitation: required their motion
+ever afterwards, and the data necessary to compute their place at any
+future time}.
+
+The mathematical methods by which the problem is solved require
+the use of the calculus, and must be sought in works on analytical
+mechanics or theoretical astronomy. Some of the results, however,
+are simple and easily stated.
+
+\nbarticle{425.} (1) In the first place the motion of the centre of gravity of
+the two bodies will not be affected by their mutual attraction, but it
+will move on uniformly through space, as if the bodies were united
+into one at that point, and their motions combined under the same
+laws which hold good in the case of the collision of inelastic bodies.
+
+\includegraphicsmid{illo141}{\textsc{Fig.~141.}---Motion of Bodies relative to their Centre of Gravity.}
+
+\begin{fineprint}
+The motion of this centre of gravity is most easily worked out graphically
+as follows: First, in \figref{illo141}{Fig.~141}, join the original places of the bodies $A$ and
+$B$ by a straight line, and mark on it $G$, the place of the centre of gravity;
+then take the positions $A'$ and $B'$ they would occupy at the end of a unit
+of time (if they did not attract each other), and mark the new position of
+the centre of gravity $G'$ on the line joining them. The line $GG'$ connecting
+the two positions of the centre of gravity will show the direction and rapidity
+of its motion; with reference to this point the two bodies will have opposite
+motions proportional to their distances from it; that is, they will swing
+around this point as if on a rod pivoted there, and will either both move
+towards it along the rod, or from it, with speeds inversely proportional
+to their masses. These relative motions \textit{with respect to the centre of gravity}
+are easily found by drawing through $G$ a line parallel to $A'B'$ and measuring
+off on it distances $GA''$ and $GB''$ respectively equal to $G'A'$ and $G'B'$.
+$AA''$ and $BB''$ will then be the two motions of $A$ and $B$ \textit{relative to their centre
+of gravity $G$}.
+\end{fineprint}
+%% -----File: 272.png---Folio 261-------
+
+\nbarticle{426.} \nbparatext{The Effect of their Mutual Attraction.}---This will cause
+them to describe similar conics around this centre of gravity; the
+size of their two orbits being inversely proportional to their masses.
+The form of the orbits and dimensions will be determined by the
+combined mass of the two bodies, and by their velocities with respect
+to the common centre of gravity.
+
+\nbarticle{427.} \nbparatext{The Orbit of the Smaller relative to the Centre of the Larger.}---It
+is convenient (though it is not necessary) to drop the consideration
+of the centre of gravity of the two bodies, and to consider
+the motion of the smaller one around the centre of the larger one.
+In reference to that point, it will move precisely as if its mass had
+been added to that of the larger body, while itself had become a mere
+particle. This relative orbit will in all respects be like the actual one
+around the centre of gravity, only magnified in the proportion of
+$M + m$ to $M$; \textit{i.e.}, if $m$ is $\frac{1}{10}$ of $M$, the \textit{actual} orbit around $G$ will
+be magnified by $\frac{11}{10}$ to produce the \textit{relative} orbit around $M$.
+
+\includegraphicsmid{illo142}{\textsc{Fig.~142.}---Elliptical Orbit determined by Projection.}
+
+\nbarticle{428.} \nbparatext{The Orbit determined by Projection.}---Suppose that in the
+figure (\figref{illo142}{Fig.~142}) the body $P$ is moving in the direction of the arrow,
+and is attracted by $S$, supposed to be at rest. $P$ will thenceforward
+move in a conic, either in an ellipse or hyperbola, according to
+its velocity, as we shall see in a moment. $S$ being at one focus of
+the curve, the other focus will be somewhere on the line $PN$, which
+makes the same angle with $PQ$ that $r$ ($SP$) does (since it is a property
+of the conics that a tangent-line at any point of the curve makes
+equal angles with the lines drawn from the two foci to that point).
+%% -----File: 273.png---Folio 262-------
+If we can find the place of the second focus $F$, or the length of the
+line $PF$ in the \figref{illo142}{figure}, the curve can at once be drawn.
+
+Now, it can be proved, though the demonstration lies beyond our
+scope, that $a$, the semi-major axis of the conic, is determined by
+the equation
+\[
+V^2 = \mu\left(\frac{2}{r} - \frac{1}{a}\right), \nbtag{(Equation 1)}
+\]
+in which $r$ is the distance $SP$, $V$ is the velocity, and $\mu$ is the attracting
+mass at $S$ expressed in proper units.
+
+\begin{fineprint}
+(See Watson's ``Theoretical Astronomy,'' p.~49; only for $\mu$ he writes $k^2(1 + m)$).
+\end{fineprint}
+
+$V$, $r$, and $\mu$ being given, of course $a$ can be found: we get
+\[
+a = \mu\frac{r}{2\mu - rV^2}. \nbtag{(Equation 2)}
+\]
+
+Then by subtracting $r$ from $2a$ we shall get $r'$, or the distance $PF$,
+if the curve is an ellipse. If it is a hyperbola, $a$ will come out negative;
+and to find $r'$ we must take $r' = 2a + r$ and measure it off to
+$F'$, on the other side of the line of motion. In either case, however,
+we easily find the other focus, and the line drawn through the foci
+will be the line of apsides; a point half-way between the foci will be
+the centre of the curve, and any line drawn through this centre will be
+a diameter. Having the two foci and the major axis $2a$, \textit{i.e.}, $AA'$
+the curve can at once be drawn.
+
+\nbarticle{429.} \nbparatext{Expression for $a$ in Terms of the ``Velocity from Infinity,'' or
+``Parabolic Velocity.''}---The expression for $a$ admits of a more convenient
+and very interesting form. It is shown in analytical mechanics
+that if, under the law of gravitation, a particle falls towards an
+attracting body whose mass is $\mu$, from one distance $s$ to another distance
+$r$, its velocity is given by the simple equation
+\[
+w^2 = 2\mu\left(\frac{1}{r} - \frac{1}{s}\right).\footnote
+  {If the difference between $s$ and $r$ is called $h$, this equation becomes
+\[
+w^2 = 2\mu\left(\frac{1}{r} - \frac{1}{r + h}\right) = 2\mu\left(\frac{h}{r^2 + rh}\right).
+\]
+Now if $h$ is very small as compared with $r$, this gives
+\[
+w^2 = \left(\frac{2\mu}{r^2}\right)h,
+\]
+which is the same as the usual expression for the velocity of a falling body at the
+earth's surface, viz., $V^2 = 2gh$,  $2g$ being replaced by the fraction $\dfrac{2\mu}{r^2}$.
+} \nbtag{(Equation 3)}
+\]
+%% -----File: 274.png---Folio 263-------
+
+If in this equation $s$ be made infinite, $w$ does not also become
+infinite (that is, a body falling from an infinite distance towards the
+sun will not acquire an infinite velocity until it actually reaches the
+centre of the sun, and $r$ becomes zero); but we get in this case
+\[
+w^2 = \frac{2\mu}{r}.
+\]
+This special value of $w$ is usually called ``the velocity from infinity
+for the distance $r$,'' or the ``\textit{parabolic velocity}'' (for a reason which
+will appear very soon). $U$ is generally used as its symbol; therefore
+\[
+U^2 = \frac{2\mu}{r}; \text{ whence }\mu = \frac{rU^2}{2}. \nbtag{(Equation 4)}
+\]
+
+Substituting this value of $\mu$ in equation 2, we get
+\[
+a = \frac{r}{2}\left(\frac{U^2}{U^2 - V^2}\right). \nbtag{(Equation 5)}
+\]
+
+\nbarticle{430.} \nbparatext{Relation between the Velocity and the Species of Conic Described.}---From
+equation~5 it is obvious how the velocity determines
+whether the orbit will be an ellipse or a hyperbola. If $V^2$ is \textit{less} than
+$U^2$, the denominator of the fraction will be positive, $a$ will also be
+positive, and the curve will be an \textit{ellipse; i.e.}, if the velocity of the
+body $P$, at the distance $r$ from the central body $S$, be less than
+the velocity acquired by the body falling from infinity to that point,
+the body will move around $S$ permanently in an ellipse.
+
+If, on the other hand, $V^2$ is \textit{greater} than $U^2$, the denominator will
+become negative, $a$ will also come out negative, and the orbit will be
+a \textit{hyperbola}. In this case $P$, after once moving past $S$ at the
+perihelion point, will go off never to return; and it will recede towards a
+a different region of space from that out of which it came, because
+the two legs of the hyperbola never become parallel. There will in
+this case be no permanent connection between the two bodies. They
+simply pass each other with a little graceful recognition of each other's
+presence by a curvature in their paths, and then part company for ever.
+
+If $V^2$ exactly equals $U^2$, the denominator of the fraction becomes
+zero, $a$ comes out infinite, and the curve is a \textit{parabola}. In this case,
+also, the body will never return; but it will recede from the sun ultimately
+towards the same point on the celestial sphere as that from
+which it appeared to come, since the two legs of the parabola tend to
+parallelism. Obviously, if a body were thus moving in a parabola,
+the slightest \textit{increase} of its velocity would transform the orbit into an
+%% -----File: 275.png---Folio 264-------
+\textit{hyperbola}, and the least \textit{diminution} into an \textit{ellipse}; the bearing of
+which remark will become evident when we come to deal with comets.
+
+\nbarticle{431.} Again, since\qquad
+$a = \dfrac{r}{2}\left(\dfrac{U^{2}}{U^{2} - V^{2}}\right)$,\\
+all bodies \textit{having the same velocity $V$, at the same distance $r$ from the
+centre of force, will have major axes of the same length for their orbits,
+no matter what may be their direction of motion}.
+
+\begin{fineprint}
+They will have the same \textit{period} also, the expression for the period being
+\[
+t = \frac{2\pi a^\frac{3}{2}}{\sqrt{\mu}}. \nbtag{(Watson, p.~46, Equation 28.)}
+\]
+\end{fineprint}
+
+If, therefore, a body moving around the sun were to explode at any
+point, all of its particles which did not receive a velocity greater than
+the ``parabolic velocity'' would come around to the same point again,
+and these which were projected with equal velocities would come
+around and meet at the same moment, however widely different their
+paths might be.
+
+\includegraphicsmid{illo143}{\textsc{Fig.~143.}---Confocal Conics described under Different Velocities of Projection.}
+%% -----File: 276.png---Folio 265-------
+
+\begin{fineprint}
+\nbarticle{432.} \figref{illo143}{Fig.~143} represents the orbits which would be described by five
+bodies projected at $O$ with different velocities along the line $OV$, the distance
+$OS$ or $r$ being taken as unity, as well as the parabolic velocity $U^{2}$. The
+squares of the velocities are assumed as given below, with the resulting
+values of $a$ and $r'$.
+\[
+{V_{1}}^{2} = \tfrac{1}{4} \text{; whence } a_{1} = \tfrac{2}{3} \text{; and } {r_{1}}' = \tfrac{1}{3}.
+\]
+This places the empty focus at $F_{1}$.
+
+For the next larger ellipse
+\begin{flalign*}
+&                              &&{V_2}^2 = \tfrac{1}{2};\ a_2 = 1;\ {r_2}' = 1.   &&\phantom{In the same way}\\
+&\indent\text{In the same way }&&{V_3}^2 = \tfrac{3}{4};\ a_3 = 2;\ {r_3}' = 3.   &&\\
+&                              &&{V_4}^2 = 1;\ a_4 = \infty ;\ {r_4}' = \infty.   \rlap{\quad\text{(Parabola.)}}\\
+&                              &&{V_5}^2 = 2;\ a_5 = -\tfrac{1}{2};\ {r_5}' = -2. \rlap{\quad\text{(Hyperbola.)}}
+\end{flalign*}
+
+\nbarticle{433.} \figref{illo144}{Fig.~144} shows how three bodies projected at $P$ with \textit{equal velocities,
+but in different directions}, indicated by the arrows, describe three different
+ellipses; all, however, having the same period, and the same length of semi-major
+axis; namely, $a = 2r$; $V^{2}$ being taken equal to $\frac{3}{4} U^{2}$.
+
+\includegraphicsmid{illo144}{\textsc{Fig.~144.}---Ellipses of the Same Periodic Time.}
+
+For a fourth body, $V^{2}$ is taken as $=\frac{1}{2} U^{2}$, and with the direction of motion
+perpendicular to $r$. This body will move in a perfect circle, $a$ coming out
+%% -----File: 277.png---Folio 266-------
+equal to $r$, when $V^{2} = \frac{1}{2} U^{2}$. In order to have \textit{circular} motion, both conditions
+must be fulfilled; namely, $V^{2}$ must equal $\frac{1}{2} U^{2}$, and the direction of motion
+must be perpendicular to the radius vector.
+
+\includegraphicsouter{illo145}{\textsc{Fig.~145.}---Van der Kolk's Theorem.}
+
+\nbarticle{434.} \nbparatext{Velocity of a Planet at Any Point in its Orbit.}---If $AA'$
+(\figref{illo145}{Fig.~145}) be the major axis
+of a planet's orbit, and $KK'$
+the diameter of a circle described
+around $S$ with $AA'$ as
+radius, then the velocity of a
+planet at any point, $N$, on its
+orbit is equal to that which it
+would have acquired by falling
+to $N$ from the point $n$ on the
+circumference of the circle.
+The demonstration is not difficult
+and may be found in
+No.~1426 of the ``Astronomische
+Nachrichten.''
+
+\includegraphicsouter{illo146}{\textsc{Fig.~146.}---Projectiles near the Earth.}
+
+\sloppy
+\nbarticle{435.} \nbparatext{Projectiles near the
+Earth.}---A good illustration
+of the principles stated above is obtained by considering the motion of
+bodies projected horizontally from the top of a tower near the earth's surface,
+supposing the air to be removed so there will be no resistance to the motion.
+
+\fussy
+The ``parabolic velocity'' due to
+the earth's attraction equals 6.94
+miles per second at the earth's surface;
+\textit{i.e.}, a body falling from the
+stars to the surface of the earth,
+\textit{drawn by the earth's attraction only},
+would have acquired this velocity
+on reaching the earth's surface.
+
+First. If a body be projected with
+a very small velocity, it would fall
+nearly straight downwards. If the
+earth were concentrated at the point
+in its centre so that the body should
+not strike its surface, it would move
+in a very long narrow ellipse having
+the centre of the earth at the further
+focus, and would return to the original point after an interval of 29.9 minutes.
+
+Second. With a larger velocity the orbit would be a wider ellipse with a
+longer period, $C$ being still at the remoter focus.
+
+Third. $V = U \sqrt{\frac{1}{2}}$, or about 4.9 miles per second. In this case the orbit of
+%% -----File: 278.png---Folio 267-------
+the body would be a \textit{perfect circle}, and the period would be 1~h.\ 24.7~m. It
+will be remembered that we found that if the earth's rotation were 17 times
+as rapid, thus completing a revolution in $1^{\text{h}}\, 24^{\text{m}}.7$, the centrifugal force
+at the equator would become equal to gravity (\artref{Art.}{154}). Also, \artref{Art.}{413} (4),
+this same time, $1^{\text{h}}\, 24^{\text{m}}.7$, was found from Kepler's third law as the period of
+a satellite revolving close to the earth's surface.
+
+Fourth. $V = U = 6.94$ miles. In this case the projectile would go off in
+a \textit{parabola}, never to return.
+
+Fifth. $V > 6.94$. In this case, also, the body would never return, but
+would pass off in a \textit{hyperbola}.
+\end{fineprint}
+
+\sloppy
+\nbarticle{436.} \nbparatext{Intensity of Solar Attraction.}---The attraction between
+the sun and the earth from some points of view looks like a very
+feeble action. It is only able, as has been before stated (\artref{Art.}{278}),
+to bend the earth out of a rectilinear course to the extent of about
+one-ninth of an inch in a second, while she is travelling nearly
+nineteen miles; and yet if it were attempted to replace by bonds
+of steel the invisible gravitation which holds the earth to the sun,
+we should find the surprising result that it would be necessary to
+cover the whole surface of the earth with wires as large as telegraph
+wires, and only about half an inch apart from each other, in order
+to get a metallic connection that could stand the strain. This ligament
+of wires would be stretched almost to the breaking point. The
+attraction of the sun for the earth expressed as tons of \textit{force} (not
+tons of \textit{mass}, of course) is \DPtypo{3,600,000}{3,600000} %for consistency with other use of commas
+millions of millions of tons (36 with seventeen ciphers); and similar stresses act through the
+apparently empty space in all directions between all the different
+pairs of bodies in the universe.
+
+\fussy
+\begin{fineprint}
+\nbarticle{436*.} \nbparatext{Note to \artref{Art.}{429}.}---It is worth noting that $U^{2}$ (the square of
+the parabolic velocity at any point) is simply \textit{twice the gravitation potential
+due to the sun's attraction at that point}. The ``potential'' may be defined as
+the \textit{energy} which would be acquired by a mass of one unit, in falling to the
+point in question from a place where the potential (and attraction) is zero,
+\textit{i.e.}, from infinity (Physics, pp.~26 and 28). Now $\frac{1}{2} m V^{2}$ is the general expression
+for the kinetic energy of a mass, $m$, moving with velocity $V$; if in
+this expression we make $m=1$, and $V=U$, we shall have, for the case in
+hand, Energy $= \frac{1}{2} U^{2}$, which equals the \textit{Potential at the point}.
+\end{fineprint}
+\chelabel{CHAPTERXI}
+
+%% -----File: 279.png---Folio 268-------
+
+\Chapter{XII}{The Problem of Three Bodies}
+\nbchapterhang{\stretchyspace
+THE PROBLEM OF THREE BODIES.---DISTURBING FORCES:
+LUNAR PERTURBATIONS AND THE TIDES.}
+
+\nbarticle{437.} \textsc{The} problem of \textit{two} bodies is completely solved; but if,
+instead of two spheres attracting each other, we have \textit{three} or more,
+given completely in respect to their positions, masses, and velocities,
+the general problem of finding their subsequent motions and predicting
+their positions at any future date transcends the present power of
+our mathematics.
+
+This problem of \textit{three} bodies is in itself just as determinate and
+capable of solution as that of two. Given the initial data,---that
+is, the \textit{positions, masses, and motions} of the three bodies at a given
+instant,---then their motions for all the future, and the positions
+they will occupy at any given date, are absolutely predetermined.
+The difficulty of the problem lies simply in the inadequacy
+of our present mathematical methods, and it is altogether probable
+that some time in the future this difficulty will be overcome---very
+possibly by the invention of new functions and numerical tables
+which shall bear some such relation to our present tables of logarithms,
+sines, etc.\ as these do to the old multiplication table of
+Pythagoras.
+
+\nbarticle{438.} But while the \textit{general} problem of \textit{three} bodies is thus intractable,
+all the special cases of it which arise in the consideration of
+the moon's motion and in the motions of the planets have been solved
+by special methods of approximation. Newton himself led the way;
+and the strongest proof of the truth of his theory of gravitation lies
+in the fact that it not only accounts for the \textit{regular} elliptic motions of
+the heavenly bodies, but also for the apparent \textit{irregularities} of these
+motions.
+
+\nbarticle{439.} \nbparatext{The Disturbing Force.}---In the case where two bodies are
+revolving around their common centre of gravity, and the third body
+is either \textit{very much smaller} than the central one, or very \textit{remote}, the
+%% -----File: 280.png---Folio 269-------
+motion of the two will be but slightly modified by the action of the
+third; and in such a case the small differences between the actual
+motion and the motion as it would be if the third body were not
+present, are technically called ``disturbances'' and ``perturbations,''\footnote
+  {The student will bear in mind that these terms (``perturbations'' and ``disturbances'')
+are mere figures of speech; that philosophically the purely elliptical
+motion of two mutually attracting bodies alone in space is no more ``\textit{regular}''
+than the (at present) incomputable motion of three or more attracting bodies.
+We have in mind a theologian of some note who once maintained that the ``perturbations''
+in the solar system are a consequence of Adam's fall. Hence the
+  caution.}
+and the force which produces them is called the ``disturbing force.''
+This disturbing force is not the \textit{attraction} of the disturbing body, but
+only a \textit{component} of that attraction, and usually only a small fraction
+of it.
+
+\textit{The disturbing force of the attracting body depends upon the difference
+of its attraction upon the two bodies it disturbs; difference either
+in amount or in direction, or in both.} For instance, if the sun
+attracted the earth and moon exactly alike (\textit{i.e., equally and along
+parallel lines}), it would not disturb their relative motions in the least,
+no matter how powerful its attraction might be. The sun's maximum
+\textit{disturbing force} on the moon, as we shall see, is only about one
+eighty-ninth of the earth's attraction; and yet the sun's \textit{attraction} for
+the moon is actually much greater than that of the earth.
+
+\begin{fineprint}
+Since the sun's mass is 330,000 times that of the earth, and its distance
+just about 389 times that of the moon from the earth, its attraction on the
+moon equals the earth's attraction $× \frac{330,000}{389^{2}} = 2.18$; \textit{i.e.}, the \textit{sun's attraction
+on the moon is more than double that of the earth}.
+\end{fineprint}
+
+\nbarticle{440.} \nbparatext{Why the Sun does not take the Moon away from the Earth.}---If
+at the time of new moon, when the moon is between the earth
+and sun, the sun attracts the moon more than twice as much as the
+earth does, it is a natural question why the sun does not draw the
+moon away entirely, and rob us of our satellite. It would do so if
+it were the case of a ``tug of war''; that is, if earth and sun were
+\textit{fixed in space}, pulling opposite ways upon the moon between them.
+But it is not so; neither sun nor earth has any \textit{foothold}, so to speak;
+but all three bodies are free to move, like chips floating on water.
+The sun attracts the \textit{earth} almost as much as he does the moon, and
+both earth and moon fall towards him freely; though of course this
+%% -----File: 281.png---Folio 270-------
+falling motion towards the sun is continually combined with whatever
+other motion the earth or moon possesses. The only effective disturbance
+is produced by the fact that, in the case considered, \textit{the new
+moon, being nearer the sun than the earth is} by about $\frac{1}{389}$ part of the
+whole distance, falls towards the sun a trifle faster than the earth,
+and so on that account the curvature of its orbit toward the earth is,
+for the time being, diminished.
+
+At the half-moon the two bodies are equally attracted towards the
+sun, but on \textit{converging lines}; and so as they fall towards the sun
+they \textit{approach} each other slightly; and for this reason, at quadrature,
+the moon's orbit is a little more curved towards the earth than it
+would be otherwise.
+
+\nbarticle{441.} \nbparatext{Diagram of the Disturbing Force.}---A very simple diagram
+enables us to find graphically the disturbing force produced by a
+third body.
+
+\begin{fineprint}
+(What follows applies \textit{verbatim et literatim} to either of the two diagrams
+of \figref{illo147}{Fig.~147}.)
+\end{fineprint}
+
+\includegraphicsmid{illo147}{\textsc{Fig.~147.}---Determination of the Disturbing Force by Graphical Construction.}
+
+Let $E$ be the earth, $M$ the moon, and $S$ the disturbing body (the
+sun in this case); and let the sun's attraction on the moon be represented
+by the line $MS$. On the same scale the attraction of the sun
+on the earth will be represented by the line $EG$, $G$ being a point so
+taken that $EG:MS = MS^{2}:ES^{2}$;  that is,---
+
+\begin{fineprint}
+\textit{The sun's attraction on the earth} is to \textit{the sun's attraction on the moon} as \textit{the
+square of the sun's distance from the moon} is to \textit{the square of the sun's distance
+from the earth}, according to the law of gravitation. ($MS$ has to do double
+duty in this proportion: in the first ratio it represents a \textit{force}; in the second,
+a \textit{distance}.)
+\begin{flalign*}
+&\indent\text{From this proportion }&
+&  EG = MS×\frac{MS^{2}}{ES^{2}}.
+&&\phantom{From this proportion }
+\end{flalign*}
+%% -----File: 282.png---Folio 271-------
+
+In \figref{illo147}{figure (\textit{a})} the moon is nearer to the sun than the earth is, and so $EG$
+comes out \textit{less} than $MS$. In \figref{illo147}{figure (\textit{b})} the reverse is the case, and therefore
+in this case $EG$ is \textit{larger} than $MS$.
+\end{fineprint}
+
+Now if the force represented by the line $MS$ \textit{were parallel and equal
+to that represented by} $EG$, there would be no disturbance, as has been
+said. If, then, we can resolve the force $MS$ into two components,
+one of which is equal and parallel to $EG$, this component will be innocent
+and harmless, and the other one will make all the disturbance.
+
+To effect this resolution, draw through $M$ the line $MK$ parallel
+and equal to $EG$. Join $KS$, and draw $ML$ parallel and equal to it.
+$ML$ \textit{is then the disturbing force on the same scale as} $MS$; \textit{i.e.}, the line
+$ML$ shows the true \textit{direction} of the disturbing force, and in \textit{amount}
+the disturbing force is equal to the \textit{sun's attraction for the moon multiplied
+by the fraction} $\left(\dfrac{ML}{MS}\right)$. The diagonal of the parallelogram
+$MLSK$ is $MS$, which represents the resultant of the two forces $MK$
+and $ML$, that form its sides.
+
+\begin{fineprint}
+For the sake of clearness the lines which represent forces in the \figref{illo147}{figures}
+are indicated by herring-bone markings.
+\end{fineprint}
+
+\nbarticle{442.} At first it seems a little strange that in \figref{illo147}{figure (\textit{b})} the disturbing
+force should be directed \textit{away from the sun}; but a  little
+reflection justifies the result. If $E$ and $M$ were connected by a rod,
+and the $E$-end of the rod were pulled towards the right more swiftly
+than the $M$-end, it is easy to see that the latter would he \textit{relatively}
+thrown to the left, as the \figref{illo147}{figure} shows.
+
+\nbarticle{443.} The sun is the only body that sensibly disturbs the moon.
+The planets, of course, act upon the moon to disturb it, but their
+mass is so small compared with that of the sun, and their distances
+so great, that in no case is their \textit{direct} action sensible. It is true,
+however, that some of the lunar perturbations are affected by the
+existence of one or two of the planets. While they cannot disturb
+the moon \textit{directly}, they do so \textit{indirectly}: they disturb the earth in
+her orbit sufficiently to make the sun's action different from what it
+would be if the planets did not exist. In this way the planets
+Venus, Mars, and Jupiter make themselves felt in the lunar theory.
+There are also a few small disturbances that depend upon the fact
+that the earth is not a perfect sphere.
+
+\nbarticle{444.} Since the distance of the sun is nearly 400 times that of the
+moon from the earth, the construction of the disturbing force $ML$,
+%% -----File: 283.png---Folio 272-------
+Fig.~147, admits of considerable simplification. It is only necessary
+to drop the perpendicular $MP$ upon the line that joins the earth and
+the sun, and take the point $L$ upon this line, so that $EL$ equals \textit{three
+times} $EP$. The line $ML$ so determined will then \textit{very approximately}
+(but not exactly) be the true disturbing force.
+
+\begin{fineprint}
+To prove this relation, let $MS$, in \figref{illo147}{Fig.~147}, be $D$, $ES = R$, $ME = r$, and
+$EP = p$, also $R = D + p$, \textit{very nearly}, $p$ being \textit{negative} when $MS > ES$. $EG$
+was taken equal to $MS×\dfrac{MS^{2}}{ES^{2}} = \dfrac{D^{3}}{R^{2}}$.
+
+Now,
+\[
+EL = GS = (ES - EG)
+     = R - \dfrac{D^{3}}{R^{2}} = \dfrac{R^{3} - D^{3}}{R^{2}}
+     = \dfrac{(D + p)^{3} - D^{3}}{(D + p)^{2}}.
+\]
+
+Developing this expression, we have
+\[
+EL = \frac{3D^{2}p + 3Dp^{2} + p^{3}}{D^{2} + 2Dp + p^{2}}.
+\]
+Since $p$ is very small compared with $D$, all the terms except the first
+nearly vanish both in numerator and denominator, and we have
+\[
+EL = \frac{3D^{2}p}{D^{2}} = 3p \text{ (very nearly).}
+\]
+\end{fineprint}
+
+\nbarticle{445.} \nbparatext{Resolution of the Disturbing Force into Components.}---In
+discussing the effect of the disturbing force it is more convenient to
+resolve it into three components known as the \textit{radial}, the \textit{tangential},
+and the \textit{orthogonal}. The first of these \textit{acts in the direction of the
+radius vector}, tending to draw the moon either towards or from the
+earth. The second, the tangential, \textit{operates to accelerate or retard
+the moon's orbital velocity}.
+
+\includegraphicsmid{illo148}{\textsc{Fig.~148.}---Radial and Tangential Components of the Disturbing Force.}
+
+\begin{fineprint}
+Fig.~148 exhibits these two components at different points of the moon's
+orbit.
+\end{fineprint}
+%% -----File: 284.png---Folio 273-------
+
+The orthogonal component has no existence in cases where the
+disturbing body lies in the plane of the disturbed orbit; but whenever
+it lies outside of that plane, the disturbing force $ML$ will generally
+also lie outside of the orbit-plane, and will have a component
+tending to \textit{draw the moving body out of the plane of its orbit}. The
+motion of the moon's node and the changes of the inclination of its
+orbit are due to this component of the sun's disturbing force, which
+could not be conveniently represented in the \figref{illo148}{figure}.
+
+\nbarticle{446.} The radial force in the case of the moon's orbit is a maximum
+at syzygies and quadratures; in fact, at quadratures the whole
+disturbing force is radial, the tangential and orthogonal components
+both vanishing. At syzygies (new moon and full moon) the radial
+force is \textit{negative}; that is, it draws the moon \textit{from} the earth, diminishing
+the earth's attraction by about \textit{one eighty-ninth}\footnote
+  {At syzygies $ML = NL_{0} = 2× EN$ (\figref{illo147}{Fig.~147}); but $EN = \dfrac{MS}{389}$. Therefore
+$ML = \dfrac{2}{389}$ of the \textit{sun's} attraction on the moon. Now the sun's attraction is 2.18
+  times the earth's; hence $NL_{0} = $ the \textit{earth's} attraction multiplied by $\dfrac{2× 2.18}{389} = \dfrac{1}{89.2}$.}
+of its whole
+amount.
+
+At quadrature or half-moon the radial force is \textit{positive}; and since
+$L$ then falls at $E$, it is represented by the line $QE$, and is just half
+what it is at syzygies; that is, it equals about \textit{one one hundred and
+seventy-eighth} of the earth's attraction.
+
+It becomes zero at four points $54°\, 44'$ on each side of the line of
+syzygies.
+
+\begin{fineprint}
+This angle is found from the condition that, the disturbing force $M_{1}L_{1}$,
+etc., in \figref{illo148}{Fig.~148}, must be perpendicular to the radius $EM_{1}$ at this point,
+which gives us $EP_{1}:P_{1}M_{1}::P_{1}M_{1}:P_{1}L_{1}$. But $P_{1}L_{1} = 2EP_{1}$; therefore
+$P_{1}{M_{1}}^{2} = 2{EP_{1}}^{2}$, and $\dfrac{P_{1}M_{1}}{EP_{1}} = \tan{M_{1}EL_{1}} = \sqrt{2}$.
+\end{fineprint}
+
+\nbarticle{447.} The \textit{tangential component} starts at zero at the time of full
+moon, rises to a maximum at the critical angle of $45°$ (having at
+that point a value of $\frac{1}{119}$ of the earth's attraction), and disappears
+again at quadratures. During the first and third quadrants this
+force is \textit{negative}; that is, it \textit{retards} the moon's motion;  in the second
+and fourth it is \textit{positive} and accelerates the motion.
+%% -----File: 285.png---Folio 274-------
+
+\nbarticle{448.} \nbparatext{Lunar Perturbations.}---So far it has been all plain sailing,
+for nothing beyond elementary mathematics is required in determining
+the disturbing force at any point in the moon's orbit; \textit{but to
+determine what will be the effect of this continually varying force after
+the lapse of a given time, upon the moon's place in the sky} is a problem
+of a very different order, and far beyond our scope. The reader who
+wishes to follow up this subject must take up the more extended
+works upon theoretical astronomy and the lunar theory. A few
+points, however, may be noted here.
+
+\nbarticle{449.} In the first place, it is found most convenient to consider
+the moon as never deviating from an elliptical orbit, but to consider
+\textit{the orbit itself as continually changing in place and form}, writhing and
+squirming, so to speak, under the disturbing forces; just as if the
+orbit were a material hoop with the moon strung upon it like a bead
+and unable to get away from it, although she can be set forward and
+backward in her motion upon it.
+
+\nbarticle{450.} In the next place, it is found possible to represent nearly all
+the perturbations by \textit{periodical formulæ}---the same values recurring
+over and over again indefinitely at regular intervals. This is because
+the sun, moon, and earth keep coming back into the same, or nearly
+the same, relative positions, and this leads to recurring values of the
+disturbing force itself, and also of its effects.
+
+\nbarticle{451.} Third, the number of these separate perturbations which
+have to be taken account of is very large. In the computation of
+the moon's \textit{longitude} in the American Ephemeris about \textit{seventy} different
+inequalities are reckoned in, and about half as many in the
+computation for the \textit{latitude}. Theoretically the number is infinite,
+but only a  certain number produce effects sensible to observation.
+It is of no use to compute disturbances that do not displace the moon
+as much as one-tenth of a second of arc; \textit{i.e.}, about 500 feet in her
+orbit.
+
+\nbarticle{452.} Fourth, in spite of all that has been done, the lunar theory
+is still incomplete, or in some way slightly erroneous. The best
+tables yet made begin to give inaccurate results after fifteen or
+twenty years, and require correction. The almanac place of the
+moon at present is not unfrequently ``out'' as much as $3''$ or $4''$ of
+arc; \textit{i.e.}, about three or four miles. Astronomers are continually at
+%% -----File: 286.png---Folio 275-------
+work on the subject, but the computations by our present methods
+are exceedingly tedious and liable to numerical error.
+
+\nbarticle{453.} The principal effects of the sun's disturbing action on the
+moon are the following:--
+
+\textit{First: Effect on Length of Month.}---Since the radial component of
+the disturbing force is \textit{negative} more than half the way round ($54°\,
+44'$ on each side of the line of syzygies) and is twice as great at
+syzygies as the positive component is at quadrature, the net result is
+that, taking the whole month through, the earth's attraction for the
+moon \textit{is lessened by nearly $\frac{1}{360}$ part}. The effect of this is to increase
+the major axis of the moon's orbit, and to make the month a
+little longer than it otherwise would be; that is, if the sun were
+taken away, the month would immediately shorten by about three
+hours.
+
+\includegraphicsouter{illo149}{\textsc{Fig.~149.}\\
+Advance of the Apsides of
+the Moon's Orbit.}
+
+\nbarticle{454.} \textit{Second: The Revolution of the Line of Apsides.}---This is
+due mainly to the \textit{radial} component of the disturbing force.    When
+the moon comes to \textit{apogee} at the time of new or full moon, the
+diminution of the earth's effective attraction for the moon causes it
+to move on farther than it would otherwise do
+before turning the corner, so to speak, the consequence
+being that the line of apsides \textit{advances}
+in the line of the moon's motion. When the
+moon passes \textit{perigee} at that time, the effect is
+reversed, and the apsides \textit{regress}; but since the
+disturbing force is greater at apogee than at
+perigee, in the long run the advancing motion
+predominates.  When perigee or apogee is
+passed at the time of \textit{quadrature}, the line of apsides
+is  also  disturbed; \textit{but} the disturbances
+thus produced \textit{exactly balance each other} in the long run. The net
+result, as has been stated before (\artref{Art.}{238}), is that the line of apsides
+completes a direct revolution once in about nine years (8.855 years---\textit{Neison}).
+It does not move forward steadily and uniformly, but
+its motion is made up of alternate advance and regression. \figref{illo149}{Fig.~149}
+illustrates this motion of the moon's apsides.
+
+\begin{fineprint}
+For a fuller discussion of the subject, see Herschel's ``Outlines of Astronomy,''
+sections 677--689; or Airy's ``Gravitation,'' pp.\ 89--100.
+\end{fineprint}
+
+\sloppy
+\nbarticle{455.} \textit{Third: The Regression of the Nodes.}---The \textit{orthogonal component
+generally} (not always) \textit{tends to draw the moon towards the plane}
+%% -----File: 287.png---Folio 276-------
+\textit{of the ecliptic}. Whenever this is the case at the time when the moon
+is passing a node, the effect (as is easily seen from \figref{illo150}{Fig.~150}) of such
+a force $P_1O_1$, acting upon the moon at $P_1$, is to shift the node backward
+from $N_1$ to $N_2$, the moon taking the new path $P_1b_1N_2$. As the
+moon is approaching the node, the inclination of its orbit is also increased;
+but as the moon leaves the node, it is again diminished, the
+path $N_2P_2$ being bent at $P_2$ back to $P_2b_2$, parallel to $P_1a_1$: so that
+while by both operations the node is made to recede from $N_1$ to $N_3$,
+the inclination suffers very little change, if the orthogonal component
+remains the same on both sides of the node.
+
+\fussy
+Since the orthogonal component vanishes twice a year,---when the
+sun is at the nodes of the moon's orbit,---and also twice a month,---when
+she is in quadrature,---the rate at which the nodes regress
+is extremely variable. In the long run it makes its backward revolution
+once in about nineteen years (Arts.~\arnref{249} and \arnref{391}). [18.5997
+years.---\textit{Neison}.]
+
+\begin{fineprint}
+See Herschel's ``Outlines of Astronomy,'' section 638 seqq.
+\end{fineprint}
+
+\includegraphicsmid{illo150}{\textsc{Fig.~150.}---Regression of the Nodes of the Moon's Orbit.}
+
+\sloppy
+\nbarticle{456.} \textit{Fourth: The Evection.}---This is an irregularity which at the
+maximum puts the moon forward or backward about $1\frac{1}{4}°$ $(1°\, 16'\, 27''.01\textit{---Neison})$,
+and has for its period the time which is occupied by the
+sun in passing from the line of apsides of the moon's orbit to the
+same line again; \textit{i.e.}, about a year and an eighth. This is the largest
+of the moon's perturbations, and was earliest discovered, having been
+detected by Hipparchus about 150 years \textsc{b.c.}, and afterwards more
+fully worked out by Ptolemy, though of course without any understanding
+of its cause. It was the only lunar perturbation known to
+the ancients. It depends upon the \textit{alternate increase and decrease of
+the eccentricity} of the moon's orbit, which is always a maximum when
+%% -----File: 288.png---Folio 277-------
+the sun is passing the moon's line of apsides, and a minimum when
+the sun is at right angles to it.
+
+\fussy
+This inequality may affect the time of an eclipse by nearly six
+hours, making it anywhere from three hours early to three hours late,
+as compared with the time at which it would otherwise occur; it was
+this circumstance which called the attention of Hipparchus to it.
+
+\begin{fineprint}
+See Herschel's ``Outlines of Astronomy,'' sections 748 seqq.
+\end{fineprint}
+
+\nbarticle{457.} \textit{Fifth: The Variation.}---This is an inequality due mainly to
+\textit{the tangential component} of the disturbing force. It has a period of
+one month, and a maximum amount of $39'\, 30''.70$, attained when
+the moon is half-way between the syzygies and quadratures, at the
+so-called ``octants.'' At the first and third octants the moon is $39\frac{1}{2}'$
+\textit{ahead of} her mean place (about an hour and twenty minutes); at the
+second and fourth she is as much \textit{behind}. This inequality was detected
+by Tycho Brahe, though there is some reason for believing
+that it had been previously discovered by the Arabian astronomer,
+Aboul Wefa, five centuries earlier. This inequality does not \textit{affect
+the time of an eclipse}, being zero both at the syzygies and quadratures,
+and therefore was not detected by the Greek astronomers.
+
+\begin{fineprint}
+See Herschel's ``Outlines of Astronomy,'' sections 705 seqq.
+\end{fineprint}
+
+\sloppy
+\nbarticle{458.} \textit{Sixth: The Annual Equation.}---The one remaining inequality
+which affects the moon's place by an amount visible to the
+naked eye, is the so-called ``annual equation.'' When the earth is
+nearer the sun than its mean distance, the sun's disturbing force is,
+of course, greater than the mean, and the month is \textit{lengthened a
+little}; during that half of the year, therefore, the moon keeps falling
+behindhand; and \textit{vice versa} during the half when the sun's distance
+exceeds the mean. The maximum amount of this inequality is
+$11'\, 9''.00$, and its period one anomalistic year.
+
+\fussy
+\begin{fineprint}
+See Herschel's ``Outlines of Astronomy,'' sections 738 seqq.
+\end{fineprint}
+
+There remains one lunar irregularity among the multitude of lesser
+ones, which is of great interest theoretically, and is still a bone of
+contention among mathematical astronomers; namely,---
+
+\nbarticle{459.} \textit{Seventh: The Secular Acceleration of the Moon's Mean Motion.}---It
+was found by Halley, early in the last century, by a comparison
+of ancient with modern eclipses, that the month is now
+%% -----File: 289.png---Folio 278-------
+certainly shorter than it was in the days of Ptolemy, and that the
+shortening has been progressive, apparently going on continuously,---\textit{in
+sæcula sæculorum},---whence the name. In 100 years the
+moon, according to the results of Laplace, gets in advance of its
+mean place about $10''$, \textit{and the advance increases with the square of
+the time}, so that in a thousand years it would gain nearly $1000''$,
+and in 2000 years $4000''$, or more than a degree. The moon at
+present is supposed to be just about a degree in advance of the position
+it would have held if it had kept on since the Christian era
+with precisely the rate of motion it then had. If this acceleration
+were to continue indefinitely, the ultimate result would be that the
+moon would fall upon the earth, as the quickened motion corresponds
+to a shortened distance.
+
+\nbarticle{460.} It was nearly 100 years after Halley's discovery before
+Laplace found its explanation in the decreasing eccentricity of the
+earth's orbit. Under the action of the other planets this orbit is now
+growing more nearly circular, without, however, changing the length
+of its major axis. Thus \textit{its area becomes larger}, and the earth's \textit{average}
+distance from the sun becomes greater (although the \textit{mean} distance,
+technically so-called, does not change, the ``mean distance'' being
+simply half the major axis). As a result of this rounding up of the
+earth's orbit, the \textit{average disturbing force of the sun is therefore diminished},
+and this diminution allows the month to come nearer the length
+it would have if there were no sun to disturb the motion; that is to
+say, the month keeps shortening little by little, and it will continue
+to do so until the eccentricity of the earth's orbit begins to increase
+again, some 25,000 years hence.
+
+\begin{fineprint}
+\nbarticle{461.} But the theoretical amount of this acceleration, about $6''$ in a
+century, does not agree with the value obtained by comparing the most
+ancient and modern eclipses, which is about $12''$; and this value, again, does
+not agree with the one derived by comparing modern observations of the
+moon with these made by the Arabians about a thousand years ago, which,
+according to recent investigations by Professor Newcomb, indicate an
+acceleration of only about $8''$.
+
+So long as the actual acceleration was considered to be $12''$, it was generally
+supposed that the discrepancy between the theoretical and observed
+result is due to \textit{a retardation of the earth's rotation by the friction of the
+tides, and a consequent lengthening of the day}. Evidently if the day and
+the seconds become a little longer, there will be fewer of them in each
+month or year, and the \textit{apparent} effect of such a change would be to shorten
+all really constant astronomical periods by one and the same percentage.
+%% -----File: 290.png---Folio 279-------
+As matters stand to-day it is hardly possible to assert with confidence
+that there is any real discrepancy to be accounted for between the theoretical
+and observed values, the latter being considerably uncertain. In Newcomb's
+``Popular Astronomy'' (pp.\ 96--102) there will be found an interesting
+and trustworthy discussion of the subject.
+
+Questions like this, and those relating to the remaining discrepancies
+between the lunar tables and the observed places of our satellite, lie on the
+very frontiers of mathematical astronomy, and can be dealt with only by
+the ablest and most skilful analysts.
+\end{fineprint}
+
+\section*{THE TIDES.}
+
+\nbarticle{462.} Just as the disturbing force due to the sun's attraction
+affects the motions of the moon in her orbit, so the disturbing
+forces due to the attractions of the moon and sun acting upon the
+fluids of the earth's surface produce \textit{the tides}. These consist of the
+regular rise and fall of the water of the ocean, the average interval
+between successive high waters being $24^\text{h}\ 51^\text{m}$, which is precisely
+the same as the average interval between two successive passages of
+the moon across the meridian. This coincidence, maintained indefinitely,
+of itself makes it certain that there must be some causal connection
+between the moon and the tides. As some one has said, the
+odd 51 minutes is the moon's ``\textit{ear-mark}.''
+
+\nbarticle{463.} \nbparatext{Definitions.}---When the water is rising, it is ``\textit{flood}'' tide;
+when falling, it is ``\textit{ebb}.'' It is ``\textit{high water}'' at the moment when
+the tide is highest, and ``\textit{low water}'' when it is lowest. ``\textit{Spring
+tides}'' are the highest tides of the month (which occur near the
+times of new and full moon), while ``\textit{neap tides}'' are the smallest,
+which occur when the moon is in quadrature. The relative heights of
+the spring and neap tides are about as 7 to 4. At the time
+of spring tides the interval between the corresponding tides of successive
+days is less than the average, being only about $24^\text{h}\ 38^\text{m}$,
+and then the tides are said to ``\textit{prime}.'' At neap tides the interval
+is $25^\text{h}\ 6^\text{m}$, which is greater than the mean, and the tides ``\textit{lag}.''
+
+\textit{The ``establishment'' of a port} is the mean interval between the
+time of high water at that port and the next preceding passage of
+the moon across the meridian. At New York, for instance, this
+``establishment'' is $8^\text{h}\ 13^\text{m}$,
+although the actual interval varies about
+22 minutes on each side of the mean at different times of the month.
+
+That the moon is largely responsible for the tides is also shown by
+the fact that the tides, at the time when the moon is in perigee, are
+nearly twenty per cent higher than those which occur when she is in
+%% -----File: 291.png---Folio 280-------
+apogee. The highest tides of all happen when a \textit{new or full moon
+occurs at the time the moon is in perigee, especially if this occurs
+about January 1st, when the earth is nearest to the sun}. Since, as
+we shall see, the ``tide-raising'' force varies inversely as the \textit{cube} of
+the distance, slight variations in the distance of the moon and sun
+from the earth make much greater variations in the height of the tide---greater
+nearly in the ratio of 3 to 1.
+
+\nbarticle{464.} \nbparatext{The Tide-Raising Force.}---This is the \textit{difference} between the
+attractions of the sun and moon (mainly the latter) on the main
+body of the earth, and the attractions of the same bodies on particles
+at different parts of the earth's surface. The tide-raising force
+is but a very small part of the whole attraction.
+
+\includegraphicsmid{illo151}{\textsc{Fig.~151.}---The Moon's Tide-Raising Force on the Earth.}
+
+The amount of this disturbing force for a particle at any point
+on the earth's surface can be found approximately by the same geometrical
+construction which was used for the lunar theory (\artref{Art.}{441}).
+Draw a line from the moon through the centre of the earth.
+At the points $A$ and $B$, \figref{illo151}{Fig.~151}, where the moon is directly over
+head or under foot, the tide-raising force is directly opposed to gravity,
+and equals nearly $\frac{1}{30}$ of the moon's whole attraction, since the line $Aa$
+represents the disturbing force on the same scale as the line from $A$
+to the moon represents the moon's attraction, and this line, $AM$, is
+about sixty times the earth's radius, while $Aa$ is just double it, because
+$Ca$ has to be taken equal to $3× CA$ (\artref{Art.}{444}).
+
+Since the moon's mass is only about $\frac{1}{80}$ of the earth's, and its distance
+is sixty radii of the earth, this \textit{lifting force} under the moon,
+\textit{expressed as a fraction of the earth's gravity}, equals
+\[
+  \tfrac{1}{30} × \tfrac{1}{80} × \tfrac{1}{3600}
+= \tfrac{1}{8640000} ;
+\]
+\textit{i.e.}, a body weighing \textit{four thousand tons} loses about \textit{one pound} of its
+weight when the moon is over head or under foot.
+
+At $D$ and $E$, anywhere on the circle of the earth's surface which
+is $90°$ from $A$ and $B$, the moon's disturbing force \textit{increases} the
+%% -----File: 292.png---Folio 281-------
+weight of a body by just half this amount, the disturbing force being
+measured by the lines $DC$ and $EC$. At a point $F$, situated anywhere
+on a circle drawn around either $A$ or $B$ with a radius of $54°\, 44'$, the
+\textit{weight} of a body is neither increased nor decreased, but it is urged
+towards $A$ or $B$ with a horizontal force expressed by the line $Ff$,
+which force is equal to about $\frac{1}{12\:000000}$ of its weight.
+
+In the same way the tidal forces at $G$ and $H$ are expressed by the
+lines $Gg$ and $Hh$.
+
+\begin{fineprint}
+\nbarticle{465.} The same result for the lifting-force directly under the moon may
+be obtained more exactly as follows. The distance from the moon to the
+centre of the earth is sixty times the earth's radius, and therefore the
+distance from the moon to the points $A$ and $B$ respectively will be 59 and
+61. The moon's attraction at $A$, $C$, and $B$, expressed as fractions of the
+earth's gravity, will be as follows:---
+{\renewcommand{\minalignsep}{9pt}
+\begin{flalign*}
+&&  \text{Attraction of moon on particle at } A
+&= g ×  \frac{\frac{1}{80}}{59^2} = 0.0000035910× g.&&
+\\
+&&  \text{Attraction of moon on particle at } C
+&= g ×  \frac{\frac{1}{80}}{60^2} = 0.0000034723× g.&&
+\\
+&&  \text{Attraction of moon on particle at } B
+&= g ×  \frac{\frac{1}{80}}{61^2} = 0.0000033593× g.&&
+\\
+&\rlap{Hence }&  A - C = 0.0000001187\ g
+&= \frac{1}{8,\!424,\!000}\ g.&&
+\\
+&&  C - B = 0.0000001130\ g
+&= \frac{1}{8,\!835,\!000}\ g.&&
+\end{flalign*}
+}
+
+This is more correct than the preceding, which is based on an approximation
+that considers the moon's distance as \textit{very large} compared with the
+earth's radius, while it is really only sixty times as great, and sixty is hardly
+a ``very large'' number in such a case.
+
+Attempts have been made to \textit{observe} directly the variations in the force of
+gravity produced by the moon's action, but they are too small to be detected
+with certainty by any experimental method yet contrived. Both Darwin
+and Zöllner found that other causes which they could not get rid of produced
+disturbances more than sufficient to mask the whole action of the moon.
+
+\nbarticle{466.} It is worth while to note in this connection that the maximum
+lifting-force due to the attraction of a distant body varies inversely as the
+\textit{cube} of its distance, as is easily shown, thus:---calling $D$ the distance of the
+disturbing body from the earth's centre, and $r$ the earth's radius, we have
+\begin{flalign*}
+&\begin{alignedat}{2}
+&\text{Attraction at } A
+&&= \frac{M}{(D - r)^2};\qquad \text{attraction at } C = \frac{M}{D^2}.\\
+&\text{Tide-raising force at } A
+ && = M\left\{ \frac{1}{(D-r)^2}	- \frac{1}{D^2}      \right\}\\
+&&& = M\left\{ \frac{2Dr - r^2}{D^2(D^2 - 2Dr + r^2)} \right\}\\
+&&&= M\left\{ \frac{2Dr - r^2}{D^4 - 2D^3r + D^2r^2} \right\}\\
+&&&  = M\left\{ \frac{2r}{D^3} \right\}, \text{ nearly},
+\end{alignedat}  &&
+\end{flalign*}
+when $r$ is a small fraction of $D$.
+\end{fineprint}
+%% -----File: 293.png---Folio 282-------
+
+\nbarticle{467.} It is very apt to puzzle the student that the moon's action
+should be a \textit{lifting} force at $B$ as well as at $A$ (\figref{illo151}{Fig.~151}). He is
+likely to think of the earth as fixed, and the moon also fixed and
+attracting the water upon the earth, in which case, of course, the
+moon's attraction, while it would decrease gravity at $A$, would increase
+it at $B$.
+
+\includegraphicsouter{illo152}{\textsc{Fig.~152.}---The Statical Theory of the Tides.}
+
+The two bodies are not fixed,
+however. Let him think of the three
+particles at $A$, $C$, and $B$, \figref{illo152}{Fig.~152},
+as unconnected with each other, and
+falling freely towards the moon;
+then it is obvious that they would
+separate; $A$ would fall faster than
+$C$, and $C$ than $B$. Now imagine
+them connected by an elastic cord.
+It is obvious that they will still draw apart until the tension of the
+cord prevents any further separation. Its tension will then measure
+the ``lifting force'' of the moon which tends to draw both the particles
+$A$ and $B$ away from $C$.
+
+\nbarticle{468.} \nbparatext{The Sun's Action.}---This is precisely like that of the moon,
+except that the sun's distance, instead of being only sixty times the
+earth's radius, is nearly 23,500 times that quantity. Since the tide-raising
+power varies as the \textit{cube} of the distance inversely, while the
+attracting force varies only with the inverse \textit{square}, it turns out that
+although the sun's attraction on the earth is nearly 200 times as
+great as that of the moon, its \textit{tide-raising power is only about two-fifths
+as much}. When the sun is over head or under foot, his disturbing
+force diminishes gravity by about $\frac{1}{19\: 600000}$.
+
+\sloppy
+\nbarticle{469.} \nbparatext{Statical Theory of the Tides.}---If the earth were wholly
+composed of water, and if it kept always the same face towards the
+moon (as the moon does towards the earth), so that every particle on
+the earth's surface was always subjected to the same disturbing force
+from the moon, then, leaving out of account the sun's action, a permanent
+tide would be raised upon the earth, distorting it into a
+lemon-shaped form with the point towards the moon. It would be
+permanently higher water at the points $A$ and $B$ (\figref{illo152}{Fig.~152}) directly
+under the moon, and low water all around the earth on the circle $90°$
+from these points, as at $D$ and $E$. The difference of the level of
+the water at $A$ and $D$ would in this case be about two feet.
+%% -----File: 294.png---Folio 283-------
+
+\fussy
+The sun's action would produce a similar tide superposed upon
+the lunar tide and having about two-fifths of the same elevation, if
+the two tide summits should coincide, the resulting elevation of the
+high water would be the sum of the two separate tides. If the sun
+were $90°$ from the moon, the waves would be in opposition, and the
+height of the tide would be decreased, the solar tide partly filling up
+the depression at the low water due to the moon's action.
+
+Suppose now the earth to be put in rotation. It is easy to see
+that these tidal waves would \textit{tend} to move over the earth's surface,
+following the moon and sun at a certain angle dependent on the
+inertia of the water, and with a westward velocity precisely equal to
+that of the earth's eastward rotation,---about a thousand miles an
+hour at the equator. But it is also evident that on account of the
+varying depth of the ocean, and the irregular form of the shores, the
+tides could not maintain this motion, and that the actual result must
+become exceedingly complicated. In fact, the statical theory becomes
+utterly unsatisfactory in regard to what actually takes place,
+and it is necessary to depend almost entirely upon the results of
+observation, using the theory merely as a guide in the discussion of
+the observations.
+
+\begin{fineprint}
+Yet while this statical theory of the tides worked out by Newton is
+certainly inadequate, and in some respects incorrect, it easily furnishes the
+explanation of some of the most prominent of the peculiarities of the tides.
+
+\nbarticle{470.} \nbparatext{The Priming and Lagging of the Tides.}---About the time
+of new and full moon, as has been stated before (\artref{Art.}{463}), the interval
+between the corresponding tides of successive days is about thirteen minutes
+less than the average of $24^\text{h}\ 51^\text{m}$, while a week later it is about as much
+longer. The reason is found in the combination of the solar and lunar tides.
+
+\begin{center}
+\begin{tabular}{c@{\qquad\qquad}c}
+  \includegfx{illo153}&
+  \includegfx{illo154}\\
+  \footnotesize\textsc{Fig.~153.} & \footnotesize\textsc{Fig.~154.}
+\end{tabular}
+\captionof*{figure}{Priming and Lagging of the Tide.}
+\end{center}
+
+On the days of new and full moon the two tides coincide, and the tide
+wave has its crest directly under the moon, or rather at the normal distance
+behind the moon which corresponds to the ``establishment'' of the port of
+observation.
+%% -----File: 295.png---Folio 284-------
+
+At quadrature the crest of the solar tide will be just $90°$ from the crest of
+the lunar wave, but it will leave the summit of the \textit{combined wave} just where
+it would be if there were no solar wave at all: evidently there is no possible
+reason why the smaller wave at $S$ and $S'$ should displace the crest of the wave
+at $L$ (\figref{illo153}{Fig.~153}) towards the right that would not also require its displacement
+towards the left; it will therefore simply \textit{lower} the wave at $L$ \textit{without displacing
+it} one way or the other. But when the solar tide wave $SS'$ (\figref{illo154}{Fig.~154})
+has its crest at $S_1$ and $S_1'$, $45°$ from $L$ and $L'$, as it will do about three
+days after new or full moon, then its combination with the lunar wave will
+make the crest of the combined wave take position at a point $X$ between the
+two crests, and about half an hour of time ahead (\textit{west}) of the lunar tide;
+so that at that time of the month high water will occur about half an hour
+\textit{earlier} than if there were no solar tide (since the tide waves travel westward).
+And this half-hour has to be gained by diminishing the interval between
+the successive tides for the three preceding days. Similar reasoning shows
+that when the solar tide crest falls at $S_2$ and $S_2'$, the combined tide wave will
+be \textit{east} of the lunar wave, and come later into port.
+
+\includegraphicsouter{illo155}{\textsc{Fig.~155.}---The Diurnal Inequality.}
+
+\nbarticle{471.} \nbparatext{Effect of the Moon's Declination and Diurnal Inequality.}---In
+high latitudes on the Pacific Ocean, twice a month, when the moon is
+farthest north or south of the celestial equator, the two tides of the day are
+very different in magnitude. When the
+moon's declination is zero, there is no
+such difference: nor is there ever any
+difference at ports which are near the
+earth's equator.
+
+Fig.~155 makes it clear why it should
+be so. When the moon's declination is
+zero, things are as in \figref{illo152}{Fig.~152} (\artref{Art.}{469}),
+and the two tides of the same
+day are sensibly equal at ports in all
+latitudes. When the moon is at her
+greatest northern declination, say $28°$,
+the two tide summits will be at $A$ and $A'$ in \figref{illo155}{Fig.~155}; the tide which
+occurs at $B$ when the moon is overhead will be great, while the tide in the
+corresponding southern latitude at $B'$ will be small. The tides which
+occur twelve hours later will be small at the northern station, then situated
+at $C$, and large at the southern station, then at $C'$. For a port on the
+equator at $E$ or $Q$ there will be no such difference. In the Atlantic Ocean
+the difference is hardly noticeable, because, as we shall see very soon, the
+tides in that ocean are mainly (not entirely) due to tide waves propagated
+into it from the Pacific and Indian Oceans around the Cape of Good Hope.
+\end{fineprint}
+
+\nbarticle{472.} \nbparatext{The Wave Theory of the Tides.}---If the earth were entirely
+covered with deep water, except a few little islands projecting here
+and there to serve for observing stations, the tide waves would run
+%% -----File: 296.png---Folio 285-------
+around the globe \textit{regularly}; and if the depth of the ocean were over
+thirteen miles, the tide crests, as can be shown, would follow the
+moon at an angle of just $90°$. It would be high water just where the
+statical theory would give low water. If the depth were (as it really
+is) much less than thirteen miles, the tide wave in the ocean could
+not keep up with the moon, and the result would be a very complicated
+one. The real state of the case is still worse. The continents
+of North and South America, with the southern antarctic continent,
+make a barrier almost complete from pole to pole, leaving only a
+narrow passage at Cape Horn; and the varying depth of the water,
+and the irregular contours of the shores are such that it is quite
+impossible to determine by theory what the course and character of
+the tide wave must be. We must depend upon observation; and
+observations are inadequate, because, with the exception of a few
+islands, our only possible tide stations are on the shores of continents
+where local circumstances largely control the phenomena.
+
+\nbarticle{473.} \nbparatext{Free and Forced Oscillations.}---If the water in the ocean is
+suddenly disturbed (as for instance, by an earthquake), and then
+left to itself, a ``free'' wave will be formed, which, if the horizontal
+dimensions of the wave are large as compared with the depth of the
+water, will travel at a rate depending solely on the depth. The velocity
+of such a free wave is given by the formula $v = \sqrt{gh}$; that is, it
+is equal \textit{to the velocity acquired by a body in falling through half the
+depth of the ocean}.
+
+\begin{fineprint}
+\begin{tabular} {@{} r@{ } c@{ } c@{ } c@{ } c@{ } c@{ } r@{} c@{ } c@{ } c@{ } c}
+Thus a depth of 25 & feet & gives & a & velocity & of & 19 &$+$& miles & per & hour.\\
+                100&``&``&``&``&``& 39&&``&``&``\\
+             10,000&``&``&``&``&``&388&&``&``&``\\
+             40,000&``&``&``&``&``&775&&``&``&``\\
+             67,200&\multicolumn{3}{@{ }l@{ }}{($12\frac{3}{4}$
+miles)}&``&``&1000&&``&``&``\\
+             90,000&``&``&``&``&``&1165&&``&``&``\\
+\end{tabular}
+\end{fineprint}
+
+Observations upon the waves caused by certain earthquakes in
+South America and Japan have thus informed us that between the
+coasts of these countries the Pacific averages between two and one-half
+and three miles in depth.
+
+\nbarticle{474.} Now, as the moon in its diurnal motion passes across the
+American continent each day, and comes over the Pacific Ocean, it
+starts such a ``parent'' wave in the Pacific, and the wave once
+started moves on nearly (but not exactly) like an earthquake wave.
+Not exactly, because the velocity of the earth's rotation being about
+%% -----File: 297.png---Folio 286-------
+1050 miles an hour at the equator, the moon runs relatively westward
+faster than the wave can naturally follow, and so for a while slightly
+accelerates it. A second tidal wave is produced daily twelve hours
+later when the moon passes \textit{underneath}. The tidal wave is thus, \textit{in
+its origin, a forced oscillation, while in its subsequent travel it is pretty
+nearly a free one}.
+
+\includegraphicssideways{illo156}{\textsc{Fig.~156.}---Map of Cotidal Lines.}
+
+\nbarticle{475.} \nbparatext{Co-Tidal lines.}---These are lines drawn upon the surface
+of the ocean connecting those places which have their high water at
+the same moment of Greenwich time. They mark the crest of the
+tide wave for each hour of Greenwich time; and if we could draw
+them with certainty upon the globe, we should have all necessary
+information as to the motion of the wave. Unfortunately we can
+obtain no direct knowledge as to the position of these lines in mid-ocean;
+we only get a few points here and there on the coasts and
+on islands, so that a great deal necessarily remains conjectural. \figref{illo156}{Fig.~156}
+is a reduced copy of such a map, borrowed with some modifications
+from that given in Guyot's ``Physical Geography.''
+
+\begin{fineprint}
+\nbarticle{476.} \nbparatext{Course of Travel of the Tidal Wave.}---On studying the map
+we find that the main or ``parent'' wave starts twice a day in the Pacific,
+off Callao, on the coast of South America. This is shown on the chart
+by a sort of oval ``eye'' in the co-tidal lines, just as a mountain summit is
+shown on a topographical chart by an ``eye'' in the contour lines. From
+this point the wave travels northwest through the deepest water of the
+Pacific, at the rate of about 850 miles per hour, reaching Kamtchatka in
+about ten hours. To the west and southwest the water is shallower and
+the travel slower,---only 400 to 600 miles per hour,---so that the wave arrives
+at New Zealand in about twelve hours. Passing on by Australia, and combining
+with the small wave which the moon raises directly in the Indian
+Ocean, the resultant tide crest reaches the Cape of Good Hope in about
+twenty-nine hours, and enters the Atlantic. Here it combines with the tide
+wave, twenty-four hours younger, which has ``backed'' into the Atlantic
+around Cape Horn, and it is modified also by the \textit{direct tide} produced by the
+moon's action upon the waters of the Atlantic. The resultant tide crest
+then travels \textit{northward} through the Atlantic at the rate of nearly 700 miles
+per hour. It is about forty hours old when it first reaches the coast of the
+United States in Florida, and our coast is so situated that it arrives at all
+the principal ports within two or three hours of that time. It is forty-one
+or forty-two hours old when it arrives at New York and Boston. To reach
+London it has to travel around the northern end of Scotland and through
+the North Sea, and is nearly sixty hours old when it arrives at that port and
+the ports of the German Ocean,---Hamburg, etc.
+
+In the great oceans there are thus three or four tide crests travelling
+%% -----File: 298.png---Folio 287-------
+%% -----File: 299.png---Folio 288-------
+simultaneously, following each other nearly in the same track, but with
+continual minor changes, owing to the variations in the relative positions of
+the sun and moon and their changing distances and declinations. If we take
+into account the tides in rivers and sounds, the number of simultaneous
+tide crests must be at least six or seven; that is, the high water at the
+extremity of its travel, up the Amazon River, for instance, must be at least
+three or four days old, reckoned from its birth in the Pacific.\footnote
+  {We are greatly indebted to Loomis's discussion of the subject in his ``Elements
+  of Astronomy.''}
+\end{fineprint}
+
+\nbarticle{477.} \nbparatext{Tides in Rivers.}---The tide wave ascends a river at a rate
+which depends upon the depth of the water, the amount of friction,
+and the swiftness of the stream. It may, and generally does, ascend
+until it comes to a \textit{rapid, where the velocity of the water is greater than
+that of the wave}. In shallow streams, however, it dies out earlier.
+
+\begin{fineprint}
+Contrary to what is usually supposed, it often ascends to an elevation far
+above that of the highest crest of the tide wave at the river's mouth. In the
+La Plata and Amazon it goes up to an elevation at least one hundred feet
+above the sea-level. The velocity of the tide wave in a river seldom exceeds
+ten or twenty miles an hour, and is usually less.
+\end{fineprint}
+
+\includegraphicsmid{illo157}{\textsc{Fig.~157.}---Increase in Height of Tide on approaching the Shore.}
+
+\nbarticle{478.} \nbparatext{Height of Tides.}---In mid-ocean the difference between high
+and low water is usually between two and three feet, as observed
+on isolated deep-water islands in the Pacific; but on the continental
+shores the height is usually much greater. As soon as the tide wave
+touches bottom, so to speak, the velocity is diminished and the height
+of the wave is increased, something as in the annexed figure (\figref{illo157}{Fig.~157}).
+Theoretically the height varies \textit{inversely as the fourth root of
+the depth}. Thus, where the water is 100 feet deep, the tide wave
+should be twice as high as at the depth of 1600 feet.
+
+Where the configuration of the shore forces the wave into a corner,
+it sometimes becomes very high. At Annapolis, on the Bay of
+Fundy, tides of seventy feet are not uncommon, and an altitude of
+100 feet is said to be occasionally attained.
+
+\begin{fineprint}
+At Bristol, England, in the mouth of the Severn the tide rises fifty feet,
+and sometimes ascends the river (as it also does the Seine, in France, and
+%% -----File: 300.png---Folio 289-------
+the Amazon) as a \textit{breaking} wave, called the ``bore'' or ``eiger'' (French,
+\textit{mascaret}), with a nearly vertical front five or six feet in height, crested with
+foam, and very dangerous to small vessels. On the east coast of Ireland,
+opposite to Bristol, the tide ranges only about two feet.
+
+In mid-ocean the water has no progressive motion, but near the land
+it has, running in at the flood to fill up the bays and cover the flats, and
+then running out again at the ebb. The velocity of these tidal \textit{currents} must
+not be confounded with that of the tide wave itself.
+\end{fineprint}
+
+\nbarticle{479.} \nbparatext{Reflection and Interference.}---The tide wave when it reaches
+the shore is not entirely destroyed, especially if the coast is bold and
+the water deep; but is partly reflected, and the reflected wave goes
+back into the ocean to meet and modify the new tide wave which is
+coming in. Of course, in such a case we get ``interferences,'' so
+that on islands in the Pacific only a few hundred miles apart we find
+great differences in the heights of the tides. At one place the direct
+waves and the waves reflected from the shores of Asia and South
+America may conspire to give a tide of three or four feet, or nearly
+double its normal value, while at another they nearly destroy each other.
+
+\begin{fineprint}
+There are places, also, which are reached by tides coming by two different
+routes. Thus on the east coast of England and Scotland the tide waves
+come both around the northern end of Scotland and through the Straits of
+Dover. In some places on this coast we have, therefore, a tide of nearly
+double height, while at others not very far away there will be hardly any
+tide at all; and at intermediate points there are sometimes \textit{four} distinct
+high waters in twenty-four hours. As a consequence of this reflection and
+interference of the tide waves it follows that if the tide-raising power were
+suddenly abolished, the tides would not immediately cease, but would continue
+to run for several days, and perhaps weeks, before they gradually died
+out.
+\end{fineprint}
+
+\nbarticle{480.} \nbparatext{Effect of the Varying Pressure of the Barometer, and of
+the Wind.}---When the barometer at a given port is lower than
+usual, the level of the water is generally higher than the average,
+at the rate of about one foot for every inch of the mercury in the
+barometer; and \textit{vice versa} when it is higher than usual.
+
+When the wind blows into the mouth of a harbor, it drives in
+the water of the ocean by its surface friction, and may raise the
+water several feet. In such cases the time of high water, contrary
+to what might at first be supposed, is \textit{delayed}, sometimes as much as
+fifteen or twenty minutes.
+
+This result depends upon the fact that the water runs into the
+harbor for a longer time than it would do if the wind were not blowing.
+%% -----File: 301.png---Folio 290-------
+The normal depth of the water on the bar is reached \textit{before}
+the predicted time, so that at the predicted time the water is deeper
+than it would be if there were no wind, but the \textit{maximum} depth is
+not attained until some time later. Of course, the results are the
+opposite when the wind blows out of the harbor: the time of high
+water comes earlier, and the depth of water on the bar at the predicted
+time of high water is less than it otherwise would be.
+
+\begin{fineprint}
+\nbarticle{481.} \nbparatext{Tides in Lakes and Inland Seas.}---These are small and difficult
+to detect. Theoretically, the range between high and low water in a
+land-locked sea should bear about the same ratio to the rise and fall of the
+tide in mid-ocean that the length of the sea does to the diameter of the earth.
+Variations in the direction of the wind and the barometric pressure cause
+continual oscillations in the water-level which, even in a quiet lake, are much
+larger than the true tides; so that it is only by taking a long series of observations,
+and discussing them with reference to the moon's position in the sky,
+that it is possible to separate the real tide from the effects of other causes.
+In Lake Michigan, at Chicago, a tide of about one and three-quarters inches
+has thus been detected, the ``establishment'' of the port being about thirty
+minutes. In Lake Erie, at Buffalo and Toledo, the tide is about three-quarters
+of an inch. On the coasts of the Mediterranean the tide averages about
+eighteen inches, attaining a height of three or four feet at the head of some
+of the bays.
+\end{fineprint}
+
+\nbarticle{482.} \nbparatext{The Rigidity of the Earth.}---Sir W.~Thomson has endeavored
+to make the tides the criterion of the rigidity of the earth's core.
+Evidently if the solid parts of the earth were fluid, there would be no
+observable tide anywhere, since the whole surface would rise and fall
+together. If the earth were semi-solid, so to speak (that is, viscous,
+and capable of yielding more or less to the forces tending to change
+its form), the tides would be observable, but to a less degree than if
+the earth's core were rigid. And with this further peculiarity---since
+a viscous body requires time to change its form, waves of \textit{short period}
+would be observable upon the semi-solid earth nearly to their full extent,
+while those of \textit{long period} would almost entirely disappear,
+owing to the slow yielding of the earth's crust. Now the actual tide
+wave, as observed, is really made up of a multitude of component
+tide waves of different periods, ranging from half a day upwards.
+According to the ``principle of forced vibrations'' every regularly
+recurring periodic change in the forces which act on the surface of
+the ocean must produce a tide of greater or less magnitude, and of
+exactly corresponding period.
+%% -----File: 302.png---Folio 291-------
+
+\begin{fineprint}
+We have, for instance, the semi-diurnal, solar, and lunar tides; then the
+two monthly tides due to the change in the moon's distance and declination,
+and the two annual tides due to the changes of the sun's distance and
+declination, not to speak of the nineteen-year tide due to the revolution of
+the moon's nodes.
+\end{fineprint}
+
+A thorough analytical discussion of thirty-three years' tidal observations
+at different parts of the world has been made under the
+direction of Sir W.~Thomson by Mr.~George Darwin, with the result
+that not only do the short waves show themselves, but \textit{the waves of
+long period are found to manifest themselves with almost their full
+theoretical value}. Thomson's conclusion is that the earth as a whole
+``\textit{must be more rigid than steel, but perhaps not quite so rigid as
+glass}.'' This result is at variance with the prevalent belief of geologists
+that the core of the earth is a molten mass, and has led to much
+discussion which we cannot deal with here.
+
+\nbarticle{483.} \nbparatext{Effect of the Tides on the Earth's Rotation.}---If the tidal
+motion consisted merely in the upward and downward motion of the
+particles of the ocean to the extent of two feet or so twice a day, it
+would involve a very trifling expenditure of energy; and this is the
+case with the mid-ocean tide. But near the land this almost insensible
+mere oscillatory motion is transformed into the bodily travelling
+of immense masses of water, which flow in upon the shallows and
+then out again to sea with a great amount of fluid friction; and this
+involves the expenditure of a very considerable amount of energy
+which is dissipated as heat. From what sources does this energy
+come? The answer is that it must he derived \textit{mainly from the
+earth's energy of rotation}, and the necessary effect is to diminish that
+energy by lessening the speed of the rotation. Compared with the
+earth's whole stock of rotational energy, however, the loss of it by
+tidal friction, even in a century, is very small, and the effect on the
+length of the day is extremely slight.
+
+\begin{fineprint}
+The reader will recall the remarks upon the subject of the secular acceleration
+of the moon's mean motion a few pages back (\artref{Art.}{461}).
+\end{fineprint}
+
+While it is certain that the tidal friction \textit{tends} to lengthen the day,
+it does not follow that the day really grows longer. There are
+counteracting causes:---for example, the earth's radiation of heat
+into space, and the consequent shrinkage of her volume.
+
+As matters stand we do not know whether, \textit{as a fact}, the day is
+really longer or shorter than it was a thousand years ago. The
+%% -----File: 303.png---Folio 292-------
+change, if any has really occurred, can  hardly be as great as $\frac{1}{1000}$
+of a second.
+
+\nbarticle{484.} \nbparatext{Effect of the Tide on the Moon's Motion.}---Not only does
+the tide diminish the \textit{earth's} energy of rotation directly by the tidal
+friction, but, theoretically, it also communicates
+a minute portion of that energy to the \textit{moon}. It
+will be seen that a tidal wave, situated as in \figref{illo158}{Fig.~158},
+would slightly accelerate the moon's motion,
+the attraction of the moon by the tidal protuberance
+$F$ being slightly greater than that of the
+tide wave at $F'$---a difference tending to draw
+it along in its orbit a little, thus increasing the
+major axis of the moon's orbit. The tendency
+is therefore to make the moon \textit{recede} from the
+earth, and to \textit{lengthen} the month.
+
+\includegraphicsmid{illo158}{\textsc{Fig.~158.}\\
+Effect of the Tide on the
+Moon's Motion.}
+
+Upon this interaction between the tides and
+the motions of the earth and moon Professor
+George Darwin has founded his theory of ``\textit{tidal
+evolution}''; namely, that the satellites of a planet,
+having separated from it millions of years ago,
+have been made to recede to their present distances
+by just such an action.
+
+\nblabel{art:484*.}% typo in article 917 refers to this note to 484 as 484*
+\begin{fineprint}
+An excellent popular statement of the theory will be found in the closing
+chapters of Ball's ``Story of the Heavens.'' The original papers of Mr.~Darwin
+in the ``Philosophical Transactions'' are of course intensely mathematical.
+\end{fineprint}
+\chelabel{CHAPTERXII}
+
+%% -----File: 304.png---Folio 293-------
+
+\Chapter{XIII}{The Planets: their Motions, Apparent and Real}
+\nbchapterhang{\stretchyspace
+THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.---THE
+PTOLEMAIC, TYCHONIC, AND COPERNICAN SYSTEMS.---THE
+ORBITS AND THEIR ELEMENTS.---PLANETARY
+PERTURBATIONS.}
+
+\nbarticle{485.} For the most part, the stars keep their relative configurations
+unchanged, however much they alter their positions in the sky from
+hour to hour. The ``dipper'' remains always a ``dipper'' in whatever
+part of the heavens it may be. But while this is true of the
+stars in general, certain of the heavenly bodies, and among them
+those that are the most conspicuous of all, form an exception. The
+sun and moon continually change their places, moving eastward
+among the stars; and certain others, which to the eye appear as very
+brilliant stars, also move,\footnote
+  {When we speak of the motion of the planets, the reader will understand that
+  the \textit{diurnal} motion is not taken into account. We speak of their motions \textit{among
+  the stars}.}
+though not in quite so simple a way.
+
+\nbarticle{486.} These bodies were named by the Greeks the ``\textit{planets}''; that
+is, ``wanderers.'' The ancient astronomers counted seven of them.
+They reckoned the sun and moon, and in addition Mercury, Venus,
+Mars, Jupiter, and Saturn.
+
+Venus and Jupiter are at all times more brilliant than any of the
+fixed stars. Mars at times, but not usually, is nearly as bright as
+Jupiter; and Saturn is brighter than all but a very few of the stars.
+Mercury is also bright, but seldom seen, because always near the sun.
+
+At present the sun and moon are not reckoned as planets; but
+the roll includes, in addition to the five other bodies known by the
+ancients, the earth itself, which Copernicus showed should be counted
+among them, and also two new bodies of great magnitude (though
+inconspicuous because of their distance) which have been discovered
+in modern times; then there is in addition a host of so-called ``\textit{asteroids}''
+which circulate in the otherwise vacant space between the
+planets Mars and Jupiter.
+%% -----File: 305.png---Folio 294-------
+
+\nbarticle{487.} The list of the planets in the order of distance from the sun
+stands thus at present: Mercury, Venus, the Earth, Mars, Jupiter,
+Saturn, Uranus, and Neptune; and between Mars and Jupiter, in the
+place where a planet would naturally be expected to revolve, there
+are at present known nearly 300 little planets, which probably represent
+a single one, somehow ``spoiled in the making,'' so to speak, or
+burst into fragments.
+
+\sloppy
+The planets are all dark bodies, shining only by reflected sunlight,---globes
+which, like the earth, revolve around the sun in
+orbits nearly circular, moving all in the same direction, and (with
+some exceptions among the asteroids) nearly in the common plane of
+the ecliptic and sun's equator. All of them but the inner two and the
+asteroids are also attended by ``satellites.'' Of these the earth has one
+(the moon), Mars two, Jupiter four, Saturn eight, Uranus four, and
+Neptune one; \textit{i.e.}, so far as at present known; for although it is hardly
+probable, it is not at all impossible that others may yet be found.
+
+\nbarticle{488.} \nbparatext{Relative Distances of Planets from the Sun: Bode's Law.}---There
+is a curious approximate relation between the distances
+of the planets from the sun, which makes it easy to remember them.
+It is usually known as Bode's Law, because Bode first brought it
+prominently into notice in 1772, though Titius of Wittenberg seems
+to have discovered and enunciated it some years earlier. The law is
+this: Write a series of 4's. To the second 4 add 3; to the third add
+$3 × 2$, or 6; to the fourth, $3 × 4$, or 12; and so on, doubling the
+added number each time, as in the accompanying scheme.
+
+\fussy
+\begin{center}
+\begin{tabular}{*{8}{c@{\qquad}}c}
+  4 & 4 & \phantom{0}4 & \phantom{0}4 & \:\phantom{0}4\:
+& \phantom{0}4 & \phantom{00}4 & \phantom{00}4 & \phantom{00}4
+\\
+& \underline{3\rule[-.5ex]{0pt}{0pt}}
+& \underline{\phantom{0}6\rule[-.5ex]{0pt}{0pt}}
+& \underline{12\rule[-.5ex]{0pt}{0pt}}
+& \underline{\:24\:\rule[-.5ex]{0pt}{0pt}}
+& \underline{48\rule[-.5ex]{0pt}{0pt}}
+& \underline{\phantom{0}96\rule[-.5ex]{0pt}{0pt}}
+& \underline{192\rule[-.5ex]{0pt}{0pt}}
+& \underline{384\rule[-.5ex]{0pt}{0pt}}
+\\
+  4 & 7& 10 & 16 & [28] & 52 & 100 & 196 & 388
+\\
+  \Mercury & \Venus & \nbEarth &
+  \Mars & \;\;\nbAsteroid &
+  \Jupiter & \Saturn & \Uranus & \Neptune
+\end{tabular}
+\end{center}
+
+The resulting numbers, divided by 10, are pretty nearly the true
+mean distances of the planets from the sun, in terms of the radius
+of the earth's orbit. In the case of Neptune, however, the law
+breaks down utterly, and is not even approximately correct.
+
+For the present, at least, the law is to be regarded as a mere
+coincidence, there being so far no reasonable explanation of any
+such numerical relation.
+
+\begin{fineprint}
+The general expression for the nth term of the series is $4 + 3 × 2^{(n-2)}$;
+but it does not hold good of the first term, which is simply 4, instead of
+being $5\frac{1}{2}$, \textit{i.e.}, $(4 + 3 × 2^{-1})$, as it should be.
+% -----File: 306.png---Folio 295-------
+
+\medskip
+{\nblabel{art:489.}%\nbarticle{489.}% Article number is within table
+\footnotesize
+\noindent\centering
+\renewcommand{\arraystretch}{1.2}
+\begin{tabular} {@{}l @{\,}|@{\,} c @{\,}|@{\,} c @{\,}|@{\,} c @{\,}|@{\,} c @{\,}|@{\,} l@{\:}l @{\,}|@{\,} c@{}}
+\multicolumn{8}{c}{\nbparatext{489. Table of Names, Distances, and Periods.}}\\[1ex]
+\hline\hline
+  \textsc{Name}. & \footnotesize\textsc{Symbol}.
+& \footnotesize\textsc{Distance}. & \footnotesize\textsc{Bode}.
+& \footnotesize\textsc{Diff}.
+& \multicolumn{2}{c|@{\,}}{\footnotesize\textsc{Sid.~Period}.}
+& \multicolumn{1}{@{}m{4em}@{}}{\centering \footnotesize\textsc{Syn.\\ Period}.}
+%& \multicolumn{1}{c}{\centering \footnotesize\textsc{Syn.\ Period}.}
+\\[.5ex]
+\hline
+  Mercury \dotfill & \Mercury & \phantom{0}0.387 & \phantom{0}0.4
+& $-0.013\ $ & \ \ $88^{\text{d}}$ & or \quad $\phantom{\frac12}3^{\text{m}}$
+& $116^{\text{d}}$\rule{0pt}{3ex}
+\\
+  Venus \dotfill & \Venus & \phantom{0}0.723 & \phantom{0}0.7
+& $+0.023\ $ & $224.7^{\text{d}}$ & or \quad $7\frac{1}{2}^{\text{m}}$
+& $584^{\text{d}}$
+\\
+  Earth \dotfill & \nbEarth & \phantom{0}1.000 & \phantom{0}1.0
+& \phantom{+}$0.000\ $ & $365\frac{1}{4}^{\text{d}}$ & or $1^{\text{y}}$ & \dots
+\\
+  Mars \dotfill & \Mars & \phantom{0}1.523 & \phantom{0}1.6
+& $-0.077\ $ & $687^{\text{d}}$ & or $1^{\text{y}}\, 10\frac{1}{2}^{\text{m}}$
+& $780^{\text{d}}$
+\\[.5ex]
+\hline
+  \multicolumn{1}{@{}m{4em}@{}|@{\,}}{Mean Asteroid}& & \phantom{0}2.650 & \phantom{0}2.8 & $-0.150\ $
+& \multicolumn{2}{c|@{\,}}{$3^{\text{y}}.1$ to $8^{\text{y}}.0$}
+& various\rule{0pt}{3ex}
+\\[.5ex]
+\hline
+  Jupiter \dotfill & \Jupiter & \phantom{0}5.202 & \phantom{0}5.2 & $+0.002\ $
+& \multicolumn{2}{c|@{\,}}{$11^{\text{y}}.9$}
+& $399^{\text{d}}\rule{0pt}{3ex}$
+\\
+Saturn \dotfill & \Saturn & \phantom{0}9.539 & 10.0 & $-0.461\ $
+& \multicolumn{2}{c|@{\,}}{$29^{\text{y}}.5$} & $378^{\text{d}}$
+\\
+Uranus \dotfill & \raisebox{\depth}{\fontseries{b}\uranus} \& \Uranus & 19.183 & 19.6 & $-0.417\ $
+& \multicolumn{2}{c|@{\,}}{$84^{\text{y}}.0$} & $370^{\text{d}}$
+\\
+Neptune \dotfill & \Neptune & 30.054 & 38.8
+& $-8.746\rlap{\,!}\ $
+& \multicolumn{2}{c|@{\,}}{$164^{\text{y}}.8$}
+& $\phantom{\frac12}367\frac{1}{2}^{\text{d}}$\\[1ex]
+\hline\hline
+\end{tabular}\\[1ex]
+\renewcommand{\arraystretch}{1}%
+}%end \footnotesize
+
+The column headed ``Bode'' gives the distance according to Bode's law; the
+column headed ``Diff.,'' the difference between the true distance and that given by
+Bode's law.
+%% -----File: 307.png---Folio 296-------
+
+\nbarticle{490.} \figref{illo159}{Fig.~159} shows the smaller orbits of the system (including the orbit
+of Jupiter) drawn to scale, the radius of the earth's orbit being taken as one
+centimetre. On this scale the diameter of Saturn's orbit would be $19^{\text{cm}}.08$,
+that of Uranus would be $38^{\text{cm}}.36$, and that of Neptune, $60^{\text{cm}}.11$. The
+nearest fixed star on the same scale would be about a mile and a quarter
+away. It will be seen that the orbits of Mercury, Mars, and Jupiter are
+quite distinctly ``out of centre'' with respect to the sun. This is intentional
+and correct. The dotted half of each orbit is that which lies below, \textit{i.e.},
+south of, the plane of the ecliptic. The place of perihelion of each planet's
+orbit is marked with a $P$. The orbits of five of the asteroids, including
+the nearest and the most remote, as well as the most eccentric, are also
+given.
+\end{fineprint}
+
+\textbf{Periods.}---The \textit{sidereal period} of a planet is the time of its revolution
+around the sun from a star to the same star again, as seen from
+the sun. The \textit{synodic period} is the time between two successive conjunctions
+of the planet with the sun, as \textit{seen from the earth}. The
+sidereal and synodic periods are connected by the same relation as
+the sidereal and synodic months (\artref{Art.}{232}); namely,---
+\[
+\frac{1}{S} = \frac{1}{P} - \frac{1}{E},
+\]
+in which $E$, $P$, and $S$ are respectively the periods of the earth and of
+the planet, and the planet's synodic period, and the numerical difference
+between $\dfrac{1}{P}$ and $\dfrac{1}{E}$ is to be taken without regard to sign. The
+two last columns of the table in \artref{Article}{489} give the approximate
+periods, both sidereal and synodic, for the different planets.
+
+\includegraphicsmid{illo159}{\textsc{Fig.~159.}---Plan of the Orbits of the Planets inside of Saturn.}
+
+\nbarticle{491.} \nbparatext{Apparent Motions.}---As viewed from a distant point on the
+line drawn through the sun, perpendicular to the plane of the ecliptic,
+the planets would be seen to travel in their nearly circular orbits
+with a regular motion. As seen from the earth the apparent motion
+is much more complicated, being made up of their real motion around
+the sun combined with an apparent motion due to the earth's own
+movement.
+
+\nbarticle{492.} \nbparatext{Law of Relative Motion.}---The motion of a body relative
+to the earth can be very simply stated. \textit{It is always the same, as if
+the body had, combined with its own motion, another motion, identical
+with that of the earth, but reversed.}
+%% -----File: 308.png---Folio 297-------
+
+\includegraphicsmid{illo160}{\textsc{Fig.~160.}---The Relative Motions of Two Bodies.}
+
+\begin{fineprint}
+The proof of this is simple. Let $E$, \figref{illo160}{Fig.~160}, be the earth, and $P$ the
+planet, its direction and distance being given by the line $EP$. Let $E$ have
+a motion which will take it to $E'$ in a unit of time, and $P$ a motion which
+will take it to $P'$ in the same time. Then at the end of a unit of time the
+distance and direction of $P$ from $E$ will be given by the line $E'P'$. But if
+we suppose $E$ to remain at rest, and give to $P$ a motion $Pe$ equal to $EE'$ but
+opposite in direction, and combine this motion with $PP'$ by drawing the
+parallelogram of motions, we shall get $P''$ for the resulting place of $P$ at
+the end of the unit of time; and because the line $EP''$ is parallel and equal
+to $E'P'$ (as follows from the construction), the point $P''$, as seen from $E$,
+would occupy, in the celestial sphere, precisely the same position as $P'$ seen
+from $E'$; since all parallel lines pierce the sphere at one and the same optical
+point (\artref{Art.}{7}).
+\end{fineprint}
+
+If, therefore, the earth moves in
+a circle, every body really at rest
+will \textit{appear} to move in a circle of
+the same size as the earth's orbit,
+but keeping in such a part of its
+circle as always to have its motion
+precisely opposite to the earth's
+own real motion at the moment.
+We shall have occasion to use this
+principle very frequently.
+
+\nbarticle{493.} \nbparatext{Effect of the Combination
+of the Earth's Motion with that of
+the Planet.}---As a consequence,
+the apparent ``\textit{geocentric}'' motion
+of a planet must be made up of two motions,---that of a body moving
+once a year around the circumference of a circle equal to the earth's
+%% -----File: 309.png---Folio 298-------
+orbit, while the centre of this circle itself goes around the sun upon
+the real orbit of the planet, and with a periodic time equal to that of
+the planet. Jupiter, for instance, appears to move as in \figref{illo161}{Fig.~161},
+making eleven loops in each revolution, the smaller circle having a
+diameter of about one-fifth that of the larger one, upon which its
+centre moves, since the diameter of Jupiter's orbit is about five times
+that of the earth's.
+
+\includegraphicsouter{illo161}{\textsc{Fig.~161.}\\
+Geocentric Motion of Jupiter from 1708 to
+1720. (Cassini.)}
+
+\nbarticle{494.} \nbparatext{Direct and Retrograde Motions of the Planets and Stationary
+Points.}---As a consequence of this looped motion we have the peculiar
+back-and-forth movement of the planets among the stars which
+has been described. Starting from the time when the sun is between
+us and the planet,---the time of superior conjunction,\footnote
+  {We give \figref{illo162}{Fig.~162} to illustrate the meaning of the different terms, \textit{Opposition},
+\textit{Quadrature}, \textit{Inferior and Superior Conjunction}, and \textit{Greatest Elongation}. $E$ is the
+position of the earth, the inner circle being the orbit of an \textit{inferior} planet, while
+the outer circle is the orbit of a \textit{superior} planet. In general, the angle $PES$ (the
+angle at the earth between lines drawn from the earth to the planet and to the
+sun) is the planet's \textit{elongation} at the moment. For a superior planet it can have
+any value from zero to $180°$; for an inferior it has a maximum value that the
+  planet cannot exceed, depending upon the diameter of its orbit.}
+as it is called,
+because the planet is then \textit{above} the sun, \textit{i.e.}, further from the earth,---the
+%% -----File: 310.png---Folio 299-------
+planet moves eastward among the stars for a certain time, continually
+increasing its longitude (and also its right ascension) until at
+last its apparent motion slackens and it becomes \textit{stationary}. The distance
+of this stationary point from the sun depends upon the size of
+the planet's orbit compared with that of the earth.
+
+Then it reverses its motion and moves westward, or ``\textit{retrogrades},''
+for a while, \textit{the middle of the arc of retrogression being passed at the
+time when the earth and planet are in line with the sun, and on the
+same side of it}. If the planet is one of the outer ones, it will then
+be opposite to the sun in the sky like the full moon, and is said to be
+``\textit{in opposition}.'' If the planet is one of the inferior planets (Venus
+or Mercury), it will then be in ``\textit{inferior conjunction},'' as it is called,
+between the earth and sun.
+
+After the planet has completed its arc of retrogression, it again
+becomes stationary, turns upon its course, and once more advances
+eastward among the stars, until the synodic period is completed by
+its re-arrival at superior conjunction.
+
+\includegraphicsmid{illo162}{\textsc{Fig.~162.}---Planetary Configurations and Aspects.}
+
+Both in the number of degrees passed over, and in the time spent
+in this motion, the eastward or ``\textit{direct}'' motion always exceeds the
+retrograde. In the case of the remoter planets the excess is small---from
+$3°$ to $10°$; in the case of the nearest ones, Mars and Venus, it
+is from $16°$ to $18°$.
+
+As observed with a \textit{sidereal clock}, all the planets come \textit{later} to the
+meridian each night when moving \textit{direct}, since their right ascension is
+then increasing; but \textit{vice versa}, of course, when they are \textit{retrograding}.
+
+\nbarticle{495.} \nbparatext{Motion of the Planets with Respect to the Sun's Place in the
+Sky. Change of Elongation.}---The visibility of a planet depends
+mainly upon its angular distance, or ``\textit{elongation},'' from the sun, because
+when near the sun the planet will be above the horizon only by
+day, and cannot usually be seen. As regards their motions, considered
+from this point of view, there is a marked difference between
+the inferior planets and the superior.
+
+\nbarticle{496.} \nbparatext{Behavior of a Superior Planet.}---The superior planets drop
+always steadily \textit{westward} with respect to the sun's place in the heavens,
+continually increasing their western elongation, or decreasing their
+eastern: they therefore \textit{invariably come earlier to the meridian every
+successive night}, as observed by a time-piece \textit{keeping solar time}.
+
+\begin{fineprint}
+Beginning at superior conjunction, the planet is then moving eastward
+among the \textit{stars} with its greatest speed; but even then its eastward motion
+%% -----File: 311.png---Folio 300-------
+is not so great as the sun's, and so the planet \textit{relatively} falls westward. After
+a while it will have fallen behind by $90°$, and will then be in western quadrature,
+and on the meridian at sunrise; at the end of half its synodic period
+it will have lost $180°$, and will be just opposite the sun at sunset, being then
+at its least possible distance from the earth, and at its greatest brilliance.
+At this time the difference between the times of its daily culminations is
+also the greatest possible, and may be as much (in the case of Mars) as six
+minutes, by which amount it arrives at the meridian earlier each successive
+night. After opposition the planet is higher in the sky each night at sunset
+until it reaches eastern quadrature, when it is $90°$ east of the sun, and
+therefore on the meridian at sunset. Thence it drops back, falling more
+and more slowly westwards towards the sun, until the synodic period is
+completed by a new conjunction.
+\end{fineprint}
+
+\nbarticle{497.} \nbparatext{Motion of an Inferior Planet.}---The inferior planets appear,
+on the other hand, to \textit{vibrate} across the sun, moving out equal distances
+on each side of it, but making the westward swing much
+quicker than the eastern.
+
+The reason of this difference is obvious from \figref{illo162}{Fig.~162}. Matters
+take place with respect to the earth, sun, and planet as if the earth
+were at rest, and the planet revolving around the sun once in a \textit{synodic}
+(not sidereal) period. Now, since the distance between the
+points of greatest elongation, $V$ and $V'$, is less through inferior conjunction
+$I$, than from $V'$ around to $V$ through $C$, the time ought to
+be correspondingly shorter, as it is.
+
+\begin{fineprint}
+At superior conjunction the planet is moving eastward \textit{faster} than the
+sun. Accordingly, it creeps out to the east from the sun's rays, becoming
+visible in the twilight as an evening star. As long as its direct motion is
+greater than the sun's it keeps receding from it until it reaches its ``\textit{greatest
+eastern elongation},'' as it is called, which could in no case be greater than $90°$,
+even if the planet's distance from the sun were almost equal to that of the
+earth. (In the case of Venus it is actually about $47°$, while for Mercury it
+ranges from $18°$ to $28°$.) Then as its eastern motion slackens the sun begins
+to overtake it, and when the planet becomes really stationary as regards its
+motion among the \textit{stars}, it appears to be slipping westward towards the \textit{sun}
+at the rate of about a degree a day. At the stationary point it begins really
+to retrograde, and \textit{adds} its motion to the sun's advance, so that from that
+point it rushes swiftly towards the inferior conjunction. It passes this and
+runs out quickly on the western side, becoming a morning star, and reaching
+its western elongation in just the same number of days that it took to
+drop from eastern elongation to inferior conjunction. When the elongation
+has been gained, the planet turns around to pursue the sun, gradually gains
+upon it, and at last overtakes it again at the next superior conjunction, having
+completed its synodic period.
+\end{fineprint}
+%% -----File: 312.png---Folio 301-------
+
+\nbarticle{499.} \nbparatext{Motions in Latitude.}---If the planets' orbits lay precisely in
+the same plane with each other and with the earth's orbit, they would
+always keep in the ecliptic. But in fact, while they never go \textit{far}
+from that circle, they do deviate north or south to the extent of
+$5°$ or $6°$, and Mercury sometimes as much as $8°$; their paths among
+the stars are consequently loops and kinks like those of \figref{illo163}{Fig.~163},
+which represent forms actually observed.
+
+\includegraphicsmid{illo163}{\textsc{Fig.~163.}---Loops of Regression.}
+
+\nbarticle{500.} \nbparatext{The Ptolemaic System.}---The ancient astronomers, for the
+most part, never doubted the fixity of the earth, and its position in
+the centre of the celestial universe, though there are some reasons to
+think that Pythagoras may have done so. Assuming this and the
+actual diurnal revolution of the heavens, Ptolemy, who flourished at
+Alexandria about 140~\textsc{a.d.}, worked out the system which bears his
+name. His \mytextgreek{Meg'alh S'untaxic} (or \textit{Almagest} in Arabic) was for fourteen
+centuries the authoritative ``Scripture of Astronomy.'' He showed
+that all the apparent motions of the planets could be accounted for by
+supposing each planet to move around the circumference of a circle
+called the ``\textit{epicycle},'' while the centre of this circle, sometimes
+called the ``\textit{fictitious planet},'' itself moved on the circumference of
+another and larger circle called the ``\textit{deferent}.'' It was as if the
+real planet was carried on the end of a crank-arm which turned
+around the fictitious planet as a centre, in such a way as to point
+towards or from the earth at times when the planet is in line with the
+sun.
+
+\begin{fineprint}
+In the case of the superior planets the revolution in the epicycle was
+made once a year, so that the ``crank-arm'' was always parallel to the line
+joining earth and sun, while the motion around the deferent occupied what
+we now call the planet's period. \figref{illo164}{Fig.~164} represents the Ptolemaic System,
+except that no attention is paid to dimensions, the ``deferents'' being spaced
+%% -----File: 313.png---Folio 302-------
+at equal distances. It will be noticed that the epicycle-radii which carry at
+their extremities the planets Mars, Jupiter, and Saturn are all always parallel
+to the line that joins the earth and sun. In the case of Venus and Mercury
+this was not so. Ptolemy supposed that the deferent circles for these planets
+lay \textit{between} the earth and the sun, and that the ``fictitious planet'' in both
+cases revolved in the \textit{deferent} once a year, always keeping exactly between
+the earth and the sun: the motion in the \textit{epicycle} in this case was completed
+in the time of the planet's period, as we now know it. He ought to have
+seen that, for these two planets, the deferent was really the orbit of the
+sun itself, as the ancient Egyptians are said to have understood.
+
+\includegraphicsmid{illo164}{\textsc{Fig.~164.}---The Ptolemaic System.}
+
+\nbarticle{501.} To account for some of the irregularities of the planets' motions
+it was necessary to suppose that both the deferent and epicycle, though
+circular, are \textit{eccentric}, the earth not being exactly in the centre of the
+deferent, nor the ``fictitious planet'' in the exact centre of the epicycle.
+In after times, when the knowledge of the planetary motions had become
+more accurate, the Arabian astronomers added epicycle upon epicycle until
+the system became very complicated. King Alphonso of Spain is said to
+have remarked to the astronomers who presented to him the Alphonsine
+tables of the planetary motions, which had been computed under his orders,
+that ``if he had been present at the creation he would have given some good
+advice.''
+
+\nbarticle{502.} Some of the ancient astronomers attempted to account for the planetary
+and stellar motions in a mechanical way by means of what were called
+the ``\textit{crystalline spheres}.'' The planet Jupiter, for instance, was supposed to
+%% -----File: 314.png---Folio 303-------
+be set like a jewel on the surface of a small globe of something like glass, and
+this itself was set in a hollow made to fit it in the thick shell of a still larger
+sphere which surrounded the earth. Thus the planets were supported and
+carried by the motions of these invisible crystalline spheres; but this idea,
+though prevalent, was by no means universally accepted.
+\end{fineprint}
+
+\sloppy
+\nbarticle{503.} \nbparatext{Copernican System.}---Copernicus (1473--1543) asserted the
+diurnal rotation of the earth on its axis, and showed that it would
+fully account for the apparent diurnal revolution of the stars. He
+also showed that nearly all the known motions of the planets could
+be accounted for by supposing them to revolve around the sun, with
+the earth as one of them, in orbits circular, but slightly out of centre.
+His system, as he left it, was \textit{nearly} that which is accepted to-day,
+and \figref{illo159}{Fig.~159} may be taken as representing it. He was, however,
+obliged to retain a few small epicycles to account for certain of the
+irregularities.
+
+\fussy
+So far, no one dared to doubt the exact circularity of celestial
+orbits. It was metaphysically improper that \textit{heavenly} bodies should
+move in any but \textit{perfect} curves, and the circle was regarded as the
+only perfect one. It was left for Kepler, some sixty-five years later
+than Copernicus, to show that the planetary orbits are \textit{elliptical}, and to
+bring the system substantially into the form in which we know it now.
+
+\begin{fineprint}
+\sloppy
+\nbarticle{504.} \nbparatext{Tychonic System.}---Tycho Brahe, who came between Copernicus
+and Kepler, found himself unable to accept the Copernican system for two
+reasons. One reason was that it was unfavorably regarded by the clergy,
+and he was a good churchman. The other was the scientific objection that
+if the earth moved around the sun, the \textit{fixed stars all ought to appear to move
+in a corresponding manner} (\artref{Art.}{492}), each star describing annually an oval
+in the heavens of the same apparent dimensions as the earth's orbit itself,
+seen from the star. Technically speaking, they ought to have an ``\textit{annual
+parallax}.'' His instruments were by far the most accurate that had so far
+been made, and he could detect no such parallax; hence he concluded, not
+illogically, but incorrectly, that the earth must be at rest. He rejected the
+Copernican system, placed the earth at the centre of the universe, according
+to the then received interpretation of Scripture, made the sun revolve around
+the earth once a year, and then (this was the peculiarity of his system) made
+all the planets except the earth revolve around the sun.
+
+\fussy
+This theory just as fully accounts for all the motions of the planets as
+the Copernican, but breaks down absolutely when it encounters the aberration
+of light, and the annual parallax of the stars, which we can \textit{now} detect
+with our modern instruments, although Tycho could not with his. The
+Tychonic system never was generally accepted; the Copernican was very
+soon firmly established by Kepler and Newton.
+\end{fineprint}
+%% -----File: 315.png---Folio 304-------
+
+\nbarticle{505.} \nbparatext{Elements of a Planet's Orbit.}---Those elements are the
+numerical quantities which must be given in order to describe the
+orbit with precision, and to furnish the means of finding the planet's
+place in the orbit at any given time, whether past or future. They
+are seven in number, as follows:---
+
+1. The semi-major axis, $a$.
+
+2. The eccentricity, $e$.
+
+3. The inclination to the ecliptic, $i$.
+
+4. The longitude of the ascending node, $\nbAscnode$.
+
+5. The longitude of the perihelion, $\pi$.
+
+6. The epoch, $E$.
+
+7. The period $P$, or daily motion $\mu$.
+
+\includegraphicsmid{illo165}{\textsc{Fig.~165.}---The Elements of a Planet's Orbit.}
+
+\nbarticle{506.} Of these, the first five pertain to the orbit itself, regarded as
+an ellipse lying in space with one focus at the sun, while two are
+necessary to determine the planet's place in the orbit.
+
+The \textit{semi-major axis}, $a$ ($CA$ in \figref{illo165}{Fig.~165}), defines the \textit{Size} of the
+orbit, and may be expressed either in ``astronomical units'' (the
+earth's mean distance from the sun is the astronomical unit) or in
+miles.
+
+The \textit{Eccentricity} defines the orbit's \textit{Form}. It is a mere numerical
+quantity, being the fraction $\dfrac{c}{a}$ (usually expressed decimally), obtained
+by dividing the distance between the sun and the centre of the orbit
+by the semi-major axis. In some computations it is convenient to
+use, instead of the decimal fraction itself, the angle $\phi$ which has $e$ for
+its sine; \textit{i.e.}, $\phi = \sin^{-1}{e}$.
+%% -----File: 316.png---Folio 305-------
+
+The third element, $i$, is the \textit{Inclination} between the plane of the
+planet's orbit and that of the earth. In the \figref{illo165}{figure} it is the angle
+$KNO$, the plane of the ecliptic being lettered $EKLM$.
+
+The fourth element, \nbAscnode (\textit{the Longitude of the ascending node}),
+defines what has been called the ``\textit{aspect}'' of the orbit-plane; \textit{i.e.},
+the direction in which it faces. The line of nodes is the line $NN'$ in
+the \figref{illo165}{figure}, the intersection of the two planes of the orbit and ecliptic;
+and the angle $\aries SN$ is the longitude of the ascending node, the
+line $S \aries$ being the line drawn from the sun to the first of Aries. The
+planet, moving around its orbit in the plane $ORBT$, and in the direction
+of the arrow, passes from the lower or southern side of the plane
+of the ecliptic to the northern at the point $n$, which, as seen from $S$,
+is in the same direction as $N$.
+
+The fifth, and last, of the elements which belong strictly to the
+orbit itself is $\pi$, the so-called \textit{Longitude of the perihelion}, which defines
+the \textit{direction} in which the major axis of the ellipse (the line $PA$)
+lies on the plane $ORBT$. It is not strictly a \textit{longitude}, but equals
+the sum of the two angles $\nbAscnode$ and $\omega$; \textit{i.e.}, $\aries SN$ (in the plane of the
+ecliptic) $+\ NSP$ (in the plane of the orbit). It is quite sufficient
+to give $\omega$ alone, and in the case of cometary orbits this is usually
+done.
+
+\nbarticle{507.} If we regard the orbit as an oval wire hoop suspended in
+space, these five elements completely define its position, form, and
+size. The \textit{plane} of the orbit is fixed by the elements numbered three
+and four, the \textit{position} of the orbit in this plane by number five, the
+\textit{form} of it by number two, and finally its \textit{magnitude}, by number one.
+
+The student will recollect that the general equation of curves
+of the second degree (the conics) in analytical geometry contains five
+constants, and therefore that number of data is enough to define such
+a curve completely.
+
+\nbarticle{508.} To determine where the planet will be at any subsequent date
+we need two more elements.
+
+Sixth. \textit{The Periodic Time},---we must have the \textit{sidereal period}, $P$,
+or else \textit{the mean daily motion}, $\mu$, which is simply $360°$ divided by
+the number of days in $P$.
+
+\sloppy
+Seventh. And finally; we must have a starting-point, the ``\textit{Epoch},''
+so-called; \textit{i.e.}, the longitude of the planet as seen from the sun, at
+some given date, usually Jan.~1st, 1850 or 1900, or else some precise
+date at which the planet passed the perihelion or node.
+%% -----File: 317.png---Folio 306-------
+
+\fussy
+\begin{fineprint}
+\nbarticle{509.} If it were not for perturbations caused by the mutual interaction
+between the planets, these elements would never change, and could be used
+directly for computing the planet's place at any date in the past or in the
+future; but, excepting $a$ and $P$, they do change on account of such interaction,
+and accordingly it is usual to add in tables of the planetary elements,
+columns headed $\Delta\nbAscnode$, $\Delta\pi$, $\Delta i$, and $\Delta e$, giving the amount by which the
+quantities $\nbAscnode$, $\pi$, $i$, and $e$ respectively change in a century.
+
+\nbarticle{510.} If Kepler's harmonic law were strictly true, we should not need
+both $a$ and $P$, because we should have
+\[
+(\text{Earth's Period})^{2}:P^{2}::1^{3}:a^{3}, \text{or } P = a^{\frac{3}{2}},
+\]
+$P$ being expressed in years and $a$ in astronomical units. But since the exact
+form of the equation is
+\[
+{P_{1}}^{2}(1 + m_{1}):{P_{2}}^{2}(1 + m_{2})::{a_{1}}^{3}:{a_{2}}^{3}\ (\text{\artref{Art.}{417}}),
+\]
+it is necessary, in cases where the highest attainable accuracy is required, to
+regard $P$ and $a$ as independent, and give them both in the tables.
+\end{fineprint}
+
+\nbarticle{511.} \nbparatext{Geocentric Place.}---Our \textit{observations} of a planet's place are
+necessarily ``\textit{geocentric},'' or earth-centred; they give us, when properly
+corrected for refraction and parallax, the planet's \textit{right ascension}
+and \textit{declination} as seen from the centre of the earth, and from them,
+if desired, the corresponding geocentric \textit{longitude} and \textit{latitude} are
+easily obtained by the method explained in \artref{Article}{180}.
+
+\begin{fineprint}
+It often happens that we want the place at some moment of time when the
+planet could not be directly observed, as, for instance, in the day time. If we
+have a series of observations of the planet made about that time, the place for
+the exact moment is readily deduced by a process of \textit{interpolation}, and with
+an accuracy actually exceeding that of any single observation of the series.
+
+Graphically it is done by simply plotting the observations actually made.
+Suppose, for instance, we want the right ascension of Mars for 8~\textsc{a.m.} on
+June~3, and have meridian-circle observations made at 10 o'clock \textsc{p.m.} on
+June~1, at $9^{\text{h}}\ 55^{\text{m}}$ on June~2, at $9^{\text{h}}~50^{\text{m}}$ on June~3, and so on. We first lay off
+the times of observation as abscissas along a horizontal line taken as the
+time-scale, and then lay off the observed right ascensions as ordinates at
+points corresponding to the times. Then we draw a smooth curve through
+the points so determined, and from this curve we can read off directly, the
+right ascension corresponding to any desired moment: similarly for the
+declination. Of course, what can be done graphically can be done still
+more accurately by computation.
+\end{fineprint}
+
+\nbarticle{512.} \nbparatext{Heliocentric Place.}---The \textit{heliocentric} place of a planet is the
+place as seen from the sun; and when we know the longitude of the
+node of a planet's orbit and its inclination, as well as the planet's distance
+%% -----File: 318.png---Folio 307-------
+from the sun, this heliocentric place can at once be deduced
+from the geocentric by a trigonometrical calculation. The process is
+rather tedious, however, and its discussion lies outside the scope of
+this work.
+
+\begin{fineprint}
+(The reader is referred to Watson's ``Theoretical Astronomy,'' p.~86. An
+elementary geometrical treatment of the reduction is also given in Loomis's
+``Treatise on Astronomy,'' p.~211.)
+\end{fineprint}
+
+\nbarticle{513.} \nbparatext{Determination of a Period of a Planet.}---This can be done
+in two ways:
+
+First. \textit{By observation of its node-passage.} When the planet is
+passing its node, it is in the plane of the ecliptic, and the earth being
+also always in that plane, the planet's latitude, \textit{both geocentric and
+heliocentric}, will be zero, no matter what may be the place of the earth
+in its orbit. (At any other point of the planet's orbit except the node
+its apparent latitude would not be thus independent of the earth's
+place, but would vary according to its distance from the earth.) If,
+then, we observe the planet at two successive passages of the same
+node, the interval between the moments when the latitude becomes
+zero will be the planet's period,---\textit{exactly}, if the node is stationary;
+very \textit{approximately}, even if the node is not absolutely stationary, as
+none of the nodes actually are.
+
+\begin{fineprint}
+There are two difficulties with this method.
+
+(\textit{a}) In the case of Uranus the period is eighty-four years, and in that of
+Neptune 164 years---too long to wait.
+
+(\textit{b}) Since the orbits all cross the ecliptic at a very small angle, so that the
+latitude remains near zero for a number of days, it is extremely difficult to
+determine the precise minute and second when it is exactly zero; and slight
+errors in the declinations observed will produce great errors in the result.
+\end{fineprint}
+
+\nbarticle{514.} Second. \textit{By the mean synodic period of the planet.} The
+synodic period is the interval between two successive oppositions or
+conjunctions of the planet, the opposition being the moment when
+the planet's longitude differs from that of the sun by $180°$.
+
+\begin{fineprint}
+This angle between the planet and sun cannot well be measured directly,
+but we can make with the meridian circle a series of observations both of
+the planet's right ascension and declination for several days before and after
+the date of opposition, and reduce the observations to latitude and longitude.
+The sun will be observed, of course, at noon, and the planet near midnight;
+but from the solar observations we can deduce the longitudes of the sun
+corresponding to the exact moments when the planet was observed. From
+these, we find the difference of longitude between the planet and the sun at
+the time of each planetary observation; and finally from these differences
+%% -----File: 319.png---Folio 308-------
+of longitude, we find the precise moment when that difference was exactly
+$180°$, or the moment of opposition. This can be ascertained within a very
+few seconds of time if the observations are good.
+\end{fineprint}
+
+Since the orbits are not strictly circular, the interval between two
+successive observations will not be the \textit{mean} synodic period, but only
+an approximation to it; but when we know it \textit{nearly}, we can compare
+oppositions many years apart, and by dividing the interval by the
+known number of entire synodic periods (which is easily determined
+when we know the approximate length of a single period) we get the
+mean synodic period very closely,---especially if the two oppositions
+occur at about the same time of the year. Having the synodic
+period, the true sidereal period at once follows from the equation
+\[
+\frac{1}{P} = \frac{1}{E} - \frac{1}{S}.
+\]
+
+\nbarticle{515.} \nbparatext{To find the Distance of a Planet in Terms of the Earth's
+Distance.}---When we know the planet's sidereal period, this is easily
+done by means of two observations of the planet's ``\textit{elongation}''
+taken at an interval equal to its periodic time. The ``elongation''
+of a planet is the difference between its longitude and that of the
+%% -----File: 320.png---Folio 309-------
+sun, and a series of meridian-circle observations of sun and planet
+will furnish these differences of longitude for any selected moment
+included within the term of observation.
+
+\includegraphicsmid{illo166}{\textsc{Fig.~166.}---Determination of the Distance of a Planet from the Sun.}
+
+To find the distance of the planet Mars, for instance, we must
+therefore have two observations separated by an interval of 687 days.
+Suppose the earth to have been at $A$ (\figref{illo166}{Fig.~166}) at the moment of the
+first observation. Then at the time of the second observation she
+will be at the point $C$, the angle $ASC$ being that which the earth
+will describe in the next $43\frac{1}{2}$ days, which is the difference between
+two complete years (or $730\frac{1}{2}$ days) and the 687-day interval between
+the two observations.
+
+The angles $SCM$ and $SAM$ are the ``elongations'' of the planet
+from the sun, and are given directly by the observations. The two
+sides $SA$ and $SC$ are also given, being the earth's distance from
+the sun at the dates of observation. Hence we can easily solve
+the quadrilateral, and find the length of $SM$, as well as the angle
+$ASM$.
+
+\begin{fineprint}
+This angle determines the planet's \textit{heliocentric} longitude at $M$, since
+we know the direction of $SA$, the longitude of the earth at the time of
+observation.
+
+The student can follow out for himself the process by which, from two
+elongations of Venus, $SAV$ and $SBV$, observed at an interval of 225 days,
+the distance of Venus from the sun (or $SV$) can be obtained.
+
+\includegraphicsouter{illo167}{\textsc{Fig.~167.}}
+
+\nbarticle{516.} In order that this method may apply with strict accuracy it is
+necessary that at the moment of observation $M$ should be in the same
+plane as $A$, $S$, and $C$; that is, at the node. If it is not so, the process will
+give us, not the true distance of the planet
+itself from the sun, but that of the ``projection''
+of this distance on the plane of
+the ecliptic; \textit{i.e.}, the distance from the sun
+to the point $m$ (\figref{illo167}{Fig.~167}), where the perpendicular
+from the planet would strike
+that plane. But when we have determined
+$Am$ and the angle $mAM$, the planet's geocentric
+latitude, we easily compute $Mm$; and from $Sm$ and $Mm$ we get the
+true distance $SM$ and the heliocentric latitude of the planet $MSm$.
+\end{fineprint}
+
+\nbarticle{517.} From a series of pairs of observations distributed around
+the planet's orbit it would evidently be possible to work out the
+orbit completely. It was in this way that Kepler showed that the
+orbits of the planets are ellipses, and deduced their distances from
+%% -----File: 321.png---Folio 310-------
+the sun; and his third, or harmonic law, was then discovered simply
+by making a comparison between the distances thus found and the
+corresponding periods.
+
+\includegraphicsouter{illo168}{\textsc{Fig.~168.}\\
+Distance of an Inferior Planet determined by
+Observations of its Greatest Elongation.}
+
+\nbarticle{518.} \nbparatext{Mean Distance of an Inferior Planet by Means of Observations
+of its Greatest Elongation.}---By observing from the earth the
+greatest elongation $SEV$ (\figref{illo168}{Fig.~168})
+of one of the inferior planets,
+its distance from the sun can very
+easily be deduced if we regard the
+orbit as a circle; for the triangle
+$SVE$ will be right-angled at $V$, and
+$SV = SE × \sin{SEV}$.
+
+\begin{fineprint}
+In the case of Venus the orbit is
+so nearly circular that the method
+answers very well, the greatest elongation
+never differing much from $47°$.
+Mercury's orbit is so eccentric that the
+distance thus obtained from a single
+elongation might be very wide of the true mean distance. Since the greatest
+elongation, $SEM$, varies all the way from $18°$ to $28°$, it would be necessary
+to observe a great many elongations, and take the average result.
+\end{fineprint}
+
+\nbarticle{519.} \nbparatext{Deduction of the Orbit of a Planet from Three Observations.}---When
+one has command of a great number of observations of a
+planet running back many years, and can select such as are convenient
+for his purpose, as Kepler could from Tycho's records, it is
+comparatively easy to find the elements of a planet's orbit; but when
+a new planet is discovered, the case is different. The problem first
+arose practically in 1801, when Ceres, the first of the asteroids, was
+discovered by Piazzi in Sicily, observed for a few weeks and then
+lost in the sun's rays at conjunction, before other astronomers could
+be notified of the discovery, in these days of slow communication,
+made slower and more uncertain by war.
+
+Gauss, then a young man at Göttingen, attacked the problem, and
+invented the method which, with slight modifications, is now universally
+used in such cases.
+
+We do not propose to enter into details, but simply say that
+\textit{three absolutely accurate observations of a planet's right ascension and
+declination are sufficient to determine its orbit}. Three observations,
+made only as accurately as is now possible, with intervals of two or
+%% -----File: 322.png---Folio 311-------
+three weeks between them, will give a very good \textit{approximation} to
+the orbit; and it can then be corrected by further observations.
+
+\begin{fineprint}
+\nbarticle{520.} Since there are \textit{five} independent coefficients in the equation of a
+conic, it is necessary to have \textit{five} conditions in order to determine them.
+\textit{Three} such conditions are given by the observations themselves; viz., the
+three directions of the planet as seen from the earth at the given moments
+of observation; a \textit{fourth} is given by the fact that the sun must be in the
+focus of the orbit; and finally, the \textit{fifth} is supplied by the ``law of equal
+areas,'' since the areas described by the radius vector between the first and
+second observations, and the second and third, must be proportional to the
+corresponding intervals of time.
+
+(The student is referred to Gauss's ``Theoria Motus,'' or to Watson's
+``Theoretical Astronomy,'' or to Oppolzer's great work on ``The Determination
+of Orbits,'' for the full development of the subject.)
+\end{fineprint}
+
+\nbarticle{521.} \nbparatext{Planetary Perturbations.}---The attraction of the planets for
+each other disturbs their otherwise elliptical motion around the sun.
+As in the case of the lunar theory the disturbing forces are, however,
+always relatively small, but not for the same reason. The sun's
+disturbing force is small because its \textit{distance from the moon is nearly
+four hundred times that of the earth}. In the planetary theory the
+disturbing bodies are often nearer to the disturbed than is the sun
+itself, as, for instance, in the disturbance of Saturn by Jupiter at
+certain points of their orbits; but the \textit{mass of the disturbing body in
+no case is as great as $\frac{1}{1000}$ part of the sun's mass}, and for this reason
+the disturbing force arising from planetary attraction is never more
+than a small fraction of the sun's attraction.
+
+\begin{fineprint}
+The greatest disturbing force which occurs in the planetary system
+(except in the case, of some of the asteroids) is that of Jupiter on Saturn
+at the time when the planets are nearest: it then amounts to $\frac{1}{128}$ of the
+sun's attraction. When these two planets are most remote from each other,
+it amounts to $\frac{1}{357}$. There is no other case where the disturbing force is as
+much as $\frac{1}{1000}$ of the sun's attraction (again excepting the asteroids disturbed
+by Jupiter).
+\end{fineprint}
+
+\nbarticle{522.} In any special case the disturbing force can he worked out
+on precisely the same principles that lie at the foundation of the
+diagram by which the sun's disturbing force upon the moon was
+found (\artref{Art.}{441}, \figref{illo147}{Fig.~147}); but the resulting diagram will look very
+differently, because the disturbing body is relatively very near the
+disturbed orbit.
+%% -----File: 323.png---Folio 312-------
+
+The planetary perturbations which result from the ``integration''
+of effects of the disturbing forces, \textit{i.e.}, from their continual action
+through long intervals of time, divide themselves into two great
+classes,---the \textit{Periodic} and the \textit{Secular}.
+
+\nbarticle{523.} \nbparatext{Periodic Perturbations.}---These are such as depend on the
+positions of the planets in their orbits, and usually run through their
+course in a few revolutions of the planets concerned. For the most
+part they are very small. Those of Mercury never amount to more
+than $15''$, as seen from the sun. Those of Venus may reach about
+$30''$, those of the earth about $1'$, and these of Mars about $2'$. The
+mutual disturbances between Jupiter and Saturn are much larger,
+amounting respectively to $28'$ and $48'$; while those of Uranus are
+again small, never exceeding $3'$, and those of Neptune are not more
+than half as great as that. In the case of the asteroids, which are
+powerfully disturbed by Jupiter, the periodical perturbations are
+enormous, sometimes as much as $5°$ or $6°$.
+
+\begin{fineprint}
+\nbarticle{524.} \nbparatext{Long Inequalities.}---The periodic inequalities of the planets are
+so small, because, as a rule, there is a nearly complete compensation effected
+at every few revolutions, so that the accelerations balance the retardations.
+The line of conjunction falls at random in different parts of the orbits, and
+when this is the case, no considerable displacement of either planet can
+take place. But when the periodic times of two planets are \textit{nearly commensurable},
+their line of conjunction will fall very near the same place in
+the two orbits for a considerable number of years, and the small unbalanced
+disturbance left over at each conjunction will then accumulate in the same
+direction for a long time. Thus, five revolutions of Jupiter roughly equal
+two of Saturn; and still more nearly, seventy-seven of Jupiter equal thirty-one
+of Saturn, in a period of 913 years. From this comes the so-called
+``\textit{long inequality}'' of Jupiter and Saturn, amounting to $28'$ in the place of
+Jupiter and $48'$ in that of Saturn, and requiring more than 900 years to
+complete its cycle.
+
+In the case of the earth and Venus there is a similar ``long inequality''
+with a period of 235 years, amounting, however, to less than $3''$ in the positions
+of either of the planets; between Uranus and Neptune there is a similar
+inequality with a period of over 4000 years, but this also is very small.
+\end{fineprint}
+
+\nbarticle{525.} \nbparatext{Secular Inequalities.}---These are inequalities which depend
+not on the position of the planets in their orbits, but \textit{on the relative
+position of the orbits} themselves, with reference to each other,---the
+way, for instance, in which the \textit{lines of nodes} and \textit{apsides} of two
+neighboring orbits lie with reference to each other. Since the planetary
+orbits change their positions very slowly, these perturbations,
+%% -----File: 324.png---Folio 313-------
+although in the strict sense of the word periodic also, are very slow
+and majestic in their march, and the periods involved are such as
+stagger the imagination. They are reckoned in myriads and hundreds
+of thousands of years. From year to year they are insignificant,
+but with the lapse of time become important.
+
+\nbarticle{526.} \nbparatext{Secular Constancy of the Periods and Mean Distances.}---It is
+a remarkable fact, demonstrated by Lagrange and La~Place
+about 100 years ago, that \textit{the mean distances and periods are entirely
+free from all such secular disturbance}. They are subject to
+slight \textit{periodic} inequalities having periods of a few years, or even
+a few hundred years: but in the \textit{long run} the two elements never
+change. They suffer no perturbations which depend on the position
+of the orbits themselves, but only such as depend on the positions
+of the planets in their orbits.
+
+\nbarticle{527.} \nbparatext{Revolution of the Nodes and Apsides.}---The \textit{nodes and perihelia},
+on the other hand, move on continuously. The lines of apsides
+of all the planets (Venus alone excepted) \textit{advance}, and the nodes of
+all without exception (except possibly some of the asteroids), \textit{regress}
+on the ecliptic.
+
+\begin{fineprint}
+\sloppy
+The quickest moving line of apsides---that of Saturn's orbit---\hspace{0pt}completes
+its revolution in 67,000 years, while that of Neptune requires 540,000. The
+swiftest line of nodes is that of Uranus, which completes its circuit in less
+than 37,000 years, while the slowest---that of Mercury---requires 166,000
+years.
+\fussy
+\end{fineprint}
+
+\nbarticle{528.} \nbparatext{The Inclinations of the Orbits.}---These are all slowly changing---some
+increasing, and others decreasing; but as La~Place and
+Leverrier have shown, all the changes are confined within narrow
+limits for all the larger planets: they oscillate, but the oscillations are
+never extensive.
+
+\begin{fineprint}
+It is not certain that this is so with the asteroids, some of which have
+inclinations to the ecliptic of $25°$ and $30°$: it is possible that some of
+\textit{these} inclinations may change by a very considerable amount.
+\end{fineprint}
+
+\nbarticle{529.} \nbparatext{The Eccentricities.}---These also are slowly changing in the
+same way as the inclinations, some increasing and some decreasing;
+and their changes also are closely restricted. The periods of the alternate
+increase and decrease are always many thousand years in length.
+
+\begin{fineprint}
+The asteroids are again to be excepted; the eccentricities of their orbits
+may change considerably.
+\end{fineprint}
+%% -----File: 325.png---Folio 314-------
+
+\nbarticle{530.} \nbparatext{Stability of the Planetary System.}---About the end of the
+eighteenth century La~Place and Lagrange succeeded in proving that
+\textit{the mutual attraction} of the planets could never destroy the system,
+nor even change the elements of the orbit of any one of the larger
+planets to an extent which would greatly alter its physical condition.
+
+The nodes and apsides revolve continuously, it is true, but that
+change is of no importance. The distances from the sun and the
+periods do not change at all in the long run; while the inclinations
+and eccentricities, as has just been said, confine their variations
+within narrow limits.
+
+\begin{fineprint}
+\nbarticle{531.} \nbparatext{The ``Invariable Plane'' of the Solar System.}---There is
+no reason, except the fact that \textit{we} live on the earth, for taking the plane of
+the \textit{earth's} orbit (the plane of the ecliptic) as the fundamental plane of the
+solar system. There is, however, in the system an ``\textit{invariable plane},'' the
+position of which remains forever unchanged by any mutual action among
+the planets, as was discovered by La~Place in 1784. This plane is defined
+by the following conditions,---\textit{that if from all the planets perpendiculars be
+drawn to it} (\textit{i.e.}, to speak technically, if the planets be ``projected'' upon it),
+\textit{and then if we multiply each planet's mass by the area which the planet's projected
+radius vector describes upon this plane in a unit of time, the sum of these
+products will be a maximum}. The ecliptic is inclined about $1\frac{1}{2}°$ to this
+invariable plane, and has its ascending node nearly in longitude $286°$.
+
+\nbarticle{532.} \nbparatext{La Place's Equations for the Inclinations and Eccentricities.}---La~Place
+demonstrated the two following equations, viz.:
+\begin{align*}
+& (1)\ \ \textstyle\sum(m\sqrt{a} × e^2) = C. &
+& (2)\ \ \textstyle\sum(m\sqrt{a} × \tan^2 i) = C'.
+\end{align*}
+Equation (1) may be thus translated: \textit{Multiply the mass of each planet by the
+square root of the semi-major axis of its orbit, and by the square of its eccentricity;
+add these products for all the planets, and the sum will be a constant
+quantity $C$, which is very small}. It follows that no eccentricity can become
+very large, since $e^2$ \textit{in the equation is essentially positive:} there can therefore
+be \textit{no counterbalancing of positive and negative eccentricities}; and if the
+eccentricity of one planet increases, that of some other planet or planets
+must correspondingly decrease.
+
+The second equation is the same, merely substituting $\tan^2i$ for $e^2$, $i$ being
+\textit{the inclination of the planet's orbit to the invariable plane}.
+
+The constant in this case also is small, though of course not the same
+as in the preceding equation. These two equations, taken in connection
+with the invariability of the periods of the major axes of the planetary
+orbits, have been called the ``Magna Charta'' of the stability of the solar
+system.
+\end{fineprint}
+%% -----File: 326.png---Folio 315-------
+
+
+
+\nbarticle{533.} It does not follow that because the \textit{mutual attractions} of the
+planets cannot seriously derange the system it is, therefore, of necessity
+securely stable. There are many other conceivable actions
+which might end in its ultimate destruction; such, for instance, as
+that of a resisting medium, or the entrance into the system of bodies
+coming from without.
+\chelabel{CHAPTERXIII}
+%% -----File: 327.png---Folio 316-------
+
+\Chapter{XIV}{The Planets themselves}
+\nbchapterhang{\stretchyspace
+THE PLANETS: METHODS OF FINDING THEIR DIAMETERS, MASSES,
+ETC.---THE ``TERRESTRIAL PLANETS'' AND ASTEROIDS.---INTRA-MERCURIAL
+PLANETS AND THE ZODIACAL LIGHT.}
+
+\textsc{In} discussing the individual peculiarities of the planets, we have
+to consider a multitude of different data; for instance, their \textit{diameters},
+their \textit{masses}, and \textit{densities}, their \textit{axial rotation}, their \textit{surface-markings},
+their \textit{reflecting power} or \textit{``albedo,''} and their \textit{satellite systems}.
+
+\nbarticle{534.} \nbparatext{Diameter.}---The apparent diameter of a planet is ascertained
+by measurement with some kind of micrometer (\artref{Art.}{73}). For this
+purpose the ``double-image'' micrometer has an advantage over the
+wire micrometer because of the effect of irradiation.
+
+\includegraphicsouter{illo169}{\textsc{Fig.~169.}\\
+Micrometer Measures of a Planet's Diameter.}
+
+\begin{fineprint}
+When we bring two wires to touch the two limits of the planet in the
+field of view of the telescope, \figref{illo169}{Fig.~169}, \textit{a}, the bright image of the planet is
+always measured too large, because every bright object appears to extend
+itself somewhat into the dark surrounding
+space, by its physiological
+action upon the retina of the
+eye. This is known as \textit{irradiation}---well
+exemplified at the time of
+new moon, when the bright crescent
+appears to be much larger
+than the ``old moon'' faintly visible
+by earth-shine. With small instruments this error is often considerable,
+varying with the personal equation of the observer, but it may be
+reduced to some extent by using as bright an illumination of the field of
+view as the object will bear.
+
+With the double-image micrometer, the observer in measuring has to
+bring in contact two discs of equal brightness, as in \figref{illo169}{Fig.~169}, \textit{b}; and in this
+case the irradiation almost vanishes at the point of contact.
+\end{fineprint}
+
+The diameter thus measured is, of course, only the \textit{apparent} diameter,
+to be expressed in seconds of arc, and varies with every change
+of distance. To get the real diameter in linear units, we have
+\[
+\mathrm{Real~diameter} = \frac{\Delta× D''}{206265},
+\]
+%% -----File: 328.png---Folio 317-------
+in which $\Delta$ is the distance of the planet from the earth, and $D''$ the
+diameter in seconds of arc. If $\Delta$ is given only in astronomical units,
+the diameter comes out, of course, in terms of that unit. To get
+the diameter in miles, we must know the value of this unit in miles;
+that is, the sun's distance from the earth.
+
+\nbarticle{535.} \nbparatext{Extent of Surface and Volume.}---Having the diameter, the
+\textit{surface}, of course, is proportional to its \textit{square}, and is equal to the
+earth's surface multiplied by $\left( \dfrac{s}{\rho} \right)^2 $, in which $s$ is the semi-diameter of
+the planet and $\rho$ that of the earth.
+
+The \textit{volume} equals $\left( \dfrac{s}{\rho} \right)^3 $ in terms of the earth's volume. (The student
+must be on his guard against confounding the \textit{volume} or \textit{bulk} of
+a planet with its \textit{mass}.)
+
+The nearer the planet, other things being equal, the more accurately
+the above data can be determined. The error of $0''.1$ in measuring
+the apparent diameter of Venus, when nearest, counts for less than
+thirteen miles in the real diameter of the planet; while in Neptune's
+case it would correspond to more than 1300 miles. The student
+must not be surprised, therefore, at finding considerable discrepancies
+in the data given for the remoter planets by different authorities.
+
+\nbarticle{536.} \nbparatext{Mass of a Planet which has a Satellite.}---In this case its
+mass is easily and accurately found by observing the period and
+distance of the satellite. We have the fundamental equation
+\[
+(M + m) = 4 \pi^2 \left( \frac{r^3}{t^2} \right),
+\]
+in which $M$ is the mass of the planet, $m$ that of its satellite, $r$ the
+radius of the orbit of the satellite, and $t$ its period.
+
+\begin{fineprint}
+The formula is derived as follows: From the law of gravitation the accelerating
+force which acts on the satellite is given by the equation
+\[
+f = \frac{M + m}{r^2}
+\]
+(\artref{Art.}{417}), in which $M$ is the mass of the planet and $m$ that of the satellite.
+From the law of circular motion (\artref{Art.}{411}, Eq.~$b$) we have
+\[
+f = 4 \pi^2 \left( \frac{r}{t^2} \right) ;
+\]
+%% -----File: 329.png---Folio 318-------
+whence (equating the two values of $f$) we have
+\[
+\frac{M + m}{r^2} = 4 \pi^2 \left( \frac{r}{t^2} \right);
+\]
+and finally
+\[
+(M + m) = 4 \pi^2 \left( \frac{r^3}{t^2} \right).
+\]
+This demonstration is strictly good only for circular orbits; but the equation
+is equally true, and can be proved for elliptical orbits, if for $r$ we put $a$, the
+semi-major axis of the satellite's orbit.
+\end{fineprint}
+
+For many purposes a proportion is more convenient than this equation,
+since the equation requires that $M$, $m$, $r$, and $t$ be expressed in
+properly chosen units in order that it may be numerically true. Converting
+the equation into a proportion, we have
+\[
+(M + m) : (M_1 + m_1) = \frac{r^3}{t^2} :  \frac{r_1^3}{t_1^2};
+\]
+or, in words, \textit{the united mass of a body and its satellite is to the united
+mass of a second body and its satellite as the cube of the distance of
+the first satellite divided by the square of its period is to the cube of the
+distance of the second satellite divided by the square of its period}. This
+enables us at once to compare the masses of any two bodies which
+have attendants revolving around them.
+
+The mass of the moon is so considerable as compared with that
+of the earth (about $\frac{1}{80}$) that it will not do to neglect it; but in all
+other cases the satellite is less than $\frac{1}{1000}$ of the mass of its primary,
+and need not be taken into account.
+
+\begin{fineprint}
+\nbarticle{537.} \nbparatext{Examples.}---(1) Required the mass of the \textit{sun} compared with
+that of the \textit{earth}. The proportion is
+\[
+(S + \text{earth} ) : (E + \text{moon}) = \frac{ (93000000)^3}{(365 \frac{1}{4})^2} : \frac{(238000)^3}{(27.4)^2}.
+\]
+The quantities in the last term of the proportion are of course the distance
+and period of \textit{the moon}; and it is to be remembered that for the period of
+the moon we must use, not the \textit{actual} sidereal period, but the period \textit{as it
+would be if the moon's motion were undisturbed},---a period about three hours
+longer.
+
+(2) Compare the mass of the earth with that of Jupiter, whose remotest
+satellite has a period of $16 \frac{3}{4}$ days, and a distance of 1,160,000 miles. We have
+\[
+(E + \text{moon} ) : (J + \text{satellite}) = \frac{(238000)^3}{(27.4)^2} : \frac{1160000)^3}{(16 \frac{3}{4})^2},
+\]
+%% -----File: 330.png---Folio 319-------
+which gives the mass of Jupiter about 316 times as great as that of the earth
+and moon together.
+\end{fineprint}
+
+\nbarticle{538.} It is customary to express the mass of a planet as a certain
+fraction of the sun's mass, and the proportion is simply
+\[
+\text{Sun} : \text{Planet} = \frac{R^3}{T^2} : \frac{r^3}{t^2},
+\]
+whence \hfill $\text{Planet's mass} = \text{Sun's mass} × \left( \dfrac{r}{R} \right)^3 × \left( \dfrac{T}{t} \right)^2$, \hfill\phantom{whence} \\
+where $T$ and $R$ are the \textit{planet's} period and distance from the sun. Since
+$R$ and $r$ can both be determined in astronomical units without any
+necessity for knowing the length of that unit in miles, the \textit{masses of
+the planets in terms of the sun's mass} are independent of any knowledge
+of the solar parallax. But to compare them with the \textit{earth}, we
+must know this parallax, since the moon's distance from the earth,
+which enters into the equations, is found by observation in \textit{miles} or
+in \textit{radii of the earth}, and not in astronomical units.
+
+In order to make use of the satellites for this purpose we must
+determine by micrometrical observations their distances from the
+planets and their periods.
+
+\nbarticle{539.} \nbparatext{Mass of a Planet which has no Satellite.}---When a planet
+has not a satellite, the determination of its mass is a very difficult and
+troublesome problem, and can be solved only by finding some perturbation
+produced by the planet, and then ascertaining, by a sort of
+``trial and error'' method, the mass which would produce that perturbation.
+Venus disturbs the earth and Mercury, and from these perturbations
+her mass is ascertained. Mercury disturbs Venus, and
+also one or two comets which come near him, and in this way we get
+a rather rough determination of his mass.
+
+\nbarticle{540.} \nbparatext{Density.}---The density of a body as compared with the
+earth is determined simply by dividing its mass by its volume; \textit{i.e.},
+\[
+\text{Density} = \frac{m}{\left( \dfrac{s}{\rho} \right)^3}.
+\]
+
+\begin{fineprint}
+For example, Jupiter's \textit{diameter} is about eleven times that of the earth
+(\textit{i.e.} $\left(\dfrac{s}{\rho} \right) = 11$), so that his \textit{volume} is $11^3$, or 1331 times the earth's. His \textit{mass},
+derived from satellite observations, is about 316 times the earth's. The
+%% -----File: 331.png---Folio 320-------
+\textit{density}, therefore, equals $\frac{316}{1331}$, or about 0.24, of the earth's density, or about
+$1 \frac{1}{3}$ times that of water, the earth's density being 5.58 (\artref{Art.}{171}).
+\end{fineprint}
+
+\sloppy
+\nbarticle{541.} \nbparatext{The Surface Gravity.}---The force of gravity on a planet's
+surface as compared with that on the surface of the earth is important
+in giving us an idea of its physical condition. If $r$ is the radius of
+the planet in terms of the earth's radius, then
+\[
+\text{Surface gravity, or }G,
+= \frac{m}{\left( \frac{s}{\rho} \right)^2},
+= \frac{m}{\left( \frac{s}{\rho} \right)^3}
+  ×   \left( \frac{s}{\rho} \right).
+\]
+\textit{i.e.}, it equals the planet's \textit{density}, multiplied by its \textit{diameter} expressed
+in terms of the earth's diameter.
+
+\fussy
+\begin{fineprint}
+For Jupiter, therefore, $G = \dfrac{316}{11^2} = 11 × \text{density} = 11 × 0.24 =2.64$ nearly.
+That is, a body at Jupiter would weigh 2.6 as much as one at the earth's
+surface.
+\end{fineprint}
+
+\sloppy
+\nbarticle{542.} \nbparatext{The Planet's Oblateness.}---The ``oblateness'' or ``polar-compression''
+is the \textit{difference} between the equatorial and polar
+diameters divided by the equatorial diameter. It is, of course,
+determined, when it is possible to determine it at all, simply by
+micrometric measurements of the difference between the greatest and
+least diameters. The quantity is always very small and the observations
+delicate.
+
+\fussy
+\nbarticle{543.} \nbparatext{The Time of Rotation}, when it can be determined, is found
+by observing the passage of some spot visible in the telescope across
+the central line of the planet's disc. In reducing the observations to
+find the interval between such transits, account has to be taken of the
+continual change in the direction of the line which joins the planet
+and the earth, and also of the variations in the distance, which will
+alter the time taken by light in coming to the earth from the body.
+
+\nbarticle{544.} \nbparatext{The Inclination of the Axis} is deduced from the same observations
+which are used in obtaining the rotation period. It is necessary
+to determine with the micrometer the paths described by different
+spots as they move across the planet's disc. It is possible to
+ascertain it with accuracy for only a very few of the planets: Mars,
+Jupiter, and Saturn are the only ones that furnish the needed data.
+
+\nbarticle{545.} \nbparatext{The Surface Peculiarities and Topography} of the surface
+are studied by the telescope. The observer makes drawings of any
+%% -----File: 332.png---Folio 321-------
+markings which he may see, and by their comparison is at last able
+to discriminate between what is temporary and what is permanent on
+the planet. Mars alone, thus far, permits us to make a map of its
+surface.
+
+\nbarticle{546.} \nbparatext{Spectroscopic Peculiarities and Albedo.}---The characteristics
+of the planet's atmosphere can be to some extent studied by
+means of the spectroscope, which in some few cases shows the presence
+of water-vapor and other absorbing media, by dark bands in
+the planet's spectrum. The \textit{``albedo,''} or reflecting power of a
+planet's surface is determined by photometric observations, comparing
+it with a real or artificial star, or with some other planet.
+
+\nbarticle{547.} \nbparatext{The Satellite System of a Planet.}---The principal data to be
+ascertained are the \textit{distances} and \textit{periods} of the satellites, and the
+observations are made by measuring the \textit{apparent} distances and
+directions of the satellites from the centre of the planet with the wire
+micrometer (\artref{Art.}{73}). Observations made at the times when the
+satellite is near its elongation are especially valuable in determining
+the distance.
+
+\begin{fineprint}
+If the planet and earth were at rest, the satellite's path would appear to
+be an ellipse, unaltered in dimensions during the whole series of observations;
+but since the earth and planet are both moving, it becomes a complicated
+problem to determine the satellite's true orbit from the \textit{ensemble} of
+observations.
+
+\nbarticle{548.} With the exception of the moon and the outer satellite of Saturn,
+all the satellites of the planetary system move almost exactly in the plane of
+the equator of the primary; and all but the moon and the seventh satellite
+of Saturn (Hyperion) move in orbits almost perfectly circular. Tisserand
+has recently shown that the equatorial protuberance of a planet compels any
+satellite which is not very remote from its primary to move nearly in the
+equatorial plane, but the almost perfect circularity of the orbits is not yet
+explained. When there are a number of satellites in a system, interesting
+problems arise in connection with their mutual disturbances; and in a few
+cases it becomes possible to determine a satellite's mass as compared with
+that of its primary. In several instances satellites show peculiar variations
+in their brightness, which are supposed to indicate that they make an axial
+rotation in the time of one revolution around the primary, in the same way
+as our moon does.
+\end{fineprint}
+
+\sloppy
+\nbarticle{549.} \nbparatext{Humboldt's Classification of the Planets.}---Humboldt has
+divided the planets into two groups: the \textit{terrestrial} planets, so called,
+%% -----File: 333.png---Folio 322-------
+and the \textit{major} planets. The terrestrial planets are Mercury, Venus,
+the earth, and Mars. They are bodies of the same order of magnitude,
+ranging from 3000 to 8000 miles in diameter, not very different
+in density (though Mercury is much denser than either of
+the others), and are probably roughly alike in physical constitution,
+and covered with water and air. But we hasten to say that
+the differences in the amount of heat and light which they receive
+from the sun, and in the force of gravity upon their surfaces, and
+probably in the density of their atmospheres, are such as to bar any
+positive conclusions as to their being the abode of life resembling the
+forms of life with which we are acquainted on the earth.
+
+\includegraphicsmid{illo170}{\textsc{Fig.~170.}---Relative Sizes of the Planets.}
+
+\nbarticle{550.} The four major planets, Jupiter, Saturn, Uranus, and Neptune,
+are much larger bodies (ranging in diameter between 30,000
+and 90,000 miles), are much less dense, and so far as we can make
+out, present to us only a surface of cloud, and may not have
+anything solid about them. There are some reasons for suspecting
+that they are at a high temperature; in fact, that Jupiter is a sort of
+\textit{semi-sun}, but this is by no means yet certain.
+
+\fussy
+As for the multitudinous asteroids, the probability is that they
+represent a single planet of the terrestrial group which, as has
+been intimated, failed for some reason in its evolution, or else has
+%% -----File: 334.png---Folio 323-------
+been broken to pieces. All of them united would not make a planet
+one-half the mass of the earth.
+
+Fig.~170 shows the relative sizes of the different planets.
+
+In what follows, all the numerical data, so far as they depend on
+the solar parallax, are determined on the assumption that that parallax
+is $8''.80$, and that the sun's mean distance is 92,897000 miles.
+
+\section*{MERCURY.}
+
+\nbarticle{551.} There is no record of the discovery of the planet. It has
+been known from remote antiquity; and we have recorded \textit{observations}
+running back to \textsc{b.c.}~264.
+
+\begin{fineprint}
+For a time the ancient astronomers seem to have failed to recognize it
+as the same body on the eastern and western sides of the sun, so that the
+Greeks had for a time two names for it,---Apollo when it was morning star,
+and Mercury when it was evening star. According to Arago, the Egyptians
+called it Set and Horus, and the Hindoos also gave it two names.
+\end{fineprint}
+
+It is so near the sun that it is comparatively seldom seen with the
+naked eye; but when near its greatest elongation it is easily enough
+visible as a brilliant star of the first magnitude low down in the twilight,
+perhaps not quite so bright as Sirius, but certainly brighter than
+Arcturus. It is usually visible for about a fortnight at each elongation,
+and is best seen in the evening at such eastern elongations as
+occur in March and April. In Northern Europe it is much more
+difficult to observe than in lower latitudes, and Copernicus is said
+never to have seen it. Tycho, however, obtained a considerable
+number of observations.
+
+\nbarticle{552.} It is exceptional in the solar system in a great variety
+of ways. It is the \textit{nearest} planet to the sun, \textit{receives the most light
+and heat}, is \textit{the swiftest} in its movement, and (excepting some of the
+asteroids) \textit{has the most eccentric orbit}, with the \textit{greatest inclination}
+to the ecliptic. It is also the \textit{smallest} in diameter (again excepting
+asteroids), has the \textit{least mass}, and the \textit{greatest density} of all the
+planets.
+
+\nbarticle{553.} \nbparatext{Distance, Light, and Heat.}---Its \textit{mean distance} from the
+sun is 36,000000 miles, but the eccentricity of its orbit is so great
+(0.205), that the sun is seven and one-half millions of miles out of
+its centre, and the actual distance of the planet from the sun ranges
+all the way from 28,500000 to 43,500000, while its velocity in its orbit
+%% -----File: 335.png---Folio 324-------
+varies from thirty-five miles a second at perihelion to only twenty-three
+at aphelion. On the average it receives 6.7 times as much light and
+heat as the earth; but the heat received at perihelion is to that at
+aphelion in the ratio of 9 to 4. For this reason there must be two
+seasons in its year due to the changing distance, even if the equator
+of the planet is parallel to the plane of its orbit, which would preclude
+seasons like our own. If the planet's equator is inclined at an angle
+like the earth's, then the seasons must be very complicated.
+
+\nbarticle{554.} \nbparatext{Period.}---The \textit{sidereal period} is very nearly 88~days, and
+the \textit{synodic period}, or the time from conjunction to conjunction again,
+is about 116~days. The greatest elongation ranges from $18°$ to $28°$,
+and occurs about twenty-two days before and after the inferior
+conjunction, or about thirty-six days before and after the superior
+conjunction. The planet's arc of retrogression is about $12°$ (considerably
+variable), and the stationary point is very near the greatest
+elongation.
+
+\nbarticle{555.} \nbparatext{Inclination.}---The inclination of the orbit to the ecliptic is
+about $7°$, but the greatest geocentric latitude (that is, the planet's
+greatest distance from the ecliptic as seen from the earth) is never
+quite so great.
+
+\nbarticle{556.} \nbparatext{Diameter, Surface, and Volume.}---\textit{The apparent diameter}
+ranges from $5''$ to about $13''$, according to its distance from us; the
+least distance from the earth being about 57,000000 miles ($93 - 36$),
+while the greatest is about 129,000000 ($93 + 36$). The \textit{real
+diameter} is very near 3000 miles, not differing from that more than
+fifty miles either way. It is not easy to measure, and the ``probable
+error'' is perhaps rather larger than would have been expected.
+With this diameter, its \textit{surface} is $\frac{1}{7}$ of the earth's, and its
+\textit{volume} $\frac{1}{18.5}$.
+
+\nbarticle{557.} \nbparatext{Mass, Density, and Surface Gravity.}---Its mass is very difficult
+to determine, since it has no satellite, and the values obtained by
+La~Place, Encke, Leverrier, and others, range all the way from $\frac{1}{9}$ of
+the earth's mass to $\frac{1}{15}$.
+
+\begin{fineprint}
+The most recent, and perhaps the most reliable, determination, although
+the largest of all, is that of Backlund, by means of the perturbations of
+Encke's comet. He makes its mass about $\frac{1}{8}$ that of the earth. This gives
+%% -----File: 336.png---Folio 325-------
+us a mean density about $2 \frac{1}{4}$ times that of the earth, or nearly $12\frac{1}{2}$ times that
+of water, a little less than that of the metal \textit{mercury}. So far as known, the
+earth stands next to it in density, but with a wide interval, so that Mercury
+is altogether exceptional in this respect, as has been said before. The result,
+however, is liable to a considerable error, on account of the uncertainty of
+the planet's mass, Leverrier's estimate of the mass and density being only
+about one-half as great as Backlund's. The \textit{superficial gravity} is about $\frac{5}{6}$ that
+at the earth's surface.
+\end{fineprint}
+
+\nbarticle{558.} Its \textit{Albedo}, or reflecting power, as determined by Zöllner is
+very low---only 0.13, somewhat inferior to that of the moon.
+
+\begin{fineprint}
+In 1878 Mr.\ Nasmyth observed the planet in the same field of view with
+Venus; and although Mercury was then not much more than half as far
+from the sun as Venus, and therefore four times as brightly illuminated, it
+appeared to be less luminous in the telescope. ``Venus was like silver,
+Mercury like zinc or lead.''
+\end{fineprint}
+
+In the proportion of light given out at its different phases, it
+behaves like the moon, flashing out strongly near the full, as if it had
+a surface of the same rough structure as that of our satellite.
+
+\sloppy
+\nbarticle{559.} \nbparatext{Telescopic Appearance and Phases.}---Seen by the telescope,
+the planet looks like a little moon, showing phases precisely similar
+to those of our satellite. At inferior conjunction the dark side is
+towards us; at superior conjunction the illuminated surface. At
+the greatest elongation it appears like a half-moon. Between superior
+conjunction and greatest elongation it is gibbous, while between
+inferior conjunctions and the elongations it shows the crescent phase.
+Fig.~171 illustrates the phases of both Mercury and Venus.
+
+\includegraphicsmid{illo171}{\textsc{Fig.~171.}---Phases of Mercury and Venus.}
+
+\fussy
+\begin{fineprint}
+For the most part, Mercury can be observed only by daylight; but when
+proper precautions are taken to screen the object-glass of the telescope from
+%% -----File: 337.png---Folio 326-------
+the direct sunlight, the observation is not difficult. The surface presents
+very little of interest. There are no markings well enough defined to give
+us any trustworthy determination of the planet's rotation, or of its geography.
+Occasionally the terminator (\textit{i.e.}, the line which separates the illuminated
+and unilluminated portions of the planet) appears to be a little
+irregular, instead of a true oval, as it should be; and at times when the phase
+is crescent-form, the points, or cusps, are a little blunted; from this some astronomers
+have inferred the existence of high mountains upon the planet.
+Schröter, a German astronomer, the contemporary of the elder Herschel,
+deduced from his observations a rotation period for the planet of 24~h.\ 5~m.;
+but later observers, with instruments certainly far more perfect, have
+not been able to verify his results, and they are now considered as of little
+weight.
+\end{fineprint}
+
+\nbarticle{560.} \nbparatext{Atmosphere.}---The evidence upon this subject is not conclusive.
+Its atmosphere, if it has one, must, however, be much less
+dense than that of Venus. No ring of light is seen surrounding the
+disc of the planet when it enters the limb of the sun at the time of
+a transit, while in the case of Venus such a ring, due to the atmospheric
+refraction, is very conspicuous. On the other hand, Huggins
+and Vogel, who have examined the spectrum of the planet, report
+that certain lines in the spectrum, due to the presence of water-vapor,
+were decidedly stronger than in the spectrum of the air (illuminated
+by sunshine), which formed the background for the planet,
+making it probable that it has an atmosphere containing water-vapor
+like the atmosphere of the earth, but probably less extensive and
+dense.
+
+\nbarticle{561.} \nbparatext{Transits.}---Usually at the time of inferior conjunction the
+planet passes north or south of the sun, the inclination of its orbit
+being $7°$; but if the conjunction occurs when the planet is very near
+its node, it will cross the disc of the sun and be visible upon it us a
+small black spot---not, however, large enough to be seen without a
+telescope, as Venus can under similar circumstances.
+
+\begin{fineprint}
+At this time we have the best opportunity for measuring the diameter of
+the planet; but unless special precautions are taken, the measured diameter
+under these circumstances is likely to be \textit{too small}, on account of the irradiation
+of the surrounding background, which encroaches upon the planet's disc.
+\end{fineprint}
+
+Since the planet's nodes are in longitudes $227°$ and $47°$, and are
+passed by the earth on May~7 and November~9, the transits can
+occur only near these days. If the orbit of the planet were strictly
+circular, the ``transit limit'' (corresponding to an ecliptic limit)
+%% -----File: 338.png---Folio 327-------
+would be $2° 10'$; but at the May transits the planet is near its
+aphelion and much nearer the earth than ordinarily, so that the limit
+is diminished, while the November limit is correspondingly increased.
+The May transits are in fact only about half as numerous as the
+November transits.
+
+\nbarticle{562.} \nbparatext{Interval between Transits.}---Twenty-two synodic periods of
+Mercury are pretty nearly equal to 7~years; 41 still more nearly
+equal 13~years; and 145 almost exactly equal 46 years. Hence,
+after a November transit, a second one is possible in 7~years, probable
+in 13~years, and practically certain in 46. For the May transits
+the repetition after 7~years is not possible, and it often fails in
+13~years.
+
+\begin{fineprint}
+The transits of the present century are the following:---
+
+\textit{May Transits.}---1832, May~5; 1845, May~8; 1878, May~6; 1891, May~9.
+
+\textit{November Transits.}---1802, Nov.~9; 1815, Nov.~11; 1822, Nov.~5; 1835,
+Nov.~7; 1848, Nov.~10; 1861, Nov.~12; 1868, Nov.~5; 1881, Nov.~7; 1894,
+Nov.~10.
+
+The first transit of Mercury ever observed was by Gassendi, Nov.~7, 1631.
+
+The transits of Mercury are of no particular astronomical importance,
+except as giving accurate determinations of the planet's place, by means of
+which its orbit can be determined. Newcomb has also recently made an
+investigation of all the recorded transits, for the purpose of testing the
+uniformity of the earth's rotation. They indicate no perceptible change in
+the length of the day.
+\end{fineprint}
+
+
+\section*{VENUS.}
+
+\nbarticle{563.} The next planet in order from the sun is Venus, the brightest
+and most conspicuous of all, the earth's twin sister in magnitude,
+density, and general constitution, if not also in age, as to which we
+have no knowledge. Like Mercury, it had two names among the
+Greeks,---\textit{Phosphorus} as morning star, and \textit{Hesperus} as evening star.
+It is so brilliant that it is easily seen by the naked eye in the daytime
+for several weeks when near its greatest elongation; sometimes it is
+bright enough to catch the eye at once, but usually it is seen by daylight
+only when one knows precisely where to look for it.
+
+\begin{fineprint}
+(It is not, however, the ''\textit{Star of Bethlehem},'' though it has of late been
+frequently taken for it.)
+\end{fineprint}
+
+\sloppy
+\nbarticle{564.} \nbparatext{Distance, Period, and Inclination of Orbit.}---Its mean distance
+from the sun is 67,200000 miles. The eccentricity of the
+%% -----File: 339.png---Folio 328-------
+orbit is the smallest in the planetary system (only 0.007), so that the
+greatest and least distances of the planet from the sun differ from
+the mean only 470,000 miles each way. Its orbital velocity is
+twenty-two miles per second.
+
+\fussy
+Its \textit{sidereal period} is 225 days, or seven and one-half months, and
+its \textit{synodic period} 584~days---a year and four months. From superior
+conjunction to elongation on either side is 220 days, while from
+inferior conjunction to elongation is only 71 or 72~days. The arc of
+retrogression is $16°$.
+
+The \textit{inclination of its orbit} is only $3 \frac{1}{2}°$.
+
+\nbarticle{565.} \nbparatext{Diameter, Surface, and Volume.}---The apparent diameter
+ranges from $67''$ at the time of inferior conjunction to only $11''$ at the
+superior. This great difference depends, of course, upon the enormous
+change in the distance of the planet from the earth. At
+inferior conjunction the planet is only 26,000000 miles from us ($93 - 67$).
+No other body ever comes so near the earth except the
+moon, and occasionally a comet. Its greatest distance at superior
+conjunction is 160,000000 miles ($93 + 67$), so that the ratio between
+the greatest distance and the least is more than 6 to 1.
+
+The \textit{real diameter} of the planet is 7700 ($\pm \: 30$) miles. Its \textit{surface,}
+as compared with that of the earth, is ninety-five per cent;
+its \textit{volume} ninety-two per cent.
+
+\nbarticle{566.} \nbparatext{Mass, Density, and Gravity.}---By means of the perturbations
+she produces upon the earth, the \textit{mass} of Venus is found to be
+seventy-eight per cent of the earth's; hence her \textit{density} is eighty-six
+per cent, and her \textit{superficial} gravity eighty-three per cent of the
+earth's.
+
+\nbarticle{567.} \nbparatext{Phases.}---The telescopic appearance of the planet is striking
+on account of her great brilliance. When about midway between
+greatest elongation and inferior conjunction she has an apparent
+diameter of $40''$, so that, with a magnifying power of only forty-five,
+she looks exactly like the moon four days old, and of precisely the
+same apparent size.
+
+\begin{fineprint}
+Very few persons, however, would think so on their first view through
+the telescope, for a novice always underrates the apparent size of a telescopic
+object: he instinctively adjusts his focus as if looking at a picture
+only a few inches away, instead of projecting the object visually into
+the sky.
+\end{fineprint}
+%% -----File: 340.png---Folio 329-------
+
+\includegraphicsmid{illo172}{\textsc{Fig.~172.}---Telescopic Appearances of Venus.}
+
+According to the theory of Ptolemy, Venus could never show us
+more than half her illuminated surface, since according to his hypothesis
+she was \textit{always between us and the supposed orbit of the sun}
+(\artref{Art.}{500}). Accordingly, when in 1610 Galileo discovered that she
+exhibited the gibbous phase as well as the crescent, it was a strong
+argument for Copernicus. Galileo announced his discovery in a curious
+way, by publishing the anagram,---
+\begin{verse}
+``Haec immatura a me jam frustra leguntur; o.\ y.''
+\end{verse}
+
+Some months later he furnished the translation,---
+\begin{verse}
+``Cynthiæ figuras æmulatur mater amorum,''
+\end{verse}
+which is formed by merely transposing the letters of the anagram.
+His object was to prevent any one from claiming to have anticipated
+him in this discovery, as had been done with respect to his discovery
+of the sun spots.
+
+\begin{fineprint}
+Fig.~172 represents the disc of the planet as seen at four points in its
+orbit. 1, 3, and 5 are taken at superior conjunction, greatest elongation,
+and near inferior conjunction respectively, while 2 and 4 are at intermediate
+points.
+\end{fineprint}
+
+\nbarticle{568.} \nbparatext{Maximum Brightness.}---The planet attains its maximum
+brilliance thirty-six days before and after inferior conjunction, at a
+%% -----File: 341.png---Folio 330-------
+distance of about $38°$ or $39°$ from the sun, when its phase is like that
+of the moon about five days old. It then casts a strong shadow,
+and, as has already been said, is easily visible by day with the
+naked eye.
+
+\sloppy
+\begin{fineprint}
+\nbarticle{569.} \nbparatext{Surface Markings.}---These are not at all conspicuous. Near
+the limb of the planet, which is always much brighter than the central parts
+(as is also the case with Mercury and Mars), they can never be well seen,
+although sometimes when Venus was in the crescent phase, intensely bright
+spots have been reported near the cusps, as at $a$ and $b$ in No.~4, \figref{illo172}{Fig.~172}.
+These may perhaps be \textit{ice-caps} like those which are seen on Mars. Near the
+``terminator,'' which is less brilliant and less sharply defined than the limb,
+irregular darkish shadings are sometimes seen, such as are indicated by the
+dotted lines in the \figref{illo172}{figures}, but without any distinct outline. They may be
+continents and oceans dimly visible, or they may be mere atmospheric
+objects; observations do not yet decide.
+\end{fineprint}
+
+\fussy
+\nbarticle{570.} \nbparatext{Rotation of the Planet.}---From the observation of such
+markings Schröter deduced the rotation period of $23^\text{h}\: 21^\text{m}$, which has
+since been partially confirmed by one or two other observers, and
+may be approximately correct. De~Vico concluded that the planet's
+equator makes an angle of about $54°$ with the plane of its orbit.
+This, however, is extremely doubtful. If the bright spots which
+have been referred to in \artref{Art.}{569} are really polar ice-caps, the
+equator cannot be very greatly inclined. (See Clerke's ``History of
+Astronomy,'' p.~299.)
+
+\begin{fineprint}
+No sensible difference has been ascertained between the different diameters
+of the planet. If it were really as much flattened at the poles as the
+earth is, there should be a difference of $0''.2$ between the polar and equatorial
+diameters as measured at the time of the planet's transit.
+\end{fineprint}
+
+\sloppy
+\nbarticle{571.} \nbparatext{Mountains.}---From certain irregularities occasionally observed
+upon the terminator, and especially from the peculiar blunted
+form of one of the cusps of the crescent, various observers have concluded
+that there are numerous high mountains upon the surface of
+the planet. Schröter assigned to some of these near the southern
+pole the extravagant altitude of twenty-five or thirty miles, but the
+evidence is entirely insufficient to warrant any confidence in the conclusion.
+
+\fussy
+\nbarticle{572.} \nbparatext{Albedo.}---According to Zöllner the \textit{Albedo} of the planet is
+$0.50$, which is about three times that of the moon, and almost four
+%% -----File: 342.png---Folio 331-------
+times that of Mercury. It is, however, exceeded by the reflecting
+power of the surfaces of Jupiter and Uranus, while that of Saturn
+appears to be about the same. This high reflecting power probably
+indicates that the surface is mostly covered with cloud, as few rocks
+or soils could match it in brightness.
+
+\nbarticle{573.} \nbparatext{Evidences of Atmosphere.}---When the planet is near the
+sun, the horns of the crescent extend notably beyond the diameter,
+and when very near the sun, a thin line of light has been seen by
+several observers, especially Professor Lyman of New Haven, to
+complete the whole circumference. This is due to refraction of sunlight
+by the planet's atmosphere, a phenomenon still better seen as
+the planet is entering upon the sun's disc at a transit, when the
+black disc is surrounded by a beautiful ring of light. From the observations
+of the transit of 1874, Watson concluded that the planet's
+atmosphere must have a depth of about fifty-five miles, that of the
+earth being usually reckoned at forty miles. Other observers in
+different ways have come to substantially the same results. Its
+atmosphere is probably from one and a half to two times as extensive
+and dense as our own, and the spectroscope shows evidence of the
+presence of water-vapor in it.
+
+\textit{Lights on Dark Portion.}---Many observers have also reported
+faint lights as visible at times on the dark portion of the planet's
+disc. These cannot be accounted for by reflection, but must originate
+on the planet's surface; they recall the Aurora Borealis and
+other electrical manifestations on the earth.
+
+\nbarticle{574.} \nbparatext{Satellites.}---No satellite is known, although in the last
+century a number of observers at various times thought they had
+found one.
+
+\begin{fineprint}
+In most cases they observed small stars near the planet, which we can
+now identify by computing the place occupied by the planet at the date of
+observation. It is not, however, \textit{impossible} that the planet may have some
+very minute and near attendants like those of Mars, which may yet be
+brought to light by means of the great telescopes of the future, or by photography.
+Of course the extreme brilliance of the planet, and the fact that
+the necessary observations can be made only in strong twilight, render the
+discovery of such objects, if they exist, very difficult.
+\end{fineprint}
+
+\nbarticle{575.} \nbparatext{Transits.}---Occasionally Venus passes between the earth
+and the sun at inferior conjunction, giving us a so-called ``\textit{transit}.''
+%% -----File: 343.png---Folio 332-------
+She is then visible (even to the naked eye) as a black spot on the
+disc, crossing it from east to west.
+
+As the inclination of the planet's orbit is but $3\frac{1}{2}°$, the ``\textit{transit
+limit}'' is small (only $1\frac{3}{4}°$), and the transits are therefore very rare
+phenomena. The sun passes the nodes of the orbit on June~5 and
+December~7, so that all transits must occur on or near those dates.
+When Venus crosses the sun's disc \textit{centrally}, the duration of the
+transit is about \textit{eight hours}. Taking the mean diameter of the sun
+as $32'$, or $\frac{1}{375}$ of a circumference, and the planet's synodic period
+as 584 days, the geocentric duration of a central transit should be
+$\frac{277}{723} × \frac{1}{675} × 584^\text{d}$, which equals 0.332 days, or $7^\text{h}\: 58^\text{m}$.
+
+When the transit track lies near the edge of the disc, the duration
+is of course correspondingly shortened.
+
+\nbarticle{576.} \nbparatext{Recurrence of Transits.}---Five synodic, or thirteen sidereal,
+revolutions of Venus are very nearly equal to eight years, the difference
+being only a little more than one day; and still more nearly, in
+fact almost exactly, 243 years are equal to 152 synodic, or 395 sidereal,
+revolutions. If, then, we have a transit at any time, we \textit{may
+have} another at the same node \textit{eight} years earlier or later. Sixteen
+years before or after it would be impossible, and no other transit
+can occur at the same node until after the lapse of \textit{two hundred and
+thirty-five} or \textit{two hundred and forty-three years}.
+
+If the planet crosses the sun nearly centrally, the transit will not
+be accompanied by another at an eight-year interval, but the planet
+will pass either north or south of the sun's disc, at the conjunctions
+next preceding and following. If, however, as is now the case, the
+transit path is near the northern or southern edge of the sun, then
+there will be a companion transit across the opposite edge of the
+disc eight years before or after. Thus, if we have a pair of \textit{June}
+transits, separated by an eight-year interval, it will be followed by
+another pair at the same node in 243 years; and a pair of \textit{December}
+transits will come in about \DPtypo{halfway}{half-way} between the two pairs of June transits.
+After a thousand years or so from the present time the transits
+will cease to come in pairs, as they have been doing for 2000 years.
+
+\nbarticle{577.} Transits of Venus have occurred or will occur on the following
+dates:---
+\begin{align*}
+\left.\begin{gathered}
+\text{December 7, 1631.}\\
+\text{December 4, 1639.}\\
+\end{gathered}\right\}
+&&&
+\left.\begin{gathered}
+\text{June 5, 1761.}\\
+\text{June 3, 1769.}\\
+\end{gathered}\right\}
+\\
+\left.\begin{gathered}
+\text{December 9, 1874.}\\
+\text{December 6, 1882.}\\
+\end{gathered}\right\}
+&&&
+\left.\begin{gathered}
+\text{June 8, 2004.}\\
+\text{June 6, 2012.}\\
+\end{gathered}\right\}
+\end{align*}
+%% -----File: 344.png---Folio 333-------
+
+The special interest in these transits consists in the use that has been
+made of them for the purpose of finding the sun's
+parallax, a subject which will be discussed later on
+(\chapref{CHAPTERXVI}{Chap.~XVI.}).
+
+\includegraphicsouter{illo173}{\textsc{Fig.~173.}\\
+Transit of Venus Tracks.}
+
+The first observed transit, in 1639, was seen by
+only two persons,---Horrox and Crabtree, in England.
+The four which have occurred since then
+have been extensively observed in all parts of the
+world where they were visible, by scientific expeditions
+sent out for the purpose by different nations. The
+transits of 1769 and 1882 were visible in the United
+States. \figref{illo173}{Fig.~173} shows the track of Venus across
+the sun's disc at the two transits of 1874 and 1882.
+
+\section*{MARS.}
+
+This planet is also prehistoric as to its discovery. It is so conspicuous
+in color and brightness, and in the extent and apparent
+capriciousness of its movement among the stars, that it could not
+have escaped the notice of the very earliest observers.
+
+\nbarticle{578.} \nbparatext{Orbit.}---Its \textit{mean distance} from the sun is 141,500000
+miles, and the \textit{eccentricity} of the orbit is so considerable (0.093) that
+this distance varies about 13,000000 miles. The \textit{light and heat}
+which it receives from the sun is somewhat less than half of that
+received by the earth. The \textit{inclination} of its orbit is small, $1° 51'$.
+The planet's \textit{sidereal} period is 687 days, or $1^\text{y}\: {10\frac{1}{2}}^\text{mo}$, which gives
+it an average orbital velocity of fifteen miles per second. Its
+\textit{synodic} period is 780~days, or $2^\text{y}\: {1\frac{2}{3}}^\text{mo}$. It is the longest in the solar
+system, that of Venus (584~days) coming next. Of the 780~days, it
+moves eastward during 710, and retrogrades during 70, through an
+arc of 18°.
+
+\sloppy
+\nbarticle{579.} At opposition its \textit{average} distance from the earth is 48,600000
+miles (141,500000 miles minus 92,900000 miles). When the
+opposition occurs near the planet's perihelion, this distance is reduced
+to 35,500000 miles; if near aphelion, it is increased to over
+61,000000. At superior conjunction the average distance from the
+earth is 234,400000 miles (141,500000 plus 92,900000).
+
+\fussy
+The apparent diameter and brilliancy of the planet, of course, vary
+enormously with these great changes of distance.
+
+\begin{fineprint}
+If we put $R$ for the planet's distance from the sun, and $\Delta$ for its distance
+from the earth, its brightness, neglecting the correction for \textit{phase}, should
+%% -----File: 345.png---Folio 334-------
+equal $\dfrac{1}{R^2 \Delta^2}$. We find from this that, taking the brightness at conjunction
+as unity (at which time the planet is about as bright as the pole-star), it is
+more than twenty-three times brighter at the \textit{average} opposition, and fifty-three
+times brighter if the opposition occurs at the planet's perihelion. At
+an unfavorable opposition Mars, as has been said, may be 61,000000 miles
+distant, and its brightness then is only about twelve times as great as at
+conjunction,---the difference between favorable and unfavorable oppositions
+being more than \textit{four to one}.
+
+These favorable oppositions occur always in the latter part of August (at
+which date the sun passes the line of apsides of the planet), and at intervals
+of fifteen or seventeen years. The last was in 1877, and the next will be in
+1892. A reference to \figref{illo159}{Fig.~159} will show how great is the difference between
+the planet's opposition distance from the earth under varying circumstances.
+\end{fineprint}
+
+\nbarticle{580.} \nbparatext{Diameter, Surface, and Volume.}---The apparent diameter of
+the planet ranges from $3''.6$ at conjunction, to $24''.5$ at a favorable
+opposition. Its \textit{real} diameter is very closely 4200 miles,---the error
+may be twenty miles one way or the other. This makes its surface
+0.28, and its volume 0.147 (equal to $\frac{1}{7}$) of the earth's.
+
+\nbarticle{581.} \nbparatext{Mass, Density, and Gravity.}---Observations upon its satellites
+give its mass as $\frac{1}{9.4}$ compared with that of the earth. This
+makes its \textit{density} 0.73 and \textit{superficial gravity} 0.38; that is, a body
+which weighs 100 pounds on the earth would have a weight of 38
+pounds on the surface of Mars.
+
+\nbarticle{582.} \nbparatext{Phases.}---Since the orbit of the planet is outside that of the
+earth, it never comes between us and the sun,
+\textit{and can never show the crescent phase}; but at
+quadrature enough of the unilluminated portion
+is turned towards the earth to make the disc
+clearly \textit{gibbous} like the moon three or four days
+from full. \figref{illo174}{Fig.~174} shows its maximum phase
+accurately drawn to scale.
+
+\includegraphicsouter{illo174}{\textsc{Fig.~174.}\\
+Greatest Phase of Mars.}
+
+\nbarticle{583.} \nbparatext{The ``Albedo'' of the Planet.}---According
+to Zöllner's observations this is 0.26, which
+is considerably higher than that of the moon
+($\frac{1}{6}$), and just double that of Mercury.
+
+\nbarticle{584.} \nbparatext{Rotation.}---The planet's time of rotation is $24^\text{h}\: 37^\text{m}\: 22^\text{s}.67$.
+This very exact determination has been made by Kaiser and Bakhuyzen,
+%% -----File: 346.png---Folio 335-------
+by comparing drawings of the planet which were made more
+than 200 years ago by Huyghens with others made recently.
+
+\sloppy
+It is obvious that observations made a few days or weeks apart will
+give the time or rotation with only approximate accuracy. Knowing
+it thus approximately, we can then determine, without fear of error,
+the \textit{whole number of rotations} between two observations separated by
+a much longer interval of time. This will give a second and closer
+approximation to the true period; and with this we can carry our
+reckoning over centuries, and thus finally determine the period within
+a very minute fraction of a second. The number given is not uncertain
+by more than $\frac{1}{50}$ of a second, if so much.
+
+\nbarticle{585.} \nbparatext{The Inclination of the Planet's Equator to the Plane of its
+Orbit.}---This is very nearly $24°\: 50'$ ($26°\: 21'$ to the \textit{ecliptic}), not very
+different from the inclination of the earth's equator; so far, therefore,
+as depends upon that circumstance, its seasons should be substantially
+the same as our own.
+
+\fussy
+\nbarticle{586.} \nbparatext{Polar Compression.}---There is a slight but sensible flattening
+of the planet at the poles. The earlier observers found for the
+polar compression values as large as $\frac{1}{40}$, and even $\frac{1}{10}$. These large
+values, however, are inconsistent with the existence of any extensive
+surface of liquid upon the planet, and more recent observations
+of the writer show the polar compression to be about $\frac{1}{220}$; which is
+almost exactly what would be expected from a planet constituted as
+we suppose Mars to be.
+
+\includegraphicsouter{illo175}{\textsc{Fig.~175.}---Telescopic Views of Mars.}
+
+\sloppy
+\nbarticle{587.} \nbparatext{Telescopic Appearance and Surface-Markings.}---The fact
+that we are able to determine the time of rotation so accurately of
+course implies the existence of identifiable markings upon the surface.
+Viewed through
+a powerful telescope,
+the planet's disc, as a
+whole, is ruddy,---in
+fact, almost rose color,---very
+bright around
+the limb, but not at the
+``terminator,'' if there is
+any considerable phase.
+The central portions of
+the disc present greenish and purplish patches of shade, for the
+most part not sharply defined, though some of the markings have
+%% -----File: 347.png---Folio 336-------
+outlines reasonably distinct. On watching the planet for only a few
+hours even, the markings pass on across the disc, and are replaced
+by others. Some of them are permanent, and recur at regular intervals
+with the same form and appearance, while others appear to
+be only clouds which for a time veil the surface below, and then
+clear away.
+
+\fussy
+By comparing drawings made when the same side is turned towards
+the earth, it is possible in a short time to ascertain what features
+really belong to the planet's geography.
+
+The polar ice-caps, brilliant white patches near the poles, form a
+marked feature in the planet's telescopic appearance. These are believed
+to be \textit{ice-caps} from the fact that the one which is near the pole
+that happens to be turned towards the sun continually diminishes in
+size, while the other increases, the process being reversed with the
+seasons of the planet.
+
+\nbarticle{588.} The principal peculiarity of the surface of Mars appears to
+be the way in which land and water are intermeshed. There seem to
+be few great oceans and continents, but there are narrow arms of the
+sea, like the Baltic and the Red Sea, penetrating and dividing the
+land masses. As to the nomenclature of the principal ``Areographic''
+features, Mr.~Proctor in his map has for the most part assigned the
+names of astronomers who have taken special interest in the study of
+%% -----File: 348.png---Folio 337-------
+the planet. Schiaparelli, on the other hand, has taken his names
+mostly from classical geography. \figref{illo176}{Fig.~176} is a reduced copy of his
+map of the planet, drawn on Mercator's Projection. The student
+will remember that in maps of this sort the polar regions are extended
+beyond their due proportion, and also that the parts near the
+poles can never be seen from the earth nearly as well as those near
+the equator, and hence that our knowledge of the details in that part
+of the planet is very limited. (See \hyperref[art:608*.]{note to \artref{Art.}{588}}, p.~\pageref{pg:347}.)
+
+\includegraphicsmid{illo176}{\textsc{Fig.~176.}---Schiaparelli's Map of Mars.}
+
+\nbarticle{589.} \nbparatext{Atmosphere.}---At one time it was supposed that the atmosphere
+of the planet is very dense, but more recent observations have
+shown that this cannot be the case. The probability is that its density
+is considerably less than that of our own atmosphere. Dr.~Huggins
+has found with the spectroscope unequivocal evidence of the presence
+of aqueous vapor. Although the planet receives from the sun less
+than half the amount of heat and light per unit of surface that the
+earth does, the climate appears, for some reason not yet discovered,
+to be much more mild than would be expected. So far as
+we can judge, the water on the planet is never frozen except very
+near the poles. The earth, as seen from Mars, would present much
+more extensive snow-caps.
+
+\nbarticle{590.} \nbparatext{Satellites.}---There are two satellites, which were discovered
+in August, 1877, by Professor Hall at Washington with the then
+new 26-inch telescope. They are exceedingly minute, and can be
+seen only with the most powerful instruments. The outer one,
+Deimos, is at a distance of 14,600 miles from the centre of the
+planet, and has a period of $30^{\text{h}} \:18^{\text{m}}$, while the inner one, Phobos,
+is at a distance of only 5800 miles, and its month is but $7^{\text{h}}\: 39^{\text{m}}$
+long, not one-third of the \textit{day} of Mars. Owing to this fact it rises in
+the \textit{west} every night for the ``\textit{Marticoli}'' (if there are any people
+there) and sets in the \textit{east}, after about $11^{\text{h}}$.
+
+Deimos does not do this; it rises in the east like other stars, but
+its orbital eastward motion among the stars is so nearly equal to its
+diurnal motion westward, that it is nearly 132 hours between two
+successive risings. This is more than four of its months, so that it
+undergoes all its changes of phase four times in the interval.
+
+Of course, both the satellites are frequently eclipsed,---the inner
+one at every revolution; and it also transits across the sun's disc at
+every new moon, as seen from some point or other on the planet's
+surface.
+%% -----File: 349.png---Folio 338-------
+
+Their orbits appear to be exactly circular, and they move exactly
+in the plane of the planet's equator; and they \textit{keep} so, maintained in
+their relation to the equator by the action of the ``equatorial bulge''
+upon the planet.
+
+\nbarticle{591.} As givers of moonlight they do not amount to much. Their
+diameters are too small to be measured with any micrometer; but
+from their apparent ``\textit{magnitude}'' (\textit{i.e.}, brightness), as seen from
+the earth, and assuming that their surfaces have the same reflective
+power as that of the planet, Professor Pickering has estimated the
+diameter of Phobos, which is the larger one, as about seven miles,
+and that of Deimos, as five or six. The light given by Phobos to
+the inhabitants of Mars would be about $\frac{1}{60}$ of our moonlight; that
+of Deimos about $\frac{1}{1200}$.
+
+The period of Phobos is by far the shortest period in the solar system.
+Next to it is that of Mimas, the inner satellite of Saturn,
+which, however, is nearly three times as long,---$22 \frac{1}{2}^\text{h}$. This rapidity
+of revolution raises important questions as to the theory of the
+development of the solar system, and requires modification of the
+views which had been held up to the time of their discovery. If
+the nebular hypothesis is true, a shortening of the satellite's period,
+or a lengthening of the planet's day, must have occurred since the
+satellite came into being, since that hypothesis will not account for the
+existence of a satellite having a period shorter than the diurnal rotation
+of its primary. \figref{illo177}{Fig.~177} is a diagram of the satellite orbits as
+%% -----File: 350.png---Folio 339-------
+they appeared from the earth in 1888. It is reduced from the
+American Nautical Almanac for that year.
+
+\includegraphicsmid{illo177}{\textsc{Fig.~177.}---Orbits of the Satellites of Mars.}
+
+\section*{THE ASTEROIDS, OR MINOR PLANETS.}
+
+\nbarticle{592.} These are a group of small planets circulating in the space
+between Mars and Jupiter. The name ``\textit{asteroid}'' was suggested
+by Sir William Herschel early in the century, when the first ones
+were discovered. The later term \textit{planetoid} is preferred by many.
+
+It was very early noticed that there is a break in the series of the
+distances of the planets from the sun. Kepler, indeed, at one time
+thought he had discovered the true law\footnote
+  {His supposed law was as follows: Imagine the sun surrounded by a hollow
+spherical shell, on which lies the orbit of the earth. Inside of this shell inscribe
+a regular \textit{icosahedron} (the twenty-sided regular solid), and within that inscribe a
+second sphere. This sphere will carry upon it the orbit of Venus. Inside of the
+sphere of Venus inscribe an \textit{octahedron} (the eight-sided solid), and the sphere
+which fits within it will carry Mercury's orbit. Next, working outwards from the
+earth's orbit, circumscribe around the earth's sphere a \textit{dodecahedron}, circumscribing
+around it another sphere, and this will carry upon it the orbit of Mars.
+Around the sphere of Mars circumscribe the \textit{tetrahedron}, or the regular pyramid.
+The corners of this solid project very far, so that the sphere circumscribed
+around the tetrahedron will be at a very great distance from the sphere of Mars.
+It carries the orbit of Jupiter. Finally, the cube, or \textit{hexahedron}, circumscribed
+around the orbit of Jupiter, gives us in the same way the orbit of Saturn. We
+thus obtain a series of distances not enormously incorrect (though by no means
+agreeing with fact even as closely us does Bode's law); and, moreover, the theory
+had the great advantage to Kepler's mind of accounting for the fact that there
+are (so far as was then known) but \textit{seven} planets, there being possible but \textit{five}
+  regular solids.}
+and the real reason why the
+planets' distances are what they are. This theory of his was broached
+twenty-two years, however, before he discovered the harmonic law,
+and he probably abandoned it when he discovered the elliptical form
+of the planets' orbits. At any rate, in later life he suggested that
+it was likely that there was a planet between Mars and Jupiter too
+small to be seen.
+
+The impression that such a planet existed gained ground when
+Bode published the law which bears his name in 1772, and it was
+still further deepened when, nine years later, in 1781, Uranus was
+discovered, and the distance of the new planet was found to conform
+to Bode's law. An association of twenty-four astronomers,
+mainly German, was immediately formed to look for the missing
+planet, who divided the zodiac between them and began the work.
+%% -----File: 351.png---Folio 340-------
+Singularly enough, however, the first discovery was made, not by a
+member of this association, but by Piazzi, the Sicilian astronomer of
+Palermo, who was then engaged upon an extensive star-catalogue.
+On January~1, 1801, he observed a seventh-magnitude star which by
+the next evening had unquestionably moved, and kept on moving.
+He observed it carefully for some six weeks, when he was taken ill;
+before he recovered, it had passed on towards superior conjunction,
+and was lost in the rays of the sun. He named it \textit{Ceres}, after the
+tutelary goddess of Sicily.
+
+\begin{fineprint}
+When at length the news reached Germany in the latter part of March
+it created a great excitement, and the problem now was to rediscover the
+lost planet. The association of planet-hunters began the search in September,
+as soon as its elongation from the sun was great enough to give any
+prospect of success. During the summer Gauss devised his new method
+of computing a planetary orbit, and computed the ephemeris of its path.
+Very soon after receiving his results, Baron Von Zach rediscovered Ceres
+on December~31, and Dr.~Olbers on the next day, just one year after it was
+first found by Piazzi.
+\end{fineprint}
+
+\nbarticle{593.} In March, 1802, Dr.~Olbers, who in looking for Ceres had
+carefully examined the small stars in the constellation of Virgo, on
+going over the ground again, found a second planet, which he named
+Pallas, a body of about the same brightness as Ceres. \textit{Two} having
+now been found, and Pallas having a very eccentric and much inclined
+orbit, he conceived the idea that they were fragments of a broken
+planet, and that other planets of the same group could probably be
+found by searching near the intersection of their two orbits. Juno,
+the third, was discovered by Harding at Lilienthal (Schröter's observatory)
+in 1804; and Vesta, the largest and brightest of the whole
+group (sometimes visible to the naked eye), was found by Olbers
+himself in 1807. The search was kept up for several years after
+this, but no more planets were found because they did not look after
+small enough stars.
+
+The fifth, Astræa, was discovered in 1845 by Hencke, an amateur
+astronomer who for fifteen years had been engaged in studying
+the smaller stars in hopes of just the reward he captured. In
+1846 no asteroid was found (the discovery of Neptune was glory
+enough for that year), but in 1847 three more were brought to light;
+and since then not a year has passed without adding from one to
+twenty to the number. The list at the end of 1888 counts up 281,
+and there is no prospect of diminution in the rate of discovery,
+though the new ones are mostly very small,---stars of the twelfth
+%% -----File: 352.png---Folio 341-------
+and thirteenth magnitudes, which require a large telescope to make
+them even visible. All the brighter ones have evidently been already
+picked up.
+
+\begin{fineprint}
+They have been discovered by comparatively a few observers. Four persons
+have found more than 20 each: Palisa, of Vienna, who stands far
+in advance of all others, has alone discovered 68, and Dr.~Peters of Clinton,
+N.Y., 52; Luther of Düsseldorf, 24; and the late Professor Watson, 22.
+The German astronomers have thus far discovered 102; the American, 78;
+the French, 66; the English, 19; and the Italians, 16.
+\end{fineprint}
+
+\nbarticle{594.} \nbparatext{Method of Search.}---The asteroid-hunter selects certain
+portions of the sky, usually near the ecliptic, and prepares charts
+covering two or three square degrees, on which he sets down all the
+stars his telescope will show. This is a very laborious operation,
+which in the future is likely to be very much facilitated by photography.
+The chart once made, he goes over the ground from time
+to time, comparing every object with the map and looking out for
+interlopers. If he finds a star which is not on his chart, the \textit{probability}
+is that it is a planet, though it may be a variable star which
+was invisible when the chart was made. He very soon settles the
+question, however, by measuring with the micrometer the distance of
+the new star from some of its neighbors, keeping up the process for
+an hour or two. If it is a planet, it will move perceptibly in that
+length of time.
+
+\begin{fineprint}
+Of course, great care must be taken to be sure that it is a \textit{new} planet, and
+not one of the multitude already known. Generally it is possible to decide
+very quickly which of the known planets will be in the neighborhood, and
+a rough computation will commonly decide at once whether the planet is
+new or not. Not always, however, and mistakes in this regard are not very
+unusual.
+\end{fineprint}
+
+These minor planets are all \textit{named}, the names being derived from
+mythology and legend. They are also designated by numbers, and
+the symbol for each planet is the number written in a circle. Thus,
+for Ceres the symbol is \textcircled{\footnotesize 1}; for Hilda, $(\overline{\underline{\text{\footnotesize 153}\rule[-.3ex]{0pt}{1.7ex}}})$; and so on.
+
+\begin{fineprint}
+A full list of them, with the elements of their orbits, is published yearly
+in the ``Annuaire du Bureau des Longitudes,'' Paris.
+\end{fineprint}
+
+\nbarticle{595.} \nbparatext{Their Orbits.}---The mean distance of the different asteroids
+from the sun varies greatly, and of course the periods are correspondingly
+different. Medusa, $(\overline{\underline{\text{\footnotesize 149}\rule[-.3ex]{0pt}{1.7ex}}})$, has the smallest mean distance
+(2.13, or 198,000000 miles), and the shortest period, $3^{\text{y}}\: 40^{\text{d}}$.
+%% -----File: 353.png---Folio 342-------
+Thule, $(\overline{\underline{\text{\footnotesize 279}\rule[-.3ex]{0pt}{1.7ex}}})$, is the remotest, with a mean distance of 4.30 or 400,000000
+miles, and a period of $8^{\text{y}}\: 313^{\text{d}}$. According to Svedstrup,
+the mean distance of the ``mean asteroid'' is 2.65 (246,000000
+miles), and its period about $4\frac{1}{3}$ years. Its distance from the earth at
+time of opposition would be, of course, 1.65, or 153,000000 miles.
+
+The inclinations of their orbits \textit{average} about $8°$; but Pallas, \textcircled{\footnotesize 2},
+has an inclination of $35°$, and Euphrosyne, $(\overline{\underline{\text{\footnotesize 31}\rule[-.3ex]{0pt}{1.7ex}}})$, of $26\frac{1}{2}°$.
+
+Several of the orbits are extremely eccentric. Æthra, $(\overline{\underline{\text{\footnotesize 132}\rule[-.3ex]{0pt}{1.7ex}}})$, has
+an almost cometary eccentricity of 0.38, and nine or ten others
+have eccentricities exceeding 0.30. They are distributed quite unequally
+in the range of distance, there being, as Kirkwood has
+pointed out, very few at such distances that their periods would be
+exactly commensurable with that of Jupiter.
+
+\nbarticle{596.} \nbparatext{Diameter and Surface.}---Very little is known as to the real
+size of these bodies. The four first discovered, and a few of the
+newer ones, show a minute but sensible disc in very powerful telescopes.
+Vesta, which is the brightest of the whole group, and is just
+visible to the naked eye when near opposition, may be from two hundred
+to four hundred miles in diameter. Pickering, by photometric
+measurements (assuming the reflecting power of the planet's surface
+to be the same as that of Mars), finds a diameter of 319 miles.
+The other three of the original four are perhaps two-thirds as large.
+As for the rest, it is hardly possible that any one of them can be
+as much as 100 miles in diameter, and the smallest, such as are now
+being discovered in such numbers, are probably less than ten miles
+through,---nothing more than ``mountains broke loose.'' The surface
+area of one of the smaller ones would hardly make a large
+Western farm.
+
+\nbarticle{597.} \nbparatext{Mass, Density, etc.}---As to the individual masses and densities
+we have no certain knowledge. It is probable that the density
+does not differ much from the density of the crust of the earth, or the
+mean density of Mars. If this is so, the mass of Vesta might possibly
+be as great as $\frac{1}{15000}$ or $\frac{1}{20000}$ of the earth. On such a planet
+the force of superficial gravity would be about $\frac{1}{25}$ to $\frac{1}{30}$ of gravity on
+the earth, and a body projected from the surface with a velocity of
+about 2000 feet a second---that of an ordinary rifle-ball---would
+fly off into space and never return to the planet, but would circulate
+around the sun as a planet on its own account. On the
+smallest asteroids, with a diameter of about ten miles, it would be
+%% -----File: 354.png---Folio 343-------
+quite possible to throw a stone from the hand with velocity enough to
+send it off into space.
+
+\nbarticle{598.} \nbparatext{Aggregate Mass.}---Although we can only estimate very
+roughly the masses of the individual members of the flock, it is possible
+to get some more certain knowledge of their \textit{aggregate} mass.
+Leverrier found a motion in the line of apsides of the orbit of Mars
+which indicates that the whole amount of matter thus distributed in
+the space between Mars and Jupiter is equal to about \textit{one-fourth} of
+the mass of the earth. This quantity of matter collected into a body
+of the same density as Mars would make a planet of about 5000 miles
+in diameter.
+
+\nbenlargepage
+\begin{fineprint}
+The united masses of these which are already known would make only a
+very small fraction of such a body. Up to August, 1880, the united bulk of
+the asteroids then discovered was estimated at $\frac{1}{4000}$ part of the earth's bulk,
+with a mass probably about $\frac{1}{6000}$ of the earth's. Presumably, therefore, the
+number of these bodies remaining undiscovered is exceedingly great---to be
+counted by thousands, if not by millions. Most of them, of course, must be
+much smaller than those which are already known.
+\end{fineprint}
+
+\nbarticle{599.} \nbparatext{Forms, Variations of Brightness, and Atmosphere.}---We
+have no definite knowledge on this point, but Dr.~Olbers observed
+in the case of Vesta certain fluctuations in her brightness which
+seemed to him to indicate that she is not a globe, but an angular
+mass,---a splinter of rock. This, however, is not confirmed by the
+more recent photometric observations of Müller or Pickering.
+
+\begin{fineprint}
+Müller examined seven of the asteroids, and found their changes of brightness
+very regular. Four of them, one of which is Vesta, behaved precisely
+like Mars, as if their surfaces were comparatively smooth, while three others,
+Ceres and Pallas among them, behaved like the moon and the planet Mercury,
+as if having a rough surface with very little atmosphere, and nearly
+cloudless. Some of the earlier observers reported evidences of an extensive
+halo around them. Later observations do not confirm it, and it is
+not likely that they carry much air with them.
+\end{fineprint}
+
+\sloppy
+\nbarticle{600.} \nbparatext{Origin.}---With respect to this we can only speculate. Two
+views have been held, as has been already intimated. One is, that the
+material, which according to the nebular hypothesis ought to have
+been concentrated to form a single planet of the class to which the
+earth belongs, has failed to be so collected, and has formed a flock
+of small separate masses. It is now very generally believed that the
+%% -----File: 355.png---Folio 344-------
+matter which at present forms the planets was once distributed in
+\textit{rings}, like the rings of Saturn. If so, this ring next outside of Mars
+would necessarily suffer violent perturbations from the nearness of
+the enormous planet Jupiter, and so would be under very different
+conditions from any of the other rings. This, as Peirce has shown,
+might account for its breaking up into many fragments.
+
+\fussy
+The other view is that a planet about the size of Mars has broken
+to pieces. It is true, as has been often urged, that this theory in its
+original form, as presented by Olbers, cannot be correct. No \textit{single}
+explosion of a planet could give rise to the present assemblage of
+orbits, nor is it possible that even the perturbations of Jupiter could
+have converted a set of orbits originally all crossing at one point
+(the point of explosion) into the present tangle. The smaller orbits
+are so small that however turned about they lie wholly inside the
+larger, and cannot be made to intersect them. If, however, we
+admit a \textit{series} of explosions, this difficulty is removed; and if we
+grant an explosion at all, there seems to be nothing improbable in
+the hypothesis that the fragments formed by the bursting of the
+parent mass would carry away within themselves the same forces and
+reactions which caused the original bursting; so that they themselves
+would be likely enough to explode at some time in their later
+history.
+
+At present opinion is divided between these two theories.
+
+\begin{fineprint}
+\nbarticle{601.} The number of these bodies already known is so great, and the
+prospect for the future is so indefinite, that astronomers are at their wits'
+end how to take care of this numerous family. To compute the orbit and
+ephemeris of one of these little rocks is more laborious (on account of the
+great perturbations produced by Jupiter) than to do the same for one of the
+major planets; and to keep track of such a minute body by observation is
+far more difficult. Until recently, the German Jahrbuch has been publishing
+the ephemerides of such as came within the range of observation each
+year; but this cannot be kept up much longer, and the probability is that
+hereafter only the larger ones, or those which present some remarkable
+peculiarity in their orbits, will be followed up. One little family of them,
+however, is ``endowed.'' Professor Watson, at his death, left a fund to the
+American National Academy of Sciences to bear the expense of taking care
+of the twenty-two which he discovered.
+\end{fineprint}
+
+\section*{INTRA-MERCURIAL PLANETS AND THE ZODIACAL LIGHT.}
+
+It is very probable, indeed almost certain, that there are masses of
+matter revolving around the sun within the orbit of Mercury.
+%% -----File: 356.png---Folio 345-------
+
+\nbarticle{602.} \nbparatext{Motion of the Perihelion of Mercury's Orbit.}---Leverrier,
+in 1859, from a discussion of all the observed transits of Mercury,
+found that the perihelion of its orbit has a movement of nearly $38''$
+a century. This is more than can be accounted for by the action of
+the known planets, and, \textit{so far as known}, could be explained only by
+the attraction of a planet, or ring of small planets, revolving inside
+this orbit nearly in its plane, with a mass about half as great as that
+of Mercury itself.
+
+\begin{fineprint}
+We say ``\textit{so far as known},'' because an alternative hypothesis has been
+proposed, viz., that the law of gravitation, though \textit{strictly true for bodies at
+rest} is not absolutely so \textit{for bodies in motion}; that when bodies are moving
+towards each other the attraction is less by a minute fraction than if they
+were at rest. The hypothesis is known as the \textit{electro-dynamic theory of gravitation},
+but has at present very little to support it. If, however, it were true,
+then the peculiar motion of the apsides of Mercury's orbit would be a
+necessary consequence.
+\end{fineprint}
+
+Subsequent investigations by a number of mathematicians have
+fully confirmed Leverrier's results; Mercury's orbit is beyond question
+affected as it would be if there were an intra-Mercurial planet,
+or a number of them.
+
+\begin{fineprint}
+\nbarticle{603.} \nbparatext{Dr.\ Lescarbault's Observation: Vulcan.}---A certain country
+physician, living some eighty miles from Paris, Dr.~Lescarbault, on the publication
+of Leverrier's result, announced that he had actually seen this planet
+crossing the sun nine months before, on the 26th of March of that year,
+1859. He was visited by Leverrier, who became satisfied of the genuineness
+of his observations, and the doctor was duly congratulated and honored as
+the discoverer of ``Vulcan,'' which name was assigned to the supposed new
+planet. An interesting account of the matter may be found in Chambers'
+``Descriptive Astronomy''; and in many of the works published from twenty
+to twenty-five years ago, as well as in some more recent ones, ``Vulcan'' is
+assigned a place in the solar system, with a distance of about 13,000000
+miles and a period of $19+$ days. Lescarbault described  it as having an
+apparent diameter of about $7''$, which would make it over 2500 miles in
+diameter.
+
+\nbarticle{604.} Nevertheless, it is nearly certain that Vulcan does not exist.
+There are various opinions which we need not here discuss as to the explanation
+of this pseudo-discovery. But the planet, if real, ought since
+1859 to have been visible on the sun's face at certain definite times which
+Leverrier calculated and published; and it has never been seen, though
+very carefully looked for. Small, round, dark objects have from time to
+%% -----File: 357.png---Folio 346-------
+time been indeed reported on the sun's disc, which in the opinion of the
+observers at the time were not sun spots; but most of these observations
+were made by amateurs with comparatively little experience, with small
+telescopes, and with no measuring apparatus by which they could certainly
+determine whether or not the spot seen moved like a planet. In most of
+these cases photographs or simultaneous observations made elsewhere by
+astronomers of established reputation, and having adequate apparatus, have
+proved that the problematical ``dots'' were really nothing but ordinary
+small sun spots, and the probability is that the same explanation applies to
+the rest.
+\end{fineprint}
+
+\nbarticle{605.} \nbparatext{Eclipse Observations.}---A planet large enough to be seen
+distinctly on the sun by a $2\frac{1}{2}$-inch telescope, such as Lescarbault
+used, would be a conspicuous object at the time of a solar eclipse,
+and most careful search has been made for the planet on such occasions;
+but so far, although \textit{stars} of the third and fourth magnitudes,
+and even of the fifth, have been clearly seen by the observers within
+a few degrees of the eclipsed sun, no planet has been found.
+
+\begin{fineprint}
+One apparent exception occurred in 1878. During the eclipse of that
+year, Professor Watson observed two \DPtypo{starlike}{star-like} objects (of the fourth magnitude),
+which he thought at the time could not be identified with any known
+stars consistently with his observations. Mr.~Swift, also, at the same eclipse,
+reported the observations of two bright points very near the sun; but these
+from his statement could not (both) have been identical with Watson's stars.
+Later investigations of Dr.~Peters have shown that the assumption of a
+very small and very likely error in Professor Watson's circle-readings
+(which were got in a very ingenious, but rather rough way, without the
+use of graduations) would enable his stars to be identified with $\theta$ and $\zeta$
+Cancri, and it is almost certain that these were the stars he saw.
+Mr.~Swift's observations remain unexplained. With this exception, the eclipse
+observations all give negative results, and astronomers generally are now
+disposed to consider the ``Vulcan question'' as settled definitely and adversely.
+\end{fineprint}
+
+\nbarticle{606.} At the same time it is extremely probable that there are a
+number, and perhaps a very great number, of \textit{intra-Mercurial asteroids}.
+A body two hundred miles in diameter near the sun would
+have an angular diameter of only about $\frac{1}{2}''$, as seen from the earth,
+and would not be easily visible on the sun's disc, except with very
+large telescopes. It would not be at all likely to be picked up accidentally.
+Objects with a diameter of not more than forty or fifty
+miles would be almost sure to escape observation, either at a transit
+or during a solar eclipse.
+%% -----File: 358.png---Folio 347-------
+
+\begin{fineprint}
+\nbarticle{607.} \nbparatext{Zodiacal Light.}---This is a faint, soft beam of light that extends
+both ways from the sun along the ecliptic. In the evening it is best seen
+in February, March, and April, because the portion of the ecliptic which
+lies east of the sun's place is then most nearly perpendicular to the western
+horizon. During the autumnal months the zodiacal light is best seen in
+the morning sky for a similar reason. In our latitudes it can seldom be
+traced more than $90°$ or $100°$ from the sun; but at high elevations within
+the tropics it is said to extend entirely across the sky, forming a complete
+ring, and there is said to be in it at the point exactly opposite to the sun a
+patch a few degrees in diameter of slightly brighter luminosity, called the
+``Gegenschein'' or ``counter-glow.''
+
+The portions of this object near the sun are reasonably bright, and even
+conspicuous at the proper seasons of the year; but the more distant portions
+in the neighborhood of the ``counter-glow'' are so extremely faint that it is
+only possible to observe them at a distance from cities and large towns, in
+places where the air is free from smoke, and where the darkness of the sky
+is not affected by the general illumination due to gas and electric lights.
+
+\nbarticle{608.} The cause of the phenomenon is not certainly known, but at present
+the theory most generally accepted attributes it to \textit{sunlight reflected by
+myriads of small meteoric bodies} which are revolving around the sun nearly in
+the plane of the ecliptic, forming a thin, flat sheet like one of Saturn's rings,
+and extending far beyond the orbit of the earth. It may be that the denser
+portion of this meteoric ring within the orbit of Mercury is the cause of the
+motion of the perihelion of that planet which Leverrier detected; it is for
+this reason that we deal with the subject here rather than in connection with
+meteors. While this theory, however, is at present more generally accepted
+than any other, it cannot he said to be established. Some are disposed to
+consider the zodiacal light as a mere extension of the sun's corona, whatever
+that may be.
+
+\nbarticle{608*.} \nblabel{pg:347}(Note to \artref{Art.}{588}.) \nbparatext{The Canals of Mars.}---According to
+Schiaparelli (and his observations are at least partially confirmed by others)
+a most characteristic feature of the planet's surface is a series of long,
+straight, narrow ``canals'' connecting the larger bodies of water. These
+canals were first seen and recognized in 1877. In 1881 they were seen again,
+and at that time nearly all of them \textit{double}. If there is not some fallacy in
+the observation, the problem as to the nature of these canals, and the cause
+of their ``gemination,'' is a very important and perplexing one. It is hoped
+that at the next favorable opposition in 1892 it may find its solution.
+\end{fineprint}
+\chelabel{CHAPTERXIV}
+%% -----File: 359.png---Folio 348-------
+
+\Chapter{XV}{The Major Planets}
+\nbchapterhang{\stretchyspace
+THE PLANETS CONTINUED.---THE MAJOR PLANETS: JUPITER,
+SATURN, URANUS, AND NEPTUNE.}
+
+\section*{JUPITER.}
+
+\nbarticle{609.} While this planet is not so brilliant as Venus at her best,
+it stands next to her in this respect, being on the average about five
+times brighter than Sirius, the brightest of the fixed stars. Jupiter,
+moreover, being a ``superior'' planet, is not confined, like Venus,
+to the neighborhood of the sun, but at the time of opposition is the
+chief ornament of the midnight sky.
+
+\nbarticle{610.} \nbparatext{Orbit.}---The orbit presents no marked peculiarities. The
+\textit{mean distance} of the planet from the sun is 483,000000 miles. The
+\textit{eccentricity} of the orbit being nearly $\frac{1}{20}$ (0.04825); the greatest and
+least distances vary by about 21,000000 miles each way, making
+the planet's greatest and least distances from the sun 504,000000
+and 462,000000 miles respectively. The average distance of the
+planet from the earth at opposition is 390,000000, while at conjunction
+it is 576,000000 miles. The minimum opposition distance is
+only 369,000000, which is obtained when the opposition occurs about
+October~6, Jupiter being in perihelion when its heliocentric longitude
+is about $12°$. At an aphelion opposition (in April) the distance
+is 42,000000 miles greater; that is, 411,000000.
+
+The relative brightness of Jupiter at an average conjunction and
+at the nearest and most remote oppositions is respectively as the
+numbers 10, 27, and 18. The average brightness at opposition is,
+therefore, more than double that at conjunction; and at an October
+opposition the planet is fifty per cent brighter than at an April one.
+The differences are considerable, but far less important than in the
+case of Mars, Venus, and Mercury.
+
+The \textit{inclination} of the orbit to the ecliptic is small,---only $1°\: 19'$.
+
+\nbarticle{611.} \nbparatext{Period.}---The \textit{sidereal period} is 11.86 years, and the \textit{synodic}
+is 399 days (a number easily remembered), a little more than a year
+and a month. The planet's orbital velocity is about eight miles a
+second.
+%% -----File: 360.png---Folio 349-------
+
+\sloppy
+\nbarticle{612.} \nbparatext{Dimensions.}---\textit{The planet's apparent diameter} varies from
+$50''$ at an October opposition (or $45 \frac{1}{2}''$ at an April one) to $32''$ at
+conjunction. The form, however, of the planet's disc is not truly
+circular, the polar diameter being about $\frac{1}{17}$ part less than the equatorial,
+so that the eye notices the oval form at once. The equatorial
+diameter in \textit{miles} is 88,200, the polar being 83,000. Its mean\footnote
+  {The mean diameter of an oblate spheroid is $\dfrac{2a + b}{3}$ not $\dfrac{a+b}{2}$. Of the three
+  axes of symmetry which cross at right angles at the planet's centre, \textit{one} is the
+  axis of rotation, and both the others are equatorial.}
+diameter, therefore, is 86,500,---almost eleven times that of the
+earth.
+
+\fussy
+This makes its surface 119 times, and its volume 1300 times, that
+of the earth. It is by far the largest of the planets in the system;
+in fact, whether we regard its bulk or its mass, larger than all the rest
+put together.
+
+\nbarticle{613.} \nbparatext{Mass, Density, etc.}---Its \textit{mass} is very accurately known,
+both by the motions of its satellites, and the perturbations of the
+asteroids. It is $\frac{1}{1048}$ of the sun's mass, or very nearly 316 times
+that of the earth. Comparing this with its volume, we find its \textit{density}
+0.24, less than $\frac{1}{4}$ the density of the earth, and almost precisely the
+same as that of the sun. Its mean \textit{superficial gravity} comes out 2.64
+times that of the earth; that is, a body on Jupiter would weigh $2 \frac{5}{8}$
+times as much as upon the surface of the earth; but on account of
+the rapid rotation of the planet and its ellipticity there is a very considerable
+difference between the force of gravity at the equator and
+at the pole, amounting to $ \frac{1}{5}$ of the equatorial gravity. (On the earth
+the difference is only $ \frac{1}{190}$.)
+
+\nbarticle{614.} \nbparatext{Phases and Albedo.}---Its orbit is so much larger than that
+of the earth that the planet shows no sensible phases, even at quadrature,
+though at that time the edge farthest from the sun shows a
+slight darkening.
+
+The reflecting power, or \textit{Albedo}, of the planet's surface is very
+high,---0.62 according to Zöllner, that of white paper being only
+0.78. The centre of the disc of this planet (and the same is also
+true of Saturn) is considerably brighter than the limb---just the
+reverse, as will be remembered, from the condition of things upon
+the moon, and upon Mars, Venus, and Mercury. This peculiarity
+of a darkened limb, in which Jupiter resembles the sun, has suggested
+%% -----File: 361.png---Folio 350-------
+the idea that it is to some extent \textit{self-luminous}. This, however,
+is not a necessary consequence, as a nearly transparent atmosphere
+overlying a uniformly reflecting surface would produce the
+same effect.
+
+The light which the planet omits, if it emits any, must be very
+feeble as compared with sunlight, since the satellites, when they are
+eclipsed by entering the shadow, become totally invisible.
+
+\nbarticle{615.} \nbparatext{Axial Rotation.}---The planet rotates on its axis in \textit{about}
+$9^{\text{h}}\: 55^{\text{m}}$. The time can be given only approximately, not because
+it is difficult to find and observe distinct markings on the planet's
+disc, but simply because different results are obtained from different
+spots, according to their nature and their distance from the
+planet's equator. Speaking generally, spots near the equator indicate
+a shorter day than those in higher latitudes, and certain small,
+sharply defined, bright, white spots, such as are often seen, give a
+quicker rotation than the dark markings in the same latitude.
+
+\begin{fineprint}
+For instance, a white spot observed near the equator in 1886 for several
+months gives $9^{\text{h}}\: 50^{\text{m}}\: 4^{\text{s}}$, while another one near latitude $60°$ gave $9^{\text{h}} \:
+55^{\text{m}}\: 12^{\text{s}}$. The great red spot has given values ranging from $9^{\text{h}}\: 55^{\text{m}} \:
+34^{\text{s}}.9$ (in 1879) to $9^{\text{h}}\: 55^{\text{m}}\: 40^{\text{s}}.7$ (in 1886),---the time of rotation as
+determined in each case, being certainly accurate within half a second.
+The progressive increase has been regular and unmistakable, and is not
+due to any possible uncertainty in the observations.
+\end{fineprint}
+
+\nbarticle{616.} \nbparatext{The Axis of Rotation and the Seasons.}---The plane of the
+equator is inclined only $3°$ to that of the orbit, so that as far as the
+sun is concerned there can be no seasons. The heat and light
+received from the sun by Jupiter are, however, only about $\frac{1}{27}$ as
+intense as the solar radiation at the earth, its distance being 5.2
+times as great.
+
+\nbarticle{617.} \nbparatext{Telescopic Appearance.}---Even in a small telescope the
+planet is a beautiful object. When near opposition a magnifying
+power of only 40 makes its apparent size equal to that of the full
+moon (though, as remarked in connection with Venus, no novice
+would receive that impression), and with a telescope of 8 or 10 inches
+aperture, and with a magnifying power of 300 or 400, the disc is
+covered with an infinite variety of beautiful and interesting details
+which rapidly shift under the observer's eye in consequence of the
+planet's swift rotation. The picture is rich in color, also, browns and
+reds predominating, in contrast with olive-greens and occasional
+%% -----File: 362.png---Folio 351-------
+purples; but to bring out the colors well and clearly requires large
+instruments. For the most part the markings are arranged in streaks
+more or less parallel to the planet's equator, as shown by \figref{illo178}{Fig.~178}.
+With a small telescope the markings usually reduce to two dark and
+comparatively well-defined belts, one on each side of the equator,
+occupying about the same regions of latitude that the trade-wind
+zones do upon the earth; and very likely in Jupiter's case similar
+aerial currents have something to do with the appearance, though
+upon Jupiter, as has been already said, the solar heat is a comparatively
+unimportant factor. The markings upon the planet are almost,
+if not entirely, \textit{atmospheric}, as is proved by the manner in which
+they change their shapes and relative positions. They are \textit{cloud
+forms}. It is hardly probable that we ever see anything upon the
+solid surface of the planet underneath, nor is it even certain that
+the planet has anything solid about it. In \figref{illo178}{Fig.~178}, the upper
+left-hand figure is from a drawing by Trouvelot made in February,
+1872; the second is by Vogel in 1880. The small one below represents
+the planet as seen in a small telescope.
+
+\includegraphicsmid{illo178}{\textsc{Fig.~178.}---Telescopic Views of Jupiter.}
+
+\nbarticle{618.} \nbparatext{The Great Red Spot.}---While most of the markings on the
+planet are evanescent, it is not so with all. There are some which
+%% -----File: 363.png---Folio 352-------
+are at least ``sub-permanent,'' and continue for years, not without
+change indeed, but with only slight changes. The ``great red spot''
+is the most remarkable instance so far. It seems to have been first
+observed by Prof.\ C.~W.\ Pritchett of Glasgow, Missouri, in July,
+1878, as a pale, pinkish, oval spot some $13''$ in length by $3''$ in
+width (30,000 miles by 7000). Within a few months it had been
+noticed by a considerable number of other observers, though at first
+it did not attract any special attention, since no one thought of it as
+likely to be permanent. The next year, however, it was by far the
+most conspicuous object on the planet. It was of a clear, strong
+brick-red color, with a length fully one-third the diameter of the
+planet and a width about one fourth of its length.
+
+\includegraphicsmid{illo179}{\textsc{Fig.~179.}---Jupiter's ``Red Spot.'' From Drawings by Mr.~Denning. 1880--85.}
+
+\begin{fineprint}
+For two or three years it remained without much change: in 1882--83
+it gradually faded out: in 1885 it had become a pinkish oval \textit{ring}, the
+central part being apparently occupied with a white cloud. In 1886 it was
+again a little stronger in color, and the same in 1887,---an object not difficult
+to see with a large telescope, but the merest ghost of what it was in
+1880. The present year (1888) its appearance is about the same as in 1887,---perhaps
+a little paler. During the ten years its form and size have varied
+very little. It lies at the southern edge of the southern equatorial belt, in latitude
+%% -----File: 364.png---Folio 353-------
+about $35°$, and for some reason the belt seems to be ``notched out'' for
+it, so that there has been always a narrow white streak separating the belt
+from the spot. Even when the spot was palest and hard to see its \textit{place} was
+always evident at once from the indentation in the outline of the belt.
+
+Such phenomena suggest abundant matter for speculation which would
+be out of place here. It must suffice to say that no satisfactory explanation
+of the phenomena has yet been presented. The unquestionable fact before
+mentioned (\artref{Art.}{615}), that the time of rotation of the spot has changed by
+more than $5^\text{s}$ in the ten years, greatly complicates the subject. \figref{illo179}{Fig.~179},
+from the drawings of Mr.~Denning, represents the appearance of the spot at
+four different dates; viz., 1,~1880, Nov.~19; 2,~1882, Oct.~30; 3,~1884,
+Feb.~6; 4,~1885, Feb.~25.
+\end{fineprint}
+
+\nbarticle{619.} \nbparatext{Temperature and Physical Constitution.}---The rapidity of
+the changes upon the visible surface implies the expenditure of a considerable
+amount of heat, and since the heat received from the sun is
+too small to account for the phenomena which we see, Zöllner, thirty
+years ago, suggested that it must come from within the planet, and
+that in all probability Jupiter is at a temperature not much short of
+incandescence,---hardly yet solidified to tiny considerable extent.
+Mr.~Proctor has given special currency to these views among English
+readers. The idea that Jupiter might be such a ``semi-sun'' is not
+at all new. Buffon, Kant, Nasmyth, and Bond all entertained and
+discussed it; but it is only since the investigations of Zöllner that it
+has become an accepted item of scientific belief. (See Clerke's
+``History of Astronomy,'' p.~335 seqq.)
+
+\nbarticle{620.} \nbparatext{Atmosphere.}---As to the composition of the planet's atmosphere,
+the spectroscope gives us rather surprisingly little information.
+We get from the planet a good solar spectrum with the solar lines
+well marked, but there are no well-defined absorption bands due to
+the action of the planet's atmosphere. There are, however, some
+\textit{shadings} in the lower red portion of the spectrum that are probably
+thus caused. The light, for the most part, seems to come from the
+upper surface of the planet's envelope of clouds without having
+penetrated to any depth.
+
+\nbarticle{621.} \nbparatext{Satellite System.}---Jupiter has four satellites,---the first
+heavenly bodies ever \textit{discovered}---the first revelation of Galileo's
+telescope. His earliest observation of them was on Jan.~7, 1610,
+and in a very few weeks he had ascertained their true character, and
+determined their periods with an accuracy which is surprising when
+we consider his means of observation. The number of the heavenly
+%% -----File: 365.png---Folio 354-------
+bodies was now no longer \textit{seven}, and the discovery excited among
+churchmen and schoolmen a great deal of angry incredulity and
+vituperation. Galileo called them ``the Medicean stars.''
+
+They are now usually known as the first, second, etc., in the order of
+distance from the primary, but they also have names which are sometimes
+used; viz., Io, Europa, Ganymede, and Callisto. Their relative
+distances range between 262,000 and 1,169,000 miles, being very
+approximately 6, 9, 15, and 26 radii of the planet. The distance of
+the first from the surface of the planet is almost exactly the same as
+that of our own moon from the surface of the earth. Their sidereal
+periods range between $1^{\text{d}}\: 18 \frac{1}{2}^{\text{h}}$ and $16^{\text{d}}\:16 \frac{1}{2}^{\text{h}}$ (accurate values in
+distances and periods are given in the table in the \hyperref[app:II]{Appendix}). The
+orbits are almost exactly circular, and lie in the plane of the planet's
+equator.
+
+\begin{fineprint}
+The satellites slightly disturb each other's motions, and from these
+disturbances their masses can be ascertained in terms of the planet's mass.
+The third, which is much the largest, has a mass of about $\frac{1}{11000}$ of the
+planet's, a little more than double the mass of our own moon. The mass
+of the first satellite appears to be a little less than $\frac{1}{5}$ as much. The second
+is somewhat larger than the first, and the fourth is about half as large as the
+third; \textit{i.e.}, it has about the mass of our own moon. The densities of the
+first and fourth appear to be not very different from that of the planet itself,
+while the densities of the second and third are considerably greater.
+
+\nbarticle{622.} \nbparatext{Relation between Mean Motions and Longitudes of the
+Satellites.}---In consequence of their mutual interaction a curious relation % [*F1: Is this text intermediate between \small and the main text size?][F2: Surely the printer wasn't that mean.]
+(discovered by~La Place) exists between the mean motions of the first three
+satellites. The mean motion is of course $360°$ divided by $T$ ($T$ being the
+satellite's period). It appears that the mean motion of the first plus twice
+the mean motion of the third equals three times that of the second, or
+\[
+\frac{1}{T_1} + \frac{2}{T_3} = \frac{3}{T_2}.
+\]
+A similar relation holds for their longitudes:
+\[
+L_1 + 2 L_3 = 3 L_2 + 180°;
+\]
+so that they cannot all three come into opposition or conjunction with the
+sun at once. These relations are permanently maintained by their mutual
+attractions: \textit{exactly} in the long run, though there are slight perturbations
+produced by the fourth satellite which disturb the arrangement slightly for
+short periods. The fourth satellite does not come into any such arrangement.
+\end{fineprint}
+%% -----File: 366.png---Folio 355-------
+
+\nbarticle{623.} \nbparatext{Diameters, etc.}---The diameter of the first satellite
+is a little more than 2400 miles; the second is almost exactly the
+size of our own moon, \textit{i.e.}, between 2100 and 2200 miles;
+and the third and fourth have diameters, respectively, of 3600 and
+3000 miles, the third, Ganymede, being much larger than either of
+his sisters. When Jupiter is in opposition, the fourth satellite is
+sometimes nearly $10 \frac{1}{2}'$ away from the planet, or
+$\frac{1}{3}$ of the moon's diameter; and in very clear air can be
+seen by a sharp eye without telescopic aid. The third, though much
+larger, never goes more than $6'$ from the planet, and it is perhaps
+doubtful whether it is ever seen with the naked eye, unless when the
+fourth happens to be close beside it. A good opera-glass
+will easily show them all as minute points of light.
+
+\nbarticle{624.} \nbparatext{Brightness.}---Since the sunlight of Jupiter is only
+$\frac{1}{27}$ as intense as ours, the moonlight made by the
+satellites is decidedly inferior to our own, although their
+reflective power appears to be higher than that of the lunar
+surface. They differ among themselves considerably in this respect.
+The fourth satellite is of an especially dark complexion, so that it
+often looks perfectly black when it passes between us and the
+planet, and is projected on the disc. The others, under similar
+circumstances, show light or dark according as they have a dark or
+light portion of the planet for a background. Even the fourth, when
+crossing the disc, is always seen
+bright while very near the planet's limb.
+
+\nbarticle{625.} \nbparatext{Markings upon the Satellites.}---The satellites show
+sensible discs when viewed with a large telescope, and all of them
+but the second sometimes show dark markings upon the surface. These
+markings, however, are only visible under the most favorable
+circumstances, and it has not been possible to determine whether
+they are atmospheric or really geographical, nor
+has it yet been possible to deduce from them the satellites' periods of rotation.
+
+\nbarticle{626.} \nbparatext{Variability.}---Galileo noticed variations in the
+brightness of the satellites at different times, and subsequent
+observers have confirmed his result. In the case of the fourth
+satellite there seems to be a regular variation depending upon the
+place of the satellite in its orbit, and suggesting that in its
+axial rotation it behaves like our own moon, keeping always the same
+side next its primary. In addition it shows other \textit{irregular}
+changes in its luminosity: so also do the other satellites according
+to nearly all authorities, though it is singular that one or two of
+the best observers do not find any such irregularity indicated by
+their instrumental\footnote{ Clerke's ``History of Astronomy,''
+p.~339.} photometric
+observations.
+
+\nbarticle{627.} \nbparatext{Eclipses and Transits.}---The satellites' orbits are so
+nearly in the plane of the planet's orbit that, excepting the
+fourth, they all pass through the shadow of the planet, and suffer
+eclipse at every
+%% -----File: 367.png---Folio 356-------
+revolution. At conjunction, also, they cast their shadows upon the
+planet, and these shadows can easily be seen in the telescope as
+black dots on the planet's disc, the satellites themselves, which
+cross the disc about the same time, being much more difficult to
+observe. The fourth satellite escapes eclipse when Jupiter is far
+from the node of its orbit. Thus, during 1888 and in the first half
+of 1889, there are no eclipses of Callisto at all.
+
+Exactly at opposition or conjunction the planet's shadow lies
+straight behind it out of our sight, so that we cannot at that time
+observe the eclipses of the satellites, but only their transits
+across the disc. Before and after these times, however, the shadow
+lies one side of the planet.
+
+\includegraphicsmid{illo180}{\textsc{Fig.~180.}---Eclipses of Jupiter's Satellites, at Western Elongation.}
+
+When the planet is at quadrature and the condition of things is as
+represented in \figref{illo180}{Fig.~180} (which is drawn to scale), the shadow
+projects so far to one side of the planet that the whole eclipse of
+all the satellites, except the first, takes place clear of the
+planet's disc,---both the disappearance and reappearance of the
+satellite being visible.
+
+\nbarticle{628.} \nbparatext{``Equation of Light.''}---The most important use that has
+been made of these eclipses has been to ascertain the time required
+by light in traversing the distance between us and the sun, the so-called
+%% -----File: 368.png---Folio 357-------
+``\textit{equation of light}.'' It was in 1675 that Roemer, the Danish
+astronomer (the inventor of the transit instrument, meridian circle,
+and prime vertical instrument,---a man nearly a century in advance of
+his day), found that the eclipses of the satellites showed a peculiar
+variation in their times of occurrence, which he explained as due to
+the time taken by light to pass through space. His bold and original
+suggestion was rejected by most astronomers for more than fifty
+years,---until long after his death,---when Bradley's discovery of
+aberration (\artref{Art.}{225}) proved the correctness of his views.
+
+\includegraphicsouter{illo181}{\textsc{Fig.~181.}\\
+Determination of the Equation of Light.}
+
+\nbarticle{629.} If the planet and earth remained at an invariable distance
+the eclipses of the satellites would recur with unvarying regularity
+(their disturbances being very slight), and the mean interval could
+be determined, and the times tabulated. But if we thus predict the
+times of eclipses for a synodic period of the planet, then, beginning
+at the time of opposition,
+it will be found that as the
+planet recedes from the earth,
+the eclipses fall constantly more
+and more behindhand, and by
+precisely the same amount for
+all four of the satellites. The
+difference between the tabulated
+and observed time continues to
+increase until the planet is near
+conjunction, when the eclipses
+are more than sixteen minutes
+late.
+
+\begin{fineprint}
+From the insufficient observations
+at his command, Roemer
+made the difference twenty-two
+minutes.
+\end{fineprint}
+
+After the conjunction, the eclipses quicken their pace and exactly
+make up all the loss; so that when opposition is reached once more,
+they are again on time.
+
+It is easy to see from \figref{illo181}{Fig.~181} that at opposition the planet is
+nearer the earth than at conjunction by just twice the radius of the
+earth's orbit; \textit{i.e.}, $ JB - JA = 2 SA$. The whole apparent retardation
+of the eclipses between opposition and conjunction, should therefore
+be exactly twice the time required for light to come from the sun to
+the earth. This time is very nearly 500 seconds, or $8^{\text{m}}\: 20^{\text{s}}$.
+%% -----File: 369.png---Folio 358-------
+
+\begin{fineprint}
+Early in the century Delambre, from all the satellite eclipses of which he
+could then secure observations, found it to be $493^{\text{s}}$. A few years ago a
+redetermination by Glasenapp of Pulkowa made it $501^{\text{s}}$, from fifteen years'
+observation of the eclipses of the first satellite. Probably this value is much
+nearer the truth than Delambre's.
+\end{fineprint}
+
+\sloppy
+\nbarticle{630.}\hspace{0em} \nbparatext{Photometric Observations of the Eclipses.}---The eclipses are
+\textit{gradual} phenomena, the obscuration of the satellite proceeding continuously
+from the time it first strikes the shadow of the planet until
+it entirely vanishes. The moment at which the satellite seems to
+disappear depends, therefore, on the state of the air and of the
+observer's eye, and upon the power of his telescope. The same is
+true of the reappearance; so that the observations are doubtful to
+the extent of from half a minute for the first satellite (which moves
+quickly), to a full minute for the fourth. Professor Pickering has
+proposed to substitute for this comparatively indefinite moment of
+disappearance or reappearance, \textit{the instant when the satellite has lost
+or regained just half its normal light}, and he determines this instant
+by a series of photometric comparisons with one of the neighboring
+uneclipsed satellites, or with the planet itself.
+\fussy
+
+\begin{fineprint}
+These comparisons are made with a special photometer devised for the
+purpose, and planned with reference to rapid reading: by merely turning a
+small button, the observer is immediately able to make the image of the
+uneclipsed satellite appear to be of the same brightness as the satellite which
+is disappearing, and the observations can be repeated very rapidly with the
+help of special contrivances for recording the times and readings. It is
+found that this instant of ``half-brightness'' can be deduced from the set of
+photometric readings with an error not much exceeding a second or two.
+Observations of this kind have now been going at Cambridge (U.~S.)\footnote
+  {Professor Pickering has more recently (August, 1888) applied \textit{photography}
+  to these observations with most gratifying success. A series of pictures is taken,
+  each with an exposure of $10^{\text{s}}$, the time being recorded on a chronograph, and
+  they determine with great precision the moment when the satellite's brightness
+  had any special value, say fifty per cent of its maximum.}
+for
+several years. A similar plan has also been devised by Cornu, and is being
+carried out at the Paris Observatory under his direction.
+
+A series of such observations covering the planet's whole period of twelve
+years, ought to give us a much more accurate determination of the light-equation
+than we now have.
+\end{fineprint}
+
+\nbarticle{631.} Until 1849 our only knowledge of the velocity of light was
+obtained by observations of Jupiter's satellites. By assuming as
+%% -----File: 370.png---Folio 359-------
+known the earth's distance from the sun, the velocity of light
+follows when we know the time occupied by light in coming from
+the sun. At present, however, the case is reversed: we can determine
+the velocity of light by two independent experimental methods,
+and with a surprising degree of accuracy; and then, knowing the
+velocity and the light-equation, we can deduce the distance of the sun.
+
+\section*{SATURN.}
+
+\nbarticle{632.} \nbparatext{The Orbit and Period.}---Saturn is the remotest of the
+ancient planets, its mean distance from the sun being 9.54 astronomical
+units, or 886,000000 miles. This distance varies by nearly
+50,000000 miles on account of the eccentricity of its orbit (0.056),
+which is a little greater than that of Jupiter.
+
+Its nearest approach to the earth at a December opposition (the
+longitude of its perihelion being $90°\: 4'$) is 744 millions of miles, and
+its greatest distance at a May conjunction is 1028 millions. It
+is so far from the sun that these changes of distance do not so
+greatly affect its apparent brightness, as in the case of the nearer
+planets, the whole range of variation from this cause being less than
+two to one; that is, at the nearest of all oppositions, the planet is
+not twice as bright as at the remotest of all conjunctions. The
+changing phases of the rings make quite as great a difference as the
+variations of distance.
+
+The \textit{orbit is inclined} to the ecliptic about $2\frac{1}{2}°$.
+
+The \textit{sidereal period} of the planet is \textit{twenty-nine and one-half years},
+the \textit{synodic period} being 378 days.
+
+The planet itself is unique among the heavenly bodies. The great
+belted globe carries with it a retinue of eight satellites, and is surrounded
+by a system of rings unlike anything else in the universe
+so far as known, the whole constituting the most beautiful and most
+interesting of all telescopic objects.
+
+\sloppy
+\nbarticle{633.} \nbparatext{Diameter, Volume, and Surface.}---The apparent mean diameter
+of the planet varies from $20''$ to $14''$ according to the distance.
+We say \textit{mean} diameter because this planet is more flattened at the
+pole than any other, its ellipticity being nearly ten per cent,
+though different observers vary somewhat in their results. The
+equatorial diameter of the planet is about 75,000 miles, and its polar
+about 68,000, the mean being very nearly 73,000, or a little more
+than nine times that of the earth. Its \textit{surface} is therefore about
+eighty-two times, and its \textit{volume} 760 times that of the earth.
+%% -----File: 371.png---Folio 360-------
+
+\fussy
+\nbarticle{634.} \nbparatext{Mass, Density, and Gravity.}---Its \textit{mass} is only ninety-five
+times the earth's mass, from which follows the remarkable fact that
+the \textit{density} of Saturn is \textit{only one-eighth that of the earth}, or \textit{only
+about five-sevenths that of water}. It is by far the least dense of all
+the planets. The \textit{superficial gravity} is 1.2.
+
+\sloppy
+\nbarticle{635.} \nbparatext{Axial Rotation.}---It revolves upon its axis in about $10^\text{h}\: 14^\text{m}$
+according to a determination of Professor Hall, made in 1876 by
+means of a white spot which suddenly appeared upon its surface,
+and continued visible for some weeks. Although the surface of the
+planet is beautifully marked with belts which often show delicate
+rose-colored tints, it is seldom that any well-defined markings present
+themselves by which the rotation can be determined.
+
+\fussy
+The \textit{inclination of the axis} is about $28°$.
+
+\nbarticle{636.} \nbparatext{Surface, Albedo, and Spectrum.}---As in the case of Jupiter,
+the edges of the disc are not quite so brilliant as the central portions,
+so that the belts appear to fade out near the limb. These
+belts are less distinct and less variable than those of Jupiter; and
+are arranged as shown in \figref{illo182}{Fig.~182}, with a very brilliant zone at
+the equator, though the engraving much exaggerates the contrast.
+The planet's pole is marked by a darkish cap of greenish hue.
+
+According to Zöllner, the \textit{Albedo}, or reflecting power of the surface
+is 0.52, almost precisely the same as that of Venus, but a little inferior
+to that of Jupiter. The \textit{spectrum} of the planet is the solar spectrum
+without any evidence of the presence of water-vapor, so far as
+can be made out, but with certain unexplained dark bands in the red
+and orange similar to these observed in the spectrum of Jupiter. The
+darkest of these bands, however, are not seen in the spectrum of the
+ring; this might have been expected, since the ring probably has
+but little atmosphere.
+
+\nbarticle{637.} \nbparatext{The Rings.}---The most remarkable peculiarity of Saturn is
+his \textit{ring-system}. The planet is surrounded by three, thin, flat, concentric
+rings like circular discs of paper pierced through the centre.
+Two of them are bright, while the third, the one nearest to the planet,
+is dusky and comparatively difficult to see. They are generally referred
+to by Struve's notation as $A$, $B$, and $C$, $A$ being the exterior
+one.
+
+\includegraphicsmid{illo182}{\textsc{Fig.~182.}---Saturn and his Rings.}
+
+For nearly fifty years this appendage of Saturn was a complete
+enigma to astronomers, Galileo, in 1610, saw with his little telescope
+%% -----File: 372.png---Folio 361-------
+that the planet appeared to have something attached to it on
+each side, and he announced the discovery that ``the outermost
+planet is triple,''---``ultimam planetam tergeminam observavi.''
+Not long afterwards the rings were edgewise to the earth so that they
+became invisible to him; and in his perplexity he inquired ``whether
+Saturn had devoured his children, according to the legend.'' Huyghens,
+%% -----File: 373.png---Folio 362-------
+in 1655, was the first to solve the problem and explain the true
+structure of the rings. Cassini,\footnote
+  {In consequence of a misunderstanding of some expressions used by Ball, an
+  English astronomer who observed Saturn in 1665--66, the discovery of the division
+  between the rings was for a time attributed to him, and statements to that effect
+  will be found in a number of important books. The original drawings belonging
+  to his paper in the Philosophical Transactions have, however, recently been
+  found, and show that he did not see the division at all, nor, indeed, even understand
+  that the appendage was a ring.}
+twenty years later, discovered that
+the ring was double,---composed of two concentric portions with a
+narrow black rift of division between them.
+
+The third, or dusky ring, $C$, is an American discovery, and was
+first brought to light by W.~C. Bond at Cambridge, U.~S., in November,
+1850. About two weeks later, but before the news had been
+published in England, it was also discovered independently by
+Dawes.
+
+\begin{fineprint}
+For a while there was some question whether it was not really a new
+formation; but an examination of old drawings shows that Herschel and
+several other astronomers had previously seen it where it crosses the planet,
+although without recognizing its character.
+
+\nbarticle{638.} \nbparatext{Dimensions of the Rings.}---The outer ring, $A$, has an exterior
+diameter of 168,000 miles, and is a little more than 10,000 miles wide. The
+division between it and ring, $B$, is about 1600 miles in width, and apparently
+perfectly uniform all around. Ring $B$ is about 16,500 miles wide, and is
+much brighter than $A$, especially at its outer edge. At the inner edge it
+becomes less brilliant, and is joined without any sharp line of demarcation
+by ring $C$, which is sometimes known as the ``\textit{gauze}'' or ``\textit{crape}'' ring,
+because it is only feebly luminous and is semi-transparent, allowing the
+edge of the planet to be seen through it. The innermost ring is nearly, perhaps
+not quite, as wide as the outer one, $A$. There is thus left a clear space
+of from 9000 to 10,000 miles in width between the planet's equator and the
+inner edge of the gauze ring, the whole ring system having an external
+diameter of 168,000 miles, and a width of between 36,000 and 37,000.
+\end{fineprint}
+
+The \textit{thickness} of the rings is very small indeed, probably not exceeding
+100 miles. If we were to construct a model of them on the
+scale of 10,000 miles to the inch, so that the outer one would be nearly
+seventeen inches in diameter, the thickness of an ordinary sheet of
+writing paper would be about in due proportion. This extreme thinness
+is proved by the appearances presented when the plane of the
+ring is directed towards the earth, as it is once in every fifteen years.
+%% -----File: 374.png---Folio 363-------
+At that time the ring becomes invisible except to the most powerful
+telescopes.
+
+\nbarticle{639.} \nbparatext{Phases of the Rings.}---The rings are parallel to the equator
+of the planet, which is inclined about $27°$ to its orbit, and about $28°$
+to the plane of the ecliptic, the two nodes of the ring being in longitude
+$166°$ and $346°$, in the constellations of Aquarius and Leo. Now
+in the planet's revolution around the sun, the plane of the planet's
+equator and of the rings always keeps parallel to itself (as shown
+in \figref{illo183}{Fig.~183}), just as does the plane of the earth's equator. Twice,
+therefore, in the planet's revolution, when the plane of the ring
+passes through the earth, we see it edgewise; and twice at its maximum
+width, when it is at the points half-way between the nodes. The
+angle of inclination being $28°$, the apparent width of the ring at the
+maximum is just about half its length. The last disappearance of
+the rings was in February, 1878; the next will be in the autumn of
+1891. Near the time of disappearance the ring appears simply as
+a thin needle of light projecting on each side of the planet to a
+distance nearly equal to its diameter. Upon this the satellites are
+threaded like beads when they pass between us and the planet.
+
+\includegraphicsmid{illo183}{\textsc{Fig.~183.}---The Phases of Saturn's Rings.}
+
+\nbarticle{640.} \nbparatext{Irregularities of Surface and Structure.}---When the rings
+are edgewise we find that there are notable irregularities upon them.
+They are not truly plane, nor quite of even thickness throughout.
+
+\begin{fineprint}
+The same thing is indicated by certain peculiarities sometimes reported
+in the form of the shadow cast by the planet on the rings; but caution must
+be used in accepting and interpreting such observations, because illusions
+%% -----File: 375.png---Folio 364-------
+are very apt to occur from the least indistinctness of vision or feebleness of
+light. The writer has usually found that the better the seeing, the fewer
+abnormal appearances were noted, and the experience of the Washington
+observers is the same.
+\end{fineprint}
+
+It can hardly be doubted that the details of the rings are continually
+changing to some extent. Thus the outer ring, $A$, is occasionally
+divided into two by a very narrow black line known as ``Encke's
+division,'' although more usually there is merely a darkish streak
+upon it, not amounting to a real ``crack'' in the surface.
+
+\nbarticle{641.} \nbparatext{Structure of the Rings.}---It is now universally admitted
+that the rings are not continuous sheets of either solid or liquid
+matter, but are composed of a swarm of separate particles, each a
+little independent moon pursuing its own path around the planet.
+The idea was suggested long ago, by J.~Cassini in 1715, and by
+Wright in 1750, but was lost sight of until Bond revived it in connection
+with his discovery of the dusky ring. Professor Benjamin
+Peirce soon afterwards demonstrated that the rings could not be continuous
+solids; and Clerk Maxwell finally showed that they can be
+neither solid nor liquid sheets, but that all the known conditions would
+be answered by supposing them to consist of a flock of separate and
+independent bodies, moving in orbits nearly circular and in one
+plane,---in fact, a swarm of meteors.
+
+\begin{fineprint}
+\nbarticle{642.} \nbparatext{Stability of the Ring.}---If the ring were solid it would certainly
+not be stable, and the least disturbance would bring it down upon
+the planet; nor is it certain that even the swarm-like structure makes it
+forever secure. It is impossible to say positively that the rings may not
+after a time be broken up. A few years ago there was much interest in a
+speculation which Struve published in 1851. All the measures which he
+could obtain up to that date appeared to show that a change was actually
+in progress, and that the inner edge of the ring was extending itself towards
+the planet. His latest series of measurements (in 1885) does not, however,
+confirm this theory. They show no considerable change since 1850, and the
+measurements of other observers agree with his in this respect.
+
+The researches of Professor Kirkwood of Indiana make it probable that
+the divisions in the ring are due to the perturbations produced by the satellites.
+They occur at distances from the planet where the period of a small
+body would be precisely commensurable with the periods of a number of
+the satellites. It will be remembered that similar gaps are found in the
+distribution of the asteroids, at points when the period of an asteroid would
+be commensurable with that of Jupiter.
+\end{fineprint}
+%% -----File: 376.png---Folio 365-------
+
+\nbarticle{643.} \nbparatext{Satellites.}---Saturn has eight\footnote
+  {Until Herschel's time it was customary to distinguish the satellites as first,
+  second, etc., in order of distance from the planet; but as Herschel's new satellites
+  were within the orbits of those which were known before, their discovery confused
+  matters, and the confusion became worse confounded when the eighth
+  appeared. They are now usually designated by names assigned by Sir John
+  Herschel as follows, beginning with the most remote, namely: Iap\u{e}tus (Hyperion),
+  Titan; Rhea, Dione, Tethys; Enceladus, Mimas. It will be noticed that
+  these names, leaving out Hyperion, which was undiscovered when they were
+  assigned, form a line and a half of a regular Latin pentameter.}
+of these attendants. The
+largest of them was discovered by Huyghens in 1655. It appears as
+a star of the ninth magnitude, and is easily observable with a three-inch
+telescope. Four others were discovered by Cassini before
+1700, two by Sir William Herschel near the end of the last century,
+and one, Hyperion, the latest addition to the planet's family, by
+Bond of Cambridge, in September, 1848, and independently by Lassell
+at Liverpool two days later.
+
+The range of the system is enormous. Iapetus has a distance of
+2,225000 miles, with a period of 79 days, nearly as long as that of
+Mercury. There is a remarkable variation in the brightness of this
+satellite. On the western side of the planet it is fully twice as bright
+as upon the eastern, which practically demonstrates that, like our own
+moon, it keeps the same face towards the planet at all times, one-half
+of its surface being much more brilliant than the other.
+
+Mimas, the nearest and smallest of the satellites, coasts around the
+edge of the ring at a distance from it of only 34,000 miles, or
+118,000 from the planet's centre, having a period of only $22\frac{1}{2}$ hours.
+This satellite is so small and so near the planet that it can be seen
+only by very large telescopes and under favorable conditions.
+
+Titan, as its name suggests, is by far the largest of the family.
+Its distance is about 770,000 miles, and its period a little less than
+16 days. It is probably 3000 or 4000 miles in diameter, and according
+to Stone, its mass is $\frac{1}{4600}$ of Saturn's.
+
+\begin{fineprint}
+\nbarticle{644.} \nbparatext{Peculiar Behavior of Hyperion.}---Hyperion has a distance
+of 934,000 miles, and a period of $21\frac{1}{4}$ days. Under the action of Titan its
+orbit is rendered considerably eccentric, and \textit{its line of apsides always keeps
+itself in the line of conjunction with Titan}, retrograding in a way which
+at first seemed to defy theoretical explanation, but turns out to be only
+a ``new case in celestial mechanics,'' and a necessary result of the disturbance
+by Titan.
+
+The orbit of Iapetus is inclined about $10°$ to the plane of the rings, but all
+%% -----File: 377.png---Folio 366-------
+the other satellites move exactly in their plane, and all the five inner ones
+move in orbits sensibly circular. The orbits of Iapetus, Hyperion, and Titan
+have a slight eccentricity. It is not at all impossible or even improbable
+that other minute satellites may yet be discovered in the great gap between
+Titan and Iapetus.
+\end{fineprint}
+
+\section*{URANUS.}
+
+\nbarticle{645.} As the satellites of Jupiter were the first heavenly bodies
+to be ``discovered,'' so Uranus was the first ``discovered'' planet,
+all the other planets then known having been known from prehistoric
+antiquity. On March 13, 1781, the elder Herschel, in sweeping
+over the heavens systematically with a seven-inch reflector made
+by himself, came upon an object which, by its disc, he saw at once
+was not an ordinary star. In a day or two he had ascertained that
+it moved, and announced the discovery as that of a \textit{comet}. After
+a short time, however, it became obvious from the computations
+of Lexell, that its orbit was nearly circular, that its distance was
+enormous, and that its path did not at all resemble that ordinarily
+taken by a comet; and within a year its planetary character was
+recognized and it was formally admitted as a new member of the
+solar system. The name of \textit{Uranus}, suggested by Bode, finally prevailed
+over other appellations (Herschel himself called it the Georgium
+Sidus, in honor of the king), with the symbol \raisebox{-1pt}{\Uranus} or {\fontseries{b}\uranus}. The former is
+still generally used by English astronomers.
+
+\begin{fineprint}
+The discovery of a new planet, a thing then utterly unprecedented, caused
+great excitement. The king knighted Herschel, gave him a pension, and
+furnished him with the funds for constructing his great forty-foot reflector
+of four feet aperture, with which he afterwards discovered the two inner
+satellites of Saturn. It was found on reckoning back from the date of
+Herschel's discovery that the planet had been several times before observed
+as a star by astronomers who narrowly missed the honor which fell to the
+more fortunate and diligent Herschel. Twelve such observations had been
+made by Lemonnier alone.
+\end{fineprint}
+
+\nbarticle{646.} \nbparatext{Orbit.}---The \textit{mean-distance} of Uranus from the sun is very
+nearly 1800 millions of miles, and the \textit{eccentricity} a trifle less than
+that of Jupiter's orbit, amounting to about 83,000000. The \textit{inclination}
+of the orbit to the plane of the ecliptic is very slight, only $46'$.
+The planet's \textit{periodic time} is 84~years, and the \textit{synodic period} (from
+opposition to opposition) $369^{\text{d}}\: 8^{\text{h}}$. The \textit{orbital-velocity} is $4 \frac{1}{3}$ miles
+per second.
+%% -----File: 378.png---Folio 367-------
+
+\sloppy
+\nbarticle{647.} \nbparatext{Appearance and Magnitude.}---Uranus is distinctly visible to
+the naked eye on a dark night as a small star of the so-called sixth
+magnitude. It is so remote, its orbit having a diameter more than
+19~times that of the earth's, that there is very little change in its
+appearance, and it makes no practical difference whether it is at
+opposition or quadrature.
+
+\fussy
+In the telescope it shows a sea-green disc of about $4''$ in apparent
+diameter, corresponding to a \textit{real diameter} of 32,000 miles. Its \textit{surface}
+is about 16~times, and its \textit{volume} about 66~times greater than
+that of the earth, so that the earth compares in size with Uranus
+about as the moon does with the earth. The \textit{mass} of Uranus is 14.6
+times that of the earth, and its \textit{density} and \textit{surface-gravity }are respectively
+0.22 and 0.90.
+
+\nbarticle{648.} \nbparatext{Albedo and Light.}---The reflecting power of the planet's
+surface is very high, its \textit{albedo}, according to Zöllner, being 0.64, even
+exceeding that of Jupiter. It is to be remembered, however, that
+sunlight at Uranus is only $\frac{1}{368}$ as intense as at the earth, and only
+about $\frac{1}{14}$ as intense as at Jupiter; so that the disc of the planet does
+not appear in the telescope even nearly as bright as a piece of Jupiter's
+disc of the same apparent size. The greenish blue tint of the planet
+is accounted for by the fact that its spectrum shows certain conspicuous
+dark bands in its lower portion, bands perhaps identical with
+these which are visible in the spectrum of Saturn, but much more
+intense. The $F$ line is also specially prominent in the spectrum of
+Uranus. These facts probably indicate a dense atmosphere.
+
+\nbarticle{649.} \nbparatext{Polar Compression, Belts, and Rotation.}---The disc of the
+planet shows a decided ellipticity---about $\frac{1}{14}$ according to the Princeton
+observations of 1883, which agree nearly with those of Schiaparelli.
+There are also sometimes visible upon the planet's disc
+certain extremely faint bands or belts, much like the belts of Jupiter
+viewed with a very small telescope. What is exceedingly singular,
+however, is that the trend of these belts seems to indicate a \textit{plane
+of rotation  not coinciding with the plane of the satellites' orbits}.
+Nearly all the observers who have seen them at all find that they are
+inclined to the satellites' orbit-plane at an angle of from $15°$ to $40°$.
+Now unless there is some error in Tisserand's investigations upon the
+motions of satellites, it is certain that the plane of these orbits must
+of necessity nearly coincide with the planet's equator. Probably the
+error lies in judging the direction of the belts, which at the best are
+at the very limit of visibility.
+%% -----File: 379.png---Folio 368-------
+
+One or two observers have assigned to the planet rotation periods
+ranging from $9^{\text{h}}$ to $12^{\text{h}}$; but it cannot be said that any determination
+of this element yet made is to be trusted.
+
+\nbarticle{650.} \nbparatext{Satellites.}---The planet has four satellites; viz., Ariel,
+Umbriel, Titania, and Oberon; Ariel being the nearest to the planet.
+The two brightest of them, Oberon and Titania, were discovered by
+Sir William Herschel a few years after the discovery of the planet.
+He observed them sufficiently to obtain a reasonably correct determination
+of their distances and periods.
+
+\begin{fineprint}
+It is not certain that he saw either of the other two, though he \textit{thought} he
+had found six satellites in all, and a few years ago a popular writer on
+astronomy actually credited the planet with \textit{eight} satellites,---the four
+whose names have been given, and four others which Herschel supposed he
+had seen.
+
+Ariel and Umbriel were first \textit{certainly} discovered by Lassell in 1851, and
+have since been satisfactorily observed by numerous large telescopes. They
+are telescopically the smallest bodies in the solar system, and the most
+difficult to see. In real size, they are, of course, much larger than the satellites
+of Mars or many of the asteroids, very likely measuring from 200 to
+500~miles in diameter; but they are ten times as far away as the asteroids,
+and illuminated by a sunlight not $\frac{1}{50}$ as brilliant as theirs.
+
+\nbarticle{651.} \nbparatext{Satellite Orbits.}---The orbits of the satellites are sensibly circular,
+and all lie in one plane, which, as has been said, \textit{ought} to be, and probably
+is, coincident with the plane of the planet's equator. They are very
+\textit{close-packed} also, Oberon having a distance of only 375,000 miles, with a
+period of $13^{\text{d}}\: 11^{\text{h}}$, while Ariel has a period of $2^{\text{d}}\: 12^{\text{h}}$, at a distance of 223,000
+miles. Titania, the largest and brightest of them, has a distance of 280,000
+miles, somewhat greater than that of the moon from the earth, with a period
+of $8^{\text{d}}\: 17^{\text{h}}$. Under favorable circumstances this satellite can be just seen with
+a telescope of eight or nine inches aperture.
+\end{fineprint}
+
+\sloppy
+\nbarticle{652.} \nbparatext{Plane of Revolution.}---The most remarkable thing about
+this satellite system remains to be mentioned. The \textit{plane of their
+orbits is inclined} $82°.2$ to the plane of the ecliptic, and in that plane
+they revolve \textit{backwards}; or we may say, what comes to the same
+thing, that their orbits are inclined to the ecliptic at an angle of
+$97°.8$, in which case their revolution is to be considered as \textit{direct}.
+
+\fussy
+\begin{fineprint}
+When the line of nodes of their orbit plane passes through the earth,
+as it did in 1840 and 1882, the orbits are seen edgewise and appear as
+straight lines. On the other hand, in 1861, they were seen almost \textit{in plan}
+%% -----File: 380.png---Folio 369-------
+as nearly perfect circles, and will be seen so again in 1903. The year 1882--83
+was a specially favorable time for determining the inclination of the orbits
+and the position of the nodes, as well as for measuring the polar compression
+of the planet.
+\end{fineprint}
+
+\section*{NEPTUNE.}
+
+\nbarticle{653.} The discovery of this planet is justly reckoned as the
+greatest triumph of mathematical astronomy. Uranus failed to move
+precisely in the path which the computers predicted for it, and was
+misguided by some unknown influence to an extent which a keen
+eye might almost see without telescopic aid. The difference between
+its observed place and that prescribed for it had become in 1845
+nearly as much as the ``intolerable'' quantity of $2'$ of arc.
+
+\begin{fineprint}
+Near the bright star Vega there are two little stars which form with it a
+small equilateral triangle, the sides of the triangle being about $1\frac{3}{4}°$ long.
+The northern one of the two little stars is the beautiful double-double star
+$\epsilon$~Lyræ, and can be seen as double by a keen eye without a telescope, the
+two companions being about $3 \frac{1}{2}'$ apart. Now the distance between the
+computed place of Uranus and its actual position was, when at its maximum,
+just a little more than half of the distance between these components
+of $\epsilon$~Lyræ, that only a keen eye can separate. One would almost say that
+such a minute discrepancy between observation and theory was hardly worth
+minding, and that to consider it \textit{``intolerable''} was what a Scotchman would
+call ``sheer pernickittyness.''
+\end{fineprint}
+
+But just these minute discrepancies constituted the data which
+were found sufficient for calculating the position of a hitherto
+unknown planet, and bringing it to light. Leverrier wrote to Galle,
+in substance: \textit{``Direct your telescope to a point on the ecliptic in the
+constellation of Aquarius, in longitude $326°$, and you will find within
+a degree of that place a new planet, looking like a star of about the
+ninth magnitude, and having a perceptible disc.''} The planet was
+found at Berlin on the night of Sept.~23, 1846, in exact accordance
+with this prediction, within half an hour after the astronomers began
+looking for it, and only about $52'$ distant from the precise point that
+Leverrier had indicated.
+
+\nbarticle{654.} So far as the mathematical operations are concerned, the
+honor is to be equally divided between two then young men,---Leverrier
+of Paris, and Adams of Cambridge, England. Each took
+up the problem, and by perfectly independent and considerably different
+methods arrived at substantially the same solution, and each
+%% -----File: 381.png---Folio 370-------
+promptly communicated the result (Adams some months earlier than
+Leverrier) to a practical astronomer provided with the necessary
+apparatus for actually detecting the planet.
+
+\begin{fineprint}
+Adams, who was then a graduate of three years' standing, a fellow and a
+tutor in his college, communicated his results to Challis, his professor of
+astronomy at Cambridge, in the autumn of 1845. Challis at once consulted
+Airy, the Astronomer Royal, but between them the matter rather lay
+in abeyance for some months, until a notice appeared of a preliminary paper
+by Leverrier, which indicated that he also had reached substantially the
+same conclusions as Adams. Then, at the urgent suggestion of Airy,
+Challis decided to begin the search at once, and to capture the planet by
+siege, so to speak. If he had had such star-maps as we now possess of the
+regions where the planet lay concealed, it would have been comparatively an
+easy operation; but as he had not, he decided to go over a space $10°$ wide
+by $30°$ long, and to go over it three times. The positions of all fixed
+stars would of course be the same at each of the three observations, but a
+planet would change its place in the meantime, and so would be surely
+detected.
+
+He began his work on July 29, including in his sweep all stars down to
+the tenth magnitude. When, on Oct.~1, he learned of the actual discovery
+of the planet, he had recorded the positions of something over 3000 stars,
+and was preparing to map them. He had already secured, as it turned out,
+three observations of the planet on July~30, Aug.~12, and Sept.~29, and
+of course it was only the question of a few weeks more or less when the
+discussion of the observations would have brought the planet to light.
+
+But while, this rather deliberate process was going on in England, Leverrier
+had revised his work, making a second approximation, and had communicated
+his results to Galle, at Berlin, substantially as above indicated. The
+Berlin astronomers had the great advantage of a new star-chart by Bremiker,
+covering that very region of the sky, and therefore did not need to enter
+upon any such tedious campaign as that begun by Challis. In less than
+half an hour they found a new star, not indicated on the map, and showing
+a sensible disc, just as Leverrier had predicted; and within twenty-four hours
+its motion proved it to be the planet.
+\end{fineprint}
+
+\nbarticle{655.} \nbparatext{Computed Elements Erroneous.}---Both Adams and Leverrier,
+besides computing the planet's position in the sky had deduced
+elements of its orbit, and a value for its mass, which turned out to be
+considerably erroneous. The reason was that they had assumed \textit{that
+the mean distance of the new planet from the sun would follow Bode's
+law}, a supposition which, as it turned out, is not even roughly true,
+although it was entirely warranted by the existing facts, since all the
+then known planets, not excepting Uranus, obey it with reasonable
+%% -----File: 382.png---Folio 371-------
+exactness. This assumption of an erroneous mean distance of thirty-eight
+astronomical units, instead of the true distance of thirty, carried
+with it errors in all the other elements of the orbit; and the computed
+elements are so wide of the truth that great authorities have maintained
+that the actual Neptune was not at all the Neptune of Leverrier
+and Adams, but an entirely different planet; and even that the
+discovery was a ``happy accident.'' It was not an accident at all,
+however. While the data and methods employed were not competent
+to determine the planet's \textit{orbit} accurately, they \textit{were} sufficient to
+determine the \textit{direction} of the unknown body, which was the one
+thing needed to insure its discovery. The computers informed the
+searchers precisely where to point their telescopes, and could do so
+again were a new case of the same kind to appear.
+
+\begin{fineprint}
+\nbarticle{656.} \nbparatext{Old Observations of Neptune.}---After a few weeks' observation
+of the new planet it became possible to compute an approximate orbit;
+and reckoning back by means of this approximate orbit, the approximate
+place on any given date for many years preceding could be found. On
+examining the observations of stars made by different astronomers in these
+regions of the sky, there were found several instances in which they had
+observed the planet; a star of the ninth magnitude in the proper place for
+Neptune being recorded in their star-catalogues, while the place is now
+vacant. These old observations, thus recovered, were of great use in determining
+the planet's orbit with accuracy.
+\end{fineprint}
+
+\nbarticle{657.} \nbparatext{The Orbit of Neptune.}---The planet's \textit{mean distance} from
+the sun is a little more than 2800,000000 of miles, instead of being
+over 3600,000000, as it should be according to Bode's law. The
+orbit, instead of being considerably eccentric, as it appeared to be
+from the computation of Adams and Leverrier, is more nearly circular
+than any other in the system except that of Venus, its \textit{eccentricity}
+being only $\frac{9}{1000}$. Even this small fraction, however, makes a variation
+of over 50,000000 of miles in the planet's distance from the sun
+at different parts of its orbit. The \textit{inclination} of the orbit is about
+$1 \frac{3}{4}°$. The \textit{period} of the planet is about 164 years, instead of 217, as
+it should have boon according to Leverrier's computed orbit. The
+\textit{orbital velocity} is about $3 \frac{1}{3}$ miles a second.
+
+\nbarticle{658.} \nbparatext{The Solar System as seen from Neptune.}---At Neptune's
+distance the sun itself has an apparent diameter of only a little more
+than $1'$ of arc,---only about the diameter of Venus when nearest us,
+and too small to be seen as a disc by the eye, if there are eyes on
+Neptune. The light and heat received from it are only $\frac{1}{900}$ part of
+%% -----File: 383.png---Folio 372-------
+what we get at the earth. Still, we must not imagine that, as compared
+with starlight or even moonlight, the Neptunian sunlight is feeble.
+
+\begin{fineprint}
+Assuming Zöllner's estimate that sunlight at the earth is 618,000 times as
+intense as the light of the full moon, we find that the sun, even at Neptune,
+gives a light equal to 687 full moons. This is about seventy-eight times the
+light of a standard candle at one metre distance, or about the light of a
+thousand candle power electric arc at a distance of $10 \frac{1}{2}$ feet---abundant for
+all visual purposes. In fact, as seen from Neptune, the sun would look very
+much like a large electric arc-lamp at a distance of a few feet. We call
+special attention to this, because erroneous statements are not unfrequently
+met with that ``at Neptune the sun would be only a first magnitude star.''
+It would really give about 44,000000 times the light of such a star.
+
+\nbarticle{659.} From Neptune the four terrestrial planets would be hopelessly
+invisible, unless with powerful telescopes and by carefully screening off
+the sunlight. Mars would never reach an elongation of $3°$ from the sun;
+the maximum elongation of the earth would be about $2°$, and that of Venus
+about $1 \frac{1}{2}°$. Jupiter, attaining an elongation of about $10°$, would probably
+be easily seen somewhat as we see Mercury. Saturn and Uranus would be
+conspicuous, though the latter is the only planet of the whole system that
+can be better seen from Neptune than it can be from the earth.
+\end{fineprint}
+
+\nbarticle{660.} \nbparatext{The Planet itself.}---Neptune appears in the telescope as a
+small star of between the eighth and ninth magnitudes, absolutely
+invisible to the naked eye, though easily seen with a good opera-glass.
+It shows a greenish disc, having an apparent diameter of about $2''.6$,
+which varies very little, since the entire range of variation in the
+planet's distance from us is only about $\frac{1}{15}$ of the whole. The real
+\textit{diameter} of the planet is about 35,500 miles (but the probable error of
+this must be fully 500 miles); the \textit{volume} is a little more than ninety
+times that of the earth. Its \textit{mass}, as determined by means of its satellite,
+is about 17~times that of the earth, and its \textit{density} 0.20.
+
+The planet's \textit{albedo}, according to Zöllner, is about forty-six per
+cent, a trifle lower than that of Saturn and Venus, and considerably
+below that of Jupiter and Uranus. There are no visible markings
+upon its surface, and nothing is known as to its rotation. The
+spectrum of the planet appears to be precisely like that of Uranus.
+The light is so feeble that the ordinary lines of the solar spectrum
+are difficult to make out, but there are a number of heavy, dark
+bands, which indicate the existence of a dense atmosphere, through
+which the light, reflected by the cloud-covered surface of the planet,
+is transmitted,---an atmosphere which appears to be identically the
+%% -----File: 384.png---Folio 373-------
+same on both Uranus and Neptune, while some of its constituents are
+probably present in Jupiter and Saturn, as shown by the principal
+dark band in the red. It is not possible as yet to identify the
+substance which produces these bands.
+
+It will be seen that Uranus and Neptune form a ``pair of twins''
+very much as the earth and Venus do; being nearly alike in magnitude,
+density, and other characteristics.
+
+\nbarticle{661.} \nbparatext{Satellite.}---Neptune has one satellite, discovered by Lassell
+within a month after the discovery of the planet itself. Its distance
+is 223,000 miles, and its period is $5^{\text{d}}\: 21^{\text{h}}\: 2^{\text{m}}.7$. Its orbit is inclined
+$34°\: 53'$, and it moves \textit{backward} in it; \textit{i.e.}, clockwise, from east
+towards the west, like the satellites of Uranus. It is a very small
+object, appearing of about the same brightness as Oberon, the outer
+satellite of Uranus. From its brightness, as compared with that
+of Neptune itself, we estimate that its diameter is about the same as
+that of our own moon, though perhaps a little larger.
+
+\begin{fineprint}
+Since Neptune is so far from the sun, and the planet has no other satellite
+of any size (none certainly comparable with this one), its motion must be
+practically undisturbed and very nearly uniform. It has been therefore proposed
+to make \textit{it a test of the uniformity of other motions in the solar system},
+such as the length of the day and the length of the month. It revolves
+rapidly enough to admit of very precise observations by large telescopes.
+It is, of course, possible that the planet has other undiscovered satellites,
+but if so, they must be very minute.
+
+\nbarticle{662.} \nbparatext{Trans-Neptunian Planet.}---Perhaps the breaking down of Bode's
+law at Neptune may be regarded as an indication that the system terminates
+with him, and that there is no remoter planet. If such a planet exists, however,
+it is sure to be found sooner or later, either by means of its disturbing
+action upon Uranus and Neptune, or else by the methods of the asteroid
+hunters, although, of course, its slow motion will render its detection in this
+way difficult.
+
+\sloppy
+In 1877, Professor Todd, from a graphical investigation of the outstanding
+differences between the computed and observed places of Uranus (after
+allowing for Neptune's action), concluded that an undiscovered planet very
+probably exists, and that its longitude was then about $170°$. He made a
+careful but unsuccessful search for it with the Washington telescope, going
+over a region about $40°$ long by $2°$ wide with a power of 400, hoping to
+recognize the planet by its disc.
+
+\fussy
+\nbarticle{663.} In the \hyperref[app:I]{Appendix}, we give tables containing the most accurate data
+of the planetary system at present available, but with renewed cautions to
+the student that these data are of very different degrees of accuracy.
+%% -----File: 385.png---Folio 374-------
+
+The \textit{distances} (in astronomical units), and the \textit{periods} of the planets
+(except perhaps some of the asteroids) are known with extreme precision;
+probably the very last figure of the table may be trusted. The other
+elements of their \textit{orbits} are also known very closely, if not quite so precisely
+as the distances and periods. The \textit{masses, in terms of the sun's mass}, stand
+next to the orbit elements in order of precision, with an error probably not
+exceeding one per cent (except, however, in the case of Mercury, the mass
+of which remains still very uncertain).
+
+The ratio of the \textit{earth's} mass to the sun's is however less accurately known,
+being at present subject to an uncertainty of at least two per cent. This is
+because its determination involves a knowledge of the solar parallax (\artref{Art.}{278*}),
+the \textit{cube} of which appears in the formula for the ratio of the masses.
+
+Of course all the masses of the planets \textit{expressed in terms of the earth's mass}
+are subject to the same uncertainty in addition to all other possible causes
+of error.
+
+When we come to the \textit{diameters, volumes}, and \textit{densities} of the planets,
+the data become less and less certain, as has been pointed out before. For
+the nearer and larger planets, say Venus, Mars, and Jupiter, they are reasonably
+satisfactory, for the remoter ones less so, and the figures for the density
+of the distant planets,---Mercury, Uranus, and Neptune, for instance,---are
+very likely subject to errors of ten or twenty per cent, if not more.
+
+\nbarticle{664.} We borrow from Herschel's ``Outlines of Astronomy'' the following
+illustration of the relative magnitudes and distances of the members of our
+system. ``Choose any well-levelled field. On it place a globe two feet in
+diameter. This will represent the sun; Mercury will be represented by a
+\textit{grain of mustard-seed} on the circumference of a circle 164 feet in diameter
+for its orbit; Venus, a \textit{pea} on a circle of 284~feet in diameter; the Earth also,
+a \textit{pea} on a circle of 430~feet; Mars, a rather large \textit{pin's-head} on a circle of
+654~feet; the asteroids, \textit{grains of sand} in orbits of 1000 to 1200 feet; Jupiter,
+a \textit{moderate-sized orange} in a circle nearly half a mile across; Saturn, a \textit{small
+orange} on a circle of four-fifths of a mile; Uranus, a \textit{full-sized cherry} or
+\textit{small plum} upon the circumference of a circle more than a mile and a half;
+and finally Neptune, a \textit{good-sized plum} on a circle about two miles and a half
+in diameter\ldots. To imitate the motions of the planets in the above-mentioned
+orbits, Mercury must describe its own diameter in 41~seconds; Venus,
+in $4^{\text{m}}\: 14^{\text{s}}$; the Earth, in 7~minutes; Mars, in $4^{\text{m}}\: 48^{\text{s}}$; Jupiter, in $2^{\text{h}}\: 56^{\text{m}}$;
+Saturn, in $3^{\text{h}}\: 13^{\text{m}}$; Uranus, in $2^{\text{h}}\: 16^{\text{m}}$; and Neptune, in $3^{\text{h}}\: 30^{\text{m}}$.'' We may add
+that on this scale the nearest star would be on the opposite side of the globe,
+at the antipodes, 8000 miles away.
+\end{fineprint}
+\chelabel{CHAPTERXV}
+
+%% -----File: 386.png---Folio 375-------
+
+\Chapter{XVI}{Determination of Sun's Parallax and Distance}
+\nbchapterhang{\stretchyspace
+THE DETERMINATION OF THE SUN'S HORIZONTAL PARALLAX
+AND DISTANCE.---TRANSITS OF VENUS AND OPPOSITIONS OF
+MARS.---GRAVITATIONAL METHODS.---DETERMINATION BY
+MEANS OF THE VELOCITY OF LIGHT.}
+
+\nbarticle{665.} \textsc{This} problem, from some points of view, is the most fundamental
+of all that are encountered by the astronomer. It is true that
+it is possible to deal with many of the subjects that present themselves
+in the science without the necessity of employing any units of length
+and mass but these that are purely astronomical, leaving for subsequent
+determination the relation between these units and the more
+familiar ones of ordinary life: we can get, so to speak, a map of the
+solar system, \textit{correct in proportion, though without a scale of miles}.
+But to give the map its real meaning and use, we must find the scale
+finally, if not at first, and until this is done we can form no true conceptions
+of the actual dimensions, masses, and distances of the heavenly
+bodies.
+
+The problem of finding the true value of the astronomical unit is
+difficult, because of the great disproportion between the size of the
+earth and the distance of the sun. The relative smallness of the
+earth limits the length of our available ``base line,'' which is less
+than $\frac{1}{12000}$ part of the distance to be determined by it. It is as if
+a person confined in an upper room with a wide prospect were set to
+determine the distance of objects ten miles or more away, without
+going outside the limits of his single window. It is hopeless to look
+for accurate results by \textit{direct} methods, such as answer well enough in
+the moon's case, and astronomers are driven to indirect ones.
+
+\begin{fineprint}
+\nbarticle{666.} \nbparatext{Historical.}---Until nearly 1700 no even reasonably accurate
+knowledge of the sun's distance had been obtained. Aristarchus, by a very
+ingenious though inaccurate method, had found, as he thought, that the
+distance of the sun was nineteen times as great as that of the moon (it is
+really 390 times as great), and Hipparchus, combining this determination of
+Aristarchus with his own knowledge of the moon's distance, estimated the
+%% -----File: 387.png---Folio 376-------
+sun's parallax at $3'$, which is more than twenty times too large. This value
+was accepted by Ptolemy, and remained undisputed for twelve centuries, until
+Kepler, from Tycho's observations of Mars, satisfied himself that the sun's
+parallax could not exceed $1'$; \textit{i.e.}, that the sun's distance must be at least as
+great as twelve or fifteen millions of miles. Between 1670 and 1680 Cassini
+proposed to determine the solar parallax by observations of Mars; for by that
+time the distance of Mars from the earth at any moment could be very accurately
+computed in astronomical units, so that the determination of the parallax
+of Mars would make known that of the sun. Observations in France,
+compared with observations made in South America by astronomers sent out
+for the purpose, showed that the parallax of Mars could not exceed $25''$, or
+that of the sun, $10''$. Cassini set it at $9''.5$, corresponding to a distance of
+86,000000 of miles,---giving the first reasonable approach to the true dimensions
+of the solar system.
+
+In 1677, and more fully in 1716, Halley explained how transits of Venus
+might be utilized to furnish a far more accurate determination of the solar
+parallax than was possible by any method before used. He died before the
+transits of 1761 and 1769 occurred, but they were both observed, the first
+not very satisfactorily, but the second with perfect success, and in the most
+widely separated parts of the globe. The results, however, were by no
+means as accordant as had been expected. Various values of the sun's parallax
+were deduced, ranging all the way from $8 \frac{1}{2}''$ to $9''$, according to the
+observations used, and the way they were treated in the discussion.
+Towards the end of the century, La~Place adopted and used for a while the
+value $8''.81$, while Delambre preferred $8''.6$.
+\end{fineprint}
+
+\nbarticle{667.} In 1822--24 Encke collected all the transit observations that
+had been published, and discussed them as a whole in an extremely
+thorough manner, deducing as a final result from the two transits of
+1761 and 1769, $8''.5776$, corresponding to a distance of $95 \frac{1}{2}$ millions
+of miles. The decimal is very imposing, and the discussion by
+which it was obtained was unquestionably thorough and laborious,
+so that his value stood unquestioned and classical for many years.
+
+The first note of dissent was heard in 1854, when Hansen, in
+publishing certain researches upon the motion of the moon, announced
+that Encke's parallax was certainly too small; he afterwards
+fixed the figure at $8''.97$, but the correction of certain numerical
+errors in his work reduced this result to $8''.92$.
+
+\begin{fineprint}
+Three or four years later Leverrier found a value of $8''.95$ from the so-called
+\textit{lunar equation} of the sun's motion; and in 1862 Foucault, from a new
+determination of the velocity of light, combined with the constant of aberration,
+got the value $8''.86$. A re-discussion of the old transit of Venus
+observations was then made by Stone, of England, who deduced from them
+%% -----File: 388.png---Folio 377-------
+a value of $8''.943$. The value of $8''.95$ was adopted by the British Nautical
+Almanac, and used in it until the issue of 1882. The corresponding distance
+of $91\frac{1}{2}$ millions of miles found its way into numerous text-books, and,
+though known to be erroneous, still holds its place in some of them.
+\end{fineprint}
+
+In 1867 Newcomb made a discussion of all the data then available,
+and obtained the value $8''.848$ (or $8''.85$ practically), which
+value is now (1888) used in all the Nautical Almanacs except the
+French. They use $8''.86$, the result of an investigation published by
+Leverrier in 1872.
+
+\nbarticle{668.} The observations of Gill on the planet Mars in 1877, and
+the new determinations of the velocity of light by Michelson and
+Newcomb in this country, as well as the investigations of Neison
+and others upon the so-called ``parallactic inequality'' of the moon,
+all point, however, to a somewhat smaller value. Professor Newcomb
+says (in 1885), ``All we can say at present is that the solar
+parallax is probably between $8''.78$ and $8''.83$, or if outside these
+limits, that it can be very little outside.''
+
+\begin{fineprint}
+It is not, however, thought worth while to change the constant used in
+the almanacs until the final reduction of the transits of 1874 and 1882 has
+been made, and until certain experiments and investigations now in progress
+have been finished. The difference between $8''.80$ and $8''.85$ is of no practical
+account for \textit{almanac} purposes, and the change would involve alterations
+in a number of the tables.
+
+Accepting Clarke's value of the earth's equatorial radius (\artref{Art.}{145}), viz.,
+$6,\!378,\!206.4^{\text{m}}$ or 3963.3 miles, we find that a solar parallax of
+\begin{center}
+\begin{tabular}{c@{ }c@{ }c@{ } c@{ } c@{ } c@{ } c@{ }c@{ = } l@{ }c@{ }}
+$8''.75 $&corresponds&to&23,573&radii&of&the&earth&93,428000&miles.\\
+$8''.80 $&   ``  & ``&23,439& `` &  ``& `` &``&92,897000& ``\\
+$8''.85 $&   ``  &``&23,307& `` &  ``& `` &  `` &92,372000& ``\\
+$8''.90 $&   ``  &``&23,196& `` &  ``& `` &  `` &91,852000& ``
+\end{tabular}
+\end{center}
+\end{fineprint}
+
+\newenvironment{nbitemize}%
+  {\begin{list}{}{\setlength{\itemsep}{0pt}%
+                  \setlength{\parsep}{0pt}%
+                  \setlength{\topsep}{2pt}}}%
+  {\end{list}}
+
+\nbarticle{669.} \nbparatext{Methods of finding the Solar Parallax and Distance.}---We
+may classify them as follows:---
+\pagebreak
+\begin{nbitemize}
+
+\item[I.] Ancient Methods.
+
+\begin{nbitemize}
+\item[(A)] Method of Aristarchus [0].
+
+\item[(B)] Method of Hipparchus [0].
+\end{nbitemize}
+
+\item[II.] Geometrical and Trigonometrical Methods, in which we attempt
+to find by angular measurements the parallax, either of the
+sun itself or of one of the nearer planets.
+%% -----File: 389.png---Folio 378-------
+
+  \begin{nbitemize}
+  \item[(\textit{A})] The direct method [0].
+
+  \item[(\textit{B})] Observations of the displacement of Mars among the stars
+at the time of opposition.
+
+    \begin{nbitemize}
+    \item[(\textit{a})] {\small Declination observations from two or more stations in widely
+different latitudes made with meridian circles or micrometer
+[25].}
+
+    \item[(\textit{b})] {\small Observations made at a single station near the equator, by
+measuring the distance of the planet east or west from
+neighboring stars, using the heliometer [90].}
+  \end{nbitemize} % small letters
+
+  \item[(\textit{C})] Declination observations of Venus [20].
+
+  \item[(\textit{D})] Observations of one of the nearer asteroids in the same
+way as Mars.
+
+    \begin{nbitemize}
+    \item[(\textit{a})] {\small Meridian observations at two stations in widely different
+latitudes [20].}
+
+    \item[(\textit{b})] {\small Heliometer observations at an equatorial station [75].}
+    \end{nbitemize}
+
+  \item[(\textit{E})] Observations of the transits of Venus at widely separated
+stations.
+
+    \begin{nbitemize}
+    \item[(\textit{a})] {\small Observations of the contacts.}
+
+      (1)\hspace{0.3em}{\small Halley's method---the ``method of \textit{durations}'' [40].}
+
+      (2)\hspace{0.3em}{\small Delisle's method---observation of \textit{absolute times} [50].}
+
+    \item[(\textit{b})] {\small Heliometer measurements of the position of the planet on
+the sun [75].}
+
+    \item[(\textit{c})] {\small Photographic methods---various [20 to 75].}
+    \end{nbitemize}
+  \end{nbitemize}
+
+\item[III.] Gravitational methods.
+
+  \begin{nbitemize}
+  \item[(\textit{A})] By the moon's parallactic inequality [70].
+
+  \item[(\textit{B})] By the lunar equation of the sun's motion [40].
+
+  \item[(\textit{C})] By the perturbations produced by the earth on Venus and
+Mars [70]; (ultimately [95]).
+  \end{nbitemize}
+
+\item[IV.] By the Velocity of Light, combined with
+
+  \begin{nbitemize}
+  \item[(\textit{A})] The light equation [80].
+
+  \item[(\textit{B})] The constant of aberration [90].
+  \end{nbitemize}
+\end{nbitemize}
+
+\begin{fineprint}
+The figures in brackets at the right are intended to express roughly the
+relative value of the different methods, on the scale of 100 for a method
+which would insure absolute accuracy.
+\end{fineprint}
+
+\nbarticle{670.} \nbparatext{Of the Ancient methods}, that of Aristarchus is so ingenious
+and simple that it really deserved to be successful. When the moon
+is exactly at the half phase, the angle at $M$ (\figref{illo184}{Fig.~184}) must be just
+%% -----File: 390.png---Folio 379-------
+$90°$, and the angle $AEM$ must equal $MSE$. If, then, we can find
+how much shorter the arc $NM$ (from new to half moon) is than $MF$
+(from half moon to full), \textit{half the difference will measure $AM$, and
+give the angle at $S$.} Aristarchus concluded that the first quarter of
+the month was just about \textit{twelve hours} shorter than the second, so
+that $AM$ was measured by
+six hours' motion of the
+moon, or a little less than
+$4°$. Hence he found $SE$,
+the distance of the sun,
+to be about nineteen times
+$EM$---a value absurdly
+wrong, Since $SE$ is in fact
+nearly 390 times $EM$. The
+real difference between the two quarters of the month is only about
+half an hour, instead of twelve hours.
+
+\includegraphicsouter{illo184}{\textsc{Fig.~184.}\\
+Aristarchus' Method of Determining the Sun's Distance.}
+
+\sloppy
+The difficulty with the method is that, owing to the ragged and
+broken character of the lunar surface, it is impossible to observe
+the moment of half moon with sufficient accuracy.
+
+\begin{fineprint}
+\nbarticle{671.}  The estimate of Hipparchus was based upon the erroneous calculation
+of Aristarchus that the sun's distance is 19~times the moon's, and the
+solar parallax, therefore, $\frac{1}{19}$ of the moon's parallax.
+
+The ``radius of the earth's shadow,'' where the moon cuts it at a lunar
+eclipse, is given, as Hipparchus knew, by the formula $\rho = P + p - S$ (\artref{Art.}{372}),
+or $P + p = \rho + S$. Assuming that $P = 19p$, we have $20p = \rho + S$. Now
+$S$, the sun's semi-diameter, is about $15'$; and from the duration of lunar
+eclipses Hipparchus found $\rho$ to be about $40'$; hence he obtained for $p$, the
+solar parallax, a value a little less than $3'$, which, as has been already mentioned,
+was accepted by Ptolemy, and by succeeding astronomers for more
+than 1500 years. (Wolf's ``History of Astronomy,'' p.~175.)
+\end{fineprint}
+
+\fussy
+\nbarticle{672.} \nbparatext{Of the Geometrical Methods}, $A$, the ``direct method'' consists
+in observing the sun's apparent declination with the meridian
+circle at two stations widely differing in latitude, just as we observe
+the moon when finding its parallax (\artref{Art.}{239}). Theoretically,
+observations of this sort might give the value of the sun's parallax
+within $\frac{1}{2}''$ or so, but the method is practically worthless, because
+the errors of observation are large as compared with the quantity
+to be determined. The sun's limb is a very bad object to point on,
+and besides, its heat disturbs the adjustments of the instrument, thus
+rendering the observations still more inaccurate.
+%% -----File: 391.png---Folio 380-------
+
+\nbarticle{673.} The first of the two methods of observing the planet Mars
+is precisely the same as this direct method of observing the sun;
+but the distance of Mars at a ``near opposition'' is only a little
+more than $\frac{1}{3}$ that of the sun, so that any error of observation
+affects the final result by only about $\frac{1}{3}$ as much; and, moreover,
+Mars is a very good object to observe, so that the errors of observation
+themselves are much lessened. The planet's distance from
+the earth having been found in astronomical units by the method
+of \artref{Art.}{515}, the determination of its distance in miles will fix the
+value of this unit, and so give us directly the sun's distance and
+parallax.
+
+The method requires two observers working at a distance from each
+other with different instruments, which is a serious disadvantage.
+
+\begin{fineprint}
+For some unexplained reason, observations of this sort seem almost invariably
+to give too large a result for the solar parallax, averaging between
+$8''.90$ and $8''.98$. The red color of the planet may possibly have something
+to do with this by affecting the astronomical refraction. This method, in
+1680, was the first to give a reasonable approximation to the sun's true
+distance, as has been mentioned before.
+
+The planet Venus can be observed in the same way, and has been once
+so observed by Gillis, 1849--52, at Santiago, Chili, in co-operation with the
+Washington observers, but the result was not very satisfactory.
+\end{fineprint}
+
+\includegraphicsmid{illo185}{\textsc{Fig.~185.}---Effect of Parallax on the Right Ascension of Mars.}
+
+\nbarticle{674.} \nbparatext{Heliometer Observations of Mars} (Method \textit{b}).---It is possible,
+however, for a \text{single} observer to obtain better results than
+can be got by two or more using the preceding method. Suppose
+that the orbital motion of Mars is suspended for a while at opposition,
+and that the planet is on or near the celestial equator;
+and also that the observer is at a station, $O$, on the earth's
+equator. When Mars is rising at $M_e$, \figref{illo185}{Fig.~185}, the horizontal
+parallax $OM_eC$ depresses the planet; that is, he appears from $O$ to
+be further east than he would if seen from $C$, the centre of the
+earth; so that the parallax then \textit{increases} the planet's right ascension.
+Twelve hours later, when he is setting, the parallax will
+%% -----File: 392.png---Folio 381-------
+throw him towards the west, \textit{diminishing} his right ascension by the
+same amount. If, then, when the planet is rising, we measure carefully
+its distance west of a star $S$, which is supposed to be just east
+of it (the distance $M_{e}S$ in \figref{illo186}{Fig.~186}), and then measure the distance
+$M_{w}S$ from the same star again when it is setting, the difference will
+give us twice the horizontal parallax. The earth's rotation will
+have performed for the observer the function of a long journey in
+transporting him from one station to another 8000 miles away in a
+straight line.
+
+\includegraphicsouter{illo186}{\textsc{Fig.~186.}\\
+Micrometric Comparison of Mars with Neighboring Stars.}
+
+\nbarticle{675.} Of course the observations are not practically limited to the
+moment when the planet is just rising, nor is it necessary that the
+star measured from should be exactly east or west of the planet.
+Measures from a \textit{number} of the
+neighboring stars, $S_{1}$, $S_{2}$, $S_{3}$, and
+$S_{4}$ would fix the positions $M_{e}$ and
+$M_{w}$ with more accuracy than measures
+from $S$ alone. Nor will the
+planet stop in its orbit to be observed,
+nor will it have a declination
+of zero, nor can the observer
+command a station exactly on the
+earth's equator. But these variations
+from the ideal conditions do
+not at all affect the principles involved;
+they simply complicate the
+calculations slightly without compromising
+their accuracy.
+
+The method has the very great
+advantage that all the observations
+are made by one person, and with
+one instrument, so that, as far as can be seen, all errors that could
+affect the result are very thoroughly eliminated.
+
+\begin{fineprint}
+\nbarticle{676.}  The most elaborate determination of the solar parallax yet made
+by this method is that of Mr.~Gill, who was sent out for the purpose by the
+Royal Astronomical Society in 1877 to Ascension Island in the Atlantic
+Ocean. His result, from 350 sets of measurements, gives a solar parallax
+of $8''.783 + 0''.015$,---a result probably very close to the truth, though possibly
+a little small. In 1892 and 1894 favorable oppositions of Mars will
+occur again, and it is quite likely that the observations will be repeated on
+a scale even more extensive.
+%% -----File: 393.png---Folio 382-------
+
+Venus cannot be observed in this way, since either her rising or setting
+is in the daytime, when the small stars cannot be seen near her; but the
+nearer asteroids can be utilized by this method. As they are more distant,
+however, than Mars, a given error in observation produces a larger final
+error in the result.
+
+\sloppy
+\nbarticle{677.} \nbparatext{The Heliometer.}---The heliometer, the instrument employed in
+these measures, is one of the most important of the modern instruments of
+precision. As its name implies, it was first designed to measure the diameter
+of the sun, but it is now used to measure any distance ranging from
+a few minutes up to one or two degrees, which it does with the same accuracy
+as that with which the filar micrometer measures distances of a few
+seconds. It is a ``double image'' micrometer, made by dividing the object-glass
+of a telescope along its diameter, as shown
+in \figref{illo187}{Fig.~187}. The two halves are so mounted that
+they can slide by each other for a distance of three
+or four inches, the separation of the centres being
+accurately measured by a delicate scale, or by a
+micrometer screw operated and read by a suitable
+arrangement from the eye-end. The instrument
+is mounted equatorially with clock-work, and the
+tube can be turned in its cradle so as to make
+the line of division of the lenses lie in any desired
+direction. When the centres of the two halves of
+the object-glass coincide, the whole acts as a single
+lens, giving but one image of each object or star
+in the field of view. As soon as the centres are separated, each half of
+the object-glass forms its own image.
+
+\fussy
+\includegraphicsouter{illo187}{\textsc{Fig.~187.}---The Heliometer.}
+
+To measure the distance from Mars to a star, the telescope tube is turned
+so that the line of centres points in the right direction, and then the semi-lenses
+are separated until one of the two images of the star comes exactly in
+the centre of one of the images of Mars; this can be done in two positions
+of the semi-lens $A$ with respect to $B$, as shown by the \figref{illo187}{figure}. We may
+either make $S_{0}$ (the image of the star formed by semi-lens $B$) coincide with
+$M_{1}$ formed by $A$, or make $S_{2}$ coincide with $M_{0}$. The whole distance from
+1 to 2 then measures twice the distance between $M$ and $S$.
+\end{fineprint}
+
+\nbarticle{678.} \nbparatext{Transit of Venus Observations.}---At the time when Venus
+passes between us and the sun, her distance from the earth is only
+some 26,000000 of miles, so that her horizontal parallax is nearly
+four times as great as that of the sun itself. At this time her apparent
+displacement upon the sun's disc, due to a change of the observer's
+station upon the earth, is the \textit{difference} between her own parallax
+due to this displacement, and that of the sun itself; and this difference
+is greater than the sun's parallax nearly in the ratio of 3 to
+%% -----File: 394.png---Folio 383-------
+1, or, more exactly, of 723 to 277. The object, then, of the observations
+of a transit is to obtain in some way a measure of the angular
+displacement of Venus on the sun's disc, corresponding to the known
+distance between the observer's stations upon the earth.
+
+\includegraphicsouter{illo188}{\textsc{Fig.~188.}---Contacts in a Transit of Venus.}
+
+\nbarticle{679.} \nbparatext{Halley's Method.}---The method proposed by Halley, who in
+1677 brought to notice the great advantages presented by a transit of
+Venus for determining the sun's parallax, was as follows: Two stations
+are chosen upon the earth's surface, as far separated \textit{in latitude}
+as possible. From them we observe the \textit{duration} of the transit;
+that is, the interval of time between its beginning and end, both of
+which must be visible at both stations. If the clock runs correctly
+during the few hours during which the
+transit lasts, this is all that is necessary.
+We do not need to know its error in
+reference to Greenwich time, nor even in
+respect to the local time, except roughly.
+This was a great advantage of the method
+in those days, before the era of chronometers,
+when the determination of the
+longitude of a place was a very difficult
+and uncertain operation. The observation
+to be made is simply to note the
+clock time at which ``\textit{contact}'' occurs,
+there being four of these contacts,---two exterior and two internal,
+at the points marked 1, 2, 3, 4, in \figref{illo188}{Fig.~188}. Halley depended
+mainly on the two internal contacts, which he supposed could be
+observed with an error not exceeding one second of time.
+
+\nbarticle{680.} \nbparatext{Computation of the Parallax.}---Having the durations of the
+transits at the two stations, and knowing the hourly angular motion
+of Venus, we have at once and very accurately the length of the two
+chords described by Venus upon the sun, expressed in seconds of
+arc. We also know the sun's semi-diameter in seconds, and hence
+%% -----File: 395.png---Folio 384-------
+in the triangles (\figref{illo189}{Fig.~189}) $Sab$ and $Sde$, we can compute the length
+(in seconds still) of $Sb$ and $Se$, the difference of which, $be$, is the
+displacement sought, due to the distance between the stations on the
+earth.\footnote
+  {In order that the method may be practically successful, it is necessary that
+  the transit track should lie near the edge of the sun's disc, for two reasons. It
+  is desirable that the duration should not be more than three or four hours,
+  while for a central transit it lasts eight hours (\artref{Art.}{575}). Moreover, if the two
+  chords were near the centre of the disc, any small error in the length of either
+  chord would produce a great error in the computed distance between them.
+  When they lie as in the \figref{illo189}{figure} (which has been the case in all recent transits),
+  the reverse is true: a considerable error in the observed length of one of the
+  chords affects their computed distance only slightly.}
+The virtual base line is, of course, not the distance between
+$B$ and $E$ as a straight line, because that line is not perpendicular to
+the line of sight from the earth to Venus, nor to the plane of the
+planet's orbit, but the true value to be used is easily found. Calling
+this base line $\beta$, we have
+\[
+p'' = (be)'' × \left( \frac{277}{723} \right) \left( \frac{r}{ \beta} \right),
+\]
+$r$ being the radius of the earth.
+
+The rotation of the earth, of course, comes in to shift the places
+of $E$ and $B$ during the transit, but this can easily be allowed for.
+
+\includegraphicsmid{illo189}{\textsc{Fig.~189.}---Halley's Method.}
+
+\nbarticle{681.} \nbparatext{The Black Drop.}---Halley expected, as has been said, that
+it would be possible to observe the instant of internal contact within
+a single second of time, but he reckoned without
+his host. At the transits of 1761 and
+1769, at most of the stations the planet at
+the time of internal contact showed a ``ligament''
+or ``black drop,'' like \figref{illo190}{Fig.~190}, instead
+of presenting the appearance of a round disc
+neatly touching the edge of the sun; and the
+time of real contact was thus made doubtful
+by $10^{\text{s}}$ or $15^{\text{s}}$.
+
+\includegraphicsouter{illo190}{\textsc{Fig.~190.}---The Black Drop.}
+
+\begin{fineprint}
+This ``ligament'' depends upon the fact that the optical edge of the
+image of a bright body is not, and in the nature of things cannot be, absolutely
+sharp in the eye or in the telescope. In the eye itself we have
+irradiation. In the telescope we have the difficulty that even in a \textit{perfect}
+instrument the image of a luminous point or line has a certain width (which
+%% -----File: 396.png---Folio 385-------
+with a given magnifying power is less for a large instrument). Moreover
+a telescope is usually more or less imperfect, and practically adds other
+defects of definition, so that whenever the limbs of two objects approach
+each other in the field of view
+of a telescope we have more or
+less distortion due to the overlapping
+of the two ``penumbras
+of imperfect definition,''---the
+same sort of effect that
+is obtained by putting the
+thumb and finger in contact,
+holding them up within two
+or three inches of the eye and
+then separating them: as they
+separate, a ``black ligament''
+will be seen between them.
+
+\includegraphicsouter{illo191}{\textsc{Fig.~191.}---Atmosphere of Venus as seen during a Transit. (Vogel, 1882.)}
+
+With modern telescopes, and
+by great care in preventing the
+sun's image from being too
+bright, so as to diminish \textit{irradiation
+in the eye} as far as possible,
+the black drop was reduced
+to reasonably small proportions
+in 1874 and 1882, and
+practice beforehand with an ``artificial transit'' enabled the observer in some
+degree to allow for its effect. But a new difficulty appeared, from which
+there seems to be absolutely no way of escape,---the \textit{planet's atmosphere
+causes it to be surrounded by a luminous ring} as it enters upon the sun's
+disc, and thus renders the time of the contact uncertain by at least five
+or six seconds. In both the transits of 1874 and 1882, differences of that
+amount continually appeared among the results of the best observers. \figref{illo191}{Fig.~191}
+shows the appearances due to this cause as observed by Vogel in 1882.
+\end{fineprint}
+
+\nbarticle{682.} \nbparatext{Delisle's Method.}---Halley's method requires the use of \textit{polar}
+stations, uncomfortable and hard to reach, and also that the weather
+should permit the observer to see \textit{both} the beginning and end of the
+transit.
+
+Delisle's method, on the other hand, utilizes two stations \textit{near the
+equator}, taken on a line roughly parallel to the planet's motion. It
+requires also that the observers \textit{should know their longitude accurately},
+so as to be able to determine the Greenwich time at any moment;
+but it does not require that they should see \textit{both} the beginning and
+end of the transit; observations of \textit{either} phase can be utilized: and
+this is a great advantage. Suppose, then (\figref{illo192}{Fig.~192}), that the
+%% -----File: 397.png---Folio 386-------
+observer $W$ on one side of the earth notes the moment of internal
+contact in Greenwich time, the planet then being at $V_{1}$. When $E$
+notes the contact (also in Greenwich time) the planet will be at $V_{2}$,
+and the angle at $D$ will be the angular diameter of the earth as seen
+from $D$; \textit{i.e., simply twice the sun's parallax}. Now the angle $D$ is at
+once determined by the time occupied by Venus in passing from $V_{1}$
+to $V_{2}$, since in 584 days (the synodic period) she moves completely
+round from the line $DW$ to the same line again. If the time from
+$V_{1}$ to $V_{2}$ were twelve minutes, we should find the angle at $D$ to be
+about $18''$.
+
+\includegraphicsmid{illo192}{\textsc{Fig.~192.}---Delisle's Method.}
+
+\nbarticle{683.} \nbparatext{Heliometer Observations.}---Instead of observing simply the
+times of contact, and leaving the rest of the transit unutilized, as in
+the two preceding methods, it is possible to make a continuous series
+of measurements of the distance and direction of the planet from the
+nearest point of the sun's limb. These measurements are best made
+with the heliometer (\artref{Art.}{677}), and give the means of determining
+the planet's apparent position upon the sun's disc at any moment
+with extreme precision. Such sets of measurements, made at widely
+separated stations, will thus furnish accurate determinations of the
+apparent displacement of the planet on the sun's disc, corresponding
+to known distances on the earth, and so will give the solar parallax.
+
+During the transit of 1882 extensive series of observations of this
+sort were made by the German parties, two of which were in the
+United States,---one at Hartford, Conn., and the other at San
+Antonio, Texas. The results have not yet (August, 1888) been
+published, but they will soon appear, and it is understood that they
+are considerably more accordant than these obtained by any other
+method of observation.
+
+\sloppy
+\nbarticle{684.} \nbparatext{Photographic Observations.}---The heliometer measurements
+cannot be made very rapidly. Under the most favorable circumstances
+a complete set requires at least fifteen minutes, so that the
+%% -----File: 398.png---Folio 387-------
+whole number obtainable during the seven or eight hours of the
+transit is quite limited. Photographs, on the other hand, can be
+made with great rapidity (if necessary, at the rate of two a minute),
+and then after the transit we can measure at leisure the position of
+the planet on the sun's disc as shown upon the plate. At first sight
+this method appears extremely promising, and in 1874 great reliance
+was placed upon it. Nearly all the parties, some fifty in number,
+were provided with elaborate photographic apparatus of various
+kinds. On the whole, however, the results, upon discussion, appear
+to be no more accordant than those obtained by other methods, so
+that in 1882 the method was generally abandoned, and used only by
+the American parties, who employed an apparatus having some
+peculiar advantages of its own.
+
+\fussy
+\begin{fineprint}
+\nbarticle{685.} \nbparatext{English, German, and French Methods.}---In 1874, the English
+parties used telescopes of six or seven inches aperture, and magnified the
+image of the sun formed by the object-glass by a combination of lenses
+applied at the eye-end. There were no special appliances for eliminating the
+distortion produced by the enlarging lenses, nor for ascertaining the exact
+orientation of the picture (that is, the direction of the image upon the plate
+with reference to north and south), nor for determining its scale.
+
+\nbenlargepage
+The Germans and Russians employed a nearly similar apparatus, but
+with the important difference that at the principal focus of the object-glass
+they inserted a plate of glass ruled with squares. These, squares are photographed
+upon the image of the sun, and furnish a very satisfactory means
+of determining the scale and distortion, if any, of the image. The object-glasses
+used by the English and the Germans had a focal length of seven
+or eight feet. The French employed object-glasses with a focal length of
+some fourteen feet, the telescope being horizontal, while the rays of the sun
+were reflected into it by a plane mirror; instead of glass plates they used the
+old-fashioned metallic daguerreotype plates, in order to avoid any possible
+``creeping'' of the collodion film, which was feared in the more modern wet-plate
+process. The French plates furnish, however, no accurate orientation
+of the picture.
+
+\includegraphicsmid{illo193}{\textsc{Fig.~193.}---American Apparatus for Photographing the Transit of Venus.}
+
+\nbarticle{686.} \nbparatext{The American Apparatus.}---The Americans used a similar plan,
+with some modifications and additions. The telescope lenses employed were
+five inches in diameter and forty feet in focal length, so that the image
+directly formed upon the plate was about $4 \frac{1}{2}$~inches in diameter, and needed
+no enlargement. The telescope was placed horizontal and in the meridian,
+its exact direction being determinable by a small transit instrument which
+was mounted in such a manner that it could look info the photograph telescope,
+as into a collimator, when the reflector was removed. The reflector
+itself was a plane mirror of unsilvered glass driven by clock-work. \figref{illo193}{Fig.~193}
+%% -----File: 399.png---Folio 388-------
+shows the arrangement of the apparatus. In front of the photographic
+plate, and close to it, was supported a glass plate ruled with squares called
+the ``reticle plate,'' and in the narrow space between this and the photograph
+plate was suspended a plumb-line of fine silver wire, the image of which
+appeared upon the plate, and gave the means of determining the orientation
+of the image with extreme
+precision. If the reflector
+were, and would continue
+to be, \textit{perfectly plane}
+through the whole operation,
+the method could
+not fail to give extremely
+accurate results; but the
+measurements and discussion
+of the observations
+seem to show that this
+mirror was actually distorted
+to a considerable
+extent by the rays of the
+sun. On the whole the
+American plates do not
+appear to be much more
+trustworthy than those
+obtained by other methods
+Fig.~194 is a reduced
+copy of one of the photographs made at Princeton during the transit
+of 1882. The black disc near the middle, with a bright spot in the centre,
+%% -----File: 400.png---Folio 389-------
+is the image of a metal disc cemented to the reticle to mark the centre lines
+of the reticle plate; 192 plates were taken during the transit, and at some of
+the stations where the weather was good the number was much greater---nearly
+300 in some cases.
+
+\includegraphicsouter{illo194}{\textsc{Fig.~194.}---Photograph of Transit of Venus.}
+
+The difficulties to be encountered are numerous. Photographic irradiation,
+or the spread of the image on the plate, slight distortion of the image by the
+lenses or mirrors employed, irregularities of atmospheric refraction, uncertainty
+as to the precise scale of the picture,---all these present themselves
+in a very formidable manner. It is obvious why this should be so, when we
+recall that on a four-inch picture of the sun's disc, $\frac{1}{10\,000}$ of an inch corresponds
+to about $\frac{1}{20}$ of a second of arc, and the whole uncertainty as to the
+solar parallax does not amount to as much as that. An image of the sun,
+therefore, in which the position of Venus upon the sun's disc cannot be
+determined accurately without an error exceeding $\frac{1}{10\,000}$ of an inch, is of
+very little value. Imperfections that would be of no account whatever in
+plates taken for any other purpose make them practically worthless for
+this.
+
+Still there is reason to hope that considering the enormous number of
+photographs made in 1874 and 1882 (certainly not less than 5000 in all),
+the result to be obtained from such a mass of material will prove to be
+worth something.
+\end{fineprint}
+
+\textbf{Gravitational Methods.}---These are too recondite to permit any
+full explanation here. We can only indicate briefly the principles
+involved.
+
+\nbarticle{687.}  (1) The first of these methods is by the \textit{moon's parallactic
+inequality}, an irregularity in the moon's motion which has received
+this name, because by means of it the sun's parallax can be determined.
+It depends upon the fact that the sun's disturbing action
+upon the moon differs sensibly from what it would be if its distance,
+instead of being less than 400 times that of the moon from the earth,
+were \textit{infinitely} great.
+
+\textit{The disturbing action upon the half of the moon's orbit which lies
+nearest the sun is greater than on the opposite half of the orbit}. The
+retarding action of the tangential force, therefore, during the first
+quarter after new moon, is perceptibly greater than the acceleration
+produced during the second quarter (\artref{Art.}{447}), so that at the first and
+third quarters respectively, the moon is a little more than $2'$ behind
+and ahead of the place she would occupy if the tangential forces were
+equal in all four quadrants of the orbit---as they would be if the sun's
+distance were infinite. This puts the moon about \textit{four minutes of time
+behindhand} at the first quarter, and as much \textit{ahead} at the third; and
+\textit{if the centre of the moon could be observed within a fraction of a second
+%% -----File: 401.png---Folio 390-------
+of arc} (as it could if she were a mere point of light like a star), the
+observations would give a very accurate determination of the sun's
+distance. The irregularities of the moon's limb, however, and the
+worse fact, that at the first quarter we observe the \textit{western} limb, while
+at the third quarter it is the \textit{eastern} one which alone is observable,
+make the result somewhat uncertain, though the method certainly
+ranks high.
+
+\nbenlargepage
+\begin{fineprint}
+\nbarticle{688.} (2) The ``\textit{lunar equation of the sun's motion}'' is, it will be remembered,
+an apparent slight monthly displacement of the sun, amounting to
+about $6''.3$, and due to the fact that both earth and moon revolve around their
+common centre of gravity. It is generally made use of (\artref{Art.}{243}) to determine
+the mass of the moon as compared with that of the earth, using as a
+datum the assumed known distance of the sun; but if we consider the mass
+of the moon as known (determined by the tides, for instance), then we can
+find the sun's parallax\footnote
+  {Putting $L$ for the maximum value of the lunar equation (about $6''.3$ of arc),
+  $P$ for the sun's parallax, and $R$ and $r$ for the distance of the moon and the semi-diameter
+  of the earth respectively, we have the equation
+\[
+P = L \left( \frac{r}{R} \right) \left( \frac{E + m}{m} \right)
+  = L \left( \frac{1}{60} \right) (81)\  \text{nearly}.
+\]
+} in terms of the lunar equation.
+
+The method is not a good one, since the solar parallax, $8''.8$, is greater
+than the quantity by means of which it is determined.
+\end{fineprint}
+
+\sloppy
+\nbarticle{689.} (3) The third method (\textit{by the earth's perturbations of Venus
+and Mars}) is one of the most important of the whole list. It depends
+upon the principle that if the \textit{mass} of the earth, as compared
+with that of the sun, be accurately known, then the \textit{distance} of the
+sun can be found at once. The reader will remember that in \artref{Art.}{278}
+the mass of the sun was found by comparing the distance
+which the earth falls towards the sun in a second (as measured by
+the curvature of her orbit) with the force of gravity at the earth's
+surface; and in the calculation the sun's distance enters as a necessary
+datum. Now, if we know independently the \textit{sun's mass compared
+with the earth's}, the distance becomes the only unknown quantity,
+and can be found from the other data.
+
+\fussy
+\begin{fineprint}
+In the same way as in \artref{Art.}{536} we have
+\[
+(S + E) = 4 \pi^2 \left( \frac{D^3}{T^2} \right),
+\]
+in which $S$ and $E$ are the masses of the sun and earth, $D$ is the mean
+%% -----File: 402.png---Folio 391-------
+distance of the earth from the sun, and $T$ the number of seconds in a year.
+Also we have for the force of gravity at the earth's surface,
+\[
+  g = \frac{E}{r^2}, \quad \mathrm{or} \quad E = gr^2,
+\]
+in which $r$ is the earth's radius.
+
+Dividing the preceding equation by this we, get
+\[
+\frac{S+E}{E} = \frac{4\pi^2}{gT^2} \left ( \frac{D^3}{r^2} \right );
+\]
+whence
+\hfill$\displaystyle
+D^3 = \left (\frac{S+E}{E}\right ) \left ( \frac{gT^2r^2}{4\pi^2}\right ).
+$\hfill \phantom{whence}
+
+If we put $\dfrac{S}{E} = M$, this becomes
+\[
+D^3 = \left( \frac{M+1}{4\pi^2}\right) gT^2r^2.
+\]
+
+In this equation everything in the second term is known when we have
+once found $M$, or the ratio between the masses of the sun and earth; $g$ is
+found by pendulum observations on the earth, $T$ is the length of the year
+in seconds, and $r$ is the earth's radius.
+\end{fineprint}
+
+Now, the disturbing force of the earth upon its next neighbors,
+Mars and Venus, depends directly upon its mass as compared with
+the sun's mass, and the ratio of the masses can be determined when
+the perturbations have been accurately ascertained; though the calculation
+is, of course, anything but simple. But the great beauty of
+the method lies in this, that as time goes on, and the effect of the
+earth upon the revolution of the nodes and apsides of the neighboring
+orbits accumulates, the \textit{determination of the earth's mass in terms
+of the sun's becomes continually and cumulatively more precise}. Even
+at present the method ranks high for accuracy,---so high that Leverrier,
+who first developed it, would have nothing to do with the transit
+of Venus observations in 1874, declaring that all such old-fashioned
+ways of getting at the sun's parallax were relatively of no value.
+The method is the \textit{``method of the future,''} and two or three hundred
+years hence will have superseded all the others,---unless indeed it
+should appear that bodies at present unknown are interfering with
+the movements of our neighboring planets, or unless it should turn
+out that the law of gravitation is not quite so simple as it is now
+supposed to be.
+
+\nbarticle{690.} \nbparatext{The Physical Method.}---The physical, or \textit{``photo-tachy-metrical''}
+method, as it has been dubbed, depends upon the fundamental
+%% -----File: 403.png---Folio 392-------
+assumption that light travels in interplanetary space with the same
+velocity as \textit{in vacuo}. This is certainly very nearly true, and probably
+exactly so, though we cannot yet prove it.
+
+By the recent experiments of Michelson and Newcomb in this
+country, following the general method of Foucault, the velocity of
+light has been ascertained with very great precision and may be
+taken as 299,860 kilometres, or 186,330 miles, with a probable error
+which cannot well be as great as twenty-five miles either way.
+
+\nbarticle{691.} \nbparatext{Sun's Distance from the Equation of Light.}---(1) ``\textit{The
+equation of light}'' is the time occupied by light in travelling between
+the sun and earth, and is determined by observation of the eclipses
+of Jupiter's satellites (\artref{Art.}{629}). By simply multiplying the velocity
+of light by this time ($499^{\text{s}} \pm 2^{\text{s}}$) we have at once the sun's distance;
+and that independent of all knowledge as to the earth's dimensions.
+The reader will remember, however, that the determination
+of this ``light-equation'' is not yet so satisfactory as desirable on
+account of the indefinite nature of the eclipse observations involved.
+
+\nbarticle{692.} \nbparatext{From the Constant of Aberration.}---(2) When we know the
+velocity of light we can also derive the sun's distance from the
+``\textit{constant of aberration},'' and this constant, $20''.492$, derived from
+star observations (\artref{Art.}{225}), is known with a considerably higher
+percentage of accuracy than the light-equation.
+
+Calling the constant $\alpha$, we have
+\[
+\tan \alpha = \frac{U}{V},
+\]
+where $U$ is the velocity of the earth in its orbit, and $V$ the velocity
+of light. Now $U$ equals the circumference of the earth's orbit
+divided by the length of the year; \textit{i.e.},
+\begin{flalign*}
+&&  U &= \frac{2 \pi D}{T}; &&\phantom{hence }
+\\
+&\text{hence }&  \tan \alpha &= \frac{2 \pi D}{VT},  &&
+\\
+&\text{and }& D &=  \frac{\tan \alpha}{2 \pi} \left( VT \right).  &&
+\end{flalign*}
+On the whole it seems likely at present that the value of the sun's distance
+thus derived is the most accurate of all. Using $\alpha = 20''.492$ and
+%% -----File: 404.png---Folio 393-------
+$V = 186,330$ miles, we have $D = 92,975,500$ miles, and taking the
+earth's equatorial radius as 3963.296 miles (Clarke, 1878), we get
+$8''.793$ as the sun's equatorial horizontal parallax.
+
+\nbarticle{693.} The reader will notice that the \textit{geometrical} methods give the
+distance of the sun \textit{directly}, apart from all hypothesis or assumption,
+except as to the accuracy of the observations themselves, and of their
+necessary corrections for refraction, etc.: the \textit{gravitational} methods,
+on the other hand, assume the exactness of the law of gravitation;
+and the \textit{physical} method assumes that light travels in space with
+the same velocity as in our terrestrial experiments. The near accordance
+of the results obtained by the different methods shows that
+these assumptions must be very nearly correct, if not absolutely so.
+\chelabel{CHAPTERXVI}
+
+%% -----File: 405.png---Folio 394-------
+\Chapter{XVII}{Comets}
+\nbchapterhang{\stretchyspace
+COMETS: THEIR NUMBER, MOTIONS, AND ORBITS.---THEIR CONSTITUENT
+PARTS AND APPEARANCE.---THEIR SPEC\-TRA.---THEIR
+PHYSICAL CONSTITUTION, AND ORIGIN.}
+
+\nbarticle{694.}  \textsc{From} time to time bodies of a very different character from
+the planets make their appearance in the heavens, remain visible
+for some weeks or months, move over a longer or shorter path among
+the stars, and then vanish. These are the \textsc{Comets}, or ``\textit{hairy stars},''
+as the word means, since the appearance of such as are bright
+enough to be visible to the naked eye is that of a star surrounded
+by a hazy cloud, and usually carrying with it a streaming trail of
+light. Some of them have been magnificent objects,---the nucleus,
+or central star, as brilliant as Venus and visible even by day, while
+the cloudy head was nearly as large as the sun itself, and the tail
+extended from the horizon to the zenith,---a train of shining substances
+long enough to reach from the earth to the sun. The majority
+of comets, however, are faint, and visible only with a telescope.
+
+\sloppy
+\begin{fineprint}
+\nbarticle{695.} \nbparatext{Superstitions.}---In ancient times these bodies were regarded
+with great alarm and aversion, being considered from the astrological point
+of view as always ominous of evil. Their appearance was supposed to
+presage war, or pestilence, or the death of princes. These notions have
+survived until very recent times with more or less vigor, but, it is hardly
+necessary to say, without the least reason. The most careful research fails
+to show any effect upon the earth produced by a comet, even of the largest
+size. There is no observable change of temperature or of any meteorological
+condition, nor any effect upon vegetable or animal life.
+\end{fineprint}
+
+\fussy
+\nbarticle{696.} \nbparatext{Number of Comets.}---Thus far we have on our lists about
+650 different comets. About 400 of these were recorded previous to
+1600, before the invention of the telescope, and must, of course,
+have been bright enough to attract the attention of the naked eye.
+Since that time the number annually observed has increased very
+greatly, for only a few of these bodies, perhaps one in ten, are
+visible without telescopic aid. Their total number must be enormous.
+Not unfrequently from five to eight are discovered in a
+%% -----File: 406.png---Folio 395-------
+single year, and there is seldom a day when one or more is not in
+sight.
+
+While telescopic comets, however, are thus numerous, brilliant
+ones are comparatively rare. Between 1500 and 1800 there were,
+according to Newcomb, 79 visible to the naked eye, or about one
+in three and three-fourths years. Humboldt enumerates 43 within
+the same period as \textit{conspicuous}; during the first half of the present
+century there were 9 such, and since 1850 there have been 11. Since,
+and including, 1880 we have had 7,---a remarkable number for so
+short a time,---and two of them, the principal comet of 1881 and
+the great comet of 1882, were unusually fine ones. In August, 1881,
+for a little time two comets were conspicuously visible to the naked
+eye at once and near together in the sky, a thing almost if not quite
+unprecedented.
+
+\nbarticle{697.} \nbparatext{Designation of Comets.}---The more remarkable ones generally
+bear the name of their discoverer, or of some astronomer who
+made important investigations relating to them,---as for instance,
+Halley's, Encke's, and Donati's comets. They are also designated
+by the year of discovery, with a Roman number indicating the \textit{order
+of discovery} in the year. A third method of designation is by year
+and letter, the letters denoting the order in which the comets of a
+given year \textit{pass the perihelion}. Thus Donati's comet was both
+comet~F and comet~VI, 1858. Comet~I is, however, not \textit{necessarily}
+comet~A, though it usually is so. In some cases the comet bears the
+name of two discoverers. Thus the Pons--Brooks comet of 1883 is
+a comet which was discovered by Pons in 1812, and at its return in
+1883 was discovered by Brooks.
+
+\begin{fineprint}
+\nbarticle{698.} \nbparatext{The Discovery of Comets.}---As a rule these bodies are first seen
+by comet-hunters, who make a business of searching for them. For such
+purposes they are usually provided with a telescope known as a ``comet-seeker,''
+having an aperture of from four to six inches, with an eye-piece of
+low power, and a large field of view. When first seen, a comet is usually
+a mere roundish patch of faintly luminous cloud, which, if really a comet,
+will reveal its true character within an hour or two by its motion. Some
+observers have found a great number of these bodies. Messier discovered
+twelve between 1794 and 1798, and Pons twenty-seven between 1800 and
+1827.
+\end{fineprint}
+
+\nbarticle{699.} \nbparatext{Duration of Visibility, and Brightness.}---The time during
+which they are visible differs very much. The great comet of 1811
+%% -----File: 407.png---Folio 396-------
+was observed for seventeen months, the longest time on record. The
+comet of 1861 was observed for a year, and the great comet of 1882
+for five months. In some cases, when a comet does not happen to be
+discovered until it is receding from the sun, it is seen only for a week
+or two.
+
+As to their \textit{brightness} they also differ widely. The great majority
+can be seen only with a telescope, although a considerable number
+reach the limit of naked-eye vision at that part of their orbit where
+they are most favorably situated. A few, as has been said above,
+become \textit{conspicuous}; and a \textit{very few}, perhaps four or five in a
+century, are so brilliant that they can be seen by the naked eye
+in full sunlight, as was the case with the great comets of 1843 and
+1882.
+
+\begin{fineprint}
+\nbarticle{700.} \nbparatext{Their Orbits.}---The ideas of the ancients as to the motions of
+these bodies were very vague. Aristotle and his school believed them to be
+nothing but earthly exhalations inflamed in the upper regions of the air,
+and therefore \textit{meteorological} phenomena rather than astronomical. Ptolemy
+accordingly omits all notice of them in the Almagest.
+
+Tycho Brahe was the first to show that they are more distant than the
+moon by comparing observations of the comet of 1577 made in different
+parts of Europe. Its position among the stars at any moment, as seen from
+his observatory at Uranienburg, was sensibly the same as that observed at
+Prague, more than 400 miles to the south. It followed that its distance
+must be much greater than that of the moon, and that its real orbit must be
+of enormous size, cutting through interplanetary space in a manner absolutely
+incompatible with the old doctrine of the crystalline spheres. He
+supposed the path to be circular, however, as befitted the motion of a
+celestial body.
+
+Kepler supposed that comets moved in straight lines; and he seems to
+have been half disposed to consider them as living creatures, travelling
+through space with will and purpose, ``like fishes in the sea.''
+
+Hevelius first, nearly a hundred years later, suggested that the orbits are
+probably \textit{parabolas}, and his pupil Doerfel \textit{proved} this to be the case in 1681
+for the comet of that year. The theory of gravitation had now appeared,
+and Newton soon worked out and published a method by which the elements
+of a comet's orbit can be determined from the observations. Immediately
+afterwards Halley, using this method and computing the parabolic
+orbits of all the comets for which he could find the needed observations,
+ascertained that a series of brilliant comets having nearly the same orbit
+had appeared at intervals of about seventy-five years. He concluded that
+these were different appearances of one and the same comet, the orbit not
+being really parabolic but \textit{elliptical}, and he predicted its return, which
+actually occurred in 1759---the first of the ``periodic comets.''
+\end{fineprint}
+%% -----File: 408.png---Folio 397-------
+
+\includegraphicsmid{illo195}{\textsc{Fig.~195.}\\The Close Coincidence of Different Species of Cometary Orbits within the Earth's Orbit.}
+
+\nbarticle{701.} \nbparatext{Determination of a Comet's Orbit.}---Strictly speaking, the
+orbit of a comet being always a conic section, like that of a planet,
+requires only three perfect observations for its determination; but it
+seldom happens that the observations\footnote
+  {Observations for the determination of a comet's place are usually made with
+  an equatorial, by measuring the apparent distance between the comet and some
+  neighboring ``comparison star'' with some form of micrometer, as indicated in
+  \artref{Art.}{129}. If the star's place is not already accurately known, it is afterwards
+  specially observed with the meridian circle of some standard observatory: this
+  observation of comparison stars forms quite an item in the regular work of such
+  an institution.}
+can be made so accurately
+as to enable us to distinguish an orbit truly parabolic from one
+slightly hyperbolic, or from an ellipse of long period. The \textit{plane
+of the orbit} and \textit{its perihelion distance}, can be made out with reasonable
+accuracy from such observations as are practically obtainable,
+but the \textit{eccentricity}, and the \textit{major axis} with its corresponding \textit{period},
+can seldom be determined with much precision from the data obtained
+at a single appearance of a comet.
+
+The reason is that a comet is visible only in that very small
+%% -----File: 409.png---Folio 398-------
+portion of its orbit which lies near the earth and sun, and, as the
+figure shows (\figref{illo195}{Fig.~195}), in this portion of the orbit, the long ellipse,
+the parabola, and the hyperbola almost coincide. Moreover, from the
+diffuse nature of a comet it is not possible to observe it with the
+same accuracy as a planet.
+
+Comets which really move in parabolas or hyperbolas visit the sun
+but once, and then recede, never to return; while these that move in
+ellipses return in regular periods, unless disturbed.
+
+It will be understood, that in a catalogue of comets' orbits, those
+which are indicated as parabolic are not \textit{strictly} so. All that can be
+said is that during the time while the comet was visible, its position
+did not deviate from the parabola given \textit{by an amount sensible to
+observation}. The chances are infinity to one against a comet's
+moving exactly in a parabola, since the least \textit{retardation} of its
+velocity would render the orbit \textit{elliptical}, and the least \textit{acceleration,
+hyperbolic}, according to the principles explained in \artref{Article}{430}.
+
+\nbarticle{702.} \nbparatext{Relative Numbers of Parabolic, Elliptical, and Hyperbolic
+Comets.}---The orbits of about 270 comets have been thus far computed.
+Of this number about 200 are sensibly parabolic, and \textit{six}
+appear to be hyperbolic, although the eccentricity exceeds unity by
+so small a quantity as to leave the matter somewhat doubtful. There
+are also a number of comets which, according to the best computations,
+appear to have orbits really elliptical, but with periods so long
+that their elliptical character cannot be positively asserted. About
+\textit{fifty} have orbits which are certainly and distinctly oval; and \textit{twenty-six}
+of these have periods which are less than one hundred years.
+\textit{Thirteen} of these periodic comets have already been actually observed
+at more than one return.
+
+\begin{fineprint}
+As to the rest of the twenty-six, some of them are expected to return
+again within a few years, and some of them have been lost,---either in the
+same way as the comet of Biela, of which we shall soon speak, or by having
+their orbits so changed by perturbations that they no longer come near
+enough to the earth to be observed. \hyperref[app:III]{Table III.}\ of the Appendix gives
+the elements of these thirteen comets taken from the ``Annuaire du Bureau
+des Longitudes'' for 1888. It will be observed that the shortest period
+is that of Encke's comet, which is only three and one-half years, while the
+period of Halley's comet exceeds seventy-six.
+
+There are three comets with computed periods ranging between seventy
+and eighty years, whose returns are looked for within the next forty years.
+There is also one comet with a period of thirty-three years which is due to
+return in 1899.
+\end{fineprint}
+%% -----File: 410.png---Folio 399-------
+
+\includegraphicsouter{illo196}{\textsc{Fig.~196.}---Orbits of Short-period Comets.}
+
+\nbarticle{703.} \figref{illo196}{Fig.~196} shows the orbits of several of the comets of short
+period,---from three to eight years. (It would cause confusion to
+insert all of them.) It will be seen that in every case the comet's
+orbit comes very near to the orbit of Jupiter, and when the orbit
+crosses that of Jupiter, one
+of the nodes is always near
+the place of apparent intersection
+(the node being
+marked on the comet's orbit
+by a short cross-line). If
+Jupiter were at that point
+of its orbit at the time when
+the comet was passing, the
+two bodies would really be
+very near to each other.
+The fact, as we shall see,
+is a very significant one,
+pointing to a connection between
+these bodies and the
+planet. It is true for \textit{all}
+the comets whose periods
+are less than eight years---for
+those not inserted in
+the diagram as well as those that are. The orbits of the seventy-five-year
+comets are similarly related to the orbit of Neptune, and the
+thirty-three-year comet passes very close to the orbit of Uranus.
+
+\nbarticle{704.} \nbparatext{Recognition of Elliptic Comets.}---Modern observations are
+so much more accurate than those made two centuries ago that it is
+now sometimes possible to determine the eccentricity and period of
+an elliptic comet by means of the observations made at a single
+appearance. Still, as a general rule, it is not safe to pronounce upon
+the ellipticity of a comet's orbit until it has been observed at least
+twice, nor always then. A comet possesses no ``\textit{personal identity},''
+so to speak, by which it can be recognized merely by looking at it,---no
+personal peculiarities like those of the planets Jupiter and Saturn.
+It is identifiable only by its path.
+
+\begin{fineprint}
+When the approximate parabolic elements of a new comet's orbit have
+been computed, we examine the catalogue of preceding comets to see if we
+can find others which resemble it; that is, which have nearly the same \textit{inclination}
+%% -----File: 411.png---Folio 400-------
+and \textit{longitude of the node} with the same \textit{perihelion distance} and \textit{perihelion
+longitude}. If so, it is \textit{probable} that we have to do with the same comet
+in both cases. But it is not \textit{certain}, and investigations, often very long and
+intricate, must be made to see whether an elliptical orbit of the necessary
+period can be reconciled with the observations, after taking into account
+the perturbations produced by planetary action. These perturbations are
+extremely troublesome to compute, and are often very great, since the comets
+not unfrequently pass near to the larger planets. In some such cases the orbit
+is completely altered. Even if the result of this investigation appears to
+show that the comets are probably identical, we are not yet absolutely safe
+in the conclusion, for we have what are known as---
+\end{fineprint}
+
+\nbarticle{705.} \nbparatext{Cometary Groups.}---These are groups of comets which pursue
+nearly the same orbits, following along one after another at a greater
+or smaller interval, as if they had once been united, or had come
+from some common source. The existence of such groups was first
+pointed out by Hoek of Utrecht in 1865. The most remarkable
+group of this sort is the one composed of the great comets of 1668,
+1843, 1880, and 1882, and there is some reason to suspect that the
+little comet visible on the picture of the corona of the Egyptian
+eclipse (\artref{Art.}{328}) also belongs to it. The bodies of this group have
+orbits very peculiar in their extremely small perihelion distance (they
+actually go within half a million miles of the sun's surface), and yet,
+although their elements are almost identical, they cannot possibly all
+be different appearances of one and the same comet.
+
+\begin{fineprint}
+So far as regards the comets of 1668 and 1843, considered alone, there
+is nothing absolutely forbidding the idea of their identity: perturbations
+might account for the differences between \textit{their} two orbits. But the comets
+of 1880 and 1882 cannot possibly be one and the same; they were both
+observed for a considerable time and accurately, and the observations of
+both are absolutely inconsistent with a period of two years or anything like
+it. In fact, for the comet of 1882 all of the different computers found
+periods ranging between 600 and 900 years.
+
+There are about half a dozen other such comet-groups now known.
+\end{fineprint}
+
+\sloppy
+\nbarticle{706.} \nbparatext{Perihelion Distances.}---These vary greatly. Eight comets
+have a perihelion distance less than six millions of miles; about
+seventy-four per cent of all that have been observed lie within the
+earth's orbit; about twenty-four per cent lie outside, but within twice
+the earth's distance from the sun; and six comets have been observed
+with a perihelion distance exceeding that limit.
+%% -----File: 412.png---Folio 401-------
+
+\fussy
+\begin{fineprint}
+A single one, the comet of 1729, had a perihelion distance exceeding four
+astronomical units,---as great as the mean distance of the remoter asteroids.
+It must have been an enormous comet to be visible from such a distance.
+It is one of the six \textit{hyperbolic} comets.
+
+Obviously, however, the distribution of comets as determined by observation,
+depends not merely on the existence of the comets themselves, but upon
+their visibility from the earth. Those comets which approach near the orbit
+of the earth have the best chance of being seen, because their conspicuousness
+increases as they approach us, so that we must not lay too much stress
+on the apparent crowding of the perihelia within the earth's orbit.
+\end{fineprint}
+
+The perihelia are not distributed equally in all \textit{directions} from the
+sun, but more than sixty per cent are within $45°$ of what is called
+``the sun's way''; \textit{i.e.}, the line in space along which the sun is travelling,
+carrying with it its attendant systems.
+
+\nbarticle{707.} \nbparatext{Orbit Planes.}---The \textit{inclinations} of the comets' orbits range
+all the way from $0°$ to $90°$. The ascending nodes are distributed all
+around the ecliptic, with a decided tendency, however, to cluster in
+two regions having a longitude of about $80°$ and $270°$.
+
+\nbarticle{708.} \nbparatext{Direction of Motion.}---With the two exceptions of Halley's
+comet, and the comet of the Leonid meteors (\artref{Art.}{786}), the elliptical
+comets which have periods less than one hundred years all move in
+the direction of the planets; and the same is true of the six hyperbolic
+comets. Of the other comets, a few more move retrograde than
+direct, but there is no decided preponderance either way.
+
+\nbarticle{709.} It is hardly necessary to point out that the fact that the
+comets move for the most part in parabolas, and that the planes of
+their orbits have no evident relation to the plane of the planetary
+motions, tends to indicate (though it falls short of demonstrating)
+that \textit{they do not in any proper sense belong to the solar system
+itself, but are merely visitors from interstellar space}. They come
+towards the sun with almost precisely the velocity they would have
+if they had simply dropped towards it from an infinite distance, and
+they leave it with a velocity which, if no force but the sun's attraction
+operates upon them, will carry them back to an unlimited
+distance, or until they encounter the attraction of some other sun.
+With one remarkable exception, their motions appear to be just what
+might be expected of ponderable masses moving in empty space
+under the law of gravitation.
+%% -----File: 413.png---Folio 402-------
+
+\nbarticle{710.} \nbparatext{Acceleration of Encke's Comet.}---The one exception referred
+to is in the case of Encke's comet which, since its first discovery in
+the last century (it was not, however, discovered to be a \textit{periodic}
+comet until 1819), has been continually quickening its speed and
+shortening its period at the rate of about two hours and a half in
+each revolution; as if it were under the action of some resistance
+to its motion. No perturbation of any known body will account for
+such an acceleration, and thus far no reasonable explanation has
+been suggested as even possible, except the one mentioned---the
+resistance of an interplanetary medium which retards its motion just
+as air retards a rifle bullet. At first sight it seems almost paradoxical
+that a \textit{resistance} should \textit{accelerate} a comet's speed; but referring
+to \artref{Article}{429} we see that since the semi-major axis of a comet's orbit
+is given by the equation
+\[
+a = \frac{r}{2} \left( \frac {U^2}{U^2 - V^2} \right),
+\]
+any diminution of $V$ will also diminish $a$; and it can be shown that
+this reduction in the \textit{size} of the orbit will be followed by an increase
+of velocity above that which the body had in the larger orbit. It is
+accelerated by being thus allowed to drop nearer to the sun, and
+gains its speed in moving inwards under the sun's attraction.
+
+\begin{fineprint}
+\nbarticle{711.} Another action of such a retarding force is to diminish the eccentricity
+of the body's orbit, making it more nearly circular. If the action were
+to go on without intermission, the result would be a spiral path winding inward
+towards the sun, upon which the comet would ultimately fall. For many
+years the behavior of Encke's comet was quoted as an absolute demonstration
+of the existence of the ``luminiferous ether.'' Since, however, no other
+comets show any such action (unless perhaps Winnecke's\footnote
+  {Oppolzer, in 1880, found that according to his computations Winnecke's
+  comet was accelerated precisely in the same way as Encke's, but by less than half
+  the amount. His result, however, is not confirmed by the recent work of Härdtl,
+  who finds no acceleration at all.}
+comet---No.~5 in
+the \hyperref[app:III]{table} in the appendix), and moreover, since according to the investigations
+of Von Asten and Backlund the acceleration of Encke's comet itself
+seems suddenly to have diminished by nearly one-half in 1868, there remains
+much doubt as to the theory of a resisting medium. It looks rather more
+probable that this acceleration is due to something else than the luminiferous
+ether---perhaps to some regularly recurring encounter of the comet with a
+cloud of meteoric matter. The fact that the \textit{planets} show no such effect is
+not surprising, since, as we shall see, they are enormously more dense than
+any comet, so that the resistance that would bring a comet to rest within a
+%% -----File: 414.png---Folio 403-------
+single year would not sensibly affect a body like our earth in centuries. The
+``resisting medium,'' if it exists at all, must have much less retarding power
+than the residual gas in one of Crookes's best vacuum tubes.
+\end{fineprint}
+
+\nbarticle{712.} \nbparatext{Physical Characteristics of Comets.}---The orbits of these
+bodies are now thoroughly understood, and their \textit{motions} are calculable
+with as much accuracy as the nature of the observations permit;
+but we find in their physical constitution some of the most perplexing
+and baffling problems in the whole range of astronomy,---apparent
+paradoxes which as yet have received no satisfactory explanation.
+While comets are evidently subject to gravitational attraction, as
+shown by their orbits, they also exhibit evidence of being acted upon
+by powerful \textit{repulsive} forces emanating from the sun. While they
+shine, in part at least, by reflected light, they are also certainly \textit{self-luminous},
+their light being developed in a way not yet satisfactorily
+explained. They are the \textbf{bulkiest} bodies known, in some cases
+thousands of times larger than the sun or stars; but they are ``airy
+nothings,'' and the smallest asteroid probably rivals the largest of
+them in actual mass.
+
+\nbarticle{713.} \nbparatext{Constituent Parts of a Comet.}---(\textit{a}) The essential part of a
+comet---that which is always present and gives it its name---is the
+\textit{coma} or nebulosity, a hazy cloud of faintly shining matter, which
+is usually nearly spherical or oval in shape, though not always so.
+
+(\textit{b}) Next we have the \textit{nucleus}, which, however, is not found in all
+comets, but commonly makes its appearance as the comet approaches
+the sun. It is a bright, more or less star-like point near the centre
+of the coma, and is the object usually pointed on in determining the
+comet's place by observation. In some cases the nucleus is double
+or even multiple; that is, instead of a single nucleus there may be
+two or more near the centre of a comet. Perhaps three comets out of
+four present a nucleus during some portion of their visibility.
+
+(\textit{c}) The \textit{tail} or \textit{train}, is a streamer of light which ordinarily accompanies
+a bright comet, and is often found even in connection
+with a telescopic comet. As the comet \textit{approaches} the sun, the tail
+follows it much as the smoke and steam from the locomotive trail after
+it. But that the tail does not really consist of matter simply \textit{left
+behind} in that way, is obvious from the fact that as the comet recedes
+from the sun, the train \textit{precedes} it instead of following. It is always
+\textit{directed away from the sun}, though its precise position and form is to
+some extent determined by the comet's motion. There is abundant
+evidence that it is a material substance in an exceedingly tenuous
+%% -----File: 415.png---Folio 404-------
+condition, which in some way is driven off from the comet and then
+repelled by some solar action. (See also \artref{Art.}{736}.)
+
+(\textit{d}) \textit{Envelopes and Jets}.---In the case of a very brilliant comet, its
+head is often veined by short jets of light which appear to be continually
+emitted by the nucleus; and sometimes instead of jets the
+nucleus throws off a series of concentric envelopes, like hollow shells,
+one within the other. These phenomena, however, are not usually
+observed in telescopic comets to any marked extent.
+%% -----File: 416.png---Folio 405-------
+
+\nbarticle{714.} \nbparatext{Dimensions of Comets.}---The volume of a comet is often
+enormous---sometimes almost beyond conception, if the tail be included
+in the estimate of bulk.
+
+As a general rule the \textit{head or coma} of a telescopic comet is from
+40,000 to 100,000 miles in diameter. A comet less than 10,000
+miles in diameter is very unusual; in fact, such a comet would be
+almost sure to escape observation. Many, however, are much larger
+than 100,000 miles. The head of the comet of 1811 at one time
+measured nearly 1,200000 miles,---more than forty per cent larger
+than the diameter of the sun itself. The comet of 1680 had a head
+600,000 miles across. The head of Donati's comet of 1858 was
+250,000 miles in diameter. The head of the great comet of 1882
+was not so bulky as many others, having had a diameter of only
+150,000 miles; but its tail was at one time 100,000000 miles in
+length.
+
+\includegraphicsmid{illo197}{\textsc{Fig.~197.}---Naked-eye View of Donati's Comet, Oct.~4, 1858. (Bond.)}
+
+\nbarticle{715.} \nbparatext{Contraction of a Comet's Head as it approaches the Sun.}---It
+is a very singular fact that the head of a comet continually and regularly
+changes its diameter as it approaches to and recedes from the
+sun; and what is more singular yet, it \textit{contracts when it approaches
+the sun}, instead of expanding, as one would naturally expect it to
+do under the action of the solar heat. No satisfactory explanation is
+known. Perhaps the one suggested by Sir John Herschel is as
+plausible as any,---that the change is optical rather than real; that
+near the sun a part of the cometary matter becomes invisible, having
+been \textit{evaporated}, perhaps, by the solar heat, just as a cloud of fog
+might be.
+
+\begin{fineprint}
+The change is especially conspicuous in Encke's comet. When this body
+first comes into sight, at a distance of about 130,000000 miles from the
+sun, it has a diameter of nearly 300,000 miles. When it is near the perihelion,
+at a distance from the sun of only 33,000000 miles, its diameter
+shrinks to 12,000 or 14,000 miles, the volume then being less than $\frac{1}{10\,000}$ of
+what it was when first seen. As it recedes it expands, and resumes its
+original dimensions. Other comets show a similar, but usually less striking,
+change.
+\end{fineprint}
+
+\sloppy
+\nbarticle{716.} \nbparatext{Dimensions of the Nucleus.}---This has a diameter ranging in
+different comets from 6000 or 8000 miles in diameter (Comet~III,
+1845) to a mere point not exceeding 100 miles. Like the head, it
+also undergoes considerable and rapid changes in diameter, though its
+changes do not appear to depend in any regular way upon the comet's
+%% -----File: 417.png---Folio 406-------
+distance from the sun, but rather upon its activity at the time. They
+are usually associated with the development of jets and envelopes.
+
+\fussy
+\nbarticle{717.} \nbparatext{Dimensions of a Comet's Tail.}---The tail of a large comet, as
+regards simple magnitude, is by far its most imposing feature. The
+length is seldom less than 10,000000 to 15,000000 miles; it frequently
+reaches from 30,000000 to 50,000000, and in several cases has been
+known to exceed 100,000000. It is usually more or less fan-shaped, so
+that at the outer extremity it is millions of miles across, being shaped
+roughly like a cone projecting behind the comet from the sun, and more
+or less bent like a horn. The volume of such a train as that of the
+comet of 1882, 100,000000 miles in length, and some 200,000 miles in
+diameter at the comet's head, with a diameter of 10,000000 at its extremity,
+exceeds the bulk of the sun itself by more than 8000 times.
+
+\nbarticle{718.} \nbparatext{The Mass of Comets.}---While the volume of comets is
+enormous, their \textit{masses} appear to be insignificant. Our knowledge
+in this respect is, however, thus far entirely \textit{negative}; that is, while in
+many cases we are able to say positively that the mass of a particular
+comet \textit{cannot have exceeded} a limit which can be named, we have
+never been able to fix a lower limit which we know it must have
+reached; it has in no case been possible to perceive any action whatever
+produced by a comet on the earth or any other body of the
+planetary system, from which we can deduce its mass; and this,
+although they have frequently come so near the earth and other
+planets that their own orbits have been entirely transformed, and if
+their masses had been as much as $\frac{1}{100\,000}$ of the earth's, they would
+have produced very appreciable effects upon the motion of the planet
+which disturbed them.
+
+\begin{fineprint}
+Lexell's comet of 1770, Biela's comet on more than one occasion, and several
+others, have come so near the earth that the length of their periods of
+revolution have been changed by the earth's attraction to the extent of
+several weeks, but in no instance has the length of the year been altered
+by a single second. One might be tempted to think that comets were possessed
+of matter without attracting power; but attraction is always \textit{mutual},
+and since the comets move according to the law of gravitation, and themselves
+suffer perturbation from attraction, there is no escape from the conclusion
+that, enormous as they are in volume, they contain very little matter.
+Some have gone so far as to say that a comet properly packed could be carried
+about in a hat-box or a man's pocket, which, of course, is an extravagant
+assertion. The probability is that the total amount of matter in a comet of
+any size, though very small as compared with its bulk, is yet to be estimated
+%% -----File: 418.png---Folio 407-------
+at many millions of tons. The earth's mass (\artref{Art.}{132}, {\scriptsize 4})
+is expressed in tons
+by 6 with twenty-one ciphers following (6000 millions, of millions, of millions
+of tons). A body, therefore, weighing only one-millionth as much as the
+earth would contain 6000 millions of millions of tons. The atmosphere of
+the earth alone constitutes about $\frac{1}{250\,000}$ the earth's mass, and contains
+more than twenty-four millions of millions of tons.
+\end{fineprint}
+
+\nbarticle{719.} The late Professor Peirce based his estimate of a comet's
+mass upon the extent of the nebulous envelope which it carries with
+it, assuming (what may be doubted, however) that this envelope is
+gaseous, and is held \textit{in equilibrium} by the attraction of solid matter in
+and near the nucleus; and on this assumption he came to the conclusion
+that the matter in and near the nucleus of an average comet must be
+equivalent in mass to an \textit{iron ball as much as $100$ miles in diameter}.
+This would be about $\frac{1}{300\,000}$ of the earth's mass. While this estimate
+is not intrinsically improbable, it cannot, however, be relied
+upon. We simply do not know anything about a comet's mass, except
+that it is exceedingly small as compared with that of the earth.
+
+\nbarticle{720.} \nbparatext{Density.}---This must necessarily be almost inconceivably
+small. If a comet 40,000 miles in diameter has a mass equal to $\frac{1}{250\,000}$
+of the earth's mass, its mean density is a little less than $\frac{1}{7000}$ of that
+of the air at the earth's surface,---much lower than that of the best air-pump
+vacuum. Near the centre of the comet the density would probably
+be greater than the mean; but near its exterior very much less. As
+for the density of its tail, when such a comet has one, that, of course,
+must be far lower yet, and much below the density of the residual gas
+left in the best vacuum we can make by any means known to science.
+
+This estimate of the density of a comet is borne out by the fact
+that small stars can be seen through the head of a comet 100,000
+miles in diameter, and even very near its nucleus, with hardly any
+perceptible diminution of their lustre. In such cases the writer has
+noticed that the image of a star is rendered a little indistinct; and
+recent observations of several astronomers have shown a very small
+apparent displacement of the star, such as might be ascribed to a
+slight refraction produced by the gaseous matter of the comet.
+
+\begin{fineprint}
+Students often find difficulty in conceiving how bodies of so infinitesimal
+density as comets can move in orbits like solid masses, and with such
+enormous velocities. They forget that \textit{in a vacuum} a feather falls as freely
+and as swiftly as a stone. Interplanetary space is a vacuum far more perfect
+than any air-pump could produce, and in it the rarest and most tenuous
+bodies move as freely as the densest.
+\end{fineprint}
+%% -----File: 419.png---Folio 408-------
+
+\nbarticle{721.} The reader, however, must bear in mind that, although the
+\textit{mean} density of a comet (that is, the quantity of matter in a cubic
+mile) is small, \textit{the density of the constituent particles of a comet need
+not necessarily be so}. The comet may be composed of small, heavy
+bodies, \textit{widely separated}, and there is some reason for thinking that this
+is the case; that, in fact, the head of a comet is a swarm of meteoric
+stones; though whether these stones are many feet in diameter, or only
+a few inches, or only a few thousandths of an inch, like particles of
+dust, no one can say. In fact, it now seems quite likely that the
+greatest portion of a comet's mass is made up of such particles of
+solid matter, carrying with them a certain quantity of enveloping gas.
+
+\sloppy
+\nbarticle{722.} \nbparatext{Light of Comets.}---There has been much discussion whether
+these bodies shine by light reflected or intrinsic. The fact that they
+become less brilliant as they recede from the sun, and finally disappear
+while they are in full sight simply on account of faintness,\footnote
+  {If a comet shone with its own independent light, like a star or a nebula,
+  then, so long as it continued to show a disc of sensible diameter, the \textit{intrinsic
+  brightness} of this disc would remain unchanged: it would only grow \textit{smaller} as it
+  receded from the earth, not \textit{fainter}.}
+and not by becoming too small to be seen, shows that their light is in
+some way derived from the sun. The further fact that the light
+shows traces of polarization also indicates the presence of reflected
+sunlight. But while the light of a comet is thus in some way attributable
+to the sun's action, the spectroscope shows that it does not
+consist, to any considerable extent, of mere reflected sunlight, like
+that of the moon or a planet.
+
+\fussy
+\begin{fineprint}
+\nbarticle{723.} If a comet shone by mere reflected light, or by any light the
+intensity of which is proportional inversely to the square of the sun's
+distance (as would naturally be the case if the light were excited directly
+by the sun's radiation, and proportional to it), we should have its apparent
+brightness at any time equal to the quantity $\dfrac{1}{ D^2\Delta^2} $, in which $D$ and $\Delta$ are
+the comet's distances from the sun and from the earth respectively. The
+brightness of a comet does, in fact, generally follow this law roughly, but
+with many and striking exceptions. The light of a comet often \textit{varies
+greatly and almost capriciously}, shining out for a few hours with a splendor
+seven or eight fold multiplied, and then falling hack to the normal state or
+even below it. The Pons--Brooks comet in 1883 furnished several remarkable
+instances of this sort (Clerke, p. 418).
+\end{fineprint}
+
+\nbenlargepage
+\sloppy
+\nbarticle{724.} \nbparatext{The Spectra of Comets.}---The spectrum of most comets
+consists of a more or less faint continuous spectrum (which may be
+%% -----File: 420.png---Folio 409-------
+due to reflected sunlight, though it is usually too faint to show the
+Fraunhofer lines) overlaid by three bright bands,---one in the yellow,
+one in the green, and the third in the blue. These bands are sharply
+defined on the lower, or less refrangible, edge, and fade out towards
+the blue end of the spectrum. A fourth band is sometimes visible
+in the violet. The green band, which is much the brightest of the
+three, in some cases is crossed by a number of fine, bright lines, and
+there are traces of similar lines in the yellow and blue bands. This
+spectrum is \textit{absolutely identical with that given by the blue base of an
+ordinary gas or candle flame}; or better, by the blue flame of a Bunsen
+burner consuming ordinary illuminating gas. Almost beyond
+question it \textit{indicates the presence in the comet of some gaseous hydrocarbon},
+which in some way is made to shine; either by a \textit{general}
+heating of the whole body to the point of luminosity (which is hardly
+probable), or by electric discharges within it, or by \textit{local} heatings due
+to collisions between the solid masses disseminated through the gaseous
+envelope; or possibly \textit{phosphorescence} due to the action of
+sunlight: or none of these surmises may be correct, and we may
+have to seek some other explanation not yet suggested.
+
+\fussy
+It is not at all certain that the temperature of the comet, considered
+as a whole, is very much elevated. Nor will it do to suppose that
+because the spectrum reveals the presence in the comet of gaseous
+hydrocarbon, this substance, therefore, composes the greater part of
+the comet's mass. The probability is that the gaseous portion of the
+comet is only a small percentage of the whole quantity of matter
+contained in it.
+
+\sloppy
+\nbarticle{725.} \nbparatext{Metallic Lines in Spectrum.}---When a comet approaches
+very near to the sun, as did Wells's comet in 1882, and a few weeks
+later the great comet of that year, the spectrum shows bright metallic
+lines in addition to the hydrocarbon bands. The lines of sodium
+and magnesium are most easily and certainly recognizable. As for
+the other lines---a multitude of which were seen by Ricco (of Palermo)
+for a few hours, in the spectrum of the great comet of 1882---they
+are probably due to iron; though that is not certain, for they were
+not seen long enough to be studied thoroughly.
+
+\includegraphicsmid{illo198}{\textsc{Fig.~198.}---Comet Spectra.\\[2ex]
+(For convenience in engraving, the \textit{dark} lines of the solar spectrum in the lowest strip of the
+figure are represented as \textit{bright}.)}
+
+\fussy
+\nbarticle{726.} \nbparatext{Anomalous Spectra.}---While most comets show the hydrocarbon
+spectrum, occasionally a different spectrum of bands appears.
+Fig.~198 shows the spectra of three comets compared with the solar
+spectrum and with that of hydrocarbon gas.
+%% -----File: 421.png---Folio 410-------
+
+\begin{fineprint}
+The first, the spectrum of Tebbutt's comet of 1881, is the usual one.
+The other two are unique. Brorsen's comet, at its later returns, showed the
+\textit{ordinary} comet spectrum, and it might perhaps be considered possible that
+an error was made in fixing the position of the bands at the first observation.
+But the peculiar spectrum of comet~C, 1877, hardly permits such
+an explanation. It was observed at Dunecht on the same night, by the
+same observers and with the same spectroscope, as another comet which
+gave the usual spectrum; so that in this case it hardly seems possible that
+the anomalous result can be a mistake, though the spectrum itself as yet
+remains unidentified and unexplained.
+
+Indeed, from some points of view, it is strange that comets, coming as
+they do from such widely separated regions of space, do not show an almost
+infinite variety of spectra; \textit{a priori} we should expect differences rather than
+uniformity.
+
+It is maintained by Mr.~Lockyer that the spectrum of a comet \textit{changes} as
+it varies its distance from the sun, the bands altering in appearance and
+shifting their position. But the evidence of this is not yet conclusive.
+\end{fineprint}
+
+\nbenlargepage
+\nbarticle{727.} \nbparatext{Development of Jets and Envelopes.}---When a comet is
+first seen at a great distance from the sun it is ordinarily a mere
+roundish, hazy patch of faint nebulosity, a little brighter near the
+centre.
+%% -----File: 422.png---Folio 411-------
+
+\includegraphicsmid{illo199}{\textsc{Fig.~199.}---Head of Donati's Comet, Oct. 6, 1858. (Bond.)}
+
+As the comet draws near the sun it brightens, and the central condensation
+becomes more conspicuous and sharply defined, or star-like.
+Then, on the side next the sun, the newly formed nucleus begins to
+emit jets and streamers of light, or to throw off more or less symmetrical
+envelopes, which follow
+each other concentrically
+at intervals of some hours, expanding
+and growing fainter
+as they ascend, until they are
+lost in the general nebulosity
+which forms the head. During
+these processes the nucleus
+continually changes in brilliancy
+and magnitude, usually
+growing smaller and brighter
+just before the liberation of
+each envelope. When jets are
+thrown off, the nucleus seems
+to oscillate, moving slightly
+from side to side; but no evidences of a continuous rotation have
+ever been discovered. The two figures, \figref{illo199}{199} and \figref{illo200}{200}, represent the
+%% -----File: 423.png---Folio 412-------
+heads of two comets which behaved quite differently. \figref{illo199}{Fig.~199} is
+the head of Donati's comet as seen on Oct.~5, 1858. This comet
+was characterized by the quiet, orderly vigor of its action. It did
+very little that was anomalous or erratic, but behaved in all respects
+with perfect propriety. The system of envelopes in the head of this
+comet was probably the most symmetrical and beautiful ever seen.
+Fig.~200 is from a drawing by Common of the head of Tebbutt's
+comet in 1881. This comet, on the other hand, was always doing
+something \textit{outré}, throwing off jets, breaking into fragments, and, in
+fact, continually exhibiting unexpected phenomena.
+
+\includegraphicsouter{illo200}{\textsc{Fig.~200.}---Tebbutt's Comet, 1881. (Common.)}
+
+\nbenlargepage
+\nbarticle{728.} \nbparatext{Formation of the Tail.}---The material which is projected
+from the nucleus of the comet, as if repelled by it, is also \textit{repelled
+by the sun}, and driven backward, still luminous, to form the train. (At least,
+this is the appearance.) \figref{illo201}{Fig.~201}
+shows the manner in which the
+tail is thus supposed to be formed.\footnote
+  {Other theories of comets' tails have been presented, and have had a certain
+  currency,---theories according to which the tail is a mere ``luminous shadow'' of
+  the comet, so to speak, or a swarm of meteors. But all these theories break
+  down in the details. They fail to account for the phenomena of jets, envelopes,
+  etc., in the head of the comet, and they furnish no mathematical determination
+  of the outlines and curvature of the tail.}
+
+The researches of Bessel, Norton,
+and especially the late investigations
+of the Russian Bredichin, have
+shown that this theory---that the
+tail is composed of matter repelled
+by both the comet and the sun---not
+only accounts for the phenomena
+in a \textit{general} way, but for almost all
+the details, and agrees mathematically
+with the observed position
+and magnitude of the tail on different
+dates.
+
+\includegraphicsouter{illo201}{\textsc{Fig.~201.}---Formation of a Comet's Tail by Matter
+expelled from the Head.}
+
+\begin{fineprint}
+The repelled particles are still subject to the sun's gravitational attraction,
+and the \textit{effective} force acting upon them is therefore the difference between
+the gravitational attraction and the electrical $(?)$ repulsion. This \textit{difference}
+may or may not be in favor of the attraction, but in any case, the sun's
+attracting force is, at least, lessened. The consequence is that those repelled
+%% -----File: 424.png---Folio 413-------
+particles, as soon as they get a little away from the comet, begin to move
+around the sun in \textit{hyperbolic}\footnote
+  {Referring to the formula for the semi-major axis of an orbit, viz.,
+\[
+a = \frac{r}{ 2} \left( \frac{U^2}{U^2-V^2} \right),
+\]
+  we see that a repulsive force acting from the sun diminishes $U$ (which measures
+  the sun's attraction), and the consequence is that if the unrepelled particles are
+  describing a parabola (in which case $U^2=V^2$) then for the \textit{repelled} particles the
+  denominator will become negative ($U$ having been made smaller than $V$ by the
+  repulsive action), and thus $a$ will also become negative, so that the orbit for a
+  repelled particle will be a hyperbola.}
+orbits which
+lie in the plane of the comet's orbit, or
+nearly so, and are perfectly amenable to
+calculation.
+
+The tail is simply an assemblage of these
+repelled particles, and, according to theory,
+ought, therefore, to be a sort of flat, hollow,
+horn-shaped cone, as represented by \figref{illo202}{Fig.~202},
+open at the large end, and rounded
+and closed at the smaller one, which contains
+the nucleus.
+\end{fineprint}
+
+\includegraphicsouter{illo202}{\textsc{Fig.~202.}\\
+A Comet's Tail as a Hollow Cone.}
+
+\nbenlargepage
+\nbarticle{729.} \nbparatext{Curvature of the Tail.}---The cone is curved as shown,
+because the particles repelled still retain their original orbital motion,
+so that they will not be arranged along a straight line drawn from
+the sun through the comet, but along a curve convex to the direction
+of the comet's motion; but the stronger the repulsion, the less will
+be the curvature. \figref{illo203}{Fig.~203} shows how the tail ought to lie as the
+%% -----File: 425.png---Folio 414-------
+comet rounds the perihelion of its orbit. According to this theory,
+the tail should be \textit{hollow}, and in the case of comets when at their
+brightest it usually seems to be so, the centre being darker than the
+edges.
+
+\includegraphicsmid{illo203}{\textsc{Fig.~203.}---A Comet's Tail at Different Points in its Orbit near Perihelion.}
+
+\includegraphicsouter{illo204}{\textsc{Fig.~204.}\\
+Bright Centred Tail of Coggia's Comet. June, 1874.}
+
+\nbarticle{730.} \nbparatext{The Central Stripe in a Comet's Tail.}---Very often, there
+is a peculiar straight, dark stripe through the axis of the tail as
+shown in Figs.~199 and 200 of the head of Donati's and Tebbutt's
+comets. It might be mistaken for the shadow of the nucleus if it
+were pointed exactly away from the sun; but it is not, usually making
+an angle of several degrees with the direction of a true shadow.
+Sometimes, however, and not very unfrequently, the tail has a \textit{bright}
+centre instead of a dark one, perhaps on
+account of the feebleness of the comet's
+own repulsive action; in fact, this seems
+to be \textit{usually} the case when the comet
+has reached a considerable distance from
+the sun in receding from it, and often it
+is so when the comet is approaching the
+sun, but is still remote, as in the case
+of Coggia's comet shown in \figref{illo204}{Fig.~204}.
+In such cases the tail is generally faint
+and ill-defined at the edge, with a central spine of light, and in some
+cases it becomes apparently a mere slender ray, of less diameter
+than the head of the comet itself. This, however, is unusual.
+The explanation of this kind of tail requires a slight modification of
+the theory, so far as to admit that the particles at first repelled by
+the front of the comet are afterwards attracted by it, though still
+repelled by the sun.
+
+\sloppy
+\nbarticle{731.} \nbparatext{Tails of Three Different Types.}---Bredichin has found that
+the tails of comets may be grouped under three types:---
+
+\fussy
+1. The long, straight rays. They are formed of matter upon which
+the sun's repulsive action is from twelve to fifteen times as great as
+the gravitational attraction, so that the particles leave the comet with
+a relative velocity of at least four or five miles a second; and this
+velocity is continually increased as they recede, until at last it becomes
+enormous, the particles travelling several millions of miles in a day.
+The straight rays which are seen in the \figref{illo205}{figure} of the tail of Donati's
+comet, tangential to the tail, are streamers of this first type; as also
+was the enormous tail of the comet of 1861.
+%% -----File: 426.png---Folio 415-------
+
+\includegraphicsouter{illo205}{\textsc{Fig.~205.}---Bredichin's Three Types of Cometary Tails.}
+
+2. The second type is the curved, plume-like train, like the
+principal tail of Donati's comet. In this type the repulsive force
+varies from 2.2 times gravity (for the particles on the fixed edge of
+the tail) to half that amount for those which form the inner edge.
+This is by far the most
+common type of cometary train.
+
+3. A few comets show
+tails of the third type,---short,
+stubby brushes
+violently curved, and
+due to matter of which
+the repulsive force is
+only a fraction of gravity,---from
+$2\frac{1}{10}$ to $2\frac{1}{2}$.
+
+\sloppy
+\begin{fineprint}
+\nbarticle{732.} According to
+Bredichin, the tails of the
+first type are probably
+composed of \textit{hydrogen},
+those of the second type
+of some \textit{hydrocarbon gas},
+and those of the third of
+\textit{iron vapor}, with probably
+an admixture of sodium
+and other materials.
+
+There has been no opportunity
+since Bredichin
+published this result to
+test the matter spectroscopically
+for tails of the
+first and third types, by
+looking for the lines of
+hydrogen and iron. The
+hydrogen tails are almost
+always very faint, and the
+tails of the third class are uncommon. Tails of the second type, which are
+brightest and most usual, do show a hydrocarbon spectrum throughout their
+entire length, and so far confirm his view.
+
+The reason for this conclusion of Bredichin is that he supposes the
+repulsive force to be a \textit{surface action}, the same for equal surfaces of any kind
+of matter; the \textit{effective} accelerating force, therefore, measured by the velocity
+it would produce, would depend upon the \textit{ratio of surface to mass} in the
+particles acted upon, and so, in his view, should be inversely proportional
+%% -----File: 427.png---Folio 416-------
+to their molecular weights. Now the molecular weights of hydrogen, of
+hydrocarbon gases, and of the vapor of iron, bear to each other just about
+the required proportion.
+\end{fineprint}
+
+\fussy
+\nbarticle{733.} \nbparatext{Nature of the Repulsive Force.}---As to this we have so far
+no absolute knowledge. While the old ``corpuscular theory of
+light'' held its ground, many thought that the apparent repulsion was
+due to the actual impact of the light corpuscles. Since the abandonment
+of this theory, others have tried to find it in the ``impulse'' of
+the light and heat waves of the ether, without, however, explaining
+how such waves could produce any such impulsive action. No
+experiments show any such carrying power of light or any pressure
+produced by its impact, although when Crookes first invented his
+radiometer he seems to have thought he had found it. On the whole,
+opinion at present strongly inclines to the view long ago suggested
+by numerous speculators, but specially worked out and enforced by
+Zöllner; namely, that the force is \textit{electrical}; and some authorities of
+such eminence as Dr.~Huggins and the late Professor Peirce have
+asserted it positively. The difficulty is that we have no evidence that
+the sun is electrically charged, nor do we know how it could acquire
+a charge. At the same time, the unquestionable \textit{magnetic} effects
+produced upon the earth by solar disturbances rather favor the belief
+that there must be also a purely electric reaction between the sun and
+its attendant bodies.
+
+\begin{fineprint}
+A singular theory has been proposed by Zenker, that the repulsion is
+due to the reaction produced by rapid evaporation on the surface of the
+little solid and liquid particles of which he supposed a comet to consist: this
+evaporation would, of course, be most rapid on the side of the particles next
+the sun, and would cause \textit{a recoil} in a manner analogous to that by which
+the so-called spheroidal state of liquids is produced on a heated surface.
+Ranyard has suggested that the cometary particles may consist principally
+of minute liquid drops or frozen ``hail-stones'' of certain hydrocarbons
+which evaporate rapidly at a very low temperature (such as rhigoline
+and its congeners).
+\end{fineprint}
+
+\nbarticle{734.} \nbparatext{State of the Matter composing the Tail.}---This also is a subject
+of speculation rather than of knowledge. Perhaps the simplest
+supposition is that we have to do with gaseous matter rarefied even
+beyond the limits of the gas contained in Crookes's tubes,---so rarefied
+that since its molecules no longer suffer frequent collisions with
+each other, it has thus lost all the peculiar \textit{mechanical} characteristics
+of a gaseous mass, and become a mere cloud of separate particles
+%% -----File: 428.png---Folio 417-------
+each particle consisting, however, of but a single molecule.
+Spectroscopically such a cloud would still be \textit{gaseous}, but from a
+mechanical point of view extremes would have met, and this most
+tenuous gas would have become a cloud of finely powdered solid.
+
+\sloppy
+\nbarticle{735.} \nbparatext{What becomes of the Matter thrown off in Comets' Tails.}---To
+this we have no certain answer at present; but if the theory
+which has been stated is true, it is clear that most of the matter so
+repelled from comets can never be re-gathered by the nucleus, but
+must be dissipated in space.
+
+\fussy
+\begin{fineprint}
+Whenever a planet meets any of the particles, it picks them up, of course,
+as it picks up meteors; and Newton long ago suggested, what has of late
+been forcibly dwelt upon by Dr.\ Sterry Hunt, that in this way the atmospheres
+of the planets may be supplied with material to take the place of the
+carbon which has been absorbed and fixed by the processes of crystallization
+and of life. Otherwise it would seem that the processes now going on upon
+the earth's surface must necessarily in the course of time deprive the atmosphere
+of all its carbonic acid.
+
+If this view is correct, it follows that such comets as have tails lose a
+portion of their substance every time that they visit the sun. It is quite conceivable,
+also, that the processes by which light is excited in the head of a
+comet may use up and render unfit for future shining, a portion of its
+material; so that, as a periodic comet grows old, it may become both smaller
+and less luminous, until finally it ceases to be observable.
+\end{fineprint}
+
+\nbarticle{736.} \nbparatext{Anomalous Tails and Streamers.}---It is not very unusual for
+comets to show tails of two different types at the same time, as, for
+instance, Donati's comet. But occasionally stranger things happen,
+and the great comet of 1744 is reported to have had six tails diverging
+like a fan. Winnecke's comet of 1877 threw out a tail \textit{laterally},
+making an angle of about $60°$ with the normal tail, and having the
+same length,---about $1°$. Pechüle's comet of 1880 (a small one),
+besides the normal tail, had another of about the same dimensions
+directed straight \textit{towards} the sun: streamers of considerable length
+so directed are not very infrequent. The great comet of 1882 presented
+a number of peculiarities, which will be mentioned in the
+more particular description of that body, which is to follow. Most
+of these anomalies are as yet entirely unexplained.
+
+\nbarticle{737.} \nbparatext{Nature of Comets.}---It is obvious from what has been said
+that we have little certain knowledge on this subject; but perhaps on
+the whole the most probable hypothesis is the one which has been
+%% -----File: 429.png---Folio 418-------
+hinted at repeatedly,---that a comet is, as Professor Newton expresses
+it, nothing but a ``\textit{sand-bank}''; \textit{i.e.}, a swarm of solid particles
+of unknown size and widely separated (say pin-heads several
+hundred feet apart), each particle carrying with it an envelope of
+gas, largely hydrocarbon, in which gas light is produced, either by
+electric discharges between the particles, or by some other light-evolving
+action\footnote
+  {Some have ascribed the light to the \textit{collisions} between the little stones of
+  which they assume the comet to be made up, forgetting that, although the \textit{absolute}
+  velocity of the comet is extremely great, the \textit{relative} velocities of its constituent
+  masses with reference to each other must be very slight---far too small
+  apparently to account for any considerable rise of temperature or evolution of
+  light in that way. It is perhaps worth considering whether gases \textit{in the mass} may
+  not become sensibly luminous at a much lower temperature than has usually
+  been supposed. It would seem not improbable \textit{a priori} that at every temperature,
+  radiations of every wave-length must be omitted in some degree; \textit{i.e.}, that at \textit{any
+  temperature above the absolute zero no body is absolutely non-luminous.}}
+due to the sun's influence.
+
+This hypothesis derives its chief plausibility from the modern discovery
+of the close relationship between meteors and comets, to be
+discussed in the \chapref{CHAPTERXVIII}{next chapter}.
+
+\nbarticle{738.} \nbparatext{Origin of Periodic Comets.}---It is obvious that the comets
+which move in parabolic orbits are, as has been said already, mere
+visitors to the solar system, and not citizens of it: but as to those
+which now move in elliptical orbits around the sun, returning as
+regularly as planets, it is a question whether we are to regard them
+as \textit{native-born}, or only as \textit{naturalized}. Did they originate in the
+system, or are they captives?
+
+\nbarticle{739.} \nbparatext{Planets' Families of Comets.}---It is quite clear that in some
+way or other many of them owe their present status in the system to
+Jupiter, Saturn, and the other planets. In \artref{Article}{703} we called attention
+to the fact that, without exception, all the short-period comets
+(\textit{i.e.}, those having periods ranging from three to eight years), pass
+very near to Jupiter's orbit at some point in their paths; and they
+are now recognized and spoken of as Jupiter's family of comets,---sixteen
+of them in all, at present known.
+
+\begin{fineprint}
+Nine of the sixteen are in the table of comets whose returns have been
+actually observed more than once. One of the others was Lexell's comet,
+which was removed from the range of observation by being thrown into a
+new and larger orbit by Jupiter in 1779; and two are comparatively recent
+discoveries, whose returns are soon expected. The other four have failed to
+%% -----File: 430.png---Folio 419-------
+be observed at second and subsequent returns for unknown reasons; quite
+possibly for the same reason, whatever that may be, that has deprived us
+of Biela's comet.
+\end{fineprint}
+
+Similarly, Saturn is at present credited with two comets, one of
+which is Tuttle's comet, given in the catalogue of periodic comets.
+Uranus stands sponsor for three,---one of them Tempel's comet,
+which is very interesting in its relation to the November meteors, and
+is expected back in 1900. Finally, Neptune has a family of six.
+Halley's comet is one of them, and two of the others have been
+observed for a second time since 1880; the other three are not due
+on their second return for some years to come.
+
+\nbarticle{740.} \nbparatext{Origin of Comets: the ``Capture'' Theory.}---The generally
+accepted theory as to the \textit{origin} of these comet families is that the
+comets which compose them have been \textit{captured} by the planets to
+which they now belong.
+
+A comet entering the system from an infinite distance, and moving
+in a parabolic orbit, when it comes near a planet will be either
+accelerated or retarded. If \textit{accelerated}, its orbit becomes \textit{hyperbolic},
+and that is the end of that comet so far as the solar system is concerned;
+it never returns for a second observation. If, on the other
+hand, it is \textit{retarded}, the orbit becomes \textit{elliptical}, and the comet will
+return at regular intervals, moving in a path which, of course, always
+passes through the point where the disturbance took place.
+
+It is true, as Mr.\ Proctor has pointed out, that the attraction of
+Jupiter, huge as is his mass, could not \textit{at one effort} transform a parabolic
+orbit into an orbit so small as that, say, of Biela's comet. But
+it is not necessary that the thing should be done at one effort. The
+comet's orbit lies near to Jupiter's, and after a lapse of time, Jupiter
+and the comet will be sure to come alongside again: the comet may
+then be sent into a hyperbolic or parabolic orbit,---the chances for
+such a result are nearly even;---but it \textit{may} also have its velocity \textit{a
+second time diminished, and its orbit made still smaller}; and this may
+be done over and over again unlimitedly, until the aphelion of the
+comet falls at such a distance within the orbit of Jupiter that the
+planet is no longer able to disturb it seriously. Given time enough,
+and comets enough, for Jupiter to work upon, and the final result
+would necessarily be a comet-family such as really exists, with the
+aphelia of their orbits near to the orbit of Jupiter, and periods
+roughly half his own. But it must be frankly admitted that the
+extent of time, and the quantity of cometary material demanded, are
+enormous.
+%% -----File: 431.png---Folio 420-------
+
+\begin{fineprint}
+It may be added that the results of the recent investigations of Professor
+Newton of New Haven upon the nature and distribution of cometary orbits
+are favorable to the hypothesis that comets come into the solar system from
+outer space, and do not originate within it.
+
+\nbarticle{741.} \nbparatext{The ``Ejection'' Theory.}---Mr.\ Proctor has suggested, and
+vigorously defended, a very different theory,---\textit{that comets are masses of matter
+which have been thrown off from the heavenly bodies by eruptions of some sort};
+that the comets of Jupiter's family, for instance, once formed a portion of
+its mass, and were at some time \textit{ejected} with a velocity sufficient to set them
+free in space; and that many of the parabolic comets may have been similarly
+ejected from our own, or from other suns. The main difficulty with
+this theory is that there is no evidence of the necessary eruptive energy in
+Jupiter, or in any of the planets. A body would have to leave the upper
+surface of Jupiter's atmosphere with a velocity exceeding thirty-five miles
+a second, in order to fulfil the conditions of the problem, and become
+independent of the parent planet.
+
+It cannot be said, however, that there is any special \textit{mechanical} difficulty
+in supposing that some of the \textit{parabolic} comets may owe their origin to
+eruptions from distant suns. Our own sun unquestionably sometimes ejects
+clouds of matter (in the form of the solar prominences) with enormous
+velocity, perhaps in some cases sufficient to send them off into space. But
+so far as we can make out from the spectroscopic evidence, the material of
+comets is not identical with that of the prominences.
+\end{fineprint}
+
+\nbarticle{742.} \nbparatext{Remarkable Comets.}---(1) \textit{Halley's Comet.} This was the
+first periodic comet whose return was predicted. Halley based his
+prediction upon the fact that he found its orbit in 1682 to be nearly
+identical with those of the comets of 1607 and 1531, which had been
+carefully observed by Kepler and Apian; and he also found records
+of the appearance of great comets in 1456, in 1301, in 1145 and
+1066, which would correspond as regards the time-intervals concerned,
+though data were wanting for an accurate calculation of their orbits.
+He noticed, of course, that the two intervals between 1531 and 1607,
+and between 1607 and 1682 were not quite equal; but he had sagacity
+enough to see that the differences were no greater than might be
+accounted for by the attractions of Jupiter and Saturn.
+
+\begin{fineprint}
+The theory of perturbation was not then sufficiently developed to make it
+possible to compute with precision just what the effect would be upon the
+next return of the comet, but he saw that the action of Jupiter would
+\textit{retard} it, and he accordingly fixed upon the early part of 1759 as the time
+at which the comet might be expected. Before that date, however, mathematics
+had so advanced that the necessary calculations could be made.
+Clairaut, as the result of a most laborious investigation, fixed upon April~13
+%% -----File: 432.png---Folio 421-------
+for the perihelion passage; but in publishing his result, he remarked that
+it might easily be a month out of the way owing to the uncertainty as to
+the masses of the planets, and the possible action of undiscovered planets
+beyond Saturn (Uranus and Neptune were then unknown). The comet
+actually came to perihelion on March~13. At this return it was best seen
+in the southern hemisphere, and at one time had a tail nearly $50°$ long. At
+its next return, in 1835, it came to the predicted time within two days.
+It did not appear on this occasion as an extremely brilliant comet, but was
+reasonably conspicuous, with a tail of the first type (hydrogen) about $15°$ in
+length.
+
+Its next return will occur in or about $1911$, but the necessary calculations
+have not yet been made to determine the date with accuracy.
+
+The most remarkable of its earlier appearances were in 1066 and 1456.
+The comet of 1066 figures on the Bayeux tapestry as a propitious omen for
+William the Conqueror (of England). In 1455 the comet, according to
+popular belief, was formally excommunicated by Pope Calixtus III. in a
+bull directed mainly against the Turks, who were then threatening eastern
+Europe. It is doubtful, however, whether such a formal bull was ever really
+promulgated.
+\end{fineprint}
+
+\nbarticle{743.} (2) \textit{Encke's Comet.} This is interesting us the first of the
+short-period comets, and also as the comet having the shortest known
+time of revolution,---only about three years and a half. Encke
+first detected its \textit{periodicity} in 1819, but it had been frequently
+observed during the preceding fifty years, and has been observed at
+almost every return since then. It is usually visible only in the
+telescope, though sometimes, under very favorable circumstances,
+it can be seen by the naked eye, with a tail a degree or two
+long. It is often irregular in form, and ``lumpy,'' and seldom
+shows a well-defined nucleus; nor does it exhibit very much that
+is interesting in the way of jets, envelopes, and other cometary
+freaks. We have already mentioned its remarkable contraction in
+volume on approaching the sun (\artref{Art.}{715}), and the progressive
+shortening of its period, which has been ascribed to a resisting
+medium (\artref{Art.}{710}).
+
+\nbarticle{744.} (3) \textit{Biella's Comet.} This is also, or rather \textit{was}, a small
+comet with a period of 6.6 years,---the second comet of short period
+in order of discovery. Its history is very interesting. It was discovered
+in 1826 by Biela, an Austrian officer, and its periodic character
+was soon detected by Gambart, whose name is connected
+with it by many French authorities. Its orbit comes within a
+very few thousand miles of the earth's orbit, the nearness varying,
+%% -----File: 433.png---Folio 422-------
+of course, from time to time, on account of perturbations. The
+approach is often so close, however, that if the comet and the earth
+were to arrive at the nearest point at the same time there would be a
+collision, and the earth would pass through the outer portions of the
+comet's head. At the return of the comet in 1832, some one started
+the report that such an encounter would occur, and in consequence
+there was something hardly short of a panic in southern France, the
+first of the since numerous ``comet-scares.'' At this time the comet
+passed the critical point about a month before the earth reached it,
+so that the two bodies were never really within 15,000000 miles of
+each other.
+
+\nbarticle{745.} At the comet's next return in 1839 it failed to be observed on
+account of its unfavorable position in the sky; but in 1846 it duly
+reappeared, and did a very strange thing, so far unprecedented. It
+\textit{divided into two!} When first seen on November~28, it presented the
+ordinary appearance of any newly discovered comet. On Dec.~19
+it had become rather pear-shaped, and ten days later it had
+divided, the duplication being first seen in New Haven, and soon
+after at Washington, some days before any European astronomer
+had noticed it.
+
+\begin{fineprint}
+The twin comets travelled along side by side for more than four months,
+at an almost unvarying distance of about 160,000 miles, without showing
+the least sign of mutual attraction or disturbance; but internally both
+comets were intensely active, each developing a nucleus very bright for a
+telescopic comet, with a tail some half a degree in length, and showing
+curious fluctuations of light, which seemed as a general rule to \textit{alternate}; \textit{i.e.},
+whenever comet~$A$ brightened up, comet~$B$ grew fainter, and \textit{vice versa}.
+During a part of the time the two comets were connected by a faint arc of
+light.
+
+When next the comet returned in August, 1852, it was under rather
+unfavorable circumstances for observation, but the twins were both seen,
+now separated by about 1,500000 miles, and travelling quietly in their
+appointed orbits. Neither of them has ever been seen again, although they
+ought to have returned five times, and more than once under favorable
+conditions for visibility.
+\end{fineprint}
+
+\nbarticle{746.} But the story is not yet ended, though the remainder perhaps
+belongs more properly in the next \chapref{CHAPTERXVIII}{chapter} of our book.
+
+On the night of Nov.~27, 1872, just as the earth was passing the
+old track of the lost comet, she encountered a wonderful meteoric
+shower. As Miss Clerke expresses it, perhaps a little too positively,
+%% -----File: 434.png---Folio 423-------
+``It became evident that Biela's comet was shedding over us the
+pulverized products of its disintegration.''\footnote
+  {``Hist.\ Ast.\ in 19th Century,'' p.~384. It is probable enough that the meteors
+  were really the product of the comet's \textit{``disintegration''}; still it is by no means
+  certain. It is, of course, beyond question that they bear \textit{some} relation to the lost
+  comet.}
+
+The same thing happened again in November, 1886, when the
+earth once more passed the comet's path.
+
+The meteors of this so-called Bielid swarm, in their motion through
+the sky, all appear to come from a point in the constellation of
+Andromeda, and are therefore sometimes called the ``Andromedes,''
+and their motion is parallel to the comet's orbit, at the point where
+it intersects our own.
+
+\nbarticle{747.} (4) \textit{Donati's Comet of $1858$.} This, on the whole, was perhaps
+the finest (though not the largest or the most extraordinary) of
+the comets of the present century, having been very favorably situated
+for observation in the October sky.
+
+\begin{fineprint}
+It was discovered at Florence as a telescopic object on June 2. It did
+not, however, become visible to the naked eye until near the end of August,
+when it began to exhibit the beautiful phenomena which have made it, so
+to speak, the normal and typical comet. The comet had an apparently
+well-defined nucleus, which varied in diameter at different times from 500
+miles to 3000. For several weeks the coma exhibited in unrivalled perfection
+the development and structure of concentric envelopes. Its tail was of the
+second or hydrocarbon type, with faint tangential streamers which belong
+to the first or hydrogen type; it had a maximum apparent length of about
+50°, and was some 5° or 6° wide at the extremity, and its real length was
+about 45,000000 miles, with a width of 10,000000. The object was kept
+under accurate observation for fully nine months, so that its orbit is unusually
+well determined as a very long ellipse, with a periodic time of nearly
+2000 years. Figs.~197 and 199 show its principal features.
+\end{fineprint}
+
+Our space permits us to cite in detail only one other comet:---
+
+\nbarticle{748.} (5) \textit{The Great Comet of $1882$}, which will always be remembered,
+not only for its beauty, but for the great variety of unusual
+phenomena it presented.
+
+\textit{Discovery and Brightness.} The comet seems to have been first
+seen as a naked-eye object by some one whose name is not given, at
+Auckland, New Zealand, on Sept.~3. By the 7th or 8th it had
+%% -----File: 435.png---Folio 424-------
+become somewhat conspicuous, and was observed both at Cordova
+(South America) and at the Cape of Good Hope. It was telegraphed
+to the northern hemisphere by Cruls, of Rio Janeiro, on Sept.~11,
+but was not seen in the north until the day when it passed its
+perihelion, Sept.~17. It was then independently discovered by
+Common, in England (who had not heard of Cruls's telegram), in
+broad daylight, within $2°$ of the sun; and the next day it was similarly
+discovered by a number of observers, especially by Thollon, at
+Nice, who observed its spectrum in full sunlight, and measured the
+displacement of the sodium lines produced by its motion. It was so
+bright that there was not the slightest difficulty in seeing it by simply
+shutting off the sun with the hand held at arm's length.
+
+\begin{fineprint}
+\nbarticle{749.} \nbparatext{Transit across the Sun's Disc.}---On the afternoon of the 17th,
+its approach to the sun's disc and actual contact with it was observed by two
+persons at the Cape of Good Hope, with the same telescope and dark glasses
+ordinarily used for observing sun spots. The comet seemed to be as bright
+as the sun's surface itself, and was followed right up to the sun's limb,
+where it vanished so utterly that the observers supposed that it had gone
+\textit{behind} the sun. On the contrary, however, it passed directly across the sun's
+disc, but absolutely invisible; there was not the least trace of it upon the
+sun's surface, so that we must suppose it to have been sensibly transparent.
+It must have traversed the disc in less than fifteen minutes, though unfriendly
+clouds prevented the observation of its exit. For four days after the perihelion
+passage it remained still visible to the eye by daylight. On Sept.~22d,
+a French observer in Paris ascended in a balloon to observe it, and succeeded
+in seeing it, but not in getting any valuable results. A few days more
+carried it so far from the sun that it became a magnificent object for the
+hours before sunrise.
+\end{fineprint}
+
+\nbarticle{750.} \nbparatext{Member of a Comet Group.}---As has been stated before
+(\artref{Art.}{705}), its orbit---at least, that portion of it within the earth's
+orbit---coincides almost exactly with the orbits of three other comets
+belonging to the same group; viz., the comets of 1668, 1843, and
+1880. The salient peculiarity of these orbits lies in the \textit{closeness of
+their approach to the sun}, the perihelion distance of each of them
+being less than 750,000 miles, so that they all passed within 300,000
+miles of the sun's surface, and with a velocity which at perihelion
+exceeded $300$ miles per second, and carried them through $180°$ of
+their orbit in less than three hours. And yet, this passage through
+the sun's coronal regions did not disturb their motion in the least, as
+is shown by the fact that the orbit of the comet of 1882, deduced
+from the observations made before the perihelion passage, agrees
+%% -----File: 436.png---Folio 425-------
+exactly with that deduced from those made after it. The inference
+as to the extreme rarity of the sun's corona is obvious. Only one
+other comet---Newton's comet of 1680---has ever approached even
+nearly as close to the sun as the four comets of this group.
+
+\includegraphicsmid{illo206}{\textsc{Fig.~206.}---Orbit of the Great Comet of 1882.}
+
+\begin{fineprint}
+The comet continued visible until March, and this long period of observation
+enabled the computers to determine the orbit with a greater degree of
+accuracy than is usual. They all agree in making it a very elongated
+ellipse, with a period ranging from 652~years (according to Morrison) to
+843~years (according to Kreutz). \figref{illo206}{Fig.~206} represents the form of the orbit
+and the way in which its plane is related to the orbit of the earth.
+\end{fineprint}
+
+\sloppy
+\nbarticle{751.} \nbparatext{Telescopic Features.}---When the comet first became telescopically
+observable in the morning sky it presented very nearly the
+normal appearance. The nucleus was sensibly circular, and there
+were a number of clearly developed, concentric envelopes in the
+head; the dark, shadow-like stripe behind the nucleus was also well
+marked. In a few days the nucleus became elongated, and finally
+stretched out into a lengthened, luminous streak some 50,000 miles
+in extent, upon which there were six or eight star-like knots of condensation.
+The largest and brightest of these knots was the third
+from the forward end of the line, and was some 5000 miles in
+%% -----File: 437.png---Folio 426-------
+diameter. This  ``string of pearls'' continued to lengthen as long as
+the comet was visible, until at last the length exceeded 100,000 miles.
+This fact, of course, made it difficult to fix upon the precise point to
+be observed in determining the comet's position.
+
+\fussy
+\includegraphicsmid{illo207}{\textsc{Fig.~207.}---The Head of the Great Comet of 1882.}
+
+When the nucleus first broke up in this way, the dark stripe behind
+it was still conspicuous, making, however, a slight angle with the line
+of nuclei. A bright streak followed the line of nuclei, and in a few
+days this seemed to encroach on and to obliterate the dark stripe, so
+that after that time the backbone of the comet's tail became bright
+instead of dark, as it had been previously. The engraving (\figref{illo207}{Fig.~207})
+represents the telescopic appearance at Princeton on Oct.~9 and 15.
+The comet continued to be visible with the telescope until it was
+more than 470,000000 miles from the earth, a distance to which no
+other such object has ever been followed, except the single comet of
+1729,---the one whose perihelion distance exceeded four times the
+earth's distance from the sun.
+
+\nbarticle{752.} \nbparatext{Tail.}---The comet was so situated that its tail was not seen
+to the best advantage, being directed nearly away from the earth, and
+never having an apparent length much exceeding $35°$. In this respect
+it has, therefore, been often surpassed by much inferior comets. The
+%% -----File: 438.png---Folio 427-------
+actual length of the tail, however, at one time exceeded 100,000000
+miles,---more than the distance of the earth from the sun. It was
+of the second or hydrocarbon type.
+
+\includegraphicsmid{illo208}{\textsc{Fig.~208.}---The ``Sheath,'' and the Attendants of the Comet of 1882.}
+
+A unique and so far utterly unexplained phenomenon was a faint,
+straight-edged beam of light, or ``\textit{sheath},'' that accompanied the
+comet, enveloping the head and projecting three or four degrees in
+front of it, as shown in the figure (\figref{illo208}{Fig.~208}). Besides this, at different
+times, three or four irregular shreds of cometary matter were
+detected by Schmidt, of Athens, and other observers, accompanying
+the comet at a distance of three or four degrees when first seen, but
+gradually receding from it, and at the same time growing fainter.
+
+\begin{fineprint}
+Possibly they may have been fragments of the tail which belonged to the
+comet before passing the perihelion, or of the matter repelled from the
+comet when near perihelion. Since the comet, in passing the perihelion,
+changed the direction of its motion by nearly $180°$ in less than three hours,
+it was, of course, physically impossible that the tail it had before the perihelion
+passage could have made the circuit of the sun in that time. Its tail
+after the perihelion passage was in no sense the same that it had before; the
+comet ran away from the old one, and developed a new one. Visible or invisible,
+the particles of the old train must have kept on their way under the combined
+%% -----File: 439.png---Folio 428-------
+action of the sun's gravitation and repulsion, and calculation can show
+whether any of its fragments could have occupied the position and had the
+motion possessed by these companion-comets.
+\end{fineprint}
+
+We close the chapter with a few remarks upon a subject which has
+been much discussed.
+
+\nbarticle{753.} \nbparatext{The Earth's Danger from Comets.}---It has been supposed
+that comets might do us harm in two ways,---either by actually striking
+the earth, or by falling into the sun, and thus producing such an
+increase of solar heat as to burn us up.
+
+As regards the possibility of a collision with a comet, it is to be
+admitted that such an event \textit{is} possible. In fact, if the earth lasts
+long enough, it is practically sure to happen; for there are several
+comets' orbits which pass nearer to the earth's orbit than the semi-diameter
+of the comet's head, and at some time the earth and comet
+will certainly come together. Such encounters will, however, be very
+rare. If we accept the estimate of Babinet, they will occur about
+once in 15,000000 years in the long run.
+
+As to the consequence of such a collision it is impossible to speak
+with confidence, for want of sure knowledge of the state of aggregation
+of the matter composing a comet. If the theory presented in this
+chapter is true, everything depends on the size of the separate solid
+particles which form the main portion of the comet's mass. If they
+weigh \textit{tons}, the bombardment experienced by the earth when struck
+by a comet would be a very serious matter: if, as seems more
+likely, they are for the most part smaller than pin-heads, the result
+would be simply a meteoric shower.
+
+\nbarticle{754.} \nbparatext{Effect of the Fall of a Comet into the Sun.}---As regards the
+effect of the fall of a comet into the sun, it may be stated that, except
+in the case of Encke's comet, there is no evidence of any action going
+on that would cause a now existing \textit{periodic} comet to strike the sun's
+surface; it is, however, undoubtedly possible that a comet may enter
+the system from without, so accurately aimed that it will hit the sun.
+
+But, if a comet actually strikes the sun, it is not likely that the least
+harm will be done. If a comet, having a mass equal to $\frac{1}{100\,000}$ of
+the earth's mass, were to strike the sun's surface with the parabolic
+velocity of nearly 400~miles a second, it would generate about as
+much heat as the sun radiates in eight or nine hours. If this were all
+instantly effective in producing increased radiation at the sun's surface
+%% -----File: 440.png---Folio 429-------
+(increasing it, say eightfold, for even a single hour), mischief
+would follow, of course. But it is almost certain that nothing of the
+sort would happen. The cometary particles would pierce the photosphere,
+and liberate their heat mostly \textit{below the solar surface}, simply
+expanding, by some slight amount, the sun's diameter, and so adding
+to its store of potential energy about as much as it ordinarily expends
+in a few hours. There might, and very likely would be, a flash of
+some kind at the solar surface, as the shower of cometary particles
+struck it, but probably nothing that the astronomer would not take
+delight in watching.
+\chelabel{CHAPTERXVII}
+%% -----File: 441.png---Folio 430-------
+
+\Chapter{XVIII}{Meteors}
+
+\nbarticle{755.} \nbparatext{Meteors.}---Occasionally bodies fall on the earth from the
+sky,---masses of stone or iron which sometimes weigh several tons.
+During its flight through the sky such a body is called \textit{a meteor}, and
+the pieces which fall from it are called \textit{meteorites}, or \textit{aerolites} (air-stones),
+or \textit{uranoliths} (heaven-stones), or simply \textit{meteoric stones}.
+
+\sloppy
+\nbarticle{756.} \nbparatext{Circumstances of their Fall.}---The circumstances which
+attend the fall of a meteorite are in most cases substantially as
+follows. If it occurs at night a ball of fire is seen, which moves
+with an apparent speed depending both on its real velocity and on the
+observer's position. If the body is coming ``head on,'' so to speak,
+the motion will be comparatively slow; so also if it is very distant.
+The fire-ball is generally followed by a luminous train, which marks
+out the path of the body, and often continues visible for a long time
+after the meteor itself has disappeared. The motion is seldom exactly
+straight, but is more or less irregular, and every here and there along
+its path the meteor seems to throw off fragments, and to change its
+course more or less abruptly. If the observer is near enough, the
+flight is accompanied by a heavy, continuous roar, accentuated
+by sharp detonations which accompany the visible explosions by
+which fragments are burst off from the principal body. The noise
+is sometimes tremendous, and heard for distances of forty or fifty
+miles.
+
+\fussy
+If the fall occurs by day the luminous appearances are, of course,
+principally wanting, and white clouds take the place of the fire-ball
+and the train.
+
+\sloppy
+\nbarticle{757.} \nbparatext{The Aerolites themselves.}---The mass that falls is sometimes
+a single piece, but more usually there are many separate fragments,
+sometimes numbering many thousands. In such cases the stones are
+mostly small, and sometimes they are mere grains of sand. Nearly
+all the aerolites that are actually seen to fall, and are found at the time,
+are masses of \textit{stone}; but a very few, perhaps three or four per cent
+of the whole number, consist of \textit{nearly pure iron}, more or less alloyed
+%% -----File: 442.png---Folio 431-------
+with nickel. There are also a good many cases of uranoliths, which
+are mainly stony, but have a considerable portion of iron disseminated
+through the mass in grains and globules; and nearly all the
+stony uranoliths contain as much as twenty or thirty per cent of iron
+in the form of sulphides or analogous compounds.
+
+\fussy
+\begin{fineprint}
+\sloppy
+\nbarticle{758.}\hspace{0em} The only \textit{iron meteors which have been actually seen} to fall so far,
+and are represented by specimens in our museums, are the following:---
+\fussy
+\begin{center}
+\begin{tabular}{r l@{\ldots\ldots\ldots} r}
+(1) & Agram, Bohemia,\dotfill               &\dotfill 1751.\\
+(2) & Dickson Co., Tennessee,\dotfill       &\dotfill 1835.\\
+(3) & Braunau, Bohemia,\dotfill             &\dotfill 1847.\\
+(4) & Tabarz, Saxony,\dotfill               &\dotfill 1854.\\
+(5) & Nejed, Arabia,\dotfill                &\dotfill 1865.\\
+(6) & Nedagollah, India,\dotfill            &\dotfill 1870.\\
+(7) & Maysville, California,\dotfill        &\dotfill 1873.\\
+(8) & Rowton, Shropshire, England,\dotfill  &\dotfill 1876.\\
+(9) & Emmett Co., Iowa,\dotfill             &\dotfill 1879.\\
+(10)& Mazapil, Mexico,\dotfill              &Nov.~27, 1885.\\
+(11)& Johnson Co., Arkansas,\dotfill        &\dotfill 1886.\\
+\end{tabular}
+\end{center}
+The Emmett County iron was mostly in small fragments, and along with
+them there were many large stones with quantities of iron included. The
+separate fragments of pure iron which reached the earth probably came by
+the breaking up of the stony masses.
+
+Besides these iron meteors which have been seen to fall, our cabinets
+contain a very large number of so-called meteoric irons; \textit{i.e.}, masses of iron
+found under such circumstances that they cannot easily be accounted for in
+any way except by supposing them to be of meteoric origin.
+
+\nbarticle{759.} The number of meteorites which have fallen since 1800 and
+been gathered into our cabinets\footnote
+  {In this country the cabinets of Amherst College and Harvard and Yale
+  Universities are especially rich in meteorites. The finest collection in the world,
+  however, is that at Vienna. The collection of the British Museum is also noteworthy,
+  as well as that at Paris.}
+is about 250. The most remarkable
+falls in the United States have been the six following: namely, Weston,
+Connecticut, 1807; Bishopsville, So.~Carolina, 1843; Cabarrus Co., No.~Carolina,
+1849; New Concord, Ohio, 1860; Amana, Iowa, 1875; and Emmett
+Co., Iowa, 1879. In the first case and the three last, several hundred
+fragments fell at the same time, ranging in size from five hundred pounds
+to half an ounce.
+
+\includegraphicsouter{illo209}{\textsc{Fig.~209.}\\Fragment of one of the Amana Meteoric Stones.}
+
+\nbarticle{760.} \textit{Twenty-four} of the sixty-seven known chemical \textit{elements} have
+been found in meteors, and not a single new one. The \textit{minerals} of
+%% -----File: 443.png---Folio 432-------
+which meteorites are composed present a great resemblance to terrestrial
+minerals of volcanic origin, but
+many of them are peculiar, and
+found in meteors only. (The
+study of these meteoric minerals
+is a very curious and important
+branch of mineralogy,
+though naturally it has not
+many votaries.) The occasional
+presence of carbon is to be specially
+noted, and in a meteor
+which recently fell in Russia the
+carbon appeared to be in a crystalline
+form, identical with the
+black diamond, though in exceedingly
+minute particles. \figref{illo209}{Fig.~209}
+is from a photograph of a
+fragment of one of the meteoric
+stones which fell at Amana, Iowa,
+in 1875. The picture is taken
+by the permission of the publishers
+from Professor Langley's
+``New Astronomy,'' where the
+body is designated as ``part of a
+comet.''
+\end{fineprint}
+
+\nbarticle{761.} \nbparatext{The Crust.}---The most characteristic external feature of an
+aerolite is the thin, black \textit{crust} that covers it, usually, but not always,
+glossy like a varnish. It is formed by the fusion of the surface in the
+meteor's swift motion through the air, and in some cases penetrates
+deeply into the mass of the meteor through fissures and veins. It is
+largely composed of oxide of iron, and is always strongly magnetic.
+The crusted surface usually exhibits pits and hollows, such as would
+be produced by thrusting the thumb into a mass of putty. These
+cavities are explained by the burning out of certain more fusible substances
+during the meteor's flight.
+
+\nbarticle{762.} \nbparatext{Magnitude.}---Of the meteors actually seen to fall the largest
+pieces found thus far weigh about 500 pounds, though the whole mass
+of the body when it first entered the atmosphere has sometimes been
+much larger, perhaps, in a few cases, amounting to two or three tons.\footnote
+  {Some of the masses of iron supposed to be of meteoric origin, but not actually
+  seen to fall, are very much larger. The iron mass from Otumpa in Mexico is said
+  to weigh fully sixteen tons. As regards some of these hypothetical meteorites,
+  however, their meteoric origin is more than questionable; such, for instance, is
+  the case with the Ovifak iron found by Nordenskiold on the coast of Greenland,
+  and exhibited at the Philadelphia Centennial Exhibition.}
+%% -----File: 444.png---Folio 433-------
+
+As seen from a distance of many miles, the meteoric fire-ball sometimes
+\textit{appears} to have a diameter as large as the moon, which would
+indicate a real diameter of several hundred feet. The great apparent
+size, however, is an illusion, partly due to irradiation, and partly,
+undoubtedly, to the fact that the meteor itself is surrounded by an
+extensive envelope of heated air and smoke which becomes luminous
+throughout. No meteor ever yet investigated would make a mass as
+large as ten feet in diameter.
+
+\nbarticle{763.} \nbparatext{Path.}---When a meteor has been observed by a number of
+persons at \textit{different points}, who have noted any data which will give
+its altitude and bearing at identified moments, the path can be computed.
+Observations from a single point are worthless for the purpose,
+since they can give no information as to the meteor's \textit{distance}.
+
+The meteor is generally first seen at an altitude of between eighty
+and 100 miles, and disappears at an altitude of between five and ten
+miles. The length of the path may be anywhere from 50 miles to
+500, according to its inclination to the earth's surface. The velocity
+is rather difficult to ascertain, but is found to range from ten to forty
+miles per second at the moment when the meteor first becomes visible,
+and diminishes to one or two miles per second, at the time when it
+disappears. The \textit{average} velocity with which meteors enter the
+atmosphere appears not to vary much from the ``parabolic velocity''
+of twenty-six miles per second, due to the sun's attraction at the
+earth's distance---a fact which, of course, indicates that these bodies,
+whatever their origin may be, are now moving in space, like the comets,
+under the sun's attraction.
+
+\sloppy
+\begin{fineprint}
+With possibly a very few exceptions in cases where the meteor \textit{glances},
+so to speak, on the earth's atmosphere, like a skipping-stone on water, a
+body which has once entered the air is sure to be brought to the ground:
+it is hardly possible that one meteor in a million should escape after becoming
+involved in the atmosphere. We mention this especially, because some
+authorities erroneously speak of it as a usual thing for the meteor to keep
+on its course, and leave the earth, after throwing off a few fragments.
+\end{fineprint}
+
+\fussy
+\nbarticle{764.} \nbparatext{Observation of Meteors.}---The object of the observation
+should be to obtain accurate estimates of the altitude and azimuth of
+the body at moments which can be identified. At night this is best
+%% -----File: 445.png---Folio 434-------
+done by noting the position of the meteor with reference to neighboring
+stars at the moments of its appearance and disappearance, or of
+the intervening explosions. In the daytime it can often be done by
+noting the position of the object with reference to trees or buildings.
+The observer should then mark the exact position where he is standing,
+so that by going there afterwards with proper instruments he can
+determine the data desired.
+
+\begin{fineprint}
+Of course, all such measurements must be given \textit{in angular units}. To
+speak of a meteor as having an altitude of twenty \textit{feet}, and pursuing a path
+100 \textit{feet} long, is meaningless, unless the size of the ``foot'' is somehow
+defined.
+\end{fineprint}
+
+The determination of the meteor's \textit{velocity} is more difficult, as it
+is seldom possible to look at a watch-face quickly enough, even in
+the daytime. The usual course is for the observer to repeat some
+familiar piece of doggerel as rapidly as possible, beginning when the
+object first becomes visible and stopping when it explodes or disappears,
+noting also the precise syllable where he stops. By repeating
+the same sentence over again before a clock it is possible to determine
+within a few tenths of a second the time occupied by the
+meteor's flight.
+
+\nbarticle{765.} \nbparatext{Explanation of the Heat and Light of a Meteor.}---These
+are due simply to the destruction of the body's velocity; its kinetic
+mass-energy of motion is transformed into heat by the friction of the
+air. If a moving body whose mass is $M$ kilograms, and its velocity
+$V$ metres per second, is stopped, the number of calories of heat
+developed is given by the equation
+\[
+Q = \frac{MV^2}{8339} \  \text{(\artref{Art.}{354}).}
+\]
+
+The quantity of heat evolved in bringing to rest a body which has a
+velocity of forty-two kilometres, or twenty-six miles a second, is enormous,
+vastly more than sufficient to fuse it even if it were made of
+the most refractory material, and hundreds of times more than would
+be produced by its combustion in oxygen if it were a mass of coal.
+
+This heat is developed all along the meteor's course, and mostly just
+upon its surface. As Sir William Thomson has shown, the thermal
+effect of the rush through the air is the same as if the meteor \textit{were
+immersed in a blow-pipe flame} having a temperature of many thousand
+degrees; and it is to be noted that this \textit{temperature is independent
+%% -----File: 446.png---Folio 435-------
+of the density of the air} through which the meteor may be passing.
+The \textit{quantity of heat} developed in a given time is greater, of course,
+where the air is dense; but the \textit{temperature} produced in the air itself,
+at the surface where it rubs against the moving body, is the same
+whether the gas be dense or rare.
+
+\begin{fineprint}
+When a moving body has a velocity of about 1500 metres per second,
+the virtual temperature of the surrounding air is about that of red heat; \textit{i.e.},
+the body becomes heated as fast as it would if it were at rest and the air
+about it were heated to that temperature. When the velocity reaches twenty
+or thirty miles per second, it is acted upon as if the surrounding gas were
+heated to the liveliest incandescence at a temperature of several thousand
+degrees. The surface is fused, and the liquefied portion is continually
+swept off by the rush of the air, condensing as it cools into the luminous
+powder that forms the train. The fused surface itself is continually
+renewed until the velocity falls below two miles a second or thereabouts,
+when it solidifies and forms the characteristic crust. As a general rule,
+therefore, the fragments are hot if found soon after their fall; but if the
+stone is a large one and falls nearly vertically, so as to have but a short path
+through the air, the heating effect will be mainly confined to its surface; and
+owing to the low conducting power of stone, the \textit{centre} may still remain
+intensely \textit{cold}, retaining nearly the temperature which it had in interplanetary
+space. It is recorded that one of the large fragments of the Dhurmsala
+(India) meteorite, which fell in 1860, was found in moist earth half an hour
+or so after the fall, \textit{coated with ice}.
+\end{fineprint}
+
+\nbarticle{766.} \nbparatext{Train.}---One unexplained feature of meteoric trains deserves
+notice. They often remain luminous for a long time, sometimes
+as much as half an hour, and are carried by the wind like
+clouds. It is impossible to suppose that such a cloud of dust remains
+\textit{incandescent from heat} for so long a time in the cold upper regions of
+the atmosphere; and the question of its enduring luminosity or phosphorescence
+is an interesting and puzzling one.
+
+\sloppy
+\nbarticle{767.} \nbparatext{Origin.}---We may at once dismiss the theories which make
+meteors to be the \textit{immediate} product of volcanic eruption on the
+earth or on the moon. They come to us for the most part, as has
+been said, from the depths of space, with the velocity of planets and
+comets, and there is no certain reason for assuming that they originated
+in any manner different from the larger heavenly bodies.
+
+\fussy
+\begin{fineprint}
+At the same time, many of them so closely resemble each other as almost
+to compel the idea of some common source; and though lunar volcanoes are
+now extinct, and no terrestrial volcano, not even Krakatão, is \textit{now} competent
+%% -----File: 447.png---Folio 436-------
+to send off its ejected missiles through the earth's atmosphere into space,
+it is not certain that this was always so. Some still maintain that these
+bodies may be fragments which were shot off millions of years ago when the
+moon's volcanoes were in full vigor and the earth was young. Since then,
+according to this view, these masses have been travelling around the sun in
+long ellipses which intersect the orbit of the earth, until at last they happen
+to come along at the right time and encounter her surface, and so return to
+the old home.
+
+As to the \textit{iron} meteors, some maintain that they are masses which have
+been ejected from our sun, or from some other star; and they fortify their
+opinion by the remarkable but unquestionable fact that these meteoric irons
+are full of ``\textit{occluded}'' hydrogen and carbon oxides which can be extracted
+from them by heating them in a proper receptacle connected with a mercurial
+pump. They argue that these gases could have been absorbed by
+the iron only when it was in the liquid state and overlaid by a dense, hot
+atmosphere containing them; and that, they say, is just the condition of the
+minute drops of iron which are supposed to form part of the photosphere of
+the sun: upon \textit{our} sun or in some other sun only could such conditions be
+found. There is no question of the sun's ability to project masses to planetary
+distances, as shown by the chromospheric eruptions which have been
+repeatedly observed by students of solar physics. And if our sun can do
+this, it is natural to suppose that other suns can do it also.\\
+\end{fineprint}
+
+However these bodies originated, it is quite certain that before they
+reach the earth they have been moving independently in space for a long
+time, just as planets and comets do. But a recent important research by
+Professor Newton has shown that more than 90 per cent of some 200
+aerolites, for the approximate determination of whose paths we have the
+data, were moving before their fall in orbits, not parabolic, but analogous
+to those of the short-period comets; and \textit{direct}, not retrograde.
+
+\nbarticle{768.} \nbparatext{Detonating Meteors, or ``Bolides,'' of which Fragments are
+not known to reach the Earth.}---Some writers discriminate between
+these meteors and aerolites, but the distinction does not seem to be
+well founded. The phenomena appear to be precisely the same, except
+that in the one case the fragments are actually found, and in the other
+they fall into the sea, the forest, or the desert; or sometimes when the
+path is nearly horizontal, and therefore long, they may be consumed
+and dissipated in the dust and vapor of the train, without reaching
+the earth's surface at all, except ultimately as impalpable dust.
+
+\nbarticle{769.} \nbparatext{Number.}---As to the number of aerolites which strike the
+earth, it is difficult to make a trustworthy estimate. Since the
+beginning of the century, at least two or three have been seen to
+%% -----File: 448.png---Folio 437-------
+fall every year, and have been added to our cabinets (see \artref{Art.}{759}),---in
+some years as many as half a dozen. This, of course,
+implies a vastly greater number which are not seen, or are not
+found. Schreibers, some years ago, estimated the number as high
+as 700 a year, and Reichenbach sets it still higher---not less than
+3000 or 4000.
+
+\section*{SHOOTING STARS.}
+
+\nbarticle{770.} A few minutes' watching on any clear, moonless night will
+be sure to reveal one or more of the swiftly moving, evanescent
+points of light that are known as ``shooting stars.'' No sound is
+ever heard from them, nor (with a single exception to be mentioned
+further on) has anything ever been known to reach the earth's surface
+from them, not even when the sky was ``as full of them as of
+snow-flakes,'' as sometimes has happened in a great meteoric shower.
+For this reason it is perhaps justifiable to allow the old distinction to
+remain between them and the bodies we have been discussing, at
+least provisionally. The difference \textit{may be}, and according to opinion
+at present prevalent very probably is, merely one of size, like that
+between boulders and grains of sand. Still there are some reasons
+for supposing that there is also a difference of constitution,---that
+while the aerolite is a solid, compact mass, the shooting star is a
+little cloud of dust and intermingled gas, like a puff of smoke.
+
+\nbarticle{771.} \nbparatext{Numbers.}---The number of these bodies is very great. A
+single watcher sees on the average from four to eight hourly. If
+observers enough are employed to guard the whole sky, so that
+none can escape unnoticed, the visible number becomes from thirty
+to sixty an hour. Since ordinarily only these are seen which are
+within two or three hundred miles of the observer, the estimated
+total daily number of these which enter the earth's atmosphere, and
+are large enough to be visible to the naked eye, rises into the millions.
+Professor Newton sets it at from 10,000000 to 15,000000, the
+average distance between them being about 200 miles.
+
+\begin{fineprint}
+There is a still larger number too small to be seen with the naked eye.
+One hardly ever works many hours with a telescope carrying a low power,
+and having a field of view as large as $15'$ in diameter, without seeing several
+of them flash across the field. In a few instances observers have reported
+\textit{dark meteors} crossing the moon's disc while they were watching it. There
+may be some question, however, as to the real nature of the objects seen in
+such a case. Birds~(?).
+\end{fineprint}
+%% -----File: 449.png---Folio 438-------
+
+\nbarticle{772.} \nbparatext{Comparative Number in Morning and Evening.}---The hourly
+number about six o'clock in the morning is fully double the hourly
+number in the evening. The obvious reason is simply that in the
+morning we are on the \textit{front of the earth} as regards its orbital
+motion, while in the evening we are in the rear; in the evening we
+only see such meteors as \textit{overtake us}; in the morning we see all that
+we either meet or overtake. If they are really moving in all directions
+alike, with the parabolic velocity corresponding to the earth's
+distance from the sun (twenty-six miles per second), theory indicates
+that the relative hourly numbers for morning and evening ought to
+be in just the observed proportion.
+
+\nbarticle{773.} \nbparatext{Brightness.}---For the most part these bodies are much like
+the stars in brightness,---a few are as brilliant as Venus or Jupiter;
+more are like stars of the first magnitude; and the majority are like
+the smaller stars. The bright ones not unfrequently show trains
+which sometimes last from five to ten minutes, when they are folded
+up and wafted away by the winds of the upper air.\footnote
+  {These air currents, at an elevation of forty miles above the earth's surface,
+  are thus observed to have, ordinarily, velocities from fifty to seventy five miles
+  per hour.}
+
+\sloppy
+\nbarticle{774.} \nbparatext{Elevation, Path, and Velocity.}---By observations made by
+two or more observers thirty or forty miles apart, it is possible to
+determine the height, path, and velocity of these bodies. It is found
+as the result of a great number of such observations that they first
+appear at an average elevation of about \textit{seventy-four miles}, and disappear
+at an average height of about \textit{fifty miles}, after traversing a
+distance of \textit{forty or fifty miles}, with an average velocity of about
+\textit{twenty-five miles} per second. They do not begin to be visible at so
+great an elevation as the aerolitic meteors, and they vanish before
+they penetrate so deeply into the atmosphere.
+
+\fussy
+\nbarticle{775.} \nbparatext{Materials.}---Occasionally it has been possible to catch a
+glimpse of the spectrum of one of the brighter shooting stars, and
+the lines of sodium and magnesium (probably) are quite conspicuous,
+along with some other lines which cannot be securely identified by
+such a hasty glance.
+
+As these bodies are completely burned up before they reach the
+earth, all we can ever hope to get of their material is the product
+of the combustion. In most places the collection and identification
+of this meteoric ashes is, of course, hopeless: but Nordenskiold
+%% -----File: 450.png---Folio 439-------
+has thought he might find it in polar snows, and others have
+thought it might be found in the material dredged up from the
+bottom of the ocean. In fact, the Swedish naturalist, by melting
+several tons of Spitzbergen snow and filtering the water, \textit{did} find in
+it a sediment containing minute globules of oxide and sulphide of
+iron: similar globules are also found in the products of deep sea
+dredging. These \textit{may be} meteoric ashes; but what we have lately
+learned from Krakatão of the distance to which smoke and fine
+volcanic dust is carried by the winds, makes it quite possible that
+the suspected material is purely terrestrial in its origin.
+
+\nbarticle{776.} \nbparatext{Probable Mass of Shooting-Stars.}---We have no very certain
+means of getting at this. We can, however, fix a provisional value
+\textit{by means of the amount of light they give}. Photometric comparisons
+between a standard star and a meteor, when we know the meteor's distance
+and the duration of its flight, enable us to ascertain how the total
+amount of light emitted by it compares with that given by a standard
+candle shining for one minute. Now, according to determinations
+made some thirty years ago by Thomson at Copenhagen, (which
+ought to be repeated,) the \textit{light given by a standard candle in a minute
+is equivalent to about twelve foot-pounds of energy}. This excludes
+all the energy of the dark, invisible radiation of the candle. Our
+observations of the meteor give us, therefore, its total \textit{luminous energy}
+in foot-pounds; and if the whole of the meteor's energy appeared as
+light, then, since $\text{\textit{Energy}} = \frac{1}{2}MV^{2}$, we could at once get its mass by
+dividing twice this luminous energy by the square of the meteor's
+velocity. Since, however, only a small portion of the meteor's
+whole energy is transformed into light, the mass obtained in this
+way would be too small, and must be multiplied by a factor which
+expresses the ratio between the \textit{total} energy and that which is \textit{purely
+luminous}. It is not likely that this factor exceeds \textit{one hundred}, or is
+less than \textit{ten}, though on this point we very much need information.
+Assuming the largest value, however, for this factor, the photometric
+observations made in 1866 and 1867 by Professors Newcomb and
+Harkness (stationed respectively at Washington and Richmond),
+showed that the majority of the meteors of those star-showers weighed
+\textit{less than a single grain}. The largest of them did not reach 100 grains,
+or about a quarter of an ounce.
+
+\begin{fineprint}
+\nbarticle{777.} \nbparatext{Growth of the Earth.}---Since the earth (in fact, every planet)
+is thus continually receiving meteoric matter, and sending nothing away
+from it, \textit{it must be constantly growing larger:} but this growth is extremely
+%% -----File: 451.png---Folio 440-------
+insignificant. The meteoric matter received daily by the earth, if we accept
+one grain as the average weight of a shooting star, would be only about
+a ton, after making a reasonable addition for occasional aerolites, if we
+multiply this estimate by one hundred, it certainly will be exceedingly
+liberal, and at that rate the amount received by the earth in a year would
+amount to the very respectable figure of 36,500 tons; and yet, even at this
+rate, assuming the specific gravity of the average meteor as three times that
+of water, it would take, about 1000,000000 \textit{years to accumulate a layer one
+inch thick over the earth's surface.}
+
+\sloppy
+\nbarticle{778.} \nbparatext{Effect on the Earth's Orbit.}---Theoretically, the encounter of
+the earth with meteors must \textit{shorten the year} in three distinct ways:---
+
+\fussy
+First. By acting as a resisting medium, and so diminishing the size
+of the earth's orbit, and indirectly accelerating its motion, in the same
+manner as is supposed to happen with Encke's comet.
+
+Second. By increasing the attraction between the earth and the sun
+through the increase of their masses.
+
+Third. By lengthening the day---the earth's rotation being made slower
+by the increase of its diameter, so that the year will contain a smaller number
+of days.
+
+The whole effect, however, of the three causes combined, does not amount
+to $\frac{1}{1000}$ second in a million of years. The diminution of the earth's
+distance from the sun, assuming that one hundred tons of meteoric matter
+fall daily, and also assuming that the meteors are moving equally in all
+directions with the parabolic velocity of twenty-six miles per second, comes
+out about $\frac{1}{20\,000}$ of an inch \textit{per annum}.
+
+Theoretically, also, the same meteoric action should produce a shortening
+of the \textit{month}, and Oppolzer investigated the subject a few years ago, to see
+what amount of meteoric matter would account for the observed \textit{lunar acceleration}
+(\artref{Art.}{459}). He found that it would require an amount immensely
+greater than really falls.
+
+\nbarticle{779.} \nbparatext{Meteoric Heat: Effect of Meteors on the Transparency of
+Space.}---Of course each meteor brings to the earth a certain amount of \textit{heat}
+developed in the destruction of its motion; and at one time if was thought
+that a very considerable percentage of the total heat: received by the earth
+might be derived from this source (See \artref{Art.}{355}, (2)). Assuming, however,
+as before, the fall of one hundred tons of meteoric matter daily with
+an average velocity of twenty miles per second relative to the earth, the
+whole amount of heat comes out about $\frac{1}{20}$ calorie \textit{per annum} for each square
+metre of the earth's surface---as much in a \textit{year} as the sun imparts to the
+same surface in about \textit{one-tenth of a second.}
+
+One other effect of meteoric matter in space should be alluded to. It
+must necessarily render space imperfectly transparent, like a thin haze.
+Less light reaches us from a remote star than if the meteors were absent.
+\end{fineprint}
+%% -----File: 452.png---Folio 441-------
+
+\sloppy
+\nbarticle{780.} \nbparatext{Meteoric Showers.}---At certain times the shooting stars,
+instead of appearing here and there in the sky at intervals of several
+minutes, and moving in all directions, appear by thousands, and even
+hundreds of thousands, for a few hours.
+
+\nbenlargepage
+\fussy
+\textit{The Radiant.}---At such times they do not move without system;
+but they all appear to diverge or ``\textit{radiate}'' from one point in the
+sky; that is, their paths produced backward all intersect at a common
+point (or nearly so), which is called ``\textit{the radiant.}'' As an old lady
+expressed it, in speaking of the meteoric shower of 1833, ``The sky
+looked like a great umbrella.'' The meteors which appear near the
+\textit{radiant} are stationary, or have paths extremely short, while those
+which appear at a distance from it have long courses. The radiant
+keeps its place among the stars unchanged during the whole continuance
+of the shower, and the shower is named accordingly. Thus we
+have the meteor shower of the ``\textit{Leonids},'' whose radiant is in the
+constellation of Leo; similarly the ``\textit{Andromedes}'' (or Bielids), the
+``\textit{Perseids},'' the ``\textit{Geminids},'' the ``\textit{Lyrids},'' etc. \figref{illo210}{Fig.~210}.\ is a chart
+of the tracks of meteors observed on the night of Nov.~13, 1866,
+showing the radiant near $\zeta$~Leonis.
+
+\includegraphicsmid{illo210}{\textsc{Fig.~210.}---The Meteoric Radiant in Leo, Nov.~13, 1866.}
+
+The simple explanation is that the radiant is purely an effect of
+%% -----File: 453.png---Folio 442-------
+perspective. The meteors are really moving, relatively to the observer,
+in lines which are sensibly straight and parallel, as are also
+the tracks of light which they leave in the air. Hence they all seem
+to diverge from one and the same perspective ``vanishing point.'' The
+position of the radiant depends entirely upon the \textit{direction} of the
+meteor's motion relative to the earth.
+
+On account of the irregular form of the meteoric particles, they are
+deflected a little one way or the other by the air, so that their paths
+do not intersect at an absolute point; neither is it likely that before
+they enter the air their paths are \textit{exactly} parallel. The consequence
+is that the radiant, instead of being a point, is an \textit{area} of some little
+size, usually less than $2°$ in diameter.
+
+\begin{fineprint}
+\nbarticle{781.} Probably the most remarkable of all meteoric showers that ever
+occurred was that which appeared in the United States on Nov.~12, 1833,
+in the early morning---a shower of Leonids. The number that fell in
+the five or six hours during which the shower lasted was estimated at
+Boston as fully 250,000. A competent observer declared that ``he never
+saw snow-flakes thicker in a storm than were the meteors in the sky at
+some moments.'' No sound was heard, nor was any particle known to
+reach the earth.
+\end{fineprint}
+
+\nbarticle{782.} \nbparatext{Dates of Showers.}---Since the meteor-swarm pursues a regular
+orbit around the sun, the earth can only encounter it when she is
+at the point where her orbit cuts the path of the meteors; and this,
+of course, must always be on the same day of the year, except as, in
+the process of time, the meteors' orbits slowly shift their positions on
+account of perturbations. The Leonid showers, therefore, always
+appear on the 18th of November (within a day or two); the Andromedes
+on the 27th or 28th of the same month; and the Perseids early
+in August.
+
+\nbarticle{783.} \nbparatext{Meteoric Rings and Swarms.}---If the meteors are scattered
+nearly uniformly around their whole orbit, so as to form \textit{a ring}, the
+shower will recur \textit{every year}; but if the flock is concentrated, it will occur
+only when the meteor group is at the meeting-place at the same time
+as the earth. The latter is the case with the Leonids and Andromedes.
+The great star-showers from these groups occur only rarely,---for the
+Leonids once in thirty-three years, and for the Andromedes (otherwise
+known as the Bielids) about once in thirteen. The Perseids are
+much more equally and widely distributed, so that they appear in
+considerable numbers every year, and are not sharply limited to a
+%% -----File: 454.png---Folio 443-------
+particular date, but are more or less abundant for a fortnight in the
+latter part of July and the first of August.
+
+\begin{fineprint}
+The meteors which belong to the same group all have a resemblance to
+each other. The Perseids are yellowish, and move with medium velocity.
+The Leonids are very swift, for we meet them almost directly, and they are
+characterized by a greenish or bluish tint, with vivid and persistent trains.
+The Andromedes are sluggish in their movements, because they simply
+overtake the earth, instead of meeting it. They are usually decidedly red in
+color and have only small trains.
+
+\nbarticle{784.} \nbparatext{The Mazapil Meteorite.}---As has been said, during these
+showers no sound is heard, no sensible heat perceived, nor do any masses
+reach the ground; with the one exception, however, that on Nov.~27, 1885,
+a piece of meteoric iron mentioned in the list given in \artref{Article}{758}, fell at
+Mazapil in Northern Mexico during the shower of Andromedes which
+occurred that evening. Whether the coincidence is accidental or not, it is
+interesting. Many high authorities speak confidently of this particular iron
+meteor as being really a piece of Biela's comet itself.
+\end{fineprint}
+
+And now we come to one of the most remarkable discoveries of
+modern astronomy,---the discovery of---
+
+\nbarticle{785.} \nbparatext{The Connection between Comets and Meteors.}---At the time
+of the great meteoric shower of 1833, Professors Olmsted and
+Twining, of New Haven, recognized the fact and meaning of the
+radiant as pointing to the existence of \textit{swarms} of meteoric particles
+revolving in regular orbits around the sun; and Olmsted at the time
+went so far as even to call the body or swarm a ``comet.'' In some
+respects, however, his views were seriously wrong, and soon received
+modification and correction from other astronomers. Erman especially
+pointed out that in some cases, at least, it would be necessary
+to suppose that the meteors were distributed in \textit{rings}, and he also
+developed methods by which the meteoric orbits could be computed
+if the necessary data could be secured. Olmsted and Twining,
+however, were the first to show that the meteors are not terrestrial
+and atmospheric, but bodies truly cosmical.
+
+The subject was taken up later by Professor Newton, of New Haven,
+who in 1864 showed by an examination of old records that
+there had been a number of great autumnal meteoric star-showers at
+intervals of just about thirty-three years, and he predicted confidently
+a shower for Nov.~13--14, 1866. As to the orbit of the meteoric
+body (or ring, according to Erman's view), he found that it might,
+%% -----File: 455.png---Folio 444-------
+consistently with what had been so far observed, have either of \textit{five}
+different orbits; one with a period of $33 \frac{1}{4}$ years, two with periods of
+one year $\pm 11$ days, and two with periods of half a year $\pm 5 \frac{1}{2}$ days.
+He considered rather most probable the period of 354 days; but he
+pointed out that the slow change that had taken place in the annual
+date of the shower\footnote
+  {In \textsc{a.d.}~902 (the ``year of the stars'' in the old Arab chronicles), the date
+  was what would be Oct.~19, in our ``new style'' reckoning. In 1202 the shower
+  occurred five days later, and in 1833 the date was Nov.~12.}
+would furnish the means of determining which
+of the orbits was the true one.
+
+This change of date indicates a slow motion of the nodes of the
+orbit of the meteoric body at the rate of about $52''$ a year. Adams,
+of Neptunian fame, made the laborious calculation of the effect of
+planetary perturbations upon each of the five different orbits suggested
+by Professor Newton, and showed that the true orbit must be
+the largest one which has a period of $33 \frac{1}{4}$ years.
+
+\nbenlargepage
+The meteoric shower occurred in 1866 as predicted, and was
+repeated in 1867, the meteor-swarm being stretched out along its
+orbit for such a distance that the procession is nearly three years in
+passing any given point.
+%% -----File: 456.png---Folio 445-------
+
+\includegraphicsmid{illo211}{\textsc{Fig.~211.}---Orbits of Meteoric Swarms which are known to be associated with Comets.}
+
+\nbarticle{786.} \nbparatext{Identification of Cometary and Meteoric Orbits.}---The researches
+of Newton and Adams had awakened lively interest in
+the subject, and Schiaparelli, of Milan, a few weeks after the Leonid
+shower, published a paper upon the Perseids, or August meteors, in
+which he brought out the remarkable fact that they were \textit{moving in
+the same path as that of the bright comet of $1862$, known as Tuttle's
+Comet}. Shortly after this Leverrier published his orbit of the
+Leonid meteors, derived from the observed position of the radiant in
+connection with the periodic time assigned by Adams; and almost
+simultaneously, but without any idea of a connection between them,
+Oppolzer published his orbit of Tempel's comet of 1866; and the
+two orbits were at once seen to be \textit{practically identical}. Now a \textit{single}
+case of such a coincidence as that pointed out by Schiaparelli, might
+possibly be accidental, but hardly \textit{two}. Then five years later, in
+1872, came the meteorite shower of the Andromedes, following in the
+track of Biela's comet; and among the more than one hundred distinct
+meteor-swarms now recognized, Professor Alexander Herschel
+finds four or five others which have a ``comet annexed,'' so to speak.
+Fig.~211 represents the orbits of four of the meteoric swarms which
+are known to be associated with comets.
+%% -----File: 457.png---Folio 446-------
+
+\begin{fineprint}
+\nbarticle{787.} In the cases of the Leonids and Andromedes the meteor-swarm
+\textit{follows} the comet. Many believe, however, that the comet itself is simply
+the thickest part of the swarm. Kirkwood and Schiaparelli have both
+pointed out that a body constituted as a comet is supposed to be, must
+almost necessarily break up in consequence of the ``tide-producing'' perturbations
+of the sun, independent of any repulsive action such as is supposed
+to be the cause of a comet's tail. They hold that these meteor-swarms are
+therefore merely the \textit{product of a comet's disintegration}.
+
+\includegraphicsmid{illo212}{\textsc{Fig.~212.}---Transformation of the Orbit of the Leonids by the Encounter with Uranus, \textsc{a.d.}~126.}
+
+The longer the comet has been in the system, the more widely scattered
+will be its particles. The Perseids are supposed, therefore, to be old inhabitants
+of the solar system, while the Leonids and Andromedes are comparatively
+new-comers. Leverrier has shown that in the year \textsc{a.d.}~126 Tempel's
+comet must have been very near to Uranus, and a natural inference is that
+it was introduced into the solar system at that time. \figref{illo212}{Fig.~212} illustrates his
+hypothesis. However these things may be, it is now certain that the connection
+between comets and meteors is a very close one, though it can hardly
+be considered certain as yet that every scattered group of meteors is the
+result of cometary \textit{disintegration}. We are not sure that when a cometary
+mass first enters the solar system from outer space, it comes in as a close-packed
+swarm.
+\end{fineprint}
+\chelabel{CHAPTERXVIII}
+
+%% -----File: 458.png---Folio 447-------
+\Chapter{XIX}{The Stars}
+{\setlength{\hfuzz}{2pt}
+\nbchapterhang{\stretchyspace
+THE STARS:\hspace{0em} THEIR NATURE AND NUMBER.---THE
+CON\-STELLATIONS.---STAR-CATALOGUES.---DESIGNATION AND
+NOMENCLATURE.---PROPER MOTIONS AND THE MOTION
+OF THE SUN IN SPACE.---STELLAR PARALLAX AND DISTANCE.}
+
+}%end hfuzz
+\nbarticle{788.} \textsc{We} enter now upon a vaster subject. Leaving the confines
+of the solar system we cross the void that makes an island\footnote
+  {That the solar system is thus isolated by a surrounding void is proved by the
+  almost undisturbed movements of Uranus and Neptune; for their perturbations
+  would betray the presence of any body, at all comparable with the sun in magnitude,
+  within a distance a thousand times as great as that between the earth and
+  sun.}
+of the
+sun's domains, and enter the universe of the stars. The nearest star,
+so far as we have yet been able to ascertain, is one whose distance is
+more than 200,000 times the radius of the earth's annual orbit; so
+remote that, seen from that star, the sun itself would appear only
+about as bright as the pole star, and from it no telescope ever yet
+constructed could render visible a single one of all the retinue of
+planets and comets that make up the solar system.
+
+\nbarticle{789.} \nbparatext{Nature of the Stars.}---As shown by their spectra the stars
+are \textit{suns}; that is, they are bodies comparable in magnitude and in
+physical condition with our own sun, shining by their own light as the
+sun does, and emitting a radiance which in many cases could not be
+distinguished from sunlight by any of its spectroscopic characteristics.
+Some of them are vastly larger and hotter than our sun, others smaller
+and cooler, for, as we shall see, they differ enormously among themselves.
+
+\nbarticle{790.} \nbparatext{Number of the Stars.}---The impression on a dark night is of
+absolute countlessness; but, in fact, the number visible to the naked
+eye is very limited, as one can easily discover by taking some definite
+area in the sky, say the ``bowl of the dipper,'' and counting the stars
+which he can see within it. He will find that the number which he
+%% -----File: 459.png---Folio 448-------
+can fairly count is surprisingly small, though by averted vision he will
+get uncertain glimpses of many more. In the whole celestial sphere
+the number bright enough to be visible to the naked eye is only from
+6000 to 7000 in a clear, moonless sky. A little haze or moonlight
+cuts out fully half of them, and of course there is a great difference
+in eyes. But the sharpest eyes could probably never fairly see more
+than 2000 or 3000 at one time, since near the horizon the smaller stars
+are invisible, and they are immensely the most numerous, fully half
+of the whole number being those which are just on the verge of visibility.
+\textit{The total number that can be seen well enough for observation
+with such instruments as were used before the invention of the telescope
+is not quite $1100$.}
+
+With even a small telescope the number is enormously increased.
+A mere opera-glass an inch a half in diameter brings out at least
+100,000. The telescope with which Argelander made his \textit{Durchmusterung}
+of more than 300,000 stars---all north of the celestial equator---had
+a diameter of only two inches and a half. The number visible
+in the great Lick\footnote
+  {Neglecting the loss of light in the lenses, the Lick telescope ought, theoretically,
+  to show stars so faint that it would take more than 30,000 of them to make
+  a star equal to the faintest that can be seen with the naked eye. (See \artref{Art.}{38}.)}
+telescope of three feet diameter is probably nearly
+100,000000.
+
+\sloppy
+\nbarticle{791.} \nbparatext{Constellations.}---In ancient times the stars were grouped by
+``constellations,'' or ``asterisms,'' partly as a matter of convenient
+reference and partly as superstition. Many of the constellations now
+recognized,---all of those in the zodiac and those about the northern
+pole,---are of prehistoric antiquity. To these groups were given fanciful
+names, mostly of persons or objects conspicuous in the mythological
+records of antiquity; a great number of them are connected in some
+way or other with the Argonautic expedition.
+
+\fussy
+\begin{fineprint}
+In some cases the eye can trace in the arrangement of the stars a vague
+resemblance to the object which gives name to the constellation; but generally
+no reason can be assigned why the constellation should be so named or
+so bounded. Of the sixty-seven constellations now usually recognized on
+celestial globes, forty-eight have come down from Ptolemy. The others
+have been formed by Hevelius, Bayer, Royer, and one or two other astronomers,
+to embrace stars not included in Ptolemy's constellations, and especially
+to furnish a nomenclature for the stars never seen by Ptolemy on
+account of their nearness to the southern pole. A considerable number of
+%% -----File: 460.png---Folio 449-------
+other constellations, which have been tentatively established at various
+times, and are sometimes found on globes and star-maps, have been given
+up as useless and impertinent.
+
+\nbarticle{792.} We present a \hyperref[tab:const]{list of the constellations}, omitting, however, some
+of the modern ones which are now not usually recognized by astronomers.
+The constellations are arranged both vertically and horizontally. The order
+in the vertical columns is determined by right ascension, as indicated by
+the Roman numbers at the left. Horizontally the arrangement is according
+to distance from the north pole, as shown by the headings of the columns.
+The number appended to each constellation gives the number of stars it
+contains, down to and including the 6th magnitude. The zodiacal constellations
+are italicized, and the modern constellations are marked by an
+asterisk.
+\begin{sidewaystable}
+\nblabel{tab:const}
+\scriptsize\centering
+\renewcommand{\arraystretch}{0.8}
+\begin{tabular}{@{}r@{ }r@{\;}| *{5}{@{\,}c@{\,}l@{ }r|} @{\,}c@{\,}l@{ }r@{}}
+\multicolumn{20}{c}{LIST OF CONSTELLATIONS, SHOWING THEIR POSITION IN THE HEAVENS.}\\[1ex]
+
+\hline\hline
+
+\multicolumn{2}{@{}c@{}|@{\,}}{\textsc{\scriptsize
+   \rlap{\raisebox{-1ex}{\;R. A.}}%
+   \raisebox{0.8ex}{\rotatebox[origin=c]{-18}{\rule{6.5em}{0.5pt}}}% Diagonal line across top left cell
+   \llap{\raisebox{1ex}{Decl.\;}}}}
+& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $-90°$ to $+50°$.\rule{0pt}{3.5ex}}}
+& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $+50°$ to $+25°$.}}
+& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $+25°$ to $0°$.}}
+& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $0°$ to $-25°$.}}
+& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $-25°$ to $-50°$.}}
+& \multicolumn{3}{c }{\textsc{\scriptsize $-50°$ to $-90°$.}}\\[1ex]
+
+\hline
+
+\multirow{2}{*}{\scriptsize h.} &
+\multirow{2}{*}{\scriptsize h.} &
+&& &&&  &&& &&& &&& &&
+\\
+\multirow{2}{*}{I,}  & \multirow{2}{*}{II,}
+& & \multirow{2}{*}{Cassiopeia,} & \multirow{2}{*}{46}
+& & Andromeda, & 18
+& & \textit{Pisces}, & 18 & & \multirow{2}{*}{Cetus,}  & 32
+& & Ph\oe nix,       & 32 & & Ph\oe nix, \textit{bis.}
+\\
+&
+&&& & & Triangulum, &  5
+& & \textit{Aries}, & 17 &&&
+&*& App.\ Sculp.    & 13 & & Hydrus, & 18
+\\[1.5ex]
+
+\multirow{2}{*}{III,} & \multirow{2}{*}{IV,}
+& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
+& & \multirow{2}{*}{Perseus,}  & \multirow{2}{*}{40}
+& & \multirow{2}{*}{\textit{Taurus},} & \multirow{2}{*}{58}
+& & \multirow{2}{*}{Eridanus,} & \multirow{2}{*}{64}
+& & \multirow{2}{*}{(Eridanus, \textit{bis.})} & &*& Horologium, & 11
+\\
+&
+&&& &&&
+&&& &&&
+&&& &*& Reticulum, & 9
+\\[1.5ex]
+
+&
+&&& &&&
+& & \multirow{2}{*}{Orion,}        & \multirow{2}{*}{37}  &&&
+&&& &*& Dorado, & 16
+\\
+V, & VI,
+&*& Camelopardus,    & 36 & & Auriga, & 35
+& & \multirow{2}{*}{\textit{Gemini},} & \multirow{2}{*}{28} & & Lepus,  & 18
+&*& Columba,         & 15 &*& Pictor, & 14
+\\
+&
+&&& &&&
+&&& &&&
+&&& &*& Mons Mensæ, & 12
+\\[1.5ex]
+
+&
+&&& &&&
+& & \multirow{2}{*}{Canis Minor,} & \multirow{2}{*}{ 6}
+& & \multirow{2}{*}{Canis Major,} & \multirow{2}{*}{27}
+&&&
+& & Argo-Navis, \textit{bis.} &
+\\
+VII, & VIII,
+& \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
+&*& Lynx, & 28
+& & \multirow{2}{*}{\textit{Cancer},} & \multirow{2}{*}{15}
+&\multirow{2}{*}{*}& \multirow{2}{*}{Monoceros,} & \multirow{2}{*}{12}
+& & Argo-Navis, & 133
+& & \quad\ (Puppis) &
+\\
+&
+&&& &&&
+&&& &&&
+&&& &*& Piscis Volans, & 9
+\\[1.5ex]
+
+\multirow{2}{*}{IX,} & \multirow{2}{*}{X,}
+& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
+&\multirow{2}{*}{*}& \multirow{2}{*}{Leo Minor,} & \multirow{2}{*}{15}
+& & \multirow{2}{*}{\textit{Leo},} & \multirow{2}{*}{47} & & Hydra, & 49
+& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
+& & Argo-Navis &
+\\
+&
+&&& &&&
+&&& &*& Sextans, & 3
+&&& & & \quad (Vela) &
+\\[1ex]
+
+&
+&&& &&&
+&&& & & \multirow{2}{*}{Crater,}    & \multirow{2}{*}{9}
+&&& & & Argo Navis &
+\\
+XI, & XII,
+& & Ursa Major, & 53 & \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
+&*& Coma Ber.   & 20 & & \multirow{2}{*}{Corvus,} & \multirow{2}{*}{8}
+& & Centaurus,  & 54 & & \ \ (Carina) &
+\\
+&
+&&& &&&
+&&& &&&
+&&& &*& Chameleon, & 13
+\\[1.5ex]
+
+&
+&&& &\multirow{2}{*}{*} & \multirow{2}{*}{Canes Venat.} & \multirow{2}{*}{15}
+&&& &&&
+&&& & & Centaurus, \textit{bis.} &
+\\
+XIII, & XIV,
+& \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
+& & \multirow{2}{*}{Boötes,} & \multirow{2}{*}{35}
+& \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
+& & \textit{Virgo}, & 39
+& & Lupus, & 34 &*& Crux, & 13
+\\
+&
+&&& &&&
+&&& &&&
+&&& &*& Musca, & 15
+\\[1.5ex]
+
+\multirow{2}{*}{XV,} & \multirow{2}{*}{XVI,}
+& & \multirow{2}{*}{Ursa Minor,} & \multirow{2}{*}{23} & & Corona Bor.     & 19
+& & \multirow{2}{*}{Serpens,}    & \multirow{2}{*}{23}
+& & \multirow{2}{*}{\textit{Libra},} & \multirow{2}{*}{23}
+& & \multirow{2}{*}{Norma,}      & \multirow{2}{*}{14}
+&\multirow{2}{*}{*}& \multirow{2}{*}{Circinus,} & \multirow{2}{*}{10}
+\\
+&
+&&& & & Hercules, & 65
+&&& &&&
+&&& &&&
+\\[1.5ex]
+
+\multirow{2}{*}{XVII,} & \multirow{2}{*}{XVIII,}
+& & \multirow{2}{*}{Draco,}  & \multirow{2}{*}{80}
+& & \multirow{2}{*}{Lyra,}             & \multirow{2}{*}{18}
+& & Aquila, & 37 & & \textit{Scorpio}, & 34
+& & \multirow{2}{*}{Ara,}    & \multirow{2}{*}{15} &*& Triangul.\ Aust.& 11
+\\
+&
+&&& &&&
+& & Sagitta, & 5 & & Ophiuchus, & 46
+&&& &*& Apus, & 15
+\\[1.5ex]
+
+&
+&&& &&&
+&\multirow{2}{*}{*}& \multirow{2}{*}{Vulpecula,} & \multirow{2}{*}{23} &&&
+&&& &*& Telescopium, & 16
+\\
+XIX, & XX,
+& \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
+& & Cygnus, & 67
+& & \multirow{2}{*}{Delphinus,} & \multirow{2}{*}{10}
+& & \textit{Sagittarius}, & 38
+& & Corona Austr. & 7 & & Pavo. & 37
+\\
+&
+&&& &&&
+&&& &&&
+&&& &*& Octans, & 22
+\\[1.5ex]
+
+\multirow{2}{*}{XXI,} & \multirow{2}{*}{XXII,}
+& & \multirow{2}{*}{Cepheus,}      & \multirow{2}{*}{44}
+&\multirow{2}{*}{*}& \multirow{2}{*}{Lacerta,} & \multirow{2}{*}{13}
+& & \multirow{2}{*}{Equuleus,}     & \multirow{2}{*}{ 5}
+& & \multirow{2}{*}{\textit{Capricornus},} & \multirow{2}{*}{22}
+& & \multirow{2}{*}{Piscis Austr.} & \multirow{2}{*}{16} &*& Indus, & 15
+\\
+&
+&&& &&&
+&&& &&&
+&&& &*& Octans &
+\\[1.5ex]
+
+\multirow{2}{*}{XXIII,} & \multirow{2}{*}{XXIV,}
+& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
+& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
+& & \multirow{2}{*}{Pegasus,} & \multirow{2}{*}{43}
+& & \multirow{2}{*}{\textit{Aquarius},} & \multirow{2}{*}{25}
+&\multirow{2}{*}{*}& \multirow{2}{*}{Grus,} & \multirow{2}{*}{30}
+&*& Toucana, & 22
+\\
+&
+&&& &&&
+&&& &&&
+&&& &*& Octans &
+\\[1.5ex]
+
+\hline
+\end{tabular}
+\end{sidewaystable}
+
+The different groups of constellations are found near the meridian at
+half-past eight o'clock, \textsc{p.m.}, on the dates indicated below.
+
+\medskip
+{\parindent=1em
+\hangindent=4em
+Group (I., II.), Dec.~1. These constellations contain no first-magnitude
+stars, but Cassiopeia, Andromeda, Aries, and Cetus include enough
+stars of the second and third magnitude to be fairly conspicuous.
+
+\hangindent=4em
+Group (III., IV.), Jan.~1. Perseus north of the zenith, and the Pleiades
+and Aldebaran in Taurus, are characteristic.
+
+\hangindent=4em
+Group (V., VI.), Feb.~1. On the whole this is the most brilliant region
+of the sky and Orion the finest constellation.
+
+\hangindent=4em
+Group (VII., VIII.), March 1. Characterized by Procyon and Sirius, the
+latter incomparably the brightest of all the fixed stars.
+
+\hangindent=4em
+Group (IX., X.), April 1. Leo is the only conspicuous constellation.
+
+\hangindent=4em
+Group (XI., XII.), May 1. A barren region, except for Ursa Major north
+of the zenith.
+
+\hangindent=4em
+Group (XIII., XIV.), June 1. Marked by Arcturus, the brightest of the
+northern stars, with the paler Spica south of the equator.
+
+\hangindent=4em
+Group (XV., XVI.), July 1. The Northern Crown and Hercules are the
+most characteristic configurations.
+
+\hangindent=4em
+Group (XVII., XVIII.), Aug.~1. Vega is nearly overhead, and the red
+Antares low down in the south, with Altair near the equator, just
+east of Ophiuchus.
+
+\hangindent=4em
+Group (XIX., XX.), Sept.~1. Cygnus is in the zenith, and Sagittarius
+low down, while the brightest part of the Milky Way lies athwart
+the meridian.
+
+\hangindent=4em
+Group (XXI., XXII.), Oct.~1. A barren region, relieved only by the
+bright star Fomalhaut of the Southern Fish near the southern
+horizon.
+
+\hangindent=4em
+Group (XXIII., XXIV.), Nov.~1. This region also is rather barren,
+though the ``great square'' of Pegasus is a notable configuration
+of stars.
+\par} %end \parindent=1em
+\end{fineprint}
+%% -----File: 461.png---Folio 450-------
+%% -----File: 462.png---Folio 451-------
+
+\nbarticle{793.} A thorough knowledge of these artificial groups, and of the
+names and locations of the stars in them, is not at all essential, even
+to an accomplished astronomer; but it is a matter of very great convenience
+to know the principal constellations, and perhaps a hundred
+of the brightest stars, well enough to be able to recognize them readily
+and to use them as points of reference. This amount of knowledge
+is easily acquired by three or four evenings' study of the sky in
+connection with a good star-map or celestial globe, taking care to
+observe on evenings at different seasons of the year, so as to command
+the whole sky.
+
+\begin{fineprint}
+At present the best star-atlas for reference is probably that of Mr.\
+Proctor. The maps of Argelander's ``Uranometria Nova'' and Heis's atlas
+(both in German) are handsomer, and for some purposes more convenient.
+There are many others, also, which are excellent. The smaller maps which
+are found in the text-books on astronomy are not on a scale sufficiently large
+to be of much scientific use (as, for instance, in the observation of meteors),
+though they answer well enough the purpose for which they were designed,
+of introducing the student to the principal star-groups.
+\end{fineprint}
+
+\nbarticle{794.} \nbparatext{Designation of Bright Stars.}---(\textit{a}) \textit{Names.} Some fifty or
+sixty of the brighter stars have names of their own in common use.
+A majority of the names belonging to stars of the first magnitude are
+of Greek or Latin origin, and significant, as, for instance, Arcturus,
+Sirius, Procyon, Regulus, etc. Some of the brightest stars, however,
+have Arabic names, as Aldebaran, Vega, and \DPtypo{Betelguese}{Betelgueze}, and the
+names of most of the smaller stars are Arabic, when they have names
+at all.
+
+(\textit{b}) \textit{Place in Constellation.} Spica is the star in the handful of
+wheat carried by Virgo; Cynosure signifies the star at the end of the
+Dog's Tail (in ancient times the constellation we now call Ursa
+Minor seems to have been a dog); Capella is the goat which
+Auriga, the charioteer, carries in his arms. Hipparchus, Ptolemy,
+and, in fact, all the older astronomers, including Tycho Brahe, used
+this clumsy method almost entirely in designating particular stars;
+speaking, for instance, of the star in the ``head of Hercules,'' or in
+the ``right knee of Boötes,'' and so on.
+
+(\textit{c}) \textit{Constellation and Letters.} In 1603 Bayer, in publishing a new
+star-map, adopted the excellent plan, ever since in vogue, of designating
+the stars in the different constellations by the letters of the Greek
+alphabet, assigned usually in order of brightness. Thus Aldebaran is
+%% -----File: 463.png---Folio 452-------
+$\alpha$~Tauri, the next brightest star in the constellation is $\beta$~Tauri, and
+so on, as long as the Greek letters hold out; then the Roman letters
+are used as long as they last; and finally, whenever it is found necessary,
+we use the numbers which Flamsteed assigned a century later.
+At present every naked-eye star can be referred to and identified by
+some letter or number in the constellation to which it belongs.
+
+(\textit{d}) \textit{Current Number in a Star-Catalogue.} Of course all the above
+methods fail for the hundreds of thousands of smaller stars. In their
+case it is usual to refer to them as number so-and-so of some well-known
+star-catalogue; as, for instance, 22,500 Ll. (Lalande), or
+2573 B.A.C. (British Association Catalogue). At present our various
+star-catalogues contain from 600,000 to 800,000 stars, so that, except
+in the Milky Way, almost any star visible in a telescope of two or
+three inches' aperture can be identified and referred to by means of
+some star-catalogue or other.
+
+\textit{Synonyms.} Of course all the brighter stars which have \textit{names} have
+also letters, and are sure to be included in every star-catalogue which
+covers their part of the sky. A given star, therefore, has often a
+large number of aliases, and in dealing with the smaller stars great
+pains must be taken to avoid mistakes arising from this cause.
+
+\section*{STAR-CATALOGUES.}
+
+\sloppy
+\nbarticle{795.} These are lists of stars arranged in regular order (at present
+usually in order of right ascension), and giving the places of the stars
+at some given epoch, either by means of their right ascensions and
+declinations, or by their (celestial) latitudes and longitudes. The
+so-called ``magnitude,'' or brightness of the star, is also ordinarily
+indicated. The first of these star-catalogues was that of Hipparchus,
+containing 1080 stars (all that are \textit{easily} visible and measurable by
+naked-eye instruments), and giving their longitudes and latitudes for
+the epoch of 125 \textsc{b.c.}
+
+\fussy
+\begin{fineprint}
+This catalogue has been preserved for us by Ptolemy in the Almagest,
+and from it he formed his own catalogue, reducing the positions of the
+stars (\textit{i.e.}, correcting for precession the positions given by Hipparchus) to
+his own epoch, about 150~\textsc{a.d.} The next of the old catalogues of any value
+is that of Ulugh Beigh made at Samarcand about 1450~\textsc{a.d.} This appears
+to have been formed from independent observations. It was followed in
+1580 by the catalogue of Tycho Brahe containing 1005 stars, the last which
+was constructed before the invention of the telescope.
+
+The modern catalogues are numerous. Some give the places of a great
+number of stars rather roughly, merely as a means of \textit{identifying} them when
+%% -----File: 464.png---Folio 453-------
+used for cometary observations or other similar purposes. To this class
+belongs Argelander's \textit{Durchmusterung} of the northern heavens, which contains
+over 324,000 stars,---the largest number in any one catalogue thus far
+published. Then there are the ``\textit{catalogues of precision},'' like the Pulkowa
+and Greenwich catalogues, which give the places of a few hundred stars as
+accurately as possible in order to furnish ``fundamental stars,'' or reference
+points in the sky. The so-called ``\textit{Zones}'' of Bessel, Argelander, and many
+others, are catalogues covering limited portions of the heavens, containing
+stars arranged in zones about a degree wide in declination, and running
+some hours in right ascension. To the practical astronomer the most useful
+catalogue is likely to be the one which is now in process of formation by the
+co-operation of various observatories under the auspices of the Astronomische
+Gesellschaft (an International Astronomical Society, with its headquarters
+in Germany). This catalogue will contain accurate places of all stars above
+the ninth magnitude in the northern sky. Most of the necessary observations
+have already been made.
+\end{fineprint}
+
+\nbenlargepage
+\sloppy
+\nbarticle{796.} \nbparatext{Determination of Star-Places.}---The observations from which
+a star-catalogue is constructed are usually made with the meridian
+circle (\artref{Art.}{63}). For the catalogues of precision, comparatively few
+stars are observed, but all with the utmost care and during several
+years, taking all possible means to eliminate instrumental and observational
+errors of every sort.
+
+\fussy
+In the more extensive catalogues most of the stars are observed only
+once or twice, and everything is made to depend upon the accuracy
+of the places of the fundamental stars, which are assumed as correct.
+The instrument in this case is used only ``differentially'' to measure
+the comparatively small differences between the right ascension and
+declination of the fundamental stars and those of the stars to be catalogued.
+
+\begin{includegraphicspage}{213}
+\includegraphicsmid{illo213}{\textsc{Fig.~213.}---The Photographic Telescope of the Henry Brothers, Paris.}
+\end{includegraphicspage}
+
+\nbarticle{797.} \nbparatext{Method of using a Catalogue.}---The catalogue contains the
+\textit{mean} right ascension and declination of its stars for the beginning
+of some given year; \textit{i.e.}, the right ascension and declination the star
+\textit{would have} at that time if there were no aberration of light and no
+irregular motion in the celestial pole to affect the position of the
+equator and equinox. To determine the actual \textit{apparent} right ascension
+and declination of a star for a given date (which is what we
+want in practice), the catalogue place must be ``\textit{reduced}'' to the
+date in question; \textit{i.e.}, it must be corrected for precession, nutation,
+and aberration.
+
+\begin{fineprint}
+The operation with modern tables and formulæ is not a very tedious one,
+involving perhaps five minutes' work, but without the catalogue places are
+%% -----File: 465.png---Folio 454-------
+%% -----File: 466.png---Folio 455-------
+useless for most purposes. \textit{Vice versa}, the observations of a fixed star with
+the meridian circle do not give its \textit{mean} right ascension and declination
+ready to go into the catalogue, but the observations must be reduced from
+\textit{apparent} place to \textit{mean} before they can be tabulated.
+\end{fineprint}
+
+\nbarticle{798.} \nbparatext{Star-Charts.}---For many purposes \textit{charts} of the stars are
+more convenient than a catalogue, as, for instance, in searching for
+new planets. The old-fashioned way of making such charts was by
+plotting the results of zone observations. The modern way, introduced
+within the last few years, is to do it by photography. The
+plan decided upon at the Paris Astronomical Congress in 1887
+contemplates the photographing of the whole sky upon glass plates
+about six inches square, each covering an area of $2°$ square (four
+square degrees), showing all stars down to the fourteenth magnitude,---a
+project which is entirely feasible, and can be accomplished in
+five or six years by the co-operation of about a dozen different observatories
+in the northern and southern hemispheres. The instruments
+are now (1888) in process of construction.
+
+\begin{fineprint}
+The figure (\figref{illo213}{Fig.~213}) is a representation of the Paris instrument of the
+Henry Brothers, which was adopted as the typical instrument for the operation.
+It has an aperture of about fourteen inches, and a length of about
+eleven feet, the object-glass being specially corrected for the photographic
+rays. A 9-inch visual telescope is enclosed in the same tube so that the observer
+can watch the position of the instrument during the whole operation.
+
+It was originally planned to give each plate 20~minutes' exposure, but
+improvements in the photographic plates since the meeting of the Congress
+now make it possible to cut down the time very materially. It will require
+about 11,000 plates of the size named to cover the whole sky, and as each
+star is to appear on two plates at least, the whole number of plates, allowing
+for overlaps, will be about 22,000. As every plate will contain upon it a
+number of well-determined catalogue stars, it will furnish the means of
+determining accurately, whenever needed, the place of any other star which
+appears upon the same plate.
+\end{fineprint}
+
+
+\section*{STAR MOTIONS.}
+
+\nbarticle{799.} The stars are ordinarily called ``\textit{fixed},'' in distinction from
+the planets or ``\textit{wanderers},'' because as compared with the sun and
+moon and planets they have no evident motion, but keep their relative
+relations and configurations unchanged. Observations made at sufficiently
+wide intervals of time, and observations with the spectroscope,
+show, however, that they are really moving, and that with velocities
+which are comparable to the motion of the earth in her orbit.
+%% -----File: 467.png---Folio 456-------
+
+If we compare the right ascension and declination of a star
+determined to-day with that determined a hundred years ago, it will
+be found different. The difference is \textit{mainly} due to precession and
+nutation, which are not motions of the stars at all, but simply
+changes in the position of the reference circles used, and due to
+alterations in the direction of the earth's axis (Arts.~\arnref{205} and \arnref{214}).
+Aberration also comes in, and this also is not a real motion of the
+stars, but only an apparent one.
+
+\nbarticle{800.} \nbparatext{Proper Motions.}---But after allowing for all these \textit{apparent}
+and \textit{common} motions, which depend upon the stars' places in the
+sky, and are sensibly the same for all stars in the same telescopic
+field of view, whatever may be their real distance from us, we find
+that most of the larger stars have a ``\textit{proper motion}'' of their own,
+(``proper'' as opposed to ``common,'') which displaces them slightly
+with reference to the stars about them. There are only a few stars
+for which this proper motion amounts to as much as $1''$ a year; perhaps
+150 such stars are now known, but the number is constantly increasing,
+as more and more of the smaller stars come to be accurately
+observed.
+
+\nbenlargepage
+\sloppy
+The maximum proper motion known is that of the seventh magnitude
+star 1830 Groombridge (\textit{i.e.}, No.~1830 in Groombridge's catalogue of
+circumpolar stars), which has an apparent drift of $7''$ annually,---enough
+to carry it completely around the heavens in 185,000 years.
+The largest known proper motions are the following:---
+\begin{center}
+\footnotesize
+\begin{tabular}{ll@{ }l|ll@{ }l} \hline \hline
+  1830, Groombridge,  & 7th mag.,   & $7''.0$
+& $\epsilon$~Indi,    & 5th mag.,   & $4''.5$
+\\
+  9352, Lacaille,     & 7th \ \ ``  & $6''.9$
+& Lalande 21,258      & 8th \ \ ``  & $4''.4$
+\\
+  32,416, Gould,      & 9th \ \ ``  & $6''.2$
+& $o_2$ Eridani       & 6th \ \ ``  & $4''.1$
+\\
+  61 Cygni,           & 6th \ \ ``  & $5''.2$
+& $\mu$~Cassiopeiæ,   & 5th \ \ ``  & $3''.8$
+\\
+  Lalande 21,185,     & 7th \ \ ``  & $4''.7$
+& $\alpha$~Centauri,  & 1st \ \ ``  & $3''.7$
+\\
+\hline \hline
+\end{tabular}
+\end{center}
+
+\fussy
+The proper motions of Arcturus ($2''.1$), and of Sirius ($1''.2$),
+are considered ``large,'' but are exceeded by a considerable number
+of stars besides these given above. Since the time of Ptolemy,
+Arcturus has moved more than a degree, and Sirius about half as
+much. Those motions were first detected by Halley in 1718.
+
+It is found, as might be expected, that the brighter stars, which as
+a class are presumably nearer than the fainter ones, have on the
+average a greater proper motion; on the \textit{average} only, however, as
+is evident from the list given above. Many smaller stars have larger
+proper motions than any bright one, for there are more of them.
+%% -----File: 468.png---Folio 457-------
+
+\nbarticle{801.} \nbparatext{Real Motions of Stars.}---The \textit{average} proper motion of the first-magnitude
+stars appears to be about $\frac{1}{4}''$ annually, and that of a sixth-magnitude
+star,---the smallest visible to the eye,---is about $\frac{1}{25}''$.
+
+\includegraphicsouter{illo214}{\textsc{Fig.~214.}\\
+Components of a Star's Proper Motion.}
+
+The proper motion of a star gives comparatively little information
+as to its real motion until we know the distance of the star and
+the true direction of the motion,
+since the proper motion \textit{as determined
+from the star-catalogues} is
+only the angular value of that part
+or component of the star's whole
+motion which is perpendicular to
+the line of sight, as is clear from
+the \figref{illo214}{figure}. When the star really moves from $A$ to $B$ (\figref{illo214}{Fig.~214}),
+it will appear, as seen from the earth, to have moved from $A$ to $b$.
+The angular value of $Ab$ as seen from the earth is the proper motion
+(usually denoted by $\mu$), as determined from the comparison of
+star-catalogues. Expressed in seconds of arc, we have
+\[
+  \mu'' = 206\,265 \left(\frac{Ab}{\text{distance}}\right).
+\]
+A body moving directly towards or from the earth has, therefore, no
+(angular) proper motion at all,---none that can be obtained from the
+comparison of star-catalogues.
+
+Since $Ab$ in miles
+\[
+= \frac{\mu'' × \text{distance}}{206\,265},
+\]
+these motions cannot be translated into miles without a knowledge
+of the star's distance; and this knowledge, as we shall see, is at
+present exceedingly limited; nor can the true motion $AB$ be found
+until we also know either the angle $BAE$ or else the line $Aa$.
+
+\begin{fineprint}
+But since $AB$ is necessarily \textit{greater} than $Ab$, it is possible in some cases
+to determine a \textit{minor} limit of velocity, which must certainly be exceeded by
+the star. In the case of 1830,~Groombridge, for instance, we have certain
+knowledge that its distance is not \textit{less} than 2,000000 times the earth's
+distance from the sun. It may be vastly greater; but it cannot be less.
+Now at that distance the observed proper motion of $7''$ a year would correspond
+to an actual velocity along the line $Ab$ of more than 200 miles a second,
+and the star \textit{may} be moving many times more swiftly. This star has sometimes
+been called the ``runaway star.''
+\end{fineprint}
+%% -----File: 469.png---Folio 458-------
+
+\nbarticle{802.} \nbparatext{Motion in the Line of Sight.}---Although the comparison of
+star-catalogues gives us no information of the body's motion towards
+or from us along the line of sight, the velocity $Aa$ in the \figref{illo214}{figure}, yet
+it is possible, when the star is reasonably bright, to determine somewhat
+roughly its rate of approach or recession by means of the spectroscope.
+If the star is approaching us, the lines in the spectroscope,
+according to Doppler's principle (\artref{Art.}{821}, \textit{note}), will be shifted
+towards the blue; and \textit{vice versa}, towards the red, if it is receding
+from us. Dr.~Huggins was the first actually to use this method of
+investigating the star movements in 1868, and by means of it he
+arrived at some very interesting results, which, however, must be
+admitted to be somewhat uncertain as regards their quantitative
+value. He found, for instance, that Sirius was receding from us at
+the rate of nineteen miles a second, and that Arcturus was rushing
+towards us at the rate of nearly sixty miles a second; and results
+of a similar character were found for a considerable number
+of other stars.
+
+Of late the investigations of this class have been carried on mainly
+at the Greenwich Observatory, usually by comparing the stellar
+spectra with hydrogen and sodium. The observations, however,
+are extremely difficult to make, for the displacements of the lines
+are very small, and in most star spectra the lines are broad and hazy
+and not well adapted for accurate measurements. The results of different
+days' observations, therefore, for a single star are sometimes
+mournfully discrepant.
+
+\includegraphicsouter{illo215}{\textsc{Fig.~215.}\\
+Displacement of $H_{\gamma}$ Line in the Spectrum $\beta$~Orionis.}
+
+\begin{fineprint}
+Vogel has recently taken up the work \textit{photographically}
+at Potsdam, with very encouraging
+results. \figref{illo215}{Fig.~215} is from one of his
+plates (a \textit{negative}), showing the displacement
+towards the red of the $H_{\gamma}$ line in the spectrum
+of $\beta$~Orionis, or \textit{Rigel}, and indicating
+a \textit{recession} of the star at a very rapid rate.
+Much is hoped in this line from the photographic
+spectrum-work of the Draper Memorial at Cambridge, of which
+more will be said a few \hyperref[pg:few]{pages} farther on.
+\end{fineprint}
+
+\sloppy
+\nbarticle{803.} \nbparatext{Star-Groups.}---Star-atlases have been constructed by Proctor
+and Flammarion, which show by arrows the direction and rate
+of the angular proper motion of the stars as far as now known. A
+moment's inspection shows that in many cases stars in the same
+neighborhood have a proper motion nearly the same in direction and
+in amount.
+%% -----File: 470.png---Folio 459-------
+
+\fussy
+\begin{fineprint}
+Thus, Flammarion has pointed out that the stars in the ``dipper'' of \textit{Ursa
+Major} have such a community of motion, except $\alpha$ and $\eta$,---the brighter of
+the pointers and the star in the end of the handle,---which are moving in
+entirely different directions, and refuse to be counted as belonging to the
+same group. \figref{illo216}{Fig.~216} shows the proper motions of the stars which compose
+this group. The same thing appears when their motion is tested by the
+spectroscope. Huggins found that the five associated stars are rapidly
+receding from the earth, while $\alpha$ is approaching us, and $\eta$, though receding,
+has a widely different rate of motion from the others.
+
+\includegraphicsmid{illo216}{\textsc{Fig.~216.}---Common Proper Motions of Stars in the ``Dipper'' of Ursa Major.}
+
+The brighter stars of the Pleiades are found in the same way to have a
+common motion.
+\end{fineprint}
+
+In fact, it appears to be the rule rather than the exception that
+stars apparently near each other are really connected as comrades,
+travelling together in groups of twos and threes, dozens or hundreds.
+They show, as Miss Clerke graphically expresses it, a distinctly
+``\textit{gregarious} tendency.''
+
+\nbarticle{804.} \nbparatext{The ``Sun's Way.''}---The proper motions of the stars are
+due partly to their own real motion, and partly also to the motion of
+our sun, which is moving swiftly through space, taking with it the
+earth and the planets. Sir William Herschel was the first to investigate
+and determine the direction of this motion a little more than 100
+years ago. The principle involved is this: that the apparent motion
+of each star is made up of its own motion combined with the motion
+of the sun \textit{reversed} (\artref{Art.}{492}). The effect must be that \textit{on the whole},
+the stars in that part of the sky towards which the sun is moving are
+separating from each other,---the \textit{intervals between them widening out},---while
+in the opposite part of the heavens \textit{they are closing up}; and
+in the intermediate part of the sky the general drift must be \textit{backward}
+with reference to the sun's (and earth's) real motion. Just as one
+walking in a park filled with people moving indiscriminately in different
+directions, would, on the whole, find that those in front of him
+%% -----File: 471.png---Folio 460-------
+appeared to grow larger,\footnote
+  {Theoretically, of course, the stars towards which we are moving must appear
+  to \textit{grow brighter} as well as to drift apart; but this change of brightness, though
+  real, is entirely imperceptible within a human lifetime.}
+and the spaces between them to open out,
+while at the sides they would drift backwards, and in the rear close
+up.
+
+\begin{fineprint}
+The spectroscope, moreover, ought to indicate this motion and undoubtedly
+will do so when the apparent motion in the line of sight has been accurately
+determined for a considerable number of stars. In fact one or two
+attempts have already been made to determine the solar motion in this way;
+in the quarter of the sky towards which the sun is moving, the star spectra
+should, on the whole, show displacement of their lines indicating \textit{approach},
+and \textit{vice versa} in the opposite quarter; and these observations will have the
+advantage of showing directly the sun's rate of motion \textit{in miles}, a result
+which is not given by investigations founded upon the angular proper
+motions of the stars. As yet, however, this spectroscopic method has not
+furnished results of any great weight.
+\end{fineprint}
+
+\nbarticle{805.} About twenty different determinations of the point in the sky
+towards which this motion of the sun is directed have been worked
+out by various astronomers, using in their discussions the angular
+proper motions of from twenty to twenty-five hundred stars. All the
+investigations  present a reasonable accordance of results, differing
+from each other only by a few degrees, and show that the \textit{sun is now
+moving towards a point in the constellation of Hercules, having a right
+ascension of about $\mathit{267}°$ and a declination of about $+\mathit{31}°$}. This point
+is known as the ``\textit{apex of the sun's way}.''
+
+\includegraphicsouter{illo217}{\textsc{Fig.~217.}\\
+The Earth's Motion in Space an affected
+by the Sun's Drift.}
+
+\nbarticle{806.} \nbparatext{Velocity of the Sun's Movement.}---This also is determined
+by the discussion, and comes out to be such as would carry the sun
+and its system about $5''$ in 100 years as seen from the average sixth
+magnitude star (the sixth magnitude is the smallest easily visible to
+the naked eye). If we knew with any certainty the distance of this
+average sixth magnitude star we could translate this motion into
+\textit{miles}; but at present this indispensable datum can be little more than
+guessed at. On the reasonable assumption adopted by Ludwig Struve
+(who has made the most recent and extensive of all the investigations
+upon the motion of the solar system), that this distance is about
+20,000000 times the astronomical unit, the velocity of the sun's
+motion in space comes out about five units per year, that is, about
+five-sixths of the earth's orbital velocity, or nearly sixteen miles per
+second; but this result must be considered as still very uncertain.
+%% -----File: 472.png---Folio 461-------
+
+\begin{fineprint}
+It is to be noted that this swift motion of the solar system, while of
+course it affects the real motion of the planets \textit{in space}, converting them into
+a sort of corkscrew spiral like the figure (\figref{illo217}{Fig.~217}), does not in the least
+affect the \textit{relative} motion of sun and planets,
+as some paradoxers have supposed it
+must.
+\end{fineprint}
+
+\nbarticle{807.} \nbparatext{The Central Sun.}---We mention
+this subject simply to say that
+there is no real foundation for the belief
+in the existence of such a body.
+The idea that the motion of our sun
+and of the other stars is a revolution
+around some great central sun is a
+very fascinating one to certain minds,
+and one that has been frequently suggested.
+It was seriously advocated
+some fifty years ago by Mädler, who
+placed this centre of the stellar universe
+at Alcyone, the principal star
+in the Pleiades.
+
+It is certainly within bounds to
+deny that any such motion has been demonstrated, and it is still less
+probable that the star Alcyone is the centre of such a motion, if the
+motion exists. So far as we can judge at present it is most likely that
+the stars are moving, not in regular closed orbits around any centre
+whatever, but rather as bees do in a swarm, each for itself, under
+the action of the predominant attraction of its nearest neighbors.
+The \textit{solar} system is an absolute monarchy with the sun supreme. The
+great \textit{stellar} system appears to be a republic, without any such central,
+unique, and dominant authority.
+
+\section*{THE PARALLAX AND DISTANCE OF THE STARS.}
+
+\nbarticle{808.} When we speak of the ``parallax'' of the moon, the sun, or
+a planet, we always mean the \textit{diurnal} parallax, \textit{i.e.,  the angular
+semi-diameter of the earth} as seen from the body in question. In
+the case of the stars, this kind of parallax is hopelessly insensible,
+never reaching an amount of $\frac{1}{20\,000}$ of a second of arc.
+
+The expression ``parallax of a \textit{star}'' always means its \textit{annual} parallax,
+that is, the semi-diameter of the earth's \textit{orbit} as seen from the
+star. Even this in the case of all stars but a very few is a mere fraction
+of a second of arc, too small to be measured.  In a few instances
+%% -----File: 473.png---Folio 462-------
+it rises to about half a second, and in the one case of our nearest
+neighbor (so far as known at present), the star $\alpha$~Centauri, it appears
+to be about $0''.9$, according to the earlier observers, or about $0''.75$,
+according to the latest determination of Gill and Elkin. In \figref{illo218}{Fig.~218}
+the angle at the star is the star's parallax.
+
+\includegraphicsmid{illo218}{\textsc{Fig.~218.}---The Annual Parallax of a Star.}
+
+In accordance with the principle of relative motion (\artref{Art.}{492}), every
+star has, superposed upon its own motion and combined with it, an
+\textit{apparent} motion equal to that of the earth but reversed. If the star is
+really at rest it must seem to travel around each year in a little orbit
+180,000000 miles in diameter, the precise counterpart of the earth's
+orbit in size and form, and having its plane parallel to the ecliptic.
+
+\begin{fineprint}
+If the star is near the pole of the ecliptic this apparent ``parallactic'' orbit
+will be viewed perpendicularly and appear as a circle; if the star is on the
+ecliptic it will be seen edgewise as a short, straight line, while in intermediate
+latitudes the parallactic orbit will appear as an ellipse. In this
+respect it is just like the ``aberrational'' orbit of a star (\artref{Art.}{220}); but
+while the aberrational orbit is of the same size for every star, having always
+a semi-major axis of $20''.492$, the size of the parallactic orbit depends upon
+the distance of the star. Moreover, in the parallactic orbit the star is
+always opposite to the earth, while in the aberrational orbit it keeps just
+$90°$ ahead of her.
+\end{fineprint}
+
+\nbarticle{809.} If we can find a way of measuring this parallactic orbit, the
+star's distance is at once determined. It equals
+\[
+  \frac{206\,265 × R}{p''},
+\]
+in which $p''$ is the parallax in seconds of arc (the apparent semi-major
+axis of the parallactic orbit), and $R$ is the earth's distance
+from the sun.
+
+The determination of stellar parallax had been attempted over and
+over again from the days of Tycho down, but without success until
+Bessel, in 1838, succeeded in demonstrating and measuring the parallax
+of the star 61~Cygni; and the next year Henderson, of
+the Cape of Good Hope, brought out that of $\alpha$~Centauri. It will
+%% -----File: 474.png---Folio 463-------
+be remembered that it was mainly on account of his failure to detect
+stellar parallax that Tycho rejected the Copernican theory and substituted
+his own (\artref{Art.}{504}). %[* is 'own' really bold?][F2: I don't think so.]
+
+\begin{fineprint}
+Roemer of Copenhagen, in 1690, thought that he had detected the effect of
+stellar parallax in his observations of the difference of right ascension between
+Sirius and Vega at different times of the year. A few years later, Horrebow,
+his successor, from his own discussion of Roemer's observations, made out
+the amount to be nearly four seconds of time or $1'$, and published his premature
+exultation in a book entitled ``Copernicus Triumphans.'' The discovery
+of \textit{aberration} by Bradley explained many abnormal results of the early
+astronomers which had been thought to arise from stellar parallax, and
+proved that the parallax must be extremely small. About the beginning
+of the present century, Brinkley of Dublin and Pond, the Astronomer Royal,
+had a lively controversy over their observations of $\alpha$~Lyræ (Vega). Brinkley
+considered that his observations indicated a parallax of nearly $3''$. Pond,
+on the other hand, from his observations deduced a minute \textit{negative} parallax,
+which, as some one has expressed it, would put the star ``somewhere on the
+other side of nowhere.'' In fact, as it turns out, Pond was nearer right than
+Brinkley, the actual parallax as deduced from the latest observations being
+only about $0''.2$. The negative parallax, like the much too large result of
+Brinkley, simply indicates the uncertainties and errors incident to the
+instruments and methods of observation then used. The periodical changes
+of temperature and air pressure continually lead to fallacious results, except
+under the most extreme precautions.
+\end{fineprint}
+
+\nbarticle{810.} \nbparatext{Methods of determining Parallax.}---The operation of measuring
+a stellar parallax is, on the whole, the most delicate in the
+whole range of practical astronomy. Two methods have been successfully
+employed so far---the \textit{absolute} and the \textit{differential}.
+
+(\textit{a}) The first method consists in making meridian observations of
+the right ascension and declination of the star in question at different
+seasons of the year, applying till known corrections for precession,
+nutation, aberration, and proper motion, and then studying the
+resulting star-places. If the star is without parallax, the places
+should be identical after the corrections have been duly applied. If
+it has parallax, the star will be found to change its right ascension
+and declination systematically, though slightly, through the year. But
+the changes of the seasons so disturb the constants of the instrument
+that the method is treacherous and uncertain. There is no possibility
+of getting rid of these temperature effects (in producing changes
+of refraction and varying expansions of the instrument itself) by
+merely multiplying observations and \textit{taking averages,} since the
+%% -----File: 475.png---Folio 464-------
+changes of temperature are themselves annually periodic, just as is
+the parallax itself.
+
+Still, in a few cases the method has proved successful. Different
+observers at different places with different instruments have found
+for a few stars fairly accordant results; as, for instance, in the case
+of $\alpha$~Centauri, already mentioned as our nearest neighbor.
+
+\nbarticle{811.} (\textit{b}) \nbparatext{The Differential Method.}---This consists in measuring
+the change of position of the star whose parallax we are seeking
+(which is supposed to be comparatively near to us), with reference to
+other small stars, which are in the same telescopic field of view, but
+are supposed to be so far beyond the principal star as to have no
+sensible parallax of their own. If the comparison stars are near
+the large one (say within two or three minutes of arc), the ordinary
+wire micrometer answers very well for the necessary measures; but
+if they are farther away, the heliometer (\artref{Art.}{677}) represents special
+and very great advantages. It was with this instrument that Bessel,
+in the case of 61~Cygni, obtained the first success in this line of
+research.
+
+The great advantage of the differential method is that it avoids
+entirely the difficulties which arise from the uncertainties as to the
+exact amount of precession, etc.; and in great measure, though not
+entirely, those arising from the effect of the seasons upon refraction
+and the condition of the instruments. On the other hand, however,
+it gives as the final result, not the absolute parallax of the star, but
+only the \textit{difference between its parallax and that of the comparison
+star}. If the work is accurate the parallax deduced \textit{cannot be too
+great; but it may be sensibly too small}, and so may make the star
+apparently too remote. This is because the parallax of the comparison
+star can never be quite zero: if the comparison star happens to have
+a parallax of its own as large as that of the principal star, there
+will be no relative parallax at all; if larger, the parallax sought will
+come out \textit{negative}.
+
+\sloppy
+\nbarticle{812.} \nbparatext{Determination of Parallax by Photography.}---Recently it
+has been attempted to press photography into the service, and
+Professor Pritchard has obtained apparently excellent results from an
+extensive series of photographs of 61~Cygni and its neighboring stars,
+made at Oxford during 1886.
+
+\fussy
+\begin{fineprint}
+Now and then a plate was found in which the sensitive film appeared to
+have slipped a little during the development of the picture, but the measurements
+%% -----File: 476.png---Folio 465-------
+at once showed up the faulty plates, so that they could be confidently
+rejected. The measurements made upon the other plates appeared
+to be just as trustworthy as measurements made upon the actual objects in
+the sky; and of course the measurement of a photographic plate at leisure
+in one's laboratory is a vastly more comfortable operation than that of
+making micrometric settings by night.
+
+\nbarticle{813.} \nbparatext{Selection of Stars.}---It is important to select for investigations
+of this kind these stars which may reasonably be supposed to be near,
+and, therefore, to have a sensible parallax. The most important indication
+of proximity is a \textit{large proper motion}, and \textit{brightness} is, of course, confirmatory.
+At the same time, while it is probable that a bright star with
+large proper motion is comparatively near, it is not certain. The small stars
+are so much more numerous than the large ones that it will be nothing surprising
+if we should find among them one or more neighbors nearer than
+$\alpha$~Centauri itself.
+\end{fineprint}
+%*this paragraph is perhaps a 'quote' as it appears to be intermediate in point size between the adjacent paragraphs..][F2: It's a quote.]
+
+\nbarticle{814.} \nbparatext{Unit of Stellar Distance.}---\textit{The Light-Year}. The ordinary
+``\textit{astronomical unit},'' or distance of the sun from the earth, is not
+sufficiently large to be convenient in expressing the distances of the
+stars. It is found more satisfactory to take us a unit the distance
+that light travels in a year, which is about 63,000 times the distance
+of the earth from the sun. A star with a parallax of $1''$ is at a distance
+of 3.262 ``\textit{light-years},'' so that the distance of any star in
+``light-years'' is expressed by the formula
+\[
+D_{y} = \frac{3.262}{p''}.
+\]
+
+\nbarticle{815.} \hyperref[app:IV]{Table IV.}\ in the Appendix, based upon that given in Houzeau's
+``Vade Mecum of Astronomy,'' but brought down to date by some
+additions and changes, gives the parallaxes, and the distances in
+light-years, of these stars whose parallaxes may be considered as
+now fairly determined.
+
+\begin{fineprint}
+The student will, of course, see that the tabulated distance in the case of
+a remote star is liable to an enormous percentage of error. Considering the
+amount of discordance between the results of different observers, it is
+extremely charitable to assume that any of the parallaxes are certain
+within $\frac{1}{50}$ of a second of arc: but in the case of a star like the pole-star,
+which appears to have a parallax of less than $0''.08$, this $\frac{1}{50}$ of a second
+is $\frac{1}{4}$ of the whole amount; so that the distance of that star is uncertain by
+at least twenty-five per cent. ($\frac{1}{50}$ of a second is the angle subtended
+by $\frac{1}{16}$ of an inch at the distance of ten miles.)
+%% -----File: 477.png---Folio 466-------
+
+A vigorous campaign has lately been organized for the purpose of obtaining
+within a reasonable time the parallaxes of a considerable number of
+stars (perhaps one or two hundred),---enough to enable us to deduce some
+general laws by statistical methods. The instruments to be used are heliometers
+of six or seven inches' aperture, one of which is at the Cape of
+Good Hope, another has just been erected at Bamberg in Germany, and a
+third is at New Haven in this country, under the charge of Dr.~Elkin.
+
+As regards the distance of stars, the parallax of which has not yet been
+measured, very little can be said with certainty. It is \textit{probable} that the
+remoter ones are so far away that light in making its journey occupies a
+thousand and perhaps many thousand years.
+
+\nbarticle{815*.} Since the above was written Dr.~Elkin has published the result
+of his observations upon ten first-magnitude stars, as follows: $\alpha$~Tauri
+(\textit{Aldebaran}), $0''.116 \pm .029$; $\alpha$~Aurigæ (\textit{Capella}), $0''.107 \pm .047$; $\alpha$~Orionis
+(\textit{Betelgueze}), $-0''.009 \pm .049$; $\alpha$~Canis Minoris (\textit{Procyon}), $0''.266 \pm .047$; $\beta$~Geminorum
+(\textit{Pollux}), $0''.068 \pm .047$; $\alpha$~Leonis (\textit{Regulus}), $0''.093 \pm .048$; $\alpha$~Bootis
+(\textit{Arcturus}), $0''.018 \pm .022$; $\alpha$~Lyræ (Vega), $0''.034 \pm .045$; $\alpha$~Aquillæ (\textit{Altair}),
+$0''.199 \pm .047$; $\alpha$~Cygni (\textit{Deneb}), $-0''.042 \pm .047$.
+
+Of course the two \textit{negative} results simply indicate that the parallax of the
+large star was less than that of the comparison stars employed. The very
+small results for Vega and Arcturus are also rather surprising. See \hyperref[app:IV]{Table IV.}\ of Appendix.
+\end{fineprint}
+\chelabel{CHAPTERXIX}
+%% -----File: 478.png---Folio 467-------
+
+\Chapter{XX}{The Light of the Stars}
+\nbchapterhang{\stretchyspace
+THE LIGHT OF THE STARS.---STAR MAGNITUDES AND PHOTOMETRY.---VARIABLE
+STARS.---STELLAR SPECTRA.\hspace{0pt}---SCINTILLATION
+OF STARS.}
+
+\nbarticle{816.} \nbparatext{Star Magnitudes.}---The term ``magnitude,'' as applied to a
+star, refers simply to its brightness. It has nothing to do with its
+apparent angular diameter. Hipparchus and Ptolemy arbitrarily
+graded the visible stars, according to their brightness, into six classes,
+the stars of the sixth magnitude being the smallest visible to the
+eye, while the first class comprises about twenty of the brightest.
+There is no assignable reason why \textit{six} classes should have been
+constituted, rather than eight or ten.
+
+After the invention of the telescope the same system was extended
+to the smaller stars, but without any general agreement or concert,
+so that the magnitudes assigned by different observers to telescopic
+stars vary enormously. Sir William Herschel, especially, used very
+high numbers: his twentieth magnitude being about the same as the
+fourteenth on the scale now generally used, which more nearly
+corresponds with that of the elder Struve.
+
+\nbarticle{817.} \nbparatext{Fractional Magnitudes.}---Of course, the stars classed together
+under one magnitude are not exactly alike in brightness, but
+shade from the brighter to the fainter, so that exactness requires the
+use of \textit{fractional} magnitudes. It is now usual to employ decimals
+giving the brightness of a star to the nearest tenth of a magnitude.
+Thus, a star of 4.3 magnitude is a shade brighter than one of 4.4,
+and so on.
+
+\begin{fineprint}
+A peculiar notation was employed by Ptolemy, and used by Argelander
+in his ``Uranometria\footnote
+  {The term ``Uranometria'' has come to mean a catalogue of \textit{naked-eye stars};
+  like the catalogues of Hipparchus, Ptolemy, and Ulugh Beigh.}
+Nova.'' It recognizes \textit{thirds} of a magnitude as the
+smallest subdivision. Thus,\enskip2,\enskip2,3,\enskip3,2,\enskip and\enskip3\enskip express the gradations
+between second and third magnitude, 2,3 being applied to a star whose
+brightness is a little inferior to the second, and 3,2 to one a little brighter
+than the third magnitude.
+%% -----File: 479.png---Folio 468-------
+
+\nbarticle{818.} \nbparatext{Stars Visible to the Naked Eye.}---Heis enumerates the
+stars clearly visible to the naked eye in the part of the sky north of
+$35°$ south declination, as follows:---
+\begin{center}
+\begin{tabularx}{\textwidth}{l@{ }c@{ }X@{ }r | l@{ }c@{ }X@{ }r}
+\hline\hline
+  1st & magnitude, & \dotfill &   14\rule{0pt}{3ex}
+& 4th & magnitude, & \dotfill &  313
+\\
+  2d  &  ``        & \dotfill &   48
+& 5th &  ``        & \dotfill &  854
+\\
+  3d  &  ``        & \dotfill &  152
+& 6th &  ``        & \dotfill & 2010
+\\
+  & \multicolumn{6}{c}{Total \dotfill\ 3391} &
+\\[1ex]
+\hline\hline
+\end{tabularx}
+\end{center}
+
+According to Newcomb, the number of stars of each magnitude is such
+that united they would give, roughly speaking, somewhere nearly the same
+amount of light as that received from the aggregate of those of the next
+brighter magnitude. But the relation is very far from exact, and seems to
+fail entirely for the fainter magnitudes below the tenth or eleventh, the
+smaller stars being less numerous than they should be. In fact, if the law
+held out perfectly, and if light was transmitted through space without loss,
+the whole sky would be a blaze of light like the surface of the sun.
+\end{fineprint}
+
+\sloppy
+\nbarticle{819.} \nbparatext{Light-Ratio and Absolute Scale of Star Magnitudes.}---It
+was found by Sir John Herschel, about fifty years ago, that the light
+given by the average star of the first magnitude is just about one
+hundred times as great as that received from one of the sixth, and
+that a corresponding ratio has been pretty nearly maintained throughout
+the scale of magnitudes, the star's of each magnitude being about
+$2 \frac{1}{2}$ times ($\root 5\of{100}$) brighter than those of the next inferior magnitude.
+The number which expresses the ratio of the light of a star to that of
+another one magnitude fainter is called the \textit{light-ratio}.
+
+\fussy
+In the star magnitudes of the maps by Argelander, Heis, and others,
+which are most used at present, the divergence from a strict uniformity
+of light-ratio is, however, sometimes serious. Some forty years ago
+it was proposed by Pogson to reform the system, by adopting a
+scale with the uniform light-ratio of $\root 5\of{100}$, adjusting the first six
+magnitudes to correspond as nearly as possible with the magnitudes
+hitherto assigned by leading authorities, and then carrying forward
+the scale indefinitely among the telescopic stars. Until recently this
+``\textit{absolute scale} of magnitude,'' as it has been called, has not been
+much used; but in the New Uranometrias lately made at Cambridge
+and Oxford it has been adopted, and astronomers generally now
+endeavor to conform to it.
+
+\nbarticle{820.} \nbparatext{Relative Brightness of Different Star Magnitudes.}---In this
+scale the light-ratio between successive magnitudes is made exactly
+%% -----File: 480.png---Folio 469-------
+$\root 5\of{100}$, \textit{or the number whose logarithm is} 0.4000, viz., 2.512. Its reciprocal
+is the number whose logarithm is 9.6000, viz., 0.3981. If $b_{1}$ is
+the brightness of a standard first-magnitude star, expressed either in
+candle-power or other convenient unit, and $b_{n}$ be the brightness of a
+star of the $n$th magnitude on this scale, we shall therefore have
+\[
+\log b_n = \log b_1 - \frac{4}{10} (n -1);
+\]
+$(n - 1)$ in this equation being the number of magnitudes \textit{between} the
+star of the first magnitude and the star of the $n$th magnitude: \textit{i.e.},
+for a star of the sixth magnitude $(n - 1)$ is 5; so that for a star of
+the sixth magnitude, the equation reads,
+\[
+\log b_6 = \log b_1 - \frac{4}{10} × 5 = \log b_1 - 2.
+\]
+With this light-ratio, every difference of five magnitudes corresponds
+to a multiplication or division of the star's light by 100; \textit{i.e.}, to make
+one star as bright as the standard star of the first magnitude it would
+require 100 of the sixth, 10,000 of the eleventh, 1,000000 of the sixteenth,
+and 100,000000 of the twenty-first magnitude.
+
+As nearly standard stars of the first magnitude on this scale we
+have $\alpha$~Aquilæ and Aldebaran ($\alpha$~Tauri). The other stars usually
+counted as of first magnitude are some of them sensibly brighter, and
+others fainter than these. The pole-star and the two ``pointers'' are
+very nearly standard stars of the \textit{second} magnitude.
+
+\sloppy
+\begin{fineprint}
+\nbarticle{821.} \nbparatext{Negative Magnitudes.}---According to this scale, stars that are
+one magnitude \textit{brighter} than those of the standard first would be of the \textit{zero}
+magnitude, (as is the case with Arcturus), and those that are brighter yet
+would be of a \textit{negative} magnitude; \textit{e.g.}, the magnitude of Sirius is $-1.13$;
+and Jupiter at opposition, in conformity to this system, is described as a
+star of nearly $-2$d magnitude, which means that it is nearly $2.51^3$, or about
+16 times \textit{brighter than} a star of the $+1$st magnitude like Aldebaran. According
+to Seidel, Jupiter at opposition is about $8\frac{1}{4}$ times as bright as Vega, which
+would make its ``magnitude'' $-2.09$, Vega being of magnitude 0.2.
+\end{fineprint}
+
+\fussy
+\nbarticle{822.} \nbparatext{Relation of Size of Telescope to the Magnitude of the
+Smallest Star Visible with it.}---If a telescope just shows a star
+of a given magnitude, then to show stars one magnitude smaller
+we require an instrument having its aperture larger in the ratio of
+$\sqrt{2.512}$ (or $\root 10\of{100}$) to 1: \textit{i.e.} as 1.59:1. Every \textit{tenfold} increase
+in the diameter of the object-glass will therefore carry the power of
+vision just \textit{five magnitudes lower.}
+%% -----File: 481.png---Folio 470-------
+
+\begin{fineprint}
+Assuming what seems to be very nearly true for normal eyes and good
+telescopes, that the \textit{minimum visible} for a one-inch aperture is a star of the
+ninth magnitude, we obtain the following little table of \textit{apertures required to
+show stars of a given magnitude}.
+\medskip
+\begin{center}
+\footnotesize
+\begin{tabular}{@{}l|c|c|c|c|c|c}
+\hline \hline
+Star Magnitude \ldots\dotfill
+&  7  &  8  &  9  &  10  &  11  &  12\rule{0pt}{3ex}  \\
+Aperture \dotfill
+& $0^\text{in}.40$ & $0^\text{in}.63$ & $1^\text{in}.00$
+& $1^\text{in}.59$ & $2^\text{in}.51$ & $3^\text{in}.98$  \\[1ex]
+\hline
+Star Magnitude \ldots\dotfill
+&  13  &  14  &  15  &  16  &  17  &  18\rule{0pt}{3ex} \\
+Aperture \dotfill
+&  $6^\text{in}.31$ & $10^\text{in}.00$ & $15^\text{in}.90$
+& $25^\text{in}.10$ & $39^\text{in}.80$ & $63^\text{in}.10$  \\[1ex]
+\hline \hline
+\end{tabular}
+\end{center}
+
+\medskip
+But on account of the increased thickness necessary in the lenses of large
+telescopes, they never quite equal their theoretical capacity as compared with
+smaller ones.
+
+The smallest stars visible by the Lick telescope (thirty-six inches aperture),
+after allowing for all the advantage of the site, are not quite a whole magnitude
+below the smallest visible with the Washington telescope; but the \textit{number}
+visible will be at least double; since the smaller stars are vastly the more
+numerous.
+\end{fineprint}
+
+\nbarticle{823.} \nbparatext{Measurement of Star Magnitudes and Brightness.}---Until
+recently all such measurements were mere eye-estimates, and even
+yet all photometric measurements depend \textit{ultimately} on the judgment
+of the eye. But it is possible by the help of instruments to aid this
+judgment very much by limiting the point to be decided, to the
+question whether two lights as seen are, or are not, exactly equal,
+or else making the decision depend on the visibility or non-visibility
+of some appearance.
+
+\nbarticle{824.} \nbparatext{Method of Sequences.}---For some purposes the unassisted
+eye is quite as good as any photometric instrument. It judges
+directly with great precision of the \textit{order of brightness} in which a
+number of objects stand. In the method of ``sequences,'' as it is
+called, the observer merely arranges the stars he is comparing, say
+to the number of fifty or so, in the order of their brightness, taking
+care that the stars in each sequence list are nearly at the same
+altitude, and seen under equally favorable circumstances. Then he
+makes a second sequence, taking care to include in it some of the
+stars that were in the first: and so on. Finally, from the whole set
+of sequences, a list can be formed, including all the stars contained in
+any of them, arranged in the order of brightness. This process gives,
+however, no determination of the light-ratio, nor of the number of
+times by which the light of the brightest exceeds that of the faintest.
+%% -----File: 482.png---Folio 471-------
+
+\begin{fineprint}
+Variable stars are still often observed in this way, the stars with which
+they are compared, being such as have their magnitudes already well determined.
+\end{fineprint}
+
+\nbarticle{825.} \DPtypo{2. }{}\nbparatext{Instrumental Methods.}---These are based on two different
+principles:---
+
+\textit{a}. The measurement is made by causing the star to \textit{disappear} by
+diminishing its light in some measurable way. This is usually referred
+to as the ``method of \textit{extinctions}.''
+
+\textit{b}. The measurement is effected by causing the light of the star to
+appear just \textit{equal} to some other standard light, by decreasing the
+brightness of the star or of the standard in some known ratio until
+they are perfectly equalized.
+
+\begin{fineprint}
+Under the first head come the photometers which act upon the principle
+of ``\textit{limiting apertures}.'' The telescope is fitted with some arrangement,
+often a so-called ``cat's-eye,'' by which the available aperture of the object-glass
+can be diminished at will, and the observation consists in determining
+with what area of object-glass the star is just visible. The method is embarrassed
+by constant errors from the fact that the greater thickness of the
+glass in the middle of the lens comes into account, and, still worse, from the
+fact that the image of the star becomes large and diffuse on account of
+diffraction when the aperture is very much reduced.
+\end{fineprint}
+
+\nbarticle{826.} \nbparatext{The Wedge Photometer.}---The method of producing the
+``extinction'' by a ``\textit{wedge}'' of dark, neutral-tinted glass is much
+better. The wedge is usually five or six inches long, by perhaps a
+quarter of an inch wide, and at the thick end cuts off light enough to
+extinguish the brightest stars that are to be observed. In the
+Pritchard form of the instrument this wedge is placed close to the
+eye at the eye-hole of the eye-piece; in some other forms it is placed
+at the principal focus of the object-glass, where micrometer wires
+would be put.
+
+In observation the wedge is pushed along promptly until the star
+just disappears, and a graduation on the edge of the slider is read.
+
+\begin{fineprint}
+The great simplicity of the instrument commends it, and if the wedge is
+a good one of really neutral glass (which is not easy to get), the results are
+remarkably accurate. But the observations are very trying to the eyes on
+account of the straining to keep in sight an object just as it is becoming
+invisible. The constant of the wedge must be carefully determined in the
+laboratory, \textit{i.e.}, what length of the wedge corresponds to a diminution
+of the light of a star by just one magnitude (cutting off 0.602 of its light).
+It is convenient to have the slider graduated into inches or millimeters on
+%% -----File: 483.png---Folio 472-------
+the one edge and magnitudes on the other. The ``Uranometria Nova Oxoniensis''
+is a catalogue of the magnitudes of the naked-eye stars to the number
+of 2784, between the pole and $10°$ south declination, observed with an
+instrument of this kind by Professor Pritchard, and published in 1885.
+\end{fineprint}
+
+\sloppy
+\nbarticle{827.} \nbparatext{Polarization Photometers.}---The instruments, however, with
+which most of the accurate photometric work upon the stars has been
+done, are such as compare the light of the star with some standard
+by means of an ``\textit{equalizing apparatus}'' based on the application of
+the principles of double refraction and polarization.
+
+\fussy
+\begin{fineprint}
+The light of either the observed star or the comparison star (real or
+artificial) is polarized by transmission through a Nicol prism, or else both
+pencils are sent through a double refracting prism. The images are viewed
+with a Nicol prism in the eye-piece; and by turning this the polarized image
+or images can be varied in brightness at pleasure, and the amount of variation
+determined by reading a small circle attached to it. In the photometers
+of Seidel and Zöllner, who observed comparatively few objects, but very
+accurately, the artificial star with which the real stars were compared was
+formed by light from a petroleum lamp, shining through a small aperture,
+and reflected to the eye by a plate of glass in the telescope tube. Professor
+Pickering, in his extensive work embodied in the ``Harvard Photometry''
+(published in 1884, and giving the magnitudes of 4260 stars) used the
+pole-star as the standard, bringing it by an ingenious arrangement into the
+same field with the star observed.
+\end{fineprint}
+
+Photometric observations in many cases require large and somewhat
+uncertain corrections, especially for the absorption of light by
+the atmosphere at different altitudes, and the final results of different
+observers naturally fail of absolute accordance. Still the agreement
+between the two catalogues of Pickering and Pritchard is remarkably
+close, generally within one or two tenths of a magnitude.
+
+\includegraphicsmid{illo219}{\textsc{Fig.~219.}---Pickering's Meridian Photometer.}
+
+\begin{fineprint}
+\nbarticle{828.} \nbparatext{The Meridian Photometer.}---This instrument, contrived and
+used by Professor Pickering in the observations of the Harvard Photometry,
+consists of a telescope with two object-glasses side by side. The telescope is
+pointed nearly east and west, and in front of each object-glass is placed a
+silvered glass mirror ($M_{1}$ and $M_{2}$, \figref{illo219}{Fig.~219}) at an angle of $45°$. One of the
+mirrors is so set as to bring the rays of the pole-star to one object-glass; the
+other mirror is capable of being turned around the optical axis of the telescope,
+in such a way as to command a star at any part of the meridian, and bring its
+light into the other object-glass. At the eye-end is placed, first (\textit{i.e.}, next the
+object-glass), a double-image prism $D$, which separates any pencil of light
+falling upon it into two, polarized at right angles to each other. The ``ordinary''
+rays come through nearly undeflected, but the ``extraordinary'' are
+%% -----File: 484.png---Folio 473-------
+bent out of their course, as indicated in the \figref{illo219}{figure}, where the pencil $A$, coming
+from object-glass No.~1, is divided into two pencils $a_{o}$ and $a_{e}$, and in the
+same way the pencil $B$, from the second objective, is divided into $b_{o}$ and $b_{e}$.
+The angle of the double-image prism $D$ is so chosen that $a_{o}$ and $b_{e}$ will be
+nearly parallel to each other, and a suitable diaphragm\footnote
+  {The diaphragm $E$ may be replaced by an eye-stop at $I$.}
+$E$ cuts off the two
+other pencils $a_{e}$ and $b_{0}$. A Nicol prism $N$ receives the two pencils that come
+through the diaphragm, and the eye views the two images through the eye-piece
+at $I$. These pencils, being polarized at right angles to each other, will
+vary in their brightness when the Nicol is turned, one of them becoming
+brighter and the other fainter; and \textit{four} positions of the Nicol can be found
+at which the images will appear equal in brightness, whatever may be the
+original ratio of brightness between the pole-star and the object observed.
+On looking into the instrument the observer sees two stars, the pole-star at
+rest, the other moving along as in a transit instrument. He simply turns
+the Nicol until the images are equalized, setting the Nicol at all the four different
+positions which will produce the effect, and reading the graduated
+circle $C$. The whole operation consumes not more than a minute, with the
+help of an assistant to record the numbers as read off. The ``Harvard Photometry''
+(usually referred to simply as ``H.~P.'') was made by means of an
+instrument with object-glasses only two inches and a half in diameter. An
+instrument with four-inch lenses is now at work in Cambridge, measuring
+the magnitudes of all the nearly 80,000 stars of Argelander's \textit{Durchmusterung},
+which are of the eighth magnitude or brighter.
+
+\nbarticle{829.} \nbparatext{Photometry by means of Photography.}---It has been found that,
+excepting a few strongly colored stars, the intensity, or more simply the size, of
+the image of a star formed upon a photographic plate may be used as a measure
+of its brightness as compared with other stars taken on the same plate,
+or on similar plates similarly exposed. The comparison becomes easier and
+more accurate if the photographic telescope is not made to follow the stars
+exactly, but is allowed to lag a little so that the star forms a ``trail.'' It will,
+therefore, be possible to use the plates of the great photographic star campaign
+to determine star magnitudes as well as positions. But, as has been intimated,
+there are some anomalies; certain stars, for instance, that are hardly visible to
+%% -----File: 485.png---Folio 474-------
+the naked eye, photograph as bright stars, and there are others---red stars---that
+are abnormally faint on the plate. The exceptions are numerous
+enough to make it necessary to use photographic magnitudes with caution.
+\end{fineprint}
+
+\nbarticle{830.} \nbparatext{Star Colors and their Effects on Photometry.}---The stars
+differ considerably in color. The majority are of a very pure white,
+like Sirius and the sun, but there are not a few of a yellowish hue,
+like Capella, or reddish, like Arcturus and Antares; and there are
+some, mostly small stars, which are as red as garnets and rubies.
+We also have, associated with larger ones in double-star systems,
+numerous small stars which are strongly green or blue; and there
+are a few large isolated stars, which, like Vega, are of a decidedly
+bluish tinge.
+
+\begin{fineprint}
+These differences of color embarrass photometric measurements made by
+either of the methods described, because it is impossible to make a red star
+look identical with a blue one by any mere increase or diminution of brightness,
+and because different observers will differ in setting the wedge of an
+extinction photometer according to the color of the star. Some eyes are
+abnormally sensitive to blue light, some to red. To the writer, for instance,
+Vega is decidedly superior to Arcturus, while the majority of observers see
+the difference as decidedly the other way.
+
+\nbarticle{831.} \nbparatext{Spectrum Photometry.}---The only completely satisfactory and
+scientific method would be to compare the spectra of the stars with some
+standard spectrum, \textit{say that of the pole-star}, dividing the spectrum into a considerable
+number of portions, and determining and recording the amount
+of light in each portion of the spectrum as compared with homologous parts
+of the standard spectrum. This, of course, would immensely increase the
+work of comparing the brightness of the stars; but it is quite feasible to do
+it for a few hundred of the brighter ones, and it would be well worth accomplishment.
+If we ever succeed in getting photographic plates equally sensitive
+to rays of all wave length, photography would answer the purpose well.
+\end{fineprint}
+
+\sloppy
+\nbarticle{832.} \nbparatext{Starlight compared with Sunlight.}---The light \textit{received} from
+a first-magnitude star like Vega is about $\frac{1}{40000\,000000}$ (one forty
+thousand millionth) of that from the sun, according to the determinations
+of Zöllner and others. The measurement is not easy, and
+must be taken as having a very considerable margin of error.
+
+\fussy
+Sirius is nearly equivalent to six of Vega, its light being about
+$\frac{1}{7000\,000000}$ of the sun's.
+
+\sloppy
+Since the light of a sixth-magnitude star is only $\frac{1}{100}$ of that of a
+standard first-magnitude, it follows that it would require 4,000000000000
+of stars of the 6th magnitude to give us sunlight.
+%% -----File: 486.png---Folio 475-------
+
+\fussy
+\nbarticle{833.} \nbparatext{Total Light of the Stars.}---Assuming what is roughly, though
+not exactly, true, that Argelander's magnitudes follow the standard
+scale, it appears that the 324,000 stars north of the equator enumerated
+in his \textit{Durchmusterung} give a light about equal to that of 240 first-magnitude
+stars; but it is noticeable that the aggregate amount of light
+given by the stars in each of the fainter magnitudes increases rapidly.
+
+\begin{fineprint}
+The following is the estimate, substantially according to Newcomb:---\\[1ex]
+\begin{tabular}{@{}r@{ }c@{ }c@{ }l@{ }l@{ }r@{ }c@{ }c@{ }r@{.}l@{ }c@{ }l@{}}
+    10 & stars& \multicolumn{4}{r@{ }}{(above the  2d} & magnitude) &=  &  6&0  & first-magnitude & stars.
+\\
+    37 &  ``& from& 2d & to & 3d  & `` & = &   7&3  & `` & ``  \\
+   122 &  ``& `` & 3d  & to & 4th & `` & = &   9&6  & `` & ``  \\
+   310 & `` & `` & 4th & to & 5th & `` & = &   9&8  & `` & ``  \\
+  1016 & `` & `` & 5th & to & 6th & `` & = &  12&7  & `` & ``  \\
+  4322 & `` & `` & 6th & to & 7th & `` & = &  21&6  & `` & ``  \\
+ 13593 & `` & `` & 7th & to & 8th & `` & = &  27&1  & `` & ``  \\
+ 57960 & `` & `` & 8th & to & 9th & `` & = &  46&   & `` & ``  \\
+247544 & `` & `` & 9th & to &$9 \frac{1}{2}$ & `` & = & 100&   & `` & ``  \\[1ex]
+\cline{9-10}
+  &\multicolumn{6}{c}{Total \dotfill} & = & 240&&& \rule{0pt}{3ex}
+\end{tabular}
+
+How much to add for the still smaller magnitudes is very uncertain. Beyond
+the tenth magnitude the number of small stars does not increase proportionately
+fast, so that if we could carry on the account of stars to the
+twentieth magnitude, it is practically certain that we should not find the total
+light of the aggregate stars of each succeeding magnitude increasing at
+any such rate as from the seventh to the tenth. Perhaps it would be a not
+unreasonable estimate to put the total starlight of the northern hemisphere
+as equivalent to about 1500 first-magnitude stars, or that of the whole sphere
+at 3000. This would make the total starlight on a clear night about $\frac{1}{60}$ of
+the light of the full moon, and about $\frac{1}{27\,000000}$ that of the sun. The light
+from the stars which are visible to the naked eye would not be as much
+as $\frac{1}{25}$ of the whole. But the above estimate of the light received from the
+extremely small stars is hardly more than a mere guess, and may hereafter
+receive important corrections.
+\end{fineprint}
+
+\nbarticle{834.} \nbparatext{Heat from the Stars.}---Attempts have been made to measure
+by a sensitive thermopile the heat received from certain stars.
+Both Huggins and Stone (about 1870) thought they had obtained
+sensible indications of heat from Arcturus and Vega; but their results
+have not since been confirmed; and unless the radiation of invisible
+energy by these stars is much greater in comparison with their light
+than is the case with the sun, it is almost certain that there must be
+some illusion in the matter. $\frac{1}{40000\,000000}$ of the sun's \textit{heat} could
+hardly be shown by any instrument known to science, and there is no
+%% -----File: 487.png---Folio 476-------
+present reason to suppose that the total \textit{heat} received from the stars
+bears a larger ratio to that received from the sun than star\textit{light} does
+to sun\textit{light}.
+
+\nbarticle{835.} \nbparatext{Amount of Light emitted by Certain Stars.}---This, of course,
+is something vastly different from that \textit{received} by us. A star may
+emit hundreds of times as much light as the sun, and yet, if the star
+is remote enough, the amount that reaches the earth will be only an
+excessively minute fraction of sunlight. If $l$ be the amount of light
+that we \textit{receive} from a star, expressed in terms of sunlight at the earth,
+then the total amount of light \textit{emitted}, $L$, is given by the simple equation,
+\[
+L = l × D^2,
+\]
+$D$ being the distance of the star in astronomical units, while $L$ is
+expressed in terms of the sun's light emission.
+
+Turning to the table of stellar parallax (Appendix, \hyperref[app:IV]{Table~IV.}),
+we find that, according to Gill \& Elkin, $D$ for Sirius equals 542,000;
+\begin{flalign*}
+&\text{whence, for Sirius, } \quad
+  L = \frac{542000^2}{7000\,000000} = 42.0;  &&
+\end{flalign*}
+that is, the light emitted by Sirius is more than forty times as much
+as that emitted by the sun.
+
+\begin{fineprint}
+If we use the value of the parallax of this star as determined by Abbe,
+namely, $p'' = 0''.273$, we shall get $L = 68$, while Gyldén's value of the parallax
+$(0''.193)$ gives $L = 155$. In either case, however, it is clear that Sirius emits
+vastly more light than the sun.
+
+Similarly for the pole-star $(p = 0''.06)$, $L = 93$; for Vega $(p = 0''.14)$, $L =
+69$; $\alpha$~Centauri $(p = 0''.75)$, $L = 1.9$; 70~Ophiuchi $(p = 0''.16)$, $L = 1$;
+61~Cygni $(p = 0''.43)$, $L = \frac{1}{15}$; 24,258 Ll.\ $(p = 0''.26)$, $L = \frac{1}{113}$.\footnote
+  {In making this calculation the magnitudes of the Harvard Photometry were
+  used.}
+\end{fineprint}
+
+The companion of Sirius is a little star of the ninth magnitude,
+which forms a double-star system with Sirius itself. The light emitted
+by this companion does not exceed $\frac{1}{12\,000}$ that of Sirius.
+
+\nbarticle{836.} \nbparatext{Causes of Differences of Brightness in Stars.}---It used to be
+thought that the stars were all very much alike in magnitude and constitution;
+%% -----File: 488.png---Folio 477-------
+not, indeed, without considerable differences, but us much
+resembling each other as do individuals of the same race. It is now
+quite certain that this is not the case, as is obvious from the short list
+of actual light emissions just given.
+
+If the stars \textit{were} all alike, all the differences of apparent brightness
+would be traceable simply to differences of \textit{distance}; but as the facts
+are, we have to admit other causes to be equally effective. The differences
+of brightness are due, \textit{first}, to difference of \textit{distance}; \textit{second},
+to difference of \textit{dimensions}, or of light-giving area; \textit{third}, to
+difference in the \textit{brilliance of the light-giving surface}, depending upon
+difference of temperature and constitution. There are stars near and
+remote, large and small, intensely incandescent and barely glowing
+with incipient or failing light.
+
+As Bessel puts it, there is no reason why there may not be ``as
+many \textit{dark} stars as bright ones.'' As we shall soon see, the companion
+of Sirius, though only giving about $\frac{1}{12\,000}$ part as much light as
+Sirius itself, is at least $\frac{1}{10}$ part as heavy; so that, \textit{mass for mass}, it
+cannot be $\frac{1}{1000}$ part as luminous.
+
+When we compare stars by the thousand, we can say of the tenth-magnitude
+stars, for instance, as compared with the fifth, that \textit{as a
+class} they are \textit{more remote}; and also, just as truly, that \textit{their average
+diameters are smaller}, and also that \textit{their surfaces are less brilliant};
+but we must be careful not to make any assertions of this sort regarding
+any one star of the tenth magnitude compared with a particular
+individual of the fifth, unless we have some absolute knowledge of
+their relative distances. The faint star may be the larger of the two,
+or the hotter, or the nearer. We must know something beyond their
+relative ``magnitudes'' before it is possible to settle such questions.
+
+\nbarticle{837.} \nbparatext{Real Diameter of Stars.}---We have no knowledge whatever
+as to the real diameter of any star. As to the apparent \textit{angular} diameter,
+we can only say negatively that it is insensible, in no case being
+known to reach $0''.01$. If there be a star of the same diameter as our
+sun, at such a distance that its parallax equals one second, its apparent
+diameter must be $\dfrac{1924''}{206265} \vphantom{\fbox{$\dfrac{0'}{0}$}}$. (The sun's mean angular diameter is
+$1924''$ (\artref{Art.}{276}).) This equals $0''.0093$---a quantity far too small
+to be reached by any direct measurement, especially since, even in the
+Lick telescope, the ``spurious'' disc of a star has a diameter of nearly
+$\frac{1}{7}''$, and in smaller telescopes is much larger (about $0''.4$ in a ten-inch
+telescope).
+%% -----File: 489.png---Folio 478-------
+
+\begin{fineprint}
+There is a theoretical connection between the diameter of the diffraction
+rings seen around the image of a star in the telescope, and the real (as
+opposed to the spurious) diameter of the image; by comparing, therefore,
+the actual size of the rings with the size they should have if the star were an
+absolute optical point, we might hope to get a determination of the star's
+diameter. But no such attempts have succeeded, and at present no way
+seems open for obtaining the desired measurement. In some cases, as we
+shall see later, the \textit{mass} of a star as compared with that of the sun can be
+found; but not its \textit{volume} or its \textit{density}, since these require a knowledge of its
+diameter.
+\end{fineprint}
+
+\section*{VARIABLE STARS.}
+
+\nbarticle{838.} A close examination shows that many stars change their
+brightness. As a general rule, these which do this, and are called
+``\textit{variable},'' are reddish in their color; while comparatively few of
+the white stars belong to this class. The variable stars may be
+classified\footnote
+  {This classification is substantially that of Professor Pickering, slightly
+  modified, however, by Houzeau.}
+as follows:---
+
+\hangitem{I} Cases of slow continuous change.
+
+\hangitem{II} Irregular fluctuations of light: alternately brightening and
+darkening without any apparent law.
+
+\hangitem{III} Temporary stars, which blaze out suddenly and then disappear.
+
+\hangitem{IV} Periodic stars of the type of $o$ Ceti, usually of long period.
+
+\hangitem{V} Periodic stars of short period, of the type of $\beta$~Lyræ.
+
+\hangitem{VI} Periodic stars in which the variation is like that which would
+be the result of an eclipse by some intervening body---the
+Algol type.
+
+\nbarticle{839.} I\@. \textsc{Gradual Changes}. On the whole, the changes in the
+brightness of the stars since the time of Hipparchus and Ptolemy
+have been surprisingly small. There has been no general increase
+or decrease in the brightness of the stars as a whole; and there are
+few cases where any individual star has altered its brightness by a
+half or even a quarter of a magnitude. The \textit{general appearance} of
+the sky is the same as it was 2000 years ago; so that notwithstanding
+all the effect of proper motions in the meantime and the whole
+amount of the variation that has taken place in the brightness of the
+stars, there is no doubt that if either of these old astronomers were to
+come to life he would immediately recognize the familiar constellations.
+
+There are a few instances, however, in which it is almost certain
+that change has taken place and is going on. In the time of Eratosthenes
+%% -----File: 490.png---Folio 479-------
+the star in the ``claw of the Scorpion'' (now $\beta$~Libræ) was
+reckoned the brightest in the constellation. At present, it is a whole
+magnitude below Antares, which is now much superior to any star in
+the vicinity. So when the two stars Castor and Pollux in the constellation
+Gemini were lettered by Bayer, the former, $\alpha$, was brighter than
+Pollux, which was lettered $\beta$; but $\beta$ is now notably the brighter of
+the two. Taking the whole heavens, we find a considerable number
+of such cases; perhaps a dozen or more.
+
+\nbarticle{840.} \nbparatext{Missing Stars.}---It is a common belief that since accurate
+star-catalogues began to be made, many stars have disappeared and
+not a few new ones have come into existence. While it would not
+do to deny absolutely that anything of the kind has ever happened,
+it is certainly unsafe to assert that it has.
+
+\begin{fineprint}
+There are a considerable number of cases where stars are now missing
+from the older catalogues \textit{as published},---nearly, if not quite, a hundred,---but
+in almost every instance examination of the original observations shows
+that the place as printed was a mistake of some sort which can now be traced,---sometimes
+a mistake of the recorder, sometimes in the reduction of the
+observation, and sometimes of the press. In a few cases the star observed
+was a planet (Uranus, Neptune, or an asteroid); and in some cases the
+missing star may have been a ``temporary star,'' as, for instance, 55~Herculis,
+which was observed by the elder Herschel. So many of the missing stars
+are now satisfactorily explained that it is natural to suppose that the few
+cases remaining are of the same sort.
+
+There, is no known instance of a \textit{new} star appearing and remaining permanently
+visible.
+\end{fineprint}
+
+\nbarticle{841.} II\@. \textsc{Stars  that exhibit Irregular Fluctuations in
+Brightness}. The most conspicuous example of this class is the
+strange star $\eta$~Argus (not visible in the United States). This star
+ranges all the way from the \textit{zero} magnitude (in 1843, when, according
+to Sir John Herschel, it was brighter than every star but Sirius) down
+to the seventh magnitude, which is its present brightness and has been
+ever since 1865.
+
+\includegraphicsmid{illo220}{\textsc{Fig.~220.}---Light Curve of $\eta$~Argus according to Loomis.}
+
+\begin{fineprint}
+Between 1677 (when it was observed by Halley as of the fourth magnitude)
+and 1800, it oscillated in brightness, so far as we can judge from
+the few observations extant, between the fourth and second magnitudes.
+About 1810, it rose rapidly in brightness, and between 1826 and 1850 it was
+never below the standard first magnitude. When brightest, in 1843, it was
+giving more than 25,000 times as much light as in 1865. A singular fact is
+that the star is in the midst of a nebula which apparently sympathizes with
+%% -----File: 491.png---Folio 480-------
+it to some extent in its fluctuations. (There are other instances of connection
+of nebulæ with variable or temporary stars, as will appear later on.)
+Fig.~220 represents the ``light-curve'' of this object from 1800 to 1870,
+according to Loomis; who, however, imagines the star to be \textit{periodic}, with
+a period of about seventy years. If so, it ought to be again increasing in
+brightness by this time.
+\end{fineprint}
+
+$\alpha$~Orionis and $\alpha$~Cassiopeiæ behave in a somewhat similar manner,
+only the whole range of variation in their brightness is less than a
+single magnitude. The catalogue of variable stars shows a considerable
+number of other similar cases.
+
+\nbarticle{842.} III\@. \textsc{Temporary Stars}. There are eleven well authenticated
+cases in which new stars have appeared \textit{temporarily},---that is,
+for a few weeks or months,---blazing up suddenly and then gradually
+fading away. The list is as follows:---
+
+\begin{fineprint}
+
+\hangitemb{1. \phantom{1}134~\textsc{b.c.}}The star of Hipparchus.
+
+\hangitemb{2. \phantom{1}329 \textsc{a.d.}}A star in Aquilla.
+
+\hangitemb{3. 1572 \textsc{a.d.}}Tycho's star in Cassiopeiæ.
+
+\hangitemb{4. 1600 \textsc{a.d.}}\textit{P} Cygni, 3d magnitude, observed by Jansen.
+
+\hangitemb{5. 1604 \textsc{a.d.}}Kepler's star in Ophiuchus.
+
+\hangitemb{6. 1670 \textsc{a.d.}}11 Vulpeculæ, 3d magnitude, observed by Anthelum.
+
+\hangitemb{7. 1848 \textsc{a.d.}}A star of the 5th magnitude, observed by Hind---also in Ophiuchus.
+
+\hangitemb{8. 1860 \textsc{a.d.}}\textit{T} Scorpii, 7th magnitude, in star cluster M~80, observed by Auwers.
+
+\hangitemb{9. 1866 \textsc{a.d.}}\textit{T} Coronæ--Borealis, 2d magnitude.
+
+\hangitemb{10. 1876 \textsc{a.d.}}Nova Cygni, 2d magnitude.
+
+\hangitemb{11. 1885 \textsc{a.d.}}A star in the nebula of Andromeda, 6th magnitude.
+%% -----File: 492.png---Folio 481-------
+
+As regards the first of these stars, we know almost nothing. Hipparchus
+has left no record of its position or brightness; but the Chinese annals mention
+a star as appearing in Scorpio at just that date, and probably the same
+object; though the Chinese observations \textit{may} refer to a comet. The appearance
+of this new star led Hipparchus to form his catalogue of stars.
+
+As to the second on the list, we know hardly more; the records do not
+even make it absolutely certain that the object was not a comet, not being
+explicit on the point of its motion.
+\end{fineprint}
+
+\nbarticle{843.} The third is justly famous. When it was first seen by Tycho
+in November, 1572, it was already brighter than Jupiter, having
+probably appeared a few days previously. It very soon became as
+bright as Venus herself, being even visible by day. Within a week
+or two it began to fail, but continued visible to the naked eye for
+fully sixteen months before it finally disappeared. It is not certain
+whether it still exists or not as a telescopic star: Tycho determined
+its position with as much accuracy as was possible to his instruments;
+and there are a number of small stars, any one of which is so near to
+Tycho's place that it might be the real object.
+
+\begin{fineprint}
+There has been an entirely unfounded notion that this star may have
+been identical with the ``Star of Bethlehem,'' it being imagined that the
+star is \textit{periodically} variable, with a period of 314 years. If so, it might have
+been expected to reappear in 1886, and it was so expected by certain persons
+``as a sign of the second coming of the Lord.'' It is difficult to see how the
+idea came to be so generally prevalent, as it certainly has been. Probably
+every astronomer of any note has received hundreds of letters on the subject.
+At Greenwich a printed circular was prepared and sent out as a reply to
+such inquiries.
+\end{fineprint}
+
+The fifth star, observed by Kepler, was nearly, though not quite,
+as bright as that of Tycho, and lasted longer---fully two years. It
+also has disappeared so that it cannot now be identified.
+
+\nbarticle{844.} The ninth star excited much interest. It blazed out between
+the 10th and 12th of May as a star of the second magnitude,
+remained at its maximum for three or four days, and then, in five or
+six weeks, faded away to its original faintness, for it now is, and was
+before the outburst, a nine and one-half magnitude star on Argelander's
+catalogue, with nothing noticeable to distinguish it from its
+neighbors. While at the maximum its spectrum was carefully studied
+by Huggins, and exhibited brightly and strongly the bright lines of
+hydrogen, just as if it were a sun like our own, but entirely covered
+with outbursting ``prominences'' of incandescent hydrogen.
+%% -----File: 493.png---Folio 482-------
+
+The tenth star had a very similar history. It also rose to its full
+brightness (second magnitude) on November~24, within \textit{four} hours
+according to Schmidt, remained at a maximum for only a day or two,
+and faded away to invisibility within a month. But it still exists
+as a very minute telescopic star of the fifteenth magnitude. It
+was also spectroscopically studied by several observers (by Vogel
+especially) with the strange result that the spectrum, which at the
+maximum was nearly continuous, though marked by the bright lines
+of hydrogen and by bands of other unknown substances, lost more
+and more of this continuous character, until at last it became a simple
+spectrum of three bright lines \textit{like that of a nebula}.
+
+\nbarticle{845.} The eleventh and last of these temporary stars was very peculiar
+in one respect; not in its brightness, for it never exceeded the six
+and one-half magnitude, but because it appeared right in the midst
+of the great nebula of Andromeda, only $12''$ or $13''$ distant from the
+nucleus. It came out suddenly like all the others, and faded away
+gradually in about six months, to absolute extinction so far as any
+existing telescope can show. It showed under photometric measurements
+many fluctuations in brightness, not losing its light smoothly
+and regularly but in a rather paroxysmal manner. Its spectrum, even
+when brightest, was simply continuous without anything more than the
+merest trace of bright lines in it. The eighth star on the list resembled
+it in the fact that it appeared in the midst of a star cluster.
+
+\nbarticle{846.} IV\@. \textsc{Variables of the $o$~Ceti Type}. These objects almost
+without exception behave like the temporary stars in remaining generally
+faint, suddenly brightening up for a short time, and then fading
+back to the original condition; but they do it \textit{periodically}. The
+periods are generally of considerable length, from six months to two
+years; but they are very apt to be considerably irregular, not
+unfrequently to the extent of several weeks.
+
+The star $o$ Ceti (often called \textit{Mira}, that is, ``the Wonderful'') may
+be taken as the type of this class. Its variability was discovered by
+Fabricius in 1596. During most of the time it remains simply a faint
+twelfth-magnitude star, but once in about eleven months it runs up to
+the fourth, third, or even the second magnitude, and then back again,
+occupying usually about 100 days in the rise and fall. Its brightness
+increases more rapidly than it fails, and it remains at its maximum
+for a week or ten days. At the maximum its spectrum is very beautiful,
+containing a large number of intensely bright lines which, however,
+%% -----File: 494.png---Folio 483-------
+are not yet certainly identified. Its light-curve is $A$, in \figref{illo221}{Fig.~221}.
+A large proportion of the known variables belong to this
+class, and many suppose that the temporary stars also belong to it,
+differing from their classmates only in the length of their periods.
+
+\nbarticle{847.} V\@. \textsc{Variables of the Type} $\beta$~\textsc{Lyræ}. In these the main
+characteristic is that there are two or more maxima and minima in
+each period, as if we were dealing with several superposed causes of
+variation. The light-curve of $\beta$~Lyræ is given by $B$, \figref{illo221}{Fig.~221}. Its
+period is about thirteen days.
+
+\nbarticle{848.} VI\@. \textsc{Variables of the Algol Type}. The sixth and last
+class consists of stars which seem to suffer a partial eclipse at short
+intervals, their light-curves ($C$, \figref{illo221}{Fig.~221}) being the reverse of the
+$o$~Ceti type. Of this type of stars, Algol, or $\beta$~Persei, may be taken
+as the most conspicuous representative. Its period is $2^\text{d}\: 20^\text{h}\: 48^\text{m}\: 55^\text{s}.4$,
+which is subject to almost no variation, except certain slow changes
+that appear to be the result of some unknown disturbance. During
+most of the time the star remains of the second magnitude. At the
+time of obscuration it loses about five-sixths of its light, falling to
+the fourth magnitude in about four and one-half hours, remaining at
+the minimum for about twenty minutes, and then in three and one-half
+hours recovering its original condition. During the minimum
+the spectrum undergoes no considerable change, though there are
+suspicions of some slight variations in its lines.
+
+\includegraphicsmid{illo221}{\textsc{Fig.~221.}---Light Curves of Variable Stars.}
+%% -----File: 495.png---Folio 484-------
+
+Only nine stars are so far known belonging to this class, and among
+them are these of the shortest period. One in Cepheus, discovered by
+Ceraski in 1880, has a period of two days and eleven hours; another
+in Ophiuchus of only twenty hours.
+
+\nbarticle{849.} \nbparatext{Explanation of Variability.}---Evidently no simple explanation
+will hold for all the different classes. For the gradually progressive
+changes no explanation need be looked for; on the contrary,
+it is surprising that such changes are no greater than they are, for
+the stars are all growing older.
+
+As for the irregular changes, no sure account can yet be given of
+them. Where the range of variation is small, as it is in most cases,
+one thinks of spots on the surface like these of our own sun, (but
+much more extensive and numerous) and running through a period
+just as our sun spots do. Let a star with such spots upon it revolve
+on its own axis, as of course it must do, and in the combination we
+have at least a possible explanation of a great proportion of all the
+known cases, both the irregular variables and the regularly periodic.
+
+\includegraphicsouter{illo222}{\textsc{Fig.~222.}---The Collision Theory of Variable Stars.}
+
+\nbarticle{850.} \nbparatext{Collision Theory.}---For the temporary stars, and those of
+the $o$~Ceti type, Mr.~Lockyer has recently (in connection with a
+much more extended subject) suggested a collision theory, illustrated
+by \figref{illo222}{Fig.~222}. The fundamental
+idea that the
+phenomena of the temporary
+stars may be
+due to collisions is not
+new. Newton long ago
+brought it out, and to
+some extent discussed
+it; but considering the
+probable diameters of
+the stars as compared
+with the distances between
+them, it seems
+impossible that collisions
+could have been
+frequent enough to account
+for the number of
+temporary stars actually observed. Mr.~Lockyer, however, imagines
+that the temporary stars, and also variable stars of the $o$~Ceti class,
+%% -----File: 496.png---Folio 485-------
+are, in their present stage of development, not compact bodies, but
+only pretty dense swarm of meteorites of considerable extent, each
+such swarm being accompanied by another smaller one revolving
+around it in an eccentric orbit, just as comets revolve around the
+sun, or as the components of double stars revolve around each other.
+He supposes that the perihelion distance is so small that the swarms
+interpenetrate and pass through each other at the perihelion, which
+could happen without disturbing the \textit{general} motion of either of the
+two meteoric flocks; but while they are thus passing, the collisions
+are immensely increased in number and violence, with a corresponding
+increase in the evolution of light. There are many good points about
+this ingenious theory, but also serious objections to it---as, for instance,
+the great irregularity of the periods of stars of this class, an irregularity
+which seems hardly consistent with such an orbital revolution.
+
+\nbarticle{851.} \nbparatext{Stellar Eclipses.}---As to the Algol type, the natural explanation
+is by means of an eclipse of some sort. The interposition of a
+more or less opaque object between the observer and the star,---a
+dark companion revolving around it,---would produce just the effect
+observed. If, however, this is really the case, the mass of Algol must
+be absolutely enormous compared with that of our sun in order to
+produce so swift a revolution in the eclipsing body. But Ceraski's
+variable of this type in Cepheus is very refractory and exhibits
+changes of period and other phenomena that are extremely difficult
+to reconcile with the idea that \textit{its} obscuration is due to an eclipse.
+
+\nbarticle{852.} \nbparatext{Number and Designation of Variables.}---The ``Catalogue
+of Variable Stars,'' of Mr.\ S.~C.\ Chandler, published in 1888, contains
+225 objects. One hundred and sixty are distinctly \textit{periodic}
+stars: in 12 cases the periodicity is perhaps uncertain, while 14 are
+certainly \textit{irregular}.
+
+\begin{fineprint}
+The remainder includes the temporary stars and some twenty-five or
+thirty stars in respect to the variations of which very little is yet known.
+\hyperref[app:VI]{Table~VI.}\ in the Appendix presents the principal data for the naked-eye
+variables which are visible in the United States.
+
+When a star is discovered to be variable which previously had no special
+appellation of its own, it is customary to designate it by one of the last
+letters in the alphabet, beginning with R. Thus R~Sagittarii is the first
+discovered variable in Sagittarius; S~Sagittarii is the second; T~Sagittarii,
+the third, and so on.
+\end{fineprint}
+
+\nbarticle{853.} \nbparatext{Range of Variation.}---In many cases the whole range is
+only a fraction of a magnitude (especially among the more newly
+%% -----File: 497.png---Folio 486-------
+discovered variables), but in a great number it extends from four
+to eight magnitudes, the maximum brightness exceeding the minimum
+by from fifty to a thousand times; and in a few cases the
+range is greater yet. Not unfrequently considerable changes of
+color accompany the changes of brightness; the star as a rule being
+whiter at its maximum, and frequently showing bright lines in its
+spectrum.
+
+\nbarticle{854.} \nbparatext{Method of Observation.}---There is no better way than that
+of comparing the star by the eye, or with the help of an opera-glass,
+with surrounding stars of about the same brightness at the time when
+its light is near the maximum or minimum; noting to which of them
+it is just equal at the moment, and also these which are a shade
+brighter or fainter.
+
+\begin{fineprint}
+It is possible for an amateur to do really valuable work in this way, by
+putting himself in relation with some observatory which is interested in the
+subject. The observations themselves require so much time that it is difficult
+for the working force in a regular observatory to attend to the matter
+properly, and outside assistance is heartily welcomed in gathering the needed
+facts. The observations themselves are not specially difficult, require no
+very great labor or mathematical skill in their reduction, and, as has been
+said, can be made without instruments; but they require patience, assiduity,
+and a keen eye.
+\end{fineprint}
+
+\section*{STAR SPECTRA.}
+
+\nbarticle{855.} In 1824 Fraunhofer, in connection with his study of the
+lines of the solar spectrum, investigated also the spectra of certain
+stars, using an apparatus essentially similar to that which is now
+employed at Cambridge. He placed a prism in front of the object-glass
+of a small telescope and looked at the stars through this, using
+a cylindrical lens in the eye-piece to widen the spectrum, which otherwise
+would be a mere line.
+
+\begin{fineprint}
+He found that Sirius, Castor, and many other stars show very few dark
+lines in their spectrum, but strong ones; while, on the other hand, the spectra
+of Pollux ($\beta$~Geminorum) and Capella resemble closely the spectrum of the
+sun. In all the spectra he recognized the $D$~line, although it was not then
+known that it had anything to do with sodium.
+\end{fineprint}
+
+\sloppy
+\nbarticle{856.} \nbparatext{Observations of Huggins and Secchi.}---Almost as soon as
+the spectroscope had taken its place as a recognized instrument of
+science it was applied by Huggins to the study of the stars, and
+%% -----File: 498.png---Folio 487-------
+Secchi followed hard in his footsteps. The former studied the spectra
+of comparatively few stars, but with all the dispersive power he
+could obtain, and in detail; while Secchi, using a much less powerful
+instrument, examined several thousand star spectra, in a more
+general way, for purposes of classification.
+
+\fussy
+Huggins identified with considerable certainty in the spectra of
+$\alpha$~Orionis (Betelgueze) and $\alpha$~Tauri (Aldebaran) a number of elements
+that are familiar on the earth, and are most of them prominent in the
+solar spectrum. He found in the former sodium, magnesium, calcium,
+iron, bismuth, and hydrogen; and in $\alpha$~Tauri he reported in
+addition, tellurium, antimony, and mercury; but these latter metals
+have not yet been verified.
+
+\nbarticle{857.} \nbparatext{Classification of Stellar Spectra.}---Secchi, in his spectroscopic
+survey, found that the 4000 stars which he observed could all
+be reduced to four classes.
+
+\includegraphicsmid{illo223}{\textsc{Fig.~223.}---Secchi's Types of Stellar Spectra.}
+
+\textit{The first class comprises the white or blue stars.} To it belong Sirius
+and Vega, and, in fact, considerably more than half of all the stars
+examined. The spectrum is characterized by the great strength of
+the hydrogen lines, which are wide, hazy bands, much like the $H$ and
+$K$~lines in the solar spectrum. Other lines are extremely faint or
+entirely absent; the $K$~line especially, which in the solar spectrum is
+especially prominent, in the spectra of most of these stars is hardly
+visible.
+
+\textit{The second class} is also numerous, and \textit{is composed of stars with a
+spectrum substantially like that of our sun.} The $H$ and $K$~lines are
+both strong. Capella and Pollux ($\beta$~Geminorum) are prominent
+examples of this class. There are certain stars which form a connecting
+link between these two first classes, stars like Procyon and $\alpha$~Aquilæ,
+which, while they show the hydrogen lines very strongly, also
+exhibit a great number of other lines between them. The first and
+second classes together embrace fully seven-eighths of all the stars
+he observed.
+
+\textit{The third class} includes most \textit{of the red and variable stars}, some 500
+in number, and the spectrum is characterized by dark \textit{bands} instead
+of lines (though lines are generally present also). These bands,
+which are probably due to carbon, shade from the blue towards the
+red; that is, they are sharply defined and darkest at the more refrangible
+edge. Occasionally in spectra of this type some of the hydrogen
+lines are bright. $\alpha$~\DPtypo{Hercules}{Herculis},
+$\alpha$~Orionis, and \textit{Mira} ($o$~Ceti) are fine
+examples of this third class.
+%% -----File: 499.png---Folio 488-------
+
+\textit{The fourth class} is composed of a very small number of stars, less
+than sixty so far as now known, mostly small red stars. This spectrum
+is also a banded one; but compared with the third class the
+bands are \textit{reversed}, that is, faced the other way, and shaded towards
+the blue. These generally show also a number of bright lines.
+None of the conspicuous stars belong to this class---none above the
+fifth magnitude. The sixth magnitude star, 152~Schjellerup, may
+be taken as its finest example ($\alpha$, $12^\text{h}\: 39^\text{m}$; $\delta$, $+40°\: 9'$, in the constellation
+of Canes Venatici). \figref{illo223}{Fig.~223} exhibits the light-curves of
+these four types of spectrum.\footnote
+  {It is difficult to represent spectra accurately by any process of engraving that
+  can be readily reproduced in a book like the present. The curve, on the other
+  hand, is easily managed, and, though it does not please the eye like the spectrum
+  itself, it is capable of conveying all the information that could be obtained from
+  the most finished engraving. \textit{Dark lines} are represented by lines running \textit{downward}
+  from the upper boundary line of the curve, and \textit{bright lines} by lines running
+  \textit{upward}, while the bands and their shading are represented by variations in the
+  contour of the curve.}
+
+\nbarticle{858.} Vogel has revised Secchi's classification of spectra as follows,
+making only three main classes, but with subdivisions:
+%% -----File: 500.png---Folio 489-------
+
+\hangitemc{I. (\textit{a})} Same as Secchi's I. The white stars.
+
+\hangitemc{(\textit{b})} Nearly continuous; all lines wanting or very faint.
+$\beta$~Orionis is the type.
+
+\hangitemc{(\textit{c})} Showing the lines of hydrogen bright, and also the
+helium line $D_3$ (\artref{Art.}{323}).
+
+\hangitemc{II. (\textit{a})} Same as Secchi's II.
+
+\hangitemc{(\textit{b})} Like II. (\textit{a}), but showing in addition bright lines which
+are \textit{not} the lines of hydrogen or helium. (This is rare.)
+
+\hangitemc{III. (\textit{a})} Same as Secchi's III.
+
+\hangitemc{(\textit{b})} Same as Secchi's IV.
+
+\begin{fineprint}
+Vogel's classification is based in part on the very doubtful assumption
+that stars of Class~I. are hottest and also youngest, while the other classes
+belong to stars which are either beginning to fail or are already far gone in
+decrepitude. But it is very far from certain that a red star is not just as
+likely to be younger than a white one, as to be older. It probably is now at
+a \textit{lower temperature}, and possesses a more extensive envelope of gases; but
+it may be increasing in temperature as well as decreasing. At any rate we
+have no certain knowledge about its age.
+
+\sloppy
+\nbarticle{859.} \nblabel{pg:few}\nbparatext{Photography of Stellar Spectra.}---As early as 1863 Huggins
+attempted to photograph the spectrum of Vega, and succeeded in getting an
+impression of the spectrum, but without any of the lines. In 1872 Dr.~Henry
+Draper of New York, working with the reflector which he had himself
+constructed, succeeded in getting an impression of the spectrum of the
+same star, showing for the first time four of its hydrogen lines. The introduction
+of the more sensitive dry plates in 1876, induced Mr.~Huggins to
+resume the subject (as did Dr.~Draper soon after), and they soon succeeded in
+getting pictures showing many lines. The spectra were about half an inch
+long by $\frac{1}{12}$ or $\frac{1}{16}$ of an inch wide. After the lamented death of Dr.~Draper
+in 1882, Professor Pickering took up the work at Cambridge (U.\,S.); and with
+such success that Mrs.~Draper, who had intended to establish and to endow
+her husband's observatory as an establishment for astro-physical research,
+and a most fitting monument to his memory, concluded to transfer the
+instruments to Cambridge, and there establish the ``Draper Memorial.''
+\end{fineprint}
+
+\nbarticle{860.} \nbparatext{The Slitless Spectroscope.}---Professor Pickering has attained
+his remarkable success by reverting to the ``slitless spectroscope,''
+arranged in the manner first used by Fraunhofer, and later revived
+by Secchi. The instrument consists of a telescope with the \textit{objective
+corrected not for the visual, but for the photographic rays}, equatorially
+mounted and carrying \textit{in front of the object-glass} one or more prisms,
+with a refracting angle of nearly $10°$, and large enough to cover the
+whole lens.
+%% -----File: 501.png---Folio 490-------
+
+\begin{fineprint}
+The refracting edge of the prism is placed east and west, so that the
+linear spectrum of a star formed on a plate at the focus of the object-glass
+runs north and south. If, now, the clock-work of the instrument is adjusted
+to follow the star exactly, the image (\textit{i.e.}, the spectrum) will be a mere line,
+broken here and there where the dark lines of the spectrum should appear.
+By merely retarding or accelerating the clock a trifle, the linear spectrum
+will drift a little sidewise upon the plate, and so will form a spectrum having
+a width depending on the amount of this drift during the time of exposure.
+If the air is calm the lines of the spectrum thus formed are as clean and
+sharp as if a slit were used; otherwise not.
+\end{fineprint}
+
+\includegraphicsouter{illo224}{\textsc{Fig.~224.}\\
+Arrangement of the Prisms in the Slitless Spectroscope.}
+
+\nbarticle{861.} The most powerful instrument used in this work at Cambridge
+is Dr.~Draper's eleven-inch photographic refractor, with four huge
+glass prisms in a box in front of the object-glass, arranged as indicated
+in \figref{illo224}{Fig.~224}. With
+this apparatus, photographic
+spectra of the brighter stars
+are now obtained having,
+before enlargement, a length
+of fully three inches from
+$F$ in the blue of the spectrum
+to the extremity of
+the ultra violet. It is a
+pity, of course, that the
+lower portions of the spectrum
+below $F$ cannot be
+reached in the same way;
+but no plates sufficiently
+sensitive to green, yellow, and red rays have yet been found. The
+exposure necessary to obtain the impression of even the most powerful
+photographic rays is from half an hour to an hour. \figref{illo225}{Fig.~225} is
+enlarged about one-third from one of these photographs of the
+spectrum of Vega, which extends far into the ultra violet.
+
+\includegraphicsmid{illo225}{\textsc{Fig.~225.}---Photographic Spectrum of Vega. Cambridge, 1887.}
+
+These spectra bear tenfold enlargement perfectly, making them
+more than two feet long by two inches in width, and then in the spectrum
+%% -----File: 502.png---Folio 491-------
+of such a star as Capella they show hundreds of lines. It is
+simply amazing that the feeble, twinkling light of a star can be made
+to produce such an autographic record of the substance and condition
+of the inconceivably distant luminary.
+
+\nbarticle{862.} \nbparatext{Peculiar Advantages of the Slitless Spectroscope.}---The
+slitless spectroscope has three great advantages. First, that it
+utilizes all the light that comes from the star to the object-glass,
+much of which in the usual form of the instrument is lost in the
+jaws of the slit: Secondly, that by taking advantage of the length
+of a large telescope, it produces a very high dispersion with even a
+single prism: Thirdly, and most important, it gives on the same
+plate and with a single exposure the spectra of all the many stars
+whose images fall upon it. With the smaller eight-inch instrument
+made at Cambridge, and one prism, as many as 100 or 150 spectra
+are sometimes taken together; as, for instance, in a spectrum photograph
+of the Pleiades.
+
+\nbarticle{863.} \nbparatext{Disadvantages of the Slitless Spectroscope.}---\textit{Per contra},
+the giving up of the slit precludes all the usual methods of identifying
+the lines by actually confronting them with comparison spectra;
+the comparison prism (\artref{Art.}{315}) cannot be used. This makes it
+extremely difficult to utilize these magnificent pictures for purposes
+of scientific measurement.
+
+\begin{fineprint}
+If it turns out that any of the lines photographed in the spectrum are of
+\textit{atmospheric} origin, the difficulty will be largely removed, as these atmospheric
+lines could be used as reference points. Professor Pickering has tried, by
+the interposition of various vapors in the path of the light within the
+telescope tube, to get lines that will answer the purpose, but thus far
+without any really satisfactory results. Until this difficulty is overcome it
+will be impossible to make these spectra yield us all of the information which
+they undoubtedly contain with respect to the motions of the stars in the line
+of sight.
+
+Vogel in his photographic work mentioned in connection with this subject
+(\artref{Art.}{802}) used an ordinary spectroscope with a slit. This arrangement
+required a very long exposure, and limited the dispersion it was possible
+to employ; but it permitted him to use a hydrogen Geissler tube placed
+within the telescope itself, to furnish a comparison spectrum.
+\end{fineprint}
+
+\nbarticle{864.} \nbparatext{Twinkling or Scintillation of the Stars.}---This is a purely
+atmospheric effect, usually violent near the horizon and almost null
+at the zenith. It differs greatly on different nights according to the
+steadiness of the air.
+%% -----File: 503.png---Folio 492-------
+
+If the spectrum of a star near the eastern horizon be examined
+with a spectroscope so held as to make the spectrum vertical, it will
+appear to be continually traversed by dark bands running through
+the spectrum from the blue end towards the red. At the western
+horizon the bands move in the opposite direction, from red to blue;
+on the meridian they merely oscillate back and forth.
+
+\begin{fineprint}
+\textbf{Cause of Scintillation.}---Authorities differ as to the exact explanation
+of scintillation, but probably it is mainly due to \textit{two} causes (optically speaking),
+both depending on the fact that the air is full of streaks of unequal
+density that are carried by the wind.
+
+(1) In the first place, light transmitted through such a medium is concentrated
+in some places and turned away from others \textit{by simple refraction}:
+so that, if the light of a star were strong enough, a white surface illuminated
+by it would look like the sandy bottom of a shallow, rippling pool
+of water illuminated by sunlight, with light and dark mottlings which
+move with the ripples on the surface. So, as we look towards the star,
+and the mottlings due to the irregularities of the air move by us, we see the
+star alternately bright and faint; in other words, it \textit{twinkles}; and if we
+look at it in a telescope we shall see that it not only twinkles, but \textit{dances,
+i.e.}, it is slightly displaced back and forth by the refraction.
+
+(2) The other cause of twinkling is ``\textit{interference}.'' Pencils of light
+coming from the star (which optically is a mere point), and feebly
+refracted by the air in the way above explained, reach the observer by
+slightly different routes, and are just in a condition to interfere. The
+result of the interference is the temporary destruction of rays of certain
+wave-lengths, and the reinforcement of others. At a given moment the
+\textit{green} rays, for instance, will be destroyed, while the red and blue will be
+abnormally intense; hence the quivering dark bands in the spectrum. If
+the star is very near the horizon, the effects are often sufficient to produce
+marked changes of color.
+\end{fineprint}
+
+\sloppy
+\nbarticle{865.} \nbparatext{Why Planets Twinkle Less than Stars.}---This is mainly
+because they have \textit{discs of sensible diameter}, so that there is a general
+unchanging \textit{average} of brightness for the sum total of all the luminous
+points of which the disc is composed. When, for instance,
+point $A$ of the disc becomes dark for a moment, point $B$, very near
+it, is just as likely to become bright; the interference conditions
+being different for the two points. The different points of the disc
+\textit{do not keep step}, so to speak, in their twinkling.
+\chelabel{CHAPTERXX}
+%% -----File: 504.png---Folio 493-------
+
+\fussy
+\Chapter{XXI}{Aggregations of Stars}
+\nbchapterhang{\stretchyspace
+DOUBLE AND MULTIPLE STARS.---ORBITS AND MASSES OF
+DOUBLE STARS.---CLUSTERS.---NEBULÆ.---THE MILKY WAY.---DISTRIBUTION
+OF STARS.---CONSTITUTION OF THE STELLAR
+UNIVERSE.---COSMOGONY AND THE NEBULAR HYPOTHESIS.}
+
+\includegraphicsmid{illo226}{\textsc{Fig.~226.}---Double and Multiple Stars.}
+
+\nbarticle{866.} \nbparatext{Double and Multiple Stars.}---The telescope shows numerous
+instances in which two stars lie very near each other, in many
+cases so near that they can be seen separate only under a
+high magnifying power. These are called ``\textit{double stars}.'' At present
+something over 10,000 such couples are known, and the number is
+continually increasing. In not a few instances we have \textit{three} stars
+%% -----File: 505.png---Folio 494-------
+together, two of which are usually very close and the third farther
+away; and there are several cases of \textit{quadruple stars}, where there
+are two pairs of stars lying close together (as in $\epsilon$~Lyræ), or a pair
+of stars with two single stars close by; and there are some cases
+where more than four form a ``\textit{multiple star}.'' \figref{illo226}{Fig.~226} represents
+a number of such double and multiple stars.
+
+\nbarticle{867.} \nbparatext{Distance, Magnitudes, and Colors.}---The apparent distances
+usually range from $30''$ to $\frac{1}{4}''$, few telescopes being able to separate
+double stars closer than $\frac{1}{4}''$.
+
+In a very large proportion of cases (perhaps about one-third of all)
+the two stars are nearly equal; in many others they are extremely
+unequal, a minute star near a large one being usually known as its
+``companion.''
+
+Not infrequently the components of a double star present a fine
+contrast of color; \textit{never, however, in cases where they are nearly equal
+in magnitude.} It is a remarkable fact, as yet wholly unexplained,
+that when we have such a contrast of color the tint of the smaller star
+always \textit{lies higher in the spectrum} than that of the larger one. The
+larger one is \textit{reddish or yellowish}, and the smaller one \textit{green or blue,}
+without a single exception
+among the many hundreds
+of such tinted couples now
+known, $\gamma$~Andromedæ and
+$\beta$~Cygni are fine examples
+for a small telescope.
+
+\includegraphicsouter{illo227}{\textsc{Fig.~227.}---Measurement of Distance and Position-Angle of a Double Star.}
+
+\nbarticle{868.}\hspace{0em} \nbparatext{Measurement of
+Double Stars.}---Such measures
+are generally made
+with a filar position-micrometer,
+essentially such
+as shown in Fig.~28 and
+29 (\artref{Art.}{73}). The quantities
+to be determined are
+the distance and position-angle
+of the couple. By
+``distance'' we mean simply
+the apparent distance in
+seconds of arc between the centres of the two star discs. \textit{The position-angle}
+of a double star is the \textit{angle made with the hour-circle}
+%% -----File: 506.png---Folio 495-------
+\textit{by the line drawn from the larger star to the smaller}, reckoning around
+from the north through the east, as shown in \figref{illo227}{Fig.~227}.
+
+\begin{fineprint}
+Photography may also be used, and promises to become a favorite method.
+The first photograph of such an object was by Bond in 1851.
+\end{fineprint}
+
+\nbarticle{869.} \nbparatext{Stars Optically and Physically Double.}---Stars may be double
+in two different ways. They may be merely \textit{optically} double,---that
+is, simply in line with each other, but one far beyond the other; or
+they may be really very near together, in which case they are said to
+be ``\textit{physically connected},'' because they are then under the influence
+of their mutual attraction, and move accordingly.
+
+\sloppy
+\nbarticle{870.} \nbparatext{Criterion for distinguishing between Physically and Optically
+Double Stars.}---This cannot be done off-hand. It requires a series of
+measurements long enough continued to determine whether the relative
+movement of the stars is in a curve or a straight line. If the stars
+are really close together their attraction will force them to describe
+curves around each other. If they are really at a great distance and
+only accidentally in line, then their proper motions, being sensibly
+uniform and rectilinear, will produce a \textit{relative} motion of the same
+kind. Taking either star as fixed, the other star will appear to pass
+it in a straight line, and with a steady, uniform drift.
+
+\fussy
+\nbarticle{871.} \nbparatext{Relative Number of Stars Optically Double, and Physically
+connected}.---Double-star observations practically began with Sir
+William Herschel only a little more than a hundred years ago.
+When he took up the subject less than 100 such pairs had been recognized,
+such as had been accidentally encountered in making observations
+of various kinds. The great majority of double stars have been
+discovered so recently that sufficient time has not yet elapsed to make
+the criterion above given effective with more than a small proportion
+of them. But it is already perfectly clear that the optically double
+stars are, as the theory of probability shows they ought to be, very
+few in number, while several hundred pairs have shown themselves
+to be physically connected, \textit{i.e.}, to be what are known as ``\textit{binary}''
+stars, or couples which revolve around their common centre of gravity.
+
+\nbarticle{872.} \nbparatext{Binary Stars.}---Sir W.~Herschel began his observations of
+double stars in the hope of ascertaining stellar parallax. He had
+supposed in the case of couples where one was large and the other
+small that the smaller one was usually a long way beyond the other
+%% -----File: 507.png---Folio 496-------
+(as \textit{sometimes} is really the fact). In this case there should be perceptible
+variations in the distance and position of the two stars during
+the course of the year; precisely such variations as those by
+which, fifty years later, Bessel succeeded in getting the parallax of
+61~Cygni (\artref{Art.}{811}). But Herschel, instead of finding the yearly
+oscillation of distance and position which he expected, found quite
+a different, and, at the time, a surprising thing,---a regular, progressive
+change, which showed that one of the stars was slowly describing
+a regular orbit around the other. To use his own expression, he
+``went out like Saul to seek his father's asses, and found a kingdom,''---the
+dominion of gravitation\footnote
+  {It is not yet fully \textit{demonstrated} that the motions of binary stars are due to
+  gravitation, though it is extremely \textit{probable}, and the burden of proof seems to
+  be shifted upon those who are disposed to doubt it. See, however, the footnote
+  to \artref{Article}{901}.}
+extended to the stars, unlimited
+by the bounds of the solar system. $\gamma$~Virginis, $\xi$~Ursæ Majoris,
+and $\zeta$~Herculis were among the most prominent of the systems
+which he pointed out.
+
+At present the number of pairs \textit{known} to be binary is at least
+200, and as many more begin to show signs of movement. (Up
+to the present time of course only the quicker moving ones are
+obvious.) About fifty have progressed so far,---having made at
+least one entire revolution or a great part of one,---that their orbits
+have been computed more or less satisfactorily.
+
+\nbarticle{873.} \nbparatext{Orbits of Binary Stars.}---The real orbit described by \textit{each}
+of such a pair of stars is always found to be an ellipse, and assuming
+the applicability of the law of gravitation, the common centre of
+gravity must be at the focus. The two ellipses are precisely similar,
+the one described by the smaller star being larger than the other in
+inverse proportion to the star's mass.
+
+So far as the \textit{relative} motion of the two bodies goes, we may regard
+either of them (usually the larger is preferred) as being at rest, and
+the other as moving around it in a \textit{relative orbit} of precisely the same
+\textit{shape} as either of the two actual orbits which are described around
+the centre of gravity. But the relative orbit is larger, having for its
+semi-major axis the \textit{sum} of the two semi-axes of the real orbits (\artref{Art.}{427}).
+
+Usually the \textit{relative} orbit is all that we can ascertain, as this alone
+can be deduced from the micrometer measures when they consist
+only of position-angles and distances measured between the two stars.
+%% -----File: 508.png---Folio 497-------
+
+\begin{fineprint}
+In a few cases where such measures have been made from small stars in
+the same field of view with the couple, but not belonging to the system, or
+when the couple has long been observed with the meridian circle, it becomes
+possible to work out separately the orbit of each star of the pair with reference
+to their common centre of gravity; then we can deduce their relative
+masses, as for instance, in the case of Sirius and its companion.
+\end{fineprint}
+
+\nbarticle{874.} \nbparatext{Calculation of the Orbit of a Binary Star.}---If the observer
+is so placed as to view the orbit perpendicularly, he will see it in its
+true form and having the larger star in its focus, while the smaller
+moves around it, describing ``equal areas in equal times.'' But if the
+observer is anywhere else, the orbit will be apparently more or less
+distorted. It will still be an ellipse (since every projection of a conic
+is also a conic), but the large star will no longer occupy its focus,
+nor will the major and minor axes be apparently at right angles to
+each other; nor will they even coincide with the longest and shortest
+diameters of the ellipse. In this distorted ellipse the smaller star will,
+however, still describe equal areas in equal times around the larger one.
+
+Theoretically five absolutely accurate observations of the position
+and distance are sufficient to determine the elements of the relative
+orbit, if we assume that the orbital motion is described under the law
+of gravitation. Practically a greater number are needed in most cases,
+because the motions are so slow and the stars so near each other that
+observation-errors of $0''.1$ (which in most calculations are of small
+account) here become important. The work requires not only labor,
+but judgment and skill, and unless the pair has completed or nearly
+completed an entire revolution the result is apt to be seriously uncertain.
+So far, as has been said, about fifty such orbits are fairly well
+determined. Catalogues, more or less complete, will be found in
+Flammarion's book on ``Double Stars,'' also in Gledhill's ``Handbook
+of Double Stars,'' and Houzeau's ``Vade Mecum.'' See
+\hyperref[app:V]{Table~V.}\ in the Appendix.
+
+\begin{fineprint}
+\nbarticle{875.} \nbparatext{Sirius and Procyon.}---The cases of these two stars are remarkable.
+In both instances the large stars have been found from meridian-circle
+observations to be slowly moving in little ellipses, although when  this
+discovery was first made neither of them was known to be double.    In 1862
+the minute companion of Sirius was discovered by Clark with the object-glass
+of the Chicago telescope, then just finished, and at that time the
+largest object-glass in the world. And this little companion was found to be
+precisely the object needed to account for the peculiar motion of Sirius itself.
+
+In the case of Procyon, the companion, if it exists, is yet to be discovered;
+not, however, because it has not been carefully looked for with the most
+powerful instruments.
+\end{fineprint}
+%% -----File: 509.png---Folio 498-------
+
+\includegraphicsouter{illo228}{\textsc{Fig.~228.}---Orbits of Binary Stars.}
+
+\nbarticle{876.} \nbparatext{Periods.}---The periods of binary stars, so far as at present
+known, vary from twelve years (the period of $\delta$~Equulei) to nearly
+1600 years, as $\zeta$~Aquarii.
+
+\begin{fineprint}
+It is possible that one or two
+others may be found with periods
+even shorter than fourteen years,
+and it is practically certain that
+as time goes on, pairs of longer
+period than 1500 years will present
+themselves. Computed periods
+much exceeding 200 years
+must, however, be received at
+present with much reserve.
+
+Fig.~228 shows the apparent
+orbits of several of the most interesting
+binaries. The figure
+for $\zeta$~Cancri (copied from Gledhill)
+is incorrect; 1878 and 1827
+should be interchanged, and the
+arrow reversed.
+\end{fineprint}
+
+\nbarticle{877.} \nbparatext{Size of the Orbits.}---The
+angular semi-major axes
+of the orbits thus far computed range from about $0''.3$ for $\delta$~Equulei,
+to $18''$ for $\alpha$~Centauri. The real dimensions are, of course, only to be
+obtained when we know the star's parallax and distance.\footnote
+  {The \textit{real} semi-axis of the orbit in astronomical units is simply the \textit{angular}
+  semi-axis divided by the parallax.}
+Fortunately
+several of the stars whose parallaxes have been determined
+are also binary stars. Assuming the data as to parallax and orbits
+given in the \hyperref[app:V]{tables} in the Appendix (mainly taken from Houzeau's
+``Vade Mecum'') we find the following short table of results:---
+\begin{center}
+\small
+\setlength{\tabcolsep}{5pt}
+\begin{tabular}{@{}l | c | c | c | c | c@{\,}l@{}}
+\hline\hline
+\multicolumn{1}{c|}{\multirow{2}{*}{\textsc{Name.}}} & Assumed   & Apparent  &  Real & \multirow{2}{*}{Period.}  & \multicolumn{2}{c}{Mass.}\\
+                                                     & Parallax. & Semi-axis.&  Semi-axis.  && \multicolumn{2}{c}{\astrosun = 1.}\\
+\hline
+$\eta$~Cassiopeiæ \dotfill        & \phantom{$''$}$0''.15$  &  \phantom{$''$}$8''.64$  &  57.6  &  $195^\text{y}.2$  &  \phantom{0}5.0\\
+\multirow{2}{7em}{Sirius \dotfill}&  0.38  & 8.53         &  22.4         &  44.0           & \phantom{0}5.8 & \multirow{2}{*}{$\Big\}$}\\
+                                  & (0.19) &  ---         & (44.8)\rlap{\footnotemark}& --- & (46.4)\\
+$\alpha$~Geminorum \dotfill       &  0.20  & 7.54         &  37.7         & \llap{9}97.0    &  0.54\\
+$\alpha$~Centauri \dotfill        &  0.75  & \llap{1}7.50 &  23.3         &  77.0           &  2.14\\
+70 Ophiuchi \dotfill              &  0.16  & 4.79         &  29.9         &  94.5           &  3.0\phantom{0}\\
+61 Cygni \dotfill                 &  0.43  & 15.40?       &  35.8\rlap{?} & 450.0?          &  0.23\\
+\hline\hline
+\end{tabular}
+\footnotetext{The distance 44.8 is computed with Gyldén's
+parallax, $0''.193$.}
+\end{center}
+%% -----File: 510.png---Folio 499-------
+
+It is obvious from the table that the double-star orbits are comparable
+with the larger orbits of the solar system; that of $\alpha$~Centauri
+being just about equal in size to that of Uranus, and that of
+$\eta$~Cassiopeiæ not quite double the size of Neptune's orbit. Of course
+the many binary stars whose distance is so great as to make their
+parallax insensible while their apparent orbits are as large as those
+given in the list must have real orbits of still vaster dimensions.
+
+\nbarticle{878.} \nbparatext{Masses of Binary Stars.}---When we know both the size of
+the orbit of a binary and its period, the mass, according to the law
+of gravitation, follows at once from the equation of \artref{Article}{536},
+\[
+M + m = 4\pi^2\frac{a^3}{t^2}.
+\]
+If $t$ and $a$ are given respectively in \textit{years} and astronomical units
+of \textit{distance}, then, by omitting the factor $4\pi^2$, $M + m$ comes out in
+\textit{terms of the sun's mass.} The final column of the little table above
+gives the \textit{masses} of the six pairs of stars as compared with the mass
+of the sun. But the student must bear in mind that the parallaxes
+of stars are so uncertain, that these results are to be accepted with a
+very large margin of error.
+
+\begin{fineprint}
+\nbarticle{879.} \nbparatext{Relation between the ``Mass-Brightness'' of Binary Stars.}---Monck
+of Dublin has recently called attention to a curious relation
+between the apparent brightness of a binary, its period and (angular) distance
+on the one hand, and its ``\textit{mass-brightness},'' or \textit{candle-power per ton}, so
+to speak, on the other,---a relation which does not involve a knowledge of
+the \textit{parallax} of the stars. He shows that if we let $l$ be the apparent brightness
+of any double star, photometrically determined, $a$ the semi-major axis
+of its orbit, (in seconds of arc), and $t$ its period in years, while $k$ is its
+``\textit{mass-brightness},'' or candle-power per ton,---then we have\footnote
+  {This is strictly true only on the assumption that either the two components
+  of the double star have the same mass-brightness, or else that the smaller one is
+  so much smaller that its motion is practically the same as if it were a mere
+  particle. If the stars of the pair are both about alike, or if they differ
+  greatly in mass, the equation is practically correct in either case. It breaks
+  down when the two stars of a pair \textit{do not differ much in mass, but do differ greatly
+  in brightness}---probably an unusual case.}
+\[
+  k_1 : k_2
+= l_1 t_1^{\frac{4}{3}} a_1^2
+: l_2 t_2^{\frac{4}{3}} a_2^2 \quad\text{or}\quad
+  \frac{k_1}{k_2}
+= \frac{l_1}{l_2} \left( \frac{t_1}{t_2} \right)^{\frac{4}{3}}
+                  \left( \frac{a_1}{a_2} \right)^2.
+\]
+(\textit{The Observatory, February,} 1887.)
+\medskip
+%% -----File: 511.png---Folio 500-------
+
+He takes $\xi$~Ursæ Majoris as a standard (\textit{i.e.}, its $k = 1$), and finds for
+$\gamma$~Leonis $k = 93$; for $\alpha$~Geminorum, 38; for Sirius, 7; for $\zeta$~Herculis, 4; for
+$\eta$~Cor.\ Bor., 1.5; for 70~Ophiuchi, 0.36; for $\eta$~Cassiopeiæ, 0.22; and for
+61~Cygni, 0.08. In other words, $\gamma$~Leonis is enormously brilliant in proportion
+to its mass, Sirius is medium, while 61~Cygni is extremely faint. Upon
+this scale the sun's \textit{mass-brightness} would be about 2, judging from its mass
+and brightness compared with that of Sirius (Arts.~\arnref{832} and \arnref{877}). (We
+have given above only a few of Mr.~Monck's numbers.)
+\end{fineprint}
+
+\nbarticle{880.} \nbparatext{Have the Stars Planets attending them?}---It is a very
+natural supposition that the minute companions which attend some of
+the larger stars may be really planetary in their nature, shining more
+or less by reflected light. As to this we can only say, that while it is
+quite possible that other stars besides our sun may have their retinues
+of planets, it is quite certain that such planets could not be seen
+by us with any existing telescope. If our sun were viewed from
+$\alpha$~Centauri, Jupiter would be a star of less than the twenty-first magnitude,
+at a distance of only $5''$ from the sun which itself would be a
+smallish first-magnitude star.
+
+\begin{fineprint}
+\nbarticle{881.} The statement can be verified as follows: Jupiter at opposition is
+certainly not equivalent in brightness to twenty stars like Vega (most
+photometric measurements make it from eight to fourteen). Assuming,
+however, that it is equal to twenty Vegas, its light received by the earth
+would be about $\frac{1}{2000\,000000}$ of the sun's. At opposition our distance
+from Jupiter is about four astronomical units, so that seen from the same
+distance as the sun, its light would be sixteen times that quantity, or
+(nearly) $\frac{1}{125\,000000}$ of the sun's.
+
+Now a ratio of 125,000000 between the light of two stars corresponds to a
+difference of $20 +$ magnitudes
+\[
+\left( \log{\text{125,000000}}\!=\!8.0969; \text{ but } \frac{8.0969}{0.4000}\!=\!20.24 \text{ magnitudes. (\artref{Art.}{820}.)}\right)
+\]
+Accordingly, if the observer were removed to such a distance that the sun
+would appear like a first-magnitude star (as would be the case from
+$\alpha$~Centauri), Jupiter would be a star of the twenty-first magnitude. According
+to \artref{Art.}{822}, it would require a 25-inch telescope to show a star of the
+sixteenth magnitude; it would therefore require an instrument with an
+aperture of 250 inches, or nearly 21 feet, to show a star five magnitudes
+fainter, even if there were no large star near to add to the difficulty.
+
+\nbarticle{882.} \nbparatext{Triple and Multiple Stars.}---There are a considerable number
+of objects of this kind, and some of them constitute physical systems. In
+the case of $\zeta$~Cancri the two larger stars revolve around their common
+%% -----File: 512.png---Folio 501-------
+centre in a nearly circular orbit less than $2''$ in diameter, and with a period
+of about sixty years; while the third star, smaller and more distant, moves
+around the closed pair in an orbit not yet well determined, but with a period
+that must be several hundred years; and in its motion there is evidence of a
+peculiar perturbation, as yet unexplained. In $\epsilon$~Lyræ we have two pairs,
+each making a very slow revolution, of periods not yet determined, but
+probably ranging from 300 to 500 years. And since the pairs have also a
+common proper motion it is practically certain that they also are physically
+connected, and revolve around their common centre of gravity in a period to
+be reckoned by millenniums---the motion during the last hundred years
+being barely perceptible. In other cases, as for instance, in the multiple
+star $\theta$~Orionis, we have a number of stars not organized in pairs, but at more
+or less equal distances from each other: we are confronted by the problem
+of $n$ bodies in its most general and unmanageable form. Nature challenges
+the mathematicians.
+\end{fineprint}
+
+\nbarticle{883.} \nbparatext{Clusters.}---There are in the sky numerous groups of stars
+containing from one hundred to many thousand members. Some of
+them are made up of stars visible separately to the naked eye, as the
+Pleiades; some of them require a small telescope to resolve them, as,
+for instance, the Præsepe in Cancer, and the group of stars in the
+sword-handle of Perseus; while others yet, even in telescopes of
+some size, look simply like wisps or balls of shining cloud, and break
+up into stars only in the most powerful instruments.
+
+\begin{fineprint}
+In a large instrument some of the telescopic clusters are magnificent objects,
+composed of thousands of stellar sparks compressed into a ball which
+is dazzlingly bright at the centre and thinning out towards the edge. In
+some of them vividly colored stars add to the beauty of the group. In the
+northern hemisphere the finest cluster is that known as 13~M Hercules
+($\alpha$, $16^\text{h}\: 37^\text{m}$ and $\delta$,  $36°\: 40'$) not very far from the ``apex of the sun's way.''
+\end{fineprint}
+
+\includegraphicsmid{illo229}{\textsc{Fig.~229.}---The Pleiades.}
+
+\nbarticle{884.} \nbparatext{The Pleiades.}---Of the naked-eye clusters the Pleiades is
+the most interesting and important. To an ordinary eye six stars are
+easily visible in it, the six largest ones indicated in the figure (\figref{illo229}{Fig.~229}).
+Eyes a little better see easily five more---these next in size
+in the \figref{illo229}{figure} (the two stars of Asterope being seen as one). A
+very small telescope (a mere opera-glass) increases the number to
+nearly a hundred; and with large instruments more than 400 are
+catalogued in the group. A few of the stars, apparently in the cluster,
+are really only accidentally on the same line of vision, and are distinguished
+by proper motions different from those of the rest of the
+group; but the great majority have proper motions nearly the same
+%% -----File: 513.png---Folio 502-------
+in amount and direction; they have also identical spectra, and therefore
+undoubtedly constitute a single system.
+
+\begin{fineprint}
+The distances and positions of the principal stars with respect to the
+central star Alcyone have been carefully measured three or four times
+during the last fifty years. The relative motions during the period have
+not proved large enough to admit of satisfactory determination, but it is
+clear that such motions exist. A curious and interesting fact is the presence
+of \textit{nebulous matter} in considerable quantity. A portion of this nebulosity
+hanging around Merope (the northeast star of the dipper-bowl in the \figref{illo229}{figure})
+was discovered many years ago; but it was reserved to photography to
+detect very recently other wisps of nebulosity attached to other stars, especially
+to Maia, and to show that the whole space is covered with streaks and
+streamers of it, emitting light of such a character as to impress the photographic
+%% -----File: 514.png---Folio 503-------
+plate much more strongly than the eye. The eye cannot see the
+nebula with the same telescope which is able to photograph it
+strongly. The \figref{illo229}{figure} shows roughly the outlines of some of the
+principal nebulous filaments.
+
+\nbarticle{885.} \nbparatext{Distance of Star-Clusters and Size of the Component Stars.}---The
+question at once arises whether clusters, such as the one mentioned
+in Hercules, are composed of stars each comparable with our sun, and
+separated by distances corresponding to the distance between the sun
+and its neighboring stars, or whether the bodies which compose the
+swarm are really very small,---mere sparks of stellar matter:
+whether the distance of the mass from us is about the same as that
+of the stars among which it seems to be set, or whether it is far
+beyond them. Forty years ago the accepted view was that the stars
+composing the clusters are no smaller than ordinary stars, and that
+the distance of the star-clusters is immensely greater than that of
+the isolated stars. There are many eloquent passages in the writings
+of that period based upon the belief that these star-clusters are
+\textit{stellar universes},---``galaxies,'' like the group of stars
+to which the writers supposed the sun to belong, but so
+inconceivably remote that in appearance they shrank to these mere
+balls of shining dust.
+
+It is now, however, quite certain that the other view is
+correct,---that star-clusters are among \textit{our} stars and form
+part of \textit{our} universe. Large and small stars are so
+associated in the same group in many cases, as to leave us no choice
+of belief in the matter. It is true that as yet no parallax has been
+detected in any star-cluster; but that is not strange, since a
+cluster is not a convenient object for observations of the kind
+necessary to the detection of parallax.
+
+\nbarticle{886.} \nbparatext{The Nebulæ.}---There are also in the sky a multitude of
+faintly shining bodies,---shreds and balls of cloudy stuff that are
+known as ``\textit{nebulæ}'' (the word meaning strictly a ``little
+cloud''). Some 10,000 or 11,000 of these objects are already
+catalogued.
+
+Two or three of them are visible to the naked eye. The nebula in the
+girdle of Andromeda is the brightest of them, in which, it will be
+remembered, the temporary star of 1885 appeared.
+
+The next brightest is the wonderful nebula of Orion, which, in the
+beauty and variety of its details, in the interesting relations of
+the included stars, the delicate tint of the filmy light, and in its
+spectroscopic interest, far exceeds the other,---indeed, all others.
+%% -----File: 515.png---Folio 504-------
+
+It is so difficult to represent these delicate objects by any
+process available to text-books, that we limit ourselves to cuts of
+two only, the great nebula of Andromeda and the ``ring-nebula'' in
+Lyra (both from original drawings), referring the reader to the
+elaborate engravings that can be found in the ``Philosophical
+Transactions,'' the ``Memoirs of the Royal Astronomical Society,''
+and other similar works, for adequate representations of other
+interesting objects of this kind.
+\end{fineprint}
+
+With a small telescope a nebula cannot be distinguished from a close
+star-cluster, and it is quite likely that the clusters and nebulæ
+shade into each other by insensible gradations. Forty years ago it
+was supposed that there was no distinction between them except that
+of mere remoteness,---that all nebulæ could be resolved into stars
+by sufficient increase of telescopic power. When Lord Rosse's great
+telescope was first erected, it was for a time reported (and the
+statement is still often met with) that it had ``resolved'' the
+Orion nebula. This was a mistake however. No telescope ever has
+resolved that nebula into stars nor ever will, for we now
+\textit{know} that it is not composed of stars.
+
+\nbarticle{887.} \nbparatext{Forms and Magnitudes of Nebulæ.}---The larger
+and brighter nebulæ are, many of them, very irregular in form,
+stretching out sprays and streamers in all directions, and
+containing dark openings or ``lanes.'' The so-called ``fish-mouth''
+in the nebula of Orion, and the dark streaks in the nebula of
+Andromeda, are striking examples. Some of these bodies are of
+enormous volume. The nebula of Orion, with its outlying streamers,
+extends over several square degrees, and the nebula of Andromeda
+covers more than one. Now, as seen from even the nearest star, the
+apparent distance of Neptune from the sun is only $30''$, and the
+diameter of its orbit $1'$. It is perfectly certain that neither of
+these nebulæ is as near as $\alpha$~Centauri, and therefore the
+cross-section of the Orion nebula, as seen from the
+%% -----File: 516.png---Folio 505-------
+earth, must be \textit{at least} many thousand times the area of
+Neptune's orbit.
+
+\includegraphicsmid{illo230}{\textsc{Fig.~230.}---The Great Nebula in Andromeda.}
+
+We do not know what is the real shape of either of these nebulæ,
+whether it is a thin, flat sheet, or a voluminous bulk; but some
+things about these two nebulæ and several others favor strongly the
+idea that their thickness does not correspond to their apparent
+area.
+
+\includegraphicsouter{illo231}{\textsc{Fig.~231.}---The Annular Nebula in Lyra.}
+
+\nbarticle{888.} \nbparatext{The Smaller Nebulæ.}---The smaller nebulæ are for
+the most part elliptical in outline, some nearly circular, others
+more elongated, and some narrow, slender streaks of light. Generally
+they are brighter at the centre, and in many cases the centre is
+occupied by a star. Indeed, there is a considerable number of
+so-called ``\textit{nebulous stars},'' that is, stars with a hazy
+envelope
+around them.
+
+There are some nebulæ which present nearly a uniform disc of light,
+and are known as ``\textit{planetary}'' nebulæ, and there are some
+which are dark in the centre and are known as ``\textit{annular}''
+or ring nebulæ. The finest of these annular nebulæ is the one in the
+constellation of Lyra, about half-way between the stars $\beta$ and
+$\gamma$; it is shown in \figref{illo231}{Fig.~231}.
+
+There are also a number of double nebulæ, and some of these exhibit
+a remarkable spiral structure when examined by telescopes of the
+largest aperture like the great reflector of Lord Rosse. The
+so-called ``\textit{whirlpool}'' nebula in the constellation ``Canes
+Venatici'' is the most striking specimen. This spiral structure,
+however, is to be made out \textit{only} in large telescopes; in
+fact, very little of the real beauty of most of these objects is
+accessible to instruments of less than 12 inches
+aperture.
+
+\sloppy
+\nbarticle{889.} \nbparatext{Variable Nebulæ.}---There are several nebulæ which
+\textit{vary in their brightness} from time to time; one especially,
+near $\epsilon$~Tauri, at times has been visible with a small
+telescope, while at other times it
+%% -----File: 517.png---Folio 506-------
+is entirely invisible even with large ones. So far no regular
+periodicity has been ascertained in such cases.
+
+\fussy
+\nbarticle{890.} \nbparatext{Their Spectra.}---One of the earliest and most remarkable
+achievements of the spectroscope was its demonstration of the fact
+that the light of many of the nebulæ proceeds mainly from
+\textit{luminous gas}. They give a spectrum of six or seven bright
+lines\footnote{ The wave-lengths of the lines are the following, in
+the order of brightness: (1) 500.4 ? (2) 495.8 ? (3) 486.1 Hydrogen
+(\textit{F}); (4) 434.0 Hydrogen$_\gamma$; (5) 410.1 Hydrogen
+(\textit{h}); (6) 587.4 Helium (?); (7) 448?} (\figref{illo232}{Fig.~232}), three of
+which are fairly conspicuous; while the rest are very faint.
+
+\includegraphicsmid{illo232}{\textsc{Fig.~232.}---Spectrum of the Gaseous Nebulæ.}
+
+\begin{fineprint}
+Three of the lines, $F$, $H_{\gamma}$, and $h$, are due to
+\textit{Hydrogen}. One, the faintest of all, recently discovered by
+Copeland, seems to be identical with the $D_3$ line of the solar
+chromosphere; but the origin of the rest remains unknown. The
+brightest line of the whole number is in the green, $\lambda$,
+500.4, and was for a while referred to nitrogen; but under closer
+examination the identification breaks down, though the student will
+find it still called ``the nitrogen line'' in many astronomical
+works.
+
+Mr. Lockyer identifies it with a ``fluting'' in the low-temperature
+spectrum of magnesium, which he has found in the spectrum of
+meteorites; but this seems rather doubtful, as the nebula line is
+fine and sharp, and does not look at all like the relic of a
+\textit{band}; nor has the coincidence been satisfactorily
+established by actually confronting the nebula spectrum with that of
+magnesium.
+
+All the nebulæ which give a gaseous spectrum at all, present this
+same spectrum entire or in part. If the nebula is faint only the
+brightest line appears, while the $H_{\gamma}$ line and the other
+fainter lines are seen only in the brightest nebulæ and under
+favorable circumstances.
+\end{fineprint}
+
+\nbarticle{891.} But not all the nebulæ by any means give a gaseous spectrum:
+those which do so---about half the whole number---are of a more or
+less distinct greenish tint, which is at once recognizable in the
+telescope. The \textit{white} nebulæ, the nebula of Andromeda at
+their head, give only a continuous and perfectly expressionless
+spectrum,
+%% -----File: 518.png---Folio 507-------
+unmarked by any lines or bands, either bright or dark. This must
+not be interpreted as showing that these nebulæ cannot be gaseous;
+for a gas \textit{under pressure} gives just such a spectrum; but so also do
+masses of solid or liquid when heated to incandescence. The spectroscope
+simply declines to testify in this case. The \textit{telescopic} evidence
+as to the nature of the white nebulæ is the same as for the green.
+They withstand all attempts at resolution, none more firmly than the
+Andromeda nebula itself, the brightest of them all.
+
+
+\nbarticle{892.} \nbparatext{Changes in the Nebulæ.}---The question has been raised
+whether some of the nebulæ have not sensibly changed, even within
+the few years since it has become possible to observe them in detail.
+It is quite certain that in important respects the early \textit{drawings} differ
+seriously from these of recent observers; but the appearance of a
+nebula depends so much upon the telescope and the circumstances
+under which it is used, the features are so delicate and indefinite, and
+the difficulty of representing them on paper is such, that very little
+reliance can be placed on discrepancies between drawings, unless supported
+by the evidence of \textit{measures} of some kind.
+
+\begin{fineprint}
+Thus far, the best authenticated instance of such a change, according
+to Professor Holden, is in the so-called ``trifid'' nebula, in Sagittarius.
+In this object there is a peculiar three-legged area of darkness which
+divides the nebula into three lobes. A bright triple star, which in the
+early part of the century was described and figured by Herschel and
+other observers as in the \textit{middle} of one of these dark lanes, is now certainly
+in the edge of the nebula itself. The star does not seem to have moved with
+reference to the neighboring stars, and it seems therefore necessary to suppose
+that the nebula itself has drifted and changed its form.
+
+As to the nebula of Orion, Professor Holden's conclusion is, that while
+the \textit{outlines} of the different features have probably undergone but little
+change, their \textit{relative brightness} and \textit{prominence} have been continually fluctuating.
+This, however, can hardly be considered certain; to settle the
+question will probably require another fifty years or so, and the comparison,
+not of drawings but of \textit{photographs}---for it is now possible to photograph
+the brighter nebulæ.
+\end{fineprint}
+
+\nbarticle{893.} \nbparatext{Photographs of Nebulæ.}---The first success in photographing
+an object of this sort was obtained in 1880 by Dr.~Henry Draper
+in experiments upon the nebula of Orion; in 1881 and 1882 he greatly
+improved upon his first essays; and not long before his death in 1882
+he obtained a really fine photograph of the beautiful object. Mr.~Common,
+of England, with his three-foot silver-on-glass reflector, and
+%% -----File: 519.png---Folio 508-------
+exposures ranging from half an hour upward, has since then obtained
+pictures of the same object, still finer, and that seem to leave little to
+be desired. They fail of perfection only because the stars which
+jewel the nebula so brilliantly as seen by the eye, become in a photograph
+mere blotches, encroaching upon and cutting out large patches
+of the most interesting portions of the nebula.
+
+\begin{fineprint}
+A number of nebulæ have also been photographed by others as well
+as by Common; but he still maintains the lead, and it is hoped that
+when his new five-foot reflector is finished, he will procure for us an
+authentic record of the present appearance of all the most important
+objects of this class.
+\end{fineprint}
+
+\nbarticle{894.} \nbparatext{Nature of the Nebulæ.}---As to the constitution of these
+clouds we can only speculate. In the green nebulæ we can say with
+confidence that hydrogen and some other gas or gases are certainly
+present, and that the gases emit most of the light that reaches us
+from such objects. But how much solid or liquid matter in the form
+of grains and drops may be included within the gaseous cloud we
+have no means of knowing.
+
+\begin{fineprint}
+The idea of Mr.~Lockyer (a part of his wide induction as to what we
+may call the ``meteoritic constitution of the universe'') is that they are
+clouds of ``sparse meteorites, the collisions of which bring about a rise
+of temperature sufficient to render luminous one of their chief constituents,---magnesium.''
+
+How far this theory will stand the test of time and future investigations
+remains to be seen. At first view it seems very doubtful whether the \textit{collisions}
+in such a body could be frequent or violent enough to account for
+its luminosity, and one is tempted to look to other causes for the source
+of light.
+\end{fineprint}
+
+\nbarticle{895.} \nbparatext{Number and Distribution of Nebulæ.}---Sir William Herschel
+was the first extensive investigator of these interesting objects,
+and left his unfinished work as a legacy to his son, Sir John Herschel,
+who completed the survey of the heavens by a residence of
+several years at the Cape of Good Hope. His ``General Catalogue''
+is the standard of reference for objects of this kind, and contains
+about 5000 of them. Between 1000 and 2000 more have been
+added since its publication, most of them extremely faint. It is
+hardly possible that any important ones remain to be found.
+
+As to their distribution, it is a curious and important fact that it
+is \textit{in contrast} to the distribution of the stars. The stars, as we shall
+%% -----File: 520.png---Folio 509-------
+soon see, gather especially in and about the Milky Way, as do also
+the star-clusters; but the nebulæ specially crowd together in regions
+as far from the Milky Way as it is possible to get. As has been
+pointed out by more than one, this shows, however, not a want of
+relation between the stars and the nebulæ, but some ``relation of
+contrariety.'' Precisely what this is, and why the nebulæ avoid the
+regions thickly starred, is not yet clear. Possibly the stars \textit{devour
+them}, that is, gather in and appropriate surrounding nebulosity so
+that it disappears from their neighborhood.
+
+\nbenlargepage
+\nbarticle{896.} \nbparatext{Distance of the Nebulæ.}---On this point we have very little
+absolute knowledge. Attempts have been made to measure the parallax
+of one or two, but so far unsuccessfully. Still it is probable,
+indeed almost certain, that they are at the same \textit{order of distance} as
+the stars. The wisps of nebulosity which photography shows attached
+to the stars in the Pleiades (and a number of similar cases
+appear elsewhere), the nebulous stars of Herschel, and numerous
+nebulæ which have a star exactly in the centre,---these compel us to
+believe that in such cases the nebulosity is really \textit{at the star}. Then
+in the southern hemisphere there are two remarkable luminous clouds
+which look like detached portions of the Milky Way (though they
+are not near it), and are known as the Nubeculæ (major and minor).
+These are made up of stars and star-clusters, and of nebulæ also,
+all swarming together, and so associated that it is not possible to
+question their real proximity to each other.
+
+\begin{fineprint}
+\nbarticle{897.} Fifty years ago a very different view prevailed. As has been said
+already, astronomers at that time very generally believed that there was no
+distinction between nebulæ and star-clusters except in regard to distance,
+the nebulæ being only clusters too remote to show the separate stars. They
+considered a nebula, therefore, as a ``universe of stars,'' like our own ``galactic
+cluster'' to which the sun belongs, but as far beyond the ``star-clusters''
+as these were believed to be beyond the isolated stars. In some respects this
+old belief strikes one as grander than the truth even. It made our vision
+penetrate more deeply into space than we now dare think it can.
+\end{fineprint}
+
+\nbarticle{898.} \nbparatext{The Galaxy, or Milky Way.}---This is a luminous belt which
+surrounds the heavens nearly in a great circle. It varies much in
+width and brightness, and for about a third of its extent, from Cygnus
+to Scorpio, is divided into two nearly parallel streams. In
+several constellations, as in Cygnus, Sagittarius, and Argo Navis,
+it is crossed by dark straight-edged bars that look as if some
+%% -----File: 521.png---Folio 510-------
+light cloud lay athwart it, and in the constellation of Centaurus there
+is a dark pear-shaped orifice,---the ``coal sack,'' as it is called.
+
+The galaxy intersects the ecliptic at two opposite points near the
+solstices, making with it an angle of about $60°$. The northern
+``galactic pole,'' as it is called, lies, according to Sir John Herschel,
+in declination $+37°$, and right ascension $12^\text{h}\: 47^\text{m}$; the southern
+``galactic pole'' is of course at the opposite point in the southern
+hemisphere. As Herschel remarks, the ``galactic plane'' ``is to
+sidereal what the ecliptic is to planetary astronomy, a plane of
+ultimate reference, the ground plan of the sidereal system.''
+
+The Milky Way is made up almost wholly of small stars from the
+eighth magnitude down. It contains also a large number of star-clusters,
+but (as has been already mentioned) very few true nebulæ.
+In some places the stars are too thickly packed for counting, especially
+in the bright knots which abound here and there.
+
+\begin{fineprint}
+(An excellent detailed description of its appearance and course may be
+found in Herschel's ``Outlines of Astronomy.'')
+\end{fineprint}
+
+\nbarticle{899.} \nbparatext{Distribution of Stars in the Sky: Star-Gauges.}---It is
+obvious that the stars are not uniformly scattered over the heavens.
+They show a decided tendency to collect in groups here and there,
+and to form connected streams; but besides this, an enumeration of
+the stars in the great star-catalogues shows that the number increases
+with considerable regularity from the galactic poles, where they are
+most sparse, towards the galactic circle, where they are most crowded.
+The ``star-gauges'' of the Herschels make this fact still more
+obvious.
+
+\begin{fineprint}
+These gauges consisted merely in the counting of the number of stars
+visible in the field of view ($15'$ in diameter) of the twenty-foot reflector.
+Sir William Herschel made 3400 of these gauges, directing the telescope to
+different parts of the sky; and his son followed up the work at the Cape of
+Good Hope. Struve's discussion of these gauges in their relation to the
+galactic circle gives the following result:---
+\begin{center}
+\begin{tabular}{r@{}p{0.5\textwidth}@{}l}
+  \scriptsize Distance from \rlap{Galaxy.}& & \scriptsize\llap{Number} of Stars in Field.\\
+  $90°$ & \dotfill &  \phantom{00}4.15  \\
+  $75°$ & \dotfill &  \phantom{00}4.68  \\
+  $60°$ & \dotfill &  \phantom{00}6.52  \\
+  $45°$ & \dotfill &  \phantom{0}10.36  \\
+  $30°$ & \dotfill &  \phantom{0}17.68  \\
+  $15°$ & \dotfill &  \phantom{0}30.30  \\
+   $0°$ & \dotfill &            122.00  \\
+\end{tabular}
+\end{center}
+\end{fineprint}
+%% -----File: 522.png---Folio 511-------
+
+\nbarticle{900.} \nbparatext{Structure of the Heavens.}---Our space does not permit a
+discussion of the untenable conclusions reached by Herschel and
+others by combining the unquestionable data derived from observation,
+with the unfounded and untrue assumptions that the stars are
+substantially of a size and spaced at approximately equal distances.
+Many of these conclusions relating to the form and dimensions of
+the Milky Way, and of the stellar universe to which our sun belongs,
+have become almost classical; but they are none the less incorrect.
+
+It is certain, however, that the faint stars as a class are smaller
+and darker and more remote than are the bright ones \textit{as a class:} and
+accepting this, we can safely draw from the star-gauges a few general
+conclusions, as follows:---
+
+\begin{fineprint}
+We present them substantially as given by Newcomb in his ``Popular
+Astronomy,'' p.~491.
+\end{fineprint}
+
+1. ``The great mass of the stars which compose this (stellar)
+system are spread out on all sides in or near a widely extended
+plane, passing through the Milky Way. In other words, the large
+majority of the stars which we can see with the telescope are contained
+in a space having the form of a round, flat disc, the diameter
+of which is eight or ten times its thickness.\DPtypo{}{''}
+
+2. ``Within this space the stars are not scattered uniformly, but
+are for the most part collected into irregular clusters or masses, with
+comparatively vacant spaces between them.'' They are ``gregarious,''
+to use Miss Clerke's expression.
+
+3. Our sun is near the centre of this disc-like space.
+
+4. The \textit{naked-eye stars} ``are scattered in this space with a near
+approach to uniformity,'' the exceptions being a few star-clusters and
+star-groups like the Pleiades and Coma Berenices.
+
+5. ``The disc described above does not represent the form of the
+stellar system, but only the limits within which it is mostly contained.''
+The circumstances are such as to ``prevent our assigning
+any more definite form to the system than we could assign to a cloud
+of dust.''
+
+6. ``On each side of the galactic region the stars are more evenly
+and thinly scattered, but probably do not extend out to a distance at
+all approaching the extent of the galactic region,'' or if they do they
+are very few in number; but it is impossible to set any definite
+boundaries.
+
+7. On each side of the galactic and stellar region we have a nebular
+region, comparatively starless, but occupied by great numbers of
+nebulæ.
+%% -----File: 523.png---Folio 512-------
+
+As to the Milky Way itself, it is not yet certain whether the stars
+which compose it are distributed pretty equally near the galactic
+circle, or whether they form something like a ring with a comparatively
+vacant space in the middle. The ring theory seems at present
+rather the more probable of the two.
+
+As to the distance of the remotest stars in the stellar system, it is
+impossible to say anything very definite, but it seems quite certain
+that it must be at least so great that light would occupy from 10,000
+to 20,000 years in traversing it. If one asks what is beyond the
+stellar system, whether the star-filled space extends indefinitely or
+not, no certain answer can be given.
+
+\nbarticle{901.} \nbparatext{Do the Stars Form a System?}---That is, do they form an
+\textit{organized unit}, in which, as in the solar system, each of the different
+members has its own function and permanently maintains its relation
+to the rest? Gravitation certainly operates, as the binary stars
+demonstrate,\footnote
+  {This is perhaps rather too strong an expression. It would be truer to say
+  that none of the observed phenomena of the binary stars \textit{contravene} the universality
+  of gravitation, and for the most part they are just what gravitation easily
+  accounts for. But they do not \textit{demonstrate} that the central force varies inversely
+  with the square of the distance, because we do not know the angle made by the
+  orbit-plane with the line of sight, except from calculations based on the assumption
+  that the central force really varies in that way. The orbits, as directly
+  observed, are consistent with several other laws of central force than the law
+  of inverse squares. (See ``Astron.\ Journal,'' Vol.~VIII., article by Prof.\ A.~Hall.)}
+and the stars are moving swiftly in various directions
+with enormous velocities, as shown by their proper motions, and by
+the spectroscope. The question is whether these motions are controlled
+by gravitation, and whether they carry the stars in \textit{orbits} that
+can be known and predicted.
+
+That the stars are organized into a system or systems \textit{of some sort}
+can hardly be doubted, for this seems to be a necessary consequence
+of their mutual attraction. But that the system is one at all after
+the pattern of the solar system, in which the different members move
+in \textit{closed orbits},---orbits that are permanent except for the slow
+changes produced by perturbation,---this is almost certainly impossible,
+as was said a few pages back.
+
+\begin{fineprint}
+\nbarticle{902.} \nbparatext{Is there a Revolution of the Whole Mass of Stars?}---A
+favorite idea has been that the mass of stars which constitutes our system
+%% -----File: 524.png---Folio 513-------
+has a slow rotation like that of a body on its axis, the plane of this general
+revolution coinciding with the plane of the galaxy. Such a general motion
+is not in any way inconsistent with the independent motions of the individual
+stars, and there is perhaps a slight inherent probability in favor of
+such a movement; but thus far we have no evidence that it really exists---indeed,
+there hardly \textit{could} be any such evidence at present, because exact
+Astronomy is not yet old enough to have gathered the necessary data.
+
+\sloppy
+\nbarticle{903.} \nbparatext{Central Suns.}---A number of speculative astronomers, Mädler
+perhaps most prominently, have held the belief that there is a ``\textit{central sun},''
+standing in some such relation to the stellar system as our sun does to the
+solar system. It is hardly necessary to say that the notion has not the
+slightest foundation, or even probability.
+
+\fussy
+Lambert supposed \textit{many} such suns as the centres of subordinate stellar
+systems, and because we cannot see them, he imagined them to be dark.
+
+If we conceive of boundaries drawn around our stellar system, and count
+all the stars within the limits as members of it, leaving out of the account all
+that fall outside, then, of course, our system so limited has at any moment
+a perfectly definite \textit{centre of gravity}. There is no reason why some particular
+star may not be very near that centre, and in that sense a ``central sun'' is
+possible; but its central position would not give it any preëminence or rule
+over its neighbors, or put it in any such relation to the rest of the stars as
+the sun bears to the planets.
+\end{fineprint}
+
+\nbarticle{904.} \nbparatext{Orbits of Sun and Stars.}---It is practically certain that the
+motions of the stars are not \textit{orbital} in any strict sense. Excepting
+stars which are in clusters, all other stars are simultaneously acted
+upon by many forces drawing in various and opposite directions;
+and these forces must in most cases be so nearly balanced that
+the resultant cannot be very large. The motions of the stars must
+consequently, as a rule, be \textit{nearly} rectilinear.
+
+Still the balancing of the forces will seldom be exact, and accordingly
+the path of a star will almost always be \textit{slightly} curved; and
+since the amount and direction of the resultant force which acts on
+the star is continually changing, the curvature of its motion will
+alter correspondingly, and the result will be a path which does not
+lie in any one plane, but is bent about in all ways like a piece of
+crooked wire. It is hardly likely, however, that the curvature of a
+star's path would, in any ordinary case, be such as could be detected
+by the observations of a single century, or even of a thousand
+years.
+
+As has been said before, in connection with the proper motions of
+the stars, the probability is that the separate stars move nearly independently,
+%% -----File: 525.png---Folio 514-------
+``like bees in a swarm.'' In the solar system the central
+power is supreme, and perturbations or deviations from the path
+which the central power prescribes are small and transient. In the
+stellar system, on the other hand, the central force, if it exists at all
+(as an attraction towards the centre of gravity of the whole mass of
+stars) is trifling. Perturbation prevails over regularity, and ``\textit{individualism}''
+is the method of the greater system of the stars, as solar
+despotism is that of the smaller system of the planets.
+
+\nbarticle{905.} \nbparatext{Cosmogony.}---Unquestionably one of the most interesting,
+and also most baffling, topics of speculation is the problem of the
+way in which the present condition of the universe came about. By
+what processes have moons and earths and Jupiters and Saturns,
+come to their present state and into their relation to the sun? What
+has been their past history, and what has the future in store for
+them? How has the sun come to his present glory and dominion?
+and in the stellar universe, what is the meaning and mutual relation
+of the various orders of bodies we see,---of the nebulæ, the star-clusters,
+and the stars themselves?
+
+In a forest, to use a comparison long ago employed by the elder
+Herschel, we see around us trees in all stages of their life history.
+There are the seedlings just sprouting from the acorn, the slender
+saplings, the sturdy oaks in their full vigor, those also that are old
+and near decay, and the prostrate trunks of the dead. Can we
+apply the analogy to the heavens, and if we can, which of the objects
+before us are to be regarded as in their infancy, and which of them
+as old and near dissolution?
+
+\nbarticle{906.} \nbparatext{Fundamental Principles of a Rational Cosmogony.}---In
+the present state of science many of the questions thus suggested
+seem to be hopelessly beyond the reach of investigation, while
+others appear like problems which time and patient work will
+solve, and others yet have already received clear and decided
+answers. In a general way it may be said that the \textit{condensation
+and aggregation of rarefied masses of matter under the force of
+gravitation; the conversion into heat of the $($potential$)$ ``energy of
+position'' destroyed by the process of condensation; the effect of this
+heat upon the contracting mass itself, and the radiation of energy into
+space and to surrounding bodies as waves of light and heat,}---these
+principles contain nearly all the explanations that can thus far be
+given of the present state of the heavenly bodies.
+%% -----File: 526.png---Folio 515-------
+
+\nbarticle{907.} \nbparatext{The Planetary System.}---We see that our planetary system
+is not a mere accidental aggregation of bodies. Masses of matter
+coming hap-hazard towards the sun would move, as comets do, in
+orbits, always conic sections to be sure, but of every degree of eccentricity
+and inclination. There are a multitude of relations actually
+observed in the planetary system which are wholly independent of
+gravitation and demand an explanation.
+
+1. The orbits are all \textit{nearly circular}.
+
+2. They are all nearly \textit{in one plane} (excepting the cases of some
+of the little asteroids).
+
+3. The revolution of all is \textit{in the same direction.}
+
+4. There is a curiously \textit{regular progression of distances} (expressed
+by Bode's law, which, however, breaks down at Neptune).
+
+5. There is a roughly regular \textit{progression of density}, increasing
+both ways from Saturn, the least dense of all the planets in the
+system.
+
+As regards the planets themselves, we have
+
+6. The \textit{plane} of the planets' rotation \textit{nearly coinciding with that of
+the orbit} (probably excepting Uranus).
+
+7. The \textit{direction} of the rotation \textit{the same as that of the orbital
+revolution} (excepting probably Uranus and Neptune).
+
+8. The \textit{plane} of \textit{orbital revolution of the satellites} coinciding nearly
+with that of the planet's rotation.
+
+9. The \textit{direction} of the satellites' revolution also coinciding \textit{with
+that of the planet's rotation}.
+
+10. The largest planets rotate most swiftly.
+
+\begin{fineprint}
+\nbarticle{908.} \nbparatext{Origin of the Nebular Hypothesis.}---Now this is evidently a
+good arrangement for a planetary system, and therefore some have inferred
+that the Deity \textit{made} it so, perfect from the first. But to one who considers
+the way in which other perfect works of nature usually come to their perfection---their
+processes of growth and development---this explanation seems
+improbable. It appears far more likely that the planetary system \textit{grew} than
+that it was \textit{built} outright.
+
+Three different philosophers in the last, century, Swedenborg, Kant, and
+La~Place (only one of them an astronomer), independently proposed essentially
+the same hypothesis to account for the system as we now know it.
+La~Place's theory, as might have been expected from his mathematical and
+scientific attainments, was the most carefully and reasonably worked out in
+detail. It was formulated before the discovery of the great principle of the
+``conservation of energy,'' and before the mechanical equivalence of heat
+with other forms of energy was known, so that in some respects it is defective,
+%% -----File: 527.png---Folio 516-------
+and even certainly wrong. In its main idea, however, that the solar
+system once existed as a nebulous mass and has reached its present state as
+the result of a series of purely physical processes, it seems certain to prove
+correct, and it forms the foundation of all the current speculations upon the
+subject.
+\end{fineprint}
+
+\nbarticle{909.} \nbparatext{La Place's Theory.}---(\textit{a}) He supposed that at some past time,
+which may be taken as the starting-point of our system's history
+(though it is not to be considered as \textit{the beginning of the existence of
+the substance} of which our system is composed), the matter now collected
+in the sun and planets was in the form of a \textit{nebula}.
+
+(\textit{b}) This nebula was a \textit{cloud of intensely heated gas}, perhaps hotter,
+as he supposed, than the sun is now.
+
+(\textit{c}) This nebula under the action of its own gravitation, assumed
+an approximately globular form with a rotation around an axis. As
+to this movement of rotation, it appears to be necessary to account
+for it by supposing that the different portions of the nebula, before
+the time which has been taken as the starting-point, had motions of
+their own. Then, unless these motions happened to be balanced in the
+most perfect and improbable manner, a motion of rotation would set
+in of itself as the nebula contracted, just as water whirls in a basin
+when drawn off by an orifice in the bottom. The velocity of this rotation
+would become continually swifter as the volume of the nebula
+diminished, the so-called ``moment of momentum'' remaining necessarily
+unchanged.
+
+\nbarticle{910.} (\textit{d}) In consequence of this rotation, the mass, instead of
+remaining spherical, would become much flattened at the poles, and as
+the rotation went on and the motion became accelerated, the time
+would come when the centrifugal force at the equator of the nebula
+would become equal to gravity, and ``rings of nebulous matter''
+would be \textit{abandoned} (not thrown off), resembling the rings of Saturn,
+which, indeed, suggested this feature of the theory.
+
+(\textit{e}) A ring would revolve for a while as a whole, but in time would
+\textit{break}, and the material would \textit{collect into a single globe}. La~Place supposed
+that the ring would revolve as if it were solid, the outer edge,
+therefore, moving more swiftly than the inner. If this were so, the
+mass formed from the collection of the matter of the ruptured ring
+\textit{would necessarily rotate in the same direction} as the ring had revolved.
+
+(\textit{f}) The planet thus formed would continue to revolve around the
+central mass, and might itself in turn abandon rings which might
+break, and so furnish it with a retinue of satellites.
+%% -----File: 528.png---Folio 517-------
+
+\nbarticle{911.} It is obvious that this theory meets completely most of the
+conditions of the problem. It explains every one of the facts just
+mentioned as demanding explanation in the solar system. Indeed, it
+explains them almost \textit{too} well; for as the theory stands it meets a
+most serious difficulty in the \textit{exceptional} cases of the planetary system,
+such as the anomalous and retrograde revolutions of the satellites of
+Uranus and Neptune. Another difficulty lies in the swift revolution
+of Phobos (\artref{Art.}{589}), the inner satellite of Mars. According to the
+unmodified nebular hypothesis, no planet or satellite could have a
+time of revolution less than the time of rotation which the central
+body would have, if expanded until its radius becomes equal to the
+radius of the satellite's orbit; still less could it have a period shorter
+than the central body \textit{now} has.
+
+\sloppy
+\nbarticle{912.} \nbparatext{Necessary Modifications.}---The principal modifications which
+seem essential to the theory in the light of our present knowledge, are
+the following. (The small letters indicate the articles of the original
+theory to which reference is made.)
+
+\fussy
+(\textit{b}) It is not probable that the original nebula could have been \textit{at a
+temperature} even nearly as high as the present temperature of the
+sun. The process of condensation of a gaseous cloud from loss of
+heat by radiation, would cause the temperature to \textit{rise}, according to
+the remarkable and almost paradoxical law of Lane. (\artref{Art.}{357}),
+until the mass had begun to liquefy or solidify. And it appears probable
+that the original nebula, instead of being \textit{purely gaseous}, was
+rather a \textit{cloud of dust} than a ``\textit{fire-mist}''; \textit{i.e.}, that it was made up of
+finely divided particles of solid or liquid matter, each particle enveloped
+in a mantle of permanent gas. Such a nebula in condensing
+would \textit{rise} in temperature at first as if purely gaseous, so that its central
+mass after a time would reach the solar stage of temperature, the
+solid and liquid particles melting and vaporizing as the mass grew
+hotter. At a subsequent stage, when yet more of the original energy
+of the mass had been dissipated by radiation, the temperature of the
+bodies which were formed from and within the nebula would fall again.
+
+\begin{fineprint}
+And yet La Place \textit{may} have been right in ascribing a high temperature to
+the original nebula. If that were really the case, the only difference would
+be that the nebula would be longer in reaching the condition of a solar
+system; but it is not \textit{necessary}, as he supposed, to assume that the original
+temperature was high, and that the matter was originally in a purely gaseous
+condition, in order to account for the present existence of such a group of
+bodies as the incandescent sun and its cool attendant planets.
+\end{fineprint}
+%% -----File: 529.png---Folio 518-------
+
+\nbarticle{913.} (\textit{d}) As regards the manner in which the planetary bodies
+were probably liberated from the parent mass, it seem to be very
+doubtful whether the matter accumulated at the equator of the rotating
+mass would usually separate itself as a ring. If a plastic mass
+in swift rotation is not absolutely homogeneous and symmetrical, it
+is more likely to become distorted by a lump formed somewhere on
+its equator, which lump may be finally detached and circulate around
+its primary. The formation of a \textit{ring}, though possible, would seem
+likely to be only a rare occurrence.
+
+La Place seems to have believed also that the outer rings must
+necessarily have been abandoned first, and the others in regular succession,
+so that the \textit{outer} planets are much the older. It seems, however,
+quite possible, and even probable, that several of the planets
+may be of about the same age, more than one ring having been
+liberated at the same time; or several planets having been formed
+from different zones of the same ring.
+
+(\textit{e}) In the case where a ring was formed, it is practically certain
+that it could not have revolved as a solid sheet; \textit{i.e.}, with the same
+angular velocity for all the particles, and with the outer portions,
+therefore, moving more swiftly than the inner. If, for instance, the
+matter which now constitutes the earth were ever distributed to form
+a ring occupying anything like half the distance from Venus to Mars,
+it must have been of a tenuity comparable only to that of a comet.
+The separate particles of such a ring could have had very little control
+over each other, and must have moved substantially as independent
+bodies; the outer ones, like remoter planets, making their circuits
+in longer periods and moving \textit{more slowly} than those near the
+inner edge of the ring.
+
+\nbarticle{914.} \nbparatext{Explanation of the Anomalous Rotation of Uranus and Neptune.}---When
+the matter of such a ring concentrates into a single
+mass, the direction of the rotation of the resultant planet depends upon
+the manner in which the matter was originally distributed in the ring.
+If the ring be nearly of the same density throughout, the resulting
+planet (which would be formed at about the middle of the ring's
+width) must have a \textit{retrograde} rotation like Uranus and Neptune.
+But if the particles of the ring are more closely packed near its
+inner edge, so that the resultant planet would be formed much within
+the middle of its width, its axial rotation must be \textit{direct.} In the
+first case, illustrated in \figref{illo233}{Fig.~233} (\textit{a}), the particles near the \textit{inner edge
+of the ring} would control the rotation, having a greater moment of
+%% -----File: 530.png---Folio 519-------
+rotation with respect to $M$, where the planet is supposed to be formed,
+than those at the outer edge. The rotation, therefore, will be
+\textit{retrograde}, on account of their greater velocity.
+
+In the other case, \figref{illo233}{Fig.~233} (\textit{b}), where the inner edge of the ring
+is densest, and the planet is formed as at $N$, much nearer the inner
+than the outer edge of the ring, the aggregate moment of rotation
+with respect to $N$ is greater for the \textit{particles beyond $N$} (because of
+their greater distance from it) than that of the swifter moving particles
+within, and this determines a \textit{direct} rotation.
+
+\includegraphicsmid{illo233}{\textsc{Fig.~233.}---Rotation of Planets formed from Rings according to the Nebular Hypothesis.}
+
+The fact that the satellites of Uranus and Neptune revolve backwards
+is not, therefore, at all a bar to the acceptance of the nebular
+hypothesis, as sometimes represented. If a new planet should ever
+be discovered outside of Neptune it is altogether probable that its
+satellites would be found to retrograde.
+
+\begin{fineprint}
+This is not the only way in which the retrograde rotation of the outer
+planets may be accounted for. There are a number of other possible
+assumptions as to the constitution of the mass thrown off from the parent
+nebula, and the manner in which its particles ultimately coalesce to form a
+planet, which lead to a similar result.
+
+\nbarticle{915.} Faye has recently propounded a modification of the nebular hypothesis
+which makes the planets of the ``terrestrial group'' (Mercury, Venus, the
+%% -----File: 531.png---Folio 520-------
+Earth, and Mars) older than the outer ones. He supposes that the planets
+were formed by local condensations (not by the formation of rings) \textit{within}
+the revolving nebula. At first, before the nebula was much condensed at the
+centre, the inward attraction would be at any point \textit{directly proportional to
+the distance of that point from the centre of gravity} of the nebula; \textit{i.e.}, the
+force could be expressed by the equation $F = ar$. After the condensation
+has gone so far that practically almost the whole of the matter is collected
+at the centre of the nebula, the force is \textit{inversely proportional to the square of
+the distance},---the ordinary law of gravitation,
+\begin{flalign*}
+&\textit{i.e., } && F = \frac{b}{r^2}.  &&\phantom{i.e. }
+\end{flalign*}
+At any intermediate time, during the gradual condensation of the nebula,
+the intensity of the central force will, therefore, be given by an expression
+having the form
+\[
+F = ar + \frac{b}{r^2},
+\]
+$r$ being the distance of the body acted upon from the centre of gravity of
+the nebula, while $a$ and $b$ are coefficients which depend upon its age; $a$ continually
+decreasing as the nebula grows older, while $b$ increases. The planets
+formed within the nebula when it was young, \textit{i.e.}, when $a$ was large and $b$
+was small, would have \textit{direct} rotation upon their axes, while those formed
+after $a$ had sensibly vanished would have a retrograde rotation; and this he
+supposes to be the case with Uranus and Neptune, which he considers
+\textit{younger} than the inner planets. Faye's work ``L'Origine du Monde,'' 1885,
+contains an excellent summary of the views and theories of the different
+astronomers who have speculated upon the cosmogony.
+\end{fineprint}
+
+\nbarticle{916.} \nbparatext{Tidal Evolution.}---Within a few years Prof.\ G.~H. Darwin
+(son of the great naturalist) has made some important investigations
+upon the effect of \textit{tidal reaction} between a central mass and a
+body revolving about it, both of them being supposed to be of such
+a nature (\textit{i.e.}, not absolutely \textit{rigid}), that tides can be raised upon
+them by their mutual attraction. We have already alluded to the
+subject in connection with the tides (\artref{Art.}{484}). He finds in this
+reaction an explanation of many puzzling facts. It appears, for
+instance, that if a planet and its satellite have ever had their times of
+rotation of the same length as the time of their orbital revolution
+around their common centre of gravity, then, starting from that
+time, either of two things might happen,---the satellite might begin
+to recede from the planet, or it might fall back to the central mass.
+The condition is one of unstable equilibrium, and the slightest cause
+might determine the subsequent course of things in either of the
+%% -----File: 532.png---Folio 521-------
+two opposite directions. Whenever the time of rotation of the
+planet is \textit{shorter} than the orbital period of the satellite (as it would
+naturally become by condensation continuing after the separation of the
+satellite), the tendency would be, as explained in \artref{Article}{484}, slightly
+to accelerate the satellite, and so to cause it continually to recede
+by an action the reverse of that produced by the hypothetical resisting
+medium which is supposed to disturb Encke's comet. This, it
+will be remembered, is thought to be the case with our moon.
+
+\nbarticle{917.} But if by any means the rotation of the planet were \textit{retarded},
+so that its day should become \textit{longer} than the period of the satellite,
+the tides produced by the satellite upon the planet will then retard
+the motion of the satellite like a resisting medium, and so will cause
+a continual shortening of its period, precisely as in the case of
+Encke's comet. If nothing intervenes, this action will in time bring
+down the satellite upon the planet's surface. Now in the case of
+Mars there is a known cause operating to retard its rotation (namely,
+the tides which are raised by the \textit{sun} upon the planet), and those
+who accept the theory of tidal evolution suggest that this was the
+cause which first made the length of the planet's day to exceed the
+period of the satellite, and so enabled the planet to establish upon
+the satellite that retardation which has shortened its little month,
+and must ultimately bring it down upon the planet.
+
+Processes such as these of tidal evolution must necessarily be
+extremely slow. How long are the periods involved, no one can yet
+estimate with any precision, but it is certain that the years are to be
+counted by the million.
+
+\begin{fineprint}
+(We have already referred the reader (\DPtypo{\artref{Art.}{484*}}{\artref{Art.}{484}}) to the last chapter of
+Ball's ``Story of the Heavens'' as containing an excellent and easily understood
+explanation of this subject.)
+\end{fineprint}
+
+\nbarticle{918.} \nbparatext{Conclusions derived from the Theory of Heat.}---As Professor
+Newcomb has said, ``Kant and La~Place seem to have arrived at
+the nebular hypothesis by reasoning \textit{forwards}. Modern science obtains
+a similar result by reasoning \textit{backwards} from actions which we
+now see going on before our eyes.''
+
+We have abundant evidence that \textit{the earth} was once at a much
+higher temperature than now. As we penetrate below the surface
+we find the temperature continually rising at a rate of about 1°\:F.
+for every fifty or sixty feet, thus indicating that at the depth of a
+few miles the temperature must be far above incandescence. Now,
+%% -----File: 533.png---Folio 522-------
+since the surface temperature is so much lower, this implies one (or
+\textit{both}) of two things,---either that heat-making processes are going
+on within the earth (which may be true to some extent), or else that
+the earth has been much hotter than it now is, and is cooling off,---and
+this seems to be a most probable supposition. It is just as
+reasonable, as Sir W.~Thomson puts it, to suppose that the earth
+has lately been intensely heated as to suppose that a warm stone that
+one picks up in the field has been lately somewhere in the fire.
+
+\nbarticle{919.} \nbparatext{Evidence derived from the Condition of the Moon and
+Planets.}---In the case of the \textit{moon} we find a body bearing upon its
+surface all the marks of past igneous action, but now in appearance
+intensely cold. The \textit{planets}, so far as we can judge from what we
+can see through the telescope, corroborate the same conclusion.
+Their testimony is not very strong, but it is at least true that nothing
+in the aspect of any of them militates against the view that they also
+are bodies cooling like the earth; and in the cases of Jupiter and
+Saturn many phenomena go to show that they are still (or at least
+\textit{now}) at a high temperature,---as might be expected of bodies of
+such an enormous mass, which, necessarily, other things being equal,
+would cool much more slowly than smaller globes like the earth.
+
+\begin{fineprint}
+The ratio of surface to mass is smaller as the diameter of a globe grows
+larger, and upon this ratio the rate of cooling of a body largely depends. In
+short, everything we can ascertain from the observation of the planets agrees
+completely with the idea that they have come to their present condition by
+\textit{cooling down from a molten or even gaseous state.}
+\end{fineprint}
+
+\nbarticle{920.} \nbparatext{The Sun's Testimony.}---In the sun we have a body steadily
+pouring forth an absolutely inconceivable amount of heat, without
+any visible source of supply. Thus far the only reasonable hypothesis
+to account for this, and for a multitude of other phenomena which
+it shows us, is the one which makes it a great cloud-mantled ball
+of incandescent gases, slowly shrinking under its own central gravity,
+converting continually a portion of its ``potential energy of position''\footnote
+  {By ``\textit{potential} energy of position'' is meant the energy due to the separated
+  condition of its particles from each other. As they fall together and towards
+  the centre in the shrinkage of the sun, they ``do work'' in precisely the same
+  way as any falling weight.}
+into the \textit{kinetic-energy} of heat, which at present is mainly radiated off
+into space.
+%% -----File: 534.png---Folio 523-------
+
+\begin{fineprint}
+We say \textit{mainly}, because it is not impossible that the sun's temperature is
+even yet slowly rising, and that the maximum has not yet been reached.
+We are not sure whether \textit{all} the heat produced by the sun's annual shrinkage
+is radiated into space, or whether a portion is retained within its mass, thus
+raising its temperature; or whether, again, it radiates \textit{more} than the amount
+thus generated, so that its temperature is slowly diminishing.
+
+\nbarticle{921.}---That the sun is really shrinking is admittedly only an inference,
+for the shrinkage must be far too slow for direct observation. Our case is
+like that of a man who, to use one of Professor Newcomb's illustrations,
+when he comes into a room and finds a clock in motion, concludes that the
+clock-weight is descending, even though its motion is too slow to be observed.
+Knowing the construction of the clock and the arrangement of its gearing,
+and the number of teeth in each of its different wheels, he states confidently
+just how many thousandths of an inch the weight sinks at each vibration of
+the pendulum; and looking into the clock-case and measuring the length of
+the space in which the weight can move, and noting its present place, he
+proceeds to state how long ago the clock was wound up, and how long it has
+yet to run. We must not push the analogy too far, but it is in some such
+way that we conclude from our measurements of the sun's annual output of
+energy in the form of heat, how fast it is shrinking, and we find that its
+diameter must diminish not far from 250 feet in a year; at least, the loss of
+potential energy corresponding to that amount of shrinkage would account
+for one year's running of the solar mechanism.
+\end{fineprint}
+
+\nbarticle{922.} \nbparatext{Age of the Solar System.}---Looking backward, then, in
+imagination we see the sun growing continually larger through the
+reversed course of time, expanding and becoming ever less and less
+dense, until at some epoch in the past it filled all the space now
+included within the largest orbit of the solar system.
+
+How long ago that was no one can say with certainty. If we could
+assume that the amount of potential energy lost by contraction, converted
+into the actual energy of heat and radiated into space, has been
+the same each year through all the intervening ages, and moreover,
+that \textit{all} the heat radiated has come from this source \textit{only}, without subsidy
+from any original store of heat contained in an original ``fire
+mist,'' or from energy derived from outside sources, then it is not difficult
+to conclude that the sun's past history must cover some 15,000000
+or 20,000000 years.
+
+But the assumption that the loss of heat has been even nearly uniform
+is extremely improbable, considering how high the present temperature
+of the sun must be as compared with that of the original
+nebula, and how the ratio of surface to solid content has increased
+with the lessening diameter.
+%% -----File: 535.png---Folio 524-------
+
+Nor is it unlikely that the sun may have received energy from other
+sources than its own contraction. Altogether it would seem that we
+must consider the 15,000000 years to be the least possible value of a
+duration which may have been many times more extended. If the
+nebular hypothesis and the theory of the solar contraction be true,
+the sun must be as old as that,---how much older no one can tell.
+
+\nbarticle{923.} \nbparatext{Future Prospects.}---Looking forward toward the future, it
+is easy to conclude also that at its present rate of radiation and contraction
+the sun must, within 5,000000 or 10,000000 years, become so
+dense that the conditions of its constitution will be radically changed,
+and to such an extent that life on the earth, as we now know life, would
+probably be impossible. If nothing intervenes to reverse the course
+of things, the sun must at last solidify and become a dark, rigid
+globe, frozen and lifeless among its lifeless family of planets. At
+least, this is the necessary consequence of what now seems to science
+to be the true account of its present activity and the story of its life.
+
+\nbarticle{924.} \nbparatext{Stars, Star-Clusters, and the Nebulæ.}---It is obvious that the
+same nebular hypothesis applies satisfactorily to the explanation of
+the relation of these different classes of bodies to each other. In fact,
+Herschel, appealing only to the law of continuity, had concluded before
+La~Place formulated his theory, that nebulæ develop sometimes into
+clusters, sometimes into double or multiple stars, and sometimes into
+single ones. He showed the existence in the sky of all the intermediate
+forms between the nebula and the finished star. For a time,
+about the middle of our century, while it was generally supposed that
+all nebulæ were nothing but star-clusters, too remote to be resolved by
+existing telescopes, his views fell rather into abeyance; but when the
+spectroscope demonstrated the substantial differences between the
+gaseous nebulæ and the star-clusters, they regained acceptance in
+their essential features; with perhaps the reservation, that many are
+disposed to believe that the rarest even of nebulous matter, instead
+of being purely gaseous, is full of solid and liquid particles like a
+cloud of fog or smoke.
+
+\nbarticle{925.} \nbparatext{The Present System not Eternal.}---One lesson seems to
+stand out clearly,---that the present system of stars and worlds is
+not an eternal one. We have before us irrefragable evidence of
+continuous, uncompensated progress, inexorable in one direction.
+The hot bodies are losing their heat, and distributing it to the cold
+ones, so that there is a steady, unremitting tendency towards a
+%% -----File: 536.png---Folio 525-------
+uniform (and therefore useless) temperature throughout the universe:
+for heat does \textit{work}, and is \textit{available} as energy \textit{only when it
+can pass from hotter to cooler bodies}, so that this warming up of
+cooler bodies at the expense of hotter ones always involves a loss,
+not of energy (for that is indestructible), but of \textit{available} energy.
+To use the technical language now usually employed, energy is
+unceasingly ``\textit{dissipated}'' by the processes which maintain the
+present life of the universe; and this dissipation of energy can
+have but one ultimate result,---that of absolute stagnation when a
+uniform temperature has been everywhere attained. If we carry our
+imagination backwards we reach at last a ``beginning of things,''
+which has no intelligible antecedent: if forwards, an end of things in
+stagnation. That by some process or other this end of things will
+result in ``new heavens and a new earth'' we can hardly doubt,
+but science has as yet no word of explanation.
+
+\nbarticle{926.} \nbparatext{Mr.~Lockyer's Meteoric Hypothesis.}---The idea that the
+heavenly bodies in their present state may have been formed by the
+aggregation of \textit{meteoric} matter, rather than by the condensation of a
+\textit{gaseous} mass, is not new, and not original with Mr.~Lockyer, as he himself
+points out. But his adoption and advocacy of the theory, and the
+support he brings to it from spectroscopic experiments on the light
+emitted by fragments of meteoric stones under different conditions,
+has given it such currency within the last two years that his name
+will always be justly associated with it. We have already referred
+to it in several places (Arts.~\arnref{850} and \arnref{894} especially).
+
+He believes that he finds in the spectra of meteorites, under various
+conditions, an explanation\footnote
+  {In some cases the explanation appears to be at least doubtful, especially in
+  instances where the presence of a sharply defined \textit{line} in the spectrum of a heavenly
+  body is referred to the degradation of a \textit{band} observed in the meteoric
+  spectrum.}
+of the spectra of comets, nebulæ, and
+all the different types of stars, as well as the spectra of the Aurora
+Borealis and the Zodiacal Light.
+
+Assuming this, he considers that \textit{nebulæ} are meteoric swarms in
+the initial stages of condensation, the separate individuals being still
+widely separated, and collisions comparatively infrequent.
+
+As aggregation goes on, the nebulæ become \textit{stars}, which run through
+a long life-history, the temperature first increasing slowly to a maximum,
+and then falling to non-luminosity. During this life-history
+the stars pass through successive stages, each stage characterized by
+%% -----File: 537.png---Folio 526-------
+its own typical spectrum; and according to these views he divides the
+stars into six spectroscopic classes, as follows:---
+
+I\@. Those which, like the nebulæ, show \textit{bright lines or flutings} in the
+spectrum, without \textit{dark} lines or bands. Vogel's II.~(\textit{b}) and I.~(\textit{c})
+(\artref{Art.}{858}) are included in this class. $\gamma$~Cassiopeiæ and 152~Schjellerup
+are types.
+
+II\@. Those which show both \textit{dark lines and dark flutings} in the spectrum.
+Vogel's III. (\textit{a}): ($\alpha$~Orionis).
+
+III\@. Those which show the \textit{fine dark lines} of metals. Vogel's
+II.~(\textit{a}) is included here \textit{in part}. $\alpha$~Aurigæ and the sun are typical.
+
+IV\@. Those whose spectra are characterized by \textit{the conspicuous hydrogen
+lines}, all other lines and markings being faint. These stars are at
+the summit of the temperature curve. Vogel's I.~(\textit{a}): Sirius and Vega.
+
+V\@. This class, on the \textit{descending} branch of the temperature curve
+(stars past middle life), should have sensibly the same spectrum as
+these of Class~III\@. We cannot be sure in which of the two classes
+our sun, for instance, should be counted.
+
+VI\@. The \textit{red stars}, their spectrum characterized by heavy absorption
+bands. These are stars verging to extinction according to Mr.~Lockyer's
+view. Vogel's III.~(\textit{b}).
+
+It is impossible to go into detail and give here Mr.~Lockyer's ingenious
+applications of the theory to explain the phenomena of the
+different classes of variable stars. (See \artref{Art.}{850}, however.)
+
+\sloppy
+This hypothesis has recently been much strengthened by a most
+interesting mathematical investigation of Prof.\ G.~Darwin, who
+shows that, if we assume a meteoric swarm comparable in dimensions
+with our solar system, composed of individual masses such
+as fall on the earth, and endowed with such velocities as meteors are
+known to have, such a swarm, seen from the distance of the stars,
+\textit{would behave like a mass composed of a continuous gas.} This is not
+strange since, according to the kinetic theory of gases, a gas is
+simply a swarm of \textit{molecules}, behaving in just the way the meteorites
+are supposed to act. But it follows that the \textit{meteoric} theory of a
+nebula does not in the least invalidate, or even to any great extent
+modify, the reasoning of La~Place in respect to the development of
+suns and systems from a \textit{gaseous} nebula. The old hypothesis has no
+quarrel with the new.
+
+While it would be premature to indorse this speculation of Mr.~Lockyer's
+as an established discovery (since there remain in it many
+obscure and doubtful points), there can be little doubt that it marks
+an epoch in the history of the science.
+\chelabel{CHAPTERXXI}
+
+%% -----File: 538.png---Folio 527-------
+\fussy
+\clearpage
+\pdfbookmark[1]{Appendix. Tables of Astronomical Data}{Appendix. Tables of Astronomical Data}
+\chapter*{APPENDIX.}
+\chslabel{APPENDIX}
+\markboth{APPENDIX.}{}
+\thispagestyle{empty}
+\nbrule\bigskip
+
+\small
+\begin{center}
+
+\nblabel{app:greek}
+\nbappheading{THE GREEK ALPHABET.}
+\medskip
+
+\begin{tabular}{@{}ll @{\qquad} ll @{\qquad} ll@{}}
+\scriptsize Letters. & \scriptsize Name. & \scriptsize Letters.  & \scriptsize Name.& \scriptsize Letters.    & \scriptsize Name.\\
+A, $\alpha$,         & Alpha.            & I, $\iota$,           & Iota.            & P, $\rho$, $\varrho$,   & Rho.\\
+B, $\beta$,          & Beta.             & K, $\kappa$,          & Kappa.           & $\Sigma$, $\sigma$, $\varsigma$, & Sigma.\\
+$\Gamma$, $\gamma$,  & Gamma.            & $\Lambda$, $\lambda$, & Lambda.          & T, $\tau$,              & Tau.\\
+$\Delta$, $\delta$,  & Delta.            & M, $\mu$,             & Mu.              & $\Upsilon$, $\upsilon$, & Upsilon.\\
+E, $\epsilon$,       & Epsilon.          & N, $\nu$,             & Nu.              & $\Phi$, $\phi$,         & Phi.\\
+Z, $\zeta$,          & Zeta.             & $\Xi$, $\xi$,         & Xi.              & X, $\chi$,              & Chi.\\
+H, $\eta$,           & Eta.              & O, $o$,               & Omicron.         & $\Psi$, $\psi$,         & Psi.\\
+$\Theta$, $\theta$, $\vartheta$, &Theta. & $\Pi$, $\pi$, $\varpi$, & Pi.            & $\Omega$, $\omega$,     & Omega.\\
+\end{tabular}
+
+\medskip
+\nbapprule
+\medskip
+
+\nblabel{app:misc}
+\nbappheading{MISCELLANEOUS SYMBOLS.}
+\medskip
+
+\begin{tabular}{l@{\hspace{4em}}l}
+  \conjunction, Conjunction.   & A.R., or $\alpha$, Right Ascension.\\
+  $\Box$, Quadrature.              & Decl., or $\delta$, Declination.\\
+  \opposition, Opposition.     & $\lambda$, Longitude (Celestial).\\
+  $\nbAscnode$, Ascending Node.    & $\beta$, Latitude (Celestial).\\
+  $\nbDescnode$, Descending Node.  & $\phi$, Latitude (Terrestrial).\\
+  \multicolumn{2}{p{.8\textwidth}}{\hangindent=2.5em $\omega$, Angle between line of nodes and line of apsides. Also
+obliquity of the ecliptic.}
+\end{tabular}
+\end{center}
+
+\DPtypo{}{\nbapprule
+\medskip}
+
+{\centering%\medskip
+\nblabel{app:spheroid}
+\nbappheading{DIMENSIONS OF THE TERRESTRIAL SPHEROID.}
+
+\footnotesize(According to Clarke's Spheroid of 1878.\\For the spheroid of 1866, see \artref{Art.}{145}.)\par}
+
+Equatorial \DPtypo{semidiameter}{semi-diameter},---
+
+{\centering $20\,926\,202 \text{ feet} = 3963.296 \text{ miles} = 6\,378\,190 \text{ metres.}$\par}
+
+Polar \DPtypo{semidiameter}{semi-diameter},---
+
+{\centering $20\,854\,895 \text{ feet} = 3949.790 \text{ miles} = 6\,356\,456 \text{ metres.}$
+
+\smallskip
+Ellipticity, or Polar Compression, $\frac{1}{293.46}$.\par}
+
+\medskip
+\nbapprule
+\medskip
+
+Length (in metres) of $1°$ of meridian in lat.\ $\phi =$\\
+  \hspace*{6em}$111\,132.09 - 556.05 \cos{2\phi} + 1.20\cos{4\phi}$.
+
+Length (in metres) of $1°$ of parallel, in lat.\ $\phi =$\\
+  \hspace*{6em}$111\,415.10 \cos{\phi} - 94.54\cos{3\phi}$.
+%% -----File: 539.png---Folio 528-------
+
+\medskip
+\begin{tabular}{@{} l@{ }r@{ }l}
+  $1°$ of lat.\ at &    pole &$= 111\,699.3$ metres $= 69.407$ miles. \\
+  $1°$ of lat.\ at & equator &$= 110\,567.2$ metres $= 68.704$ miles.
+\end{tabular}
+
+\medskip
+These formulæ correspond to the Clarke Spheroid of 1866, used by
+the U.S. Coast and Geodetic Survey.
+
+\medskip
+\nbapprule
+
+{\centering\medskip
+\nblabel{app:time}
+\nbappheading{TIME CONSTANTS.}\par}
+
+\medskip
+\begin{tabular}{@{} l@{ }l}
+  The sidereal   day &$= 23^\text{h}\: 56^\text{m}\:  4^\text{s}.090$ of mean solar time.\\
+  The mean solar day &$= 24^\text{h}\:  3^\text{m}\: 56^\text{s}.556$ of sidereal time.
+\end{tabular}
+
+\medskip
+To reduce a time-interval expressed in units of \textit{mean solar time} to
+\textit{units of sidereal time}, multiply by 1.00273791; Log.\ of $0.00273791
+= [7.4374191]$.
+
+\medskip
+To reduce a time-interval expressed in units of \textit{sidereal time} to
+units of \textit{mean solar time}, multiply by $0.99726957 = (1 - 0.00273043)$;
+Log.\ $0.00273043 = [7.4362316]$.
+
+\medskip
+\begin{tabular}{@{} l@{}l@{}l@{}l@{ } l@{\:}l@{\:}l@{\:} r@{.}l}
+Tropical year (Lev&errier, red&uced to 19&00),
+& $365^\text{d}$ & $5^\text{h}$ & $48^\text{m}$ & $45^\text{s}$&51.
+\\
+Sidereal year    &  ``  &  ``  &  ``
+& 365 & 6 &\phantom{0}9 & 8&97.
+\\
+Anomalistic \rlap{year} &  ``  &  ``  &  ``
+& 365 & 6 &13 & 48&09.
+\end{tabular}
+
+\medskip
+\begin{tabular}{@{} l l@{\:}l@{\:}l@{\:} r}
+Mean synodical month (new moon to new),
+& $29^\text{d}$ & $12^\text{h}$ & $44^\text{m}$ & $2^\text{s}.684$.
+\\
+Sidereal month, \dotfill
+&   27 & \phantom{0}7 & 43 &  11.545.
+\\
+Tropical month (equinox to equinox),\dotfill
+&    27 & \phantom{0}7  &  43 &  4.68.
+\\
+Anomalistic month (perigee to perigee),\dotfill
+&  27 &  13 &  18 &  37.44.
+\\
+Nodical month (node to node),  \dotfill
+& 27 & \phantom{0}5 & \phantom{0}5 &  35.81.
+\end{tabular}
+
+\bigskip
+\nbapprule
+\medskip
+
+Obliquity of the ecliptic (Leverrier),\\
+  \hspace*{\fill}$23°\: 27'\: 08''.0 - 0''.4757 (t-1900)$.
+
+Constant of precession (Struve), \hfill $50''.264 + 0''.000227\, (t-1900)$.
+
+Constant of notation (Peters), \hfill \PadTo[l]{$50''.264 + 0''.000227\, (t-1900)$.}{$9''.223$.}
+
+Constant of aberration (Nyrén), \hfill \PadTo[l]{$50''.264 + 0''.000227\, (t-1900)$.}{$20''.492$.}
+
+\clearpage
+%% -----File: 540.png---Folio 529-------
+
+\begin{sidewaystable}
+\centering
+\tiny
+\noindent
+\renewcommand{\arraystretch}{1.1}
+\nblabel{app:I}
+\begin{tabular}{l|l|c| D{,}{}{-1}|D{,}{}{-1}|D{.}{.}{-1}|D{,}{}{-1}| c|c| *{3}{D{,}{}{-1}|} D{,}{}{-1}}
+\multicolumn{13}{c}{\footnotesize TABLE I.---PRINCIPAL ELEMENTS OF THE SOLAR SYSTEM.}\\[2ex]
+\hline \hline
+  \multicolumn{2}{c|}{\textsc{Name.}}
+& \multicolumn{1}{m{1em}|}{\rotatebox{90}{\textsc{Symbol.}}}
+& \multicolumn{1}{m{5.5em}|}{\centering Semi-Major Axis of Orbit.}
+& \multicolumn{1}{m{4em}|}{\centering Mean Dist.\\ Millions of Miles.}
+& \multicolumn{1}{m{6em}|}{\centering \rule{0pt}{3ex}Sidereal Period\\ (mean solar days).\rule[-2ex]{0pt}{3ex}}
+& \multicolumn{1}{m{3em}|}{\centering Period in Years.}
+& \multicolumn{1}{m{5em}|}{\centering Orbit-Velocity\\ (miles per second).}
+& \multicolumn{1}{m{4em}|}{\centering Eccen\-tricity.}
+& \multicolumn{1}{m{5em}|}{\centering Inclination to Ecliptic.}
+& \multicolumn{1}{m{5em}|}{\centering Longitude of Ascending Node.}
+& \multicolumn{1}{m{5em}|}{\centering Longitude of Perihelion.}
+& \multicolumn{1}{m{5em} }{\centering Longitude at Epoch, Jan.~1, 1850.}
+\\
+\hline
+\multirow{4}{*}{\rotatebox{90}{\parbox{4.55em}{\centering Terrestrial Planets.}}}
+& Mercury\rule{0pt}{3ex} & \Mercury
+& 0.38,7099 & 3,6.0 & 87.96926 & 0,.24
+& 23 to 35 & .20560
+&   7\rlap{°}\ 0,0\rlap{$'$}\ \phantom{0}8''
+&  46\rlap{°}\ 3,3\rlap{$'$}\ \phantom{0}9''
+&  75\rlap{°}\ \phantom{0},7\rlap{$'$}\ 14''
+& 327\rlap{°}\ 1,5\rlap{$'$}\ 20''
+\\
+& Venus & \Venus
+& 0.72,3332 & 6,7.2 & 224.7008 & 0,.62
+& 21.9 & .00684
+& 3\ 2,3\ 35 & 75\ 1,9\ 52 & 129\ 2,7\ 15 & 245\ 3,3\ 15
+\\
+& The Earth & $\oplus$
+& 1.00,0000 & 9,2.9 & 365.2564 & 1,.00
+& 18.5 & .01677
+& 0\ 0,0\ 00 & 0\ 0,0\ 00 & 100\ 2,1\ 22 & 100\ 4,6\ 44
+\\
+& Mars & \Mars
+& 1.52,3691 & 14,1.5 & 686.9505 & 1,.88
+& 15.0 & .09326
+& 1\ 5,1\ \phantom{0}2 & 48\ 2,3\ 53 & 333\ 1,7\ 54 & 103\ 2,5\ \phantom{0}3  \\[1ex]
+\hline
+
+& Ceres\rule{0pt}{3ex} & \textcircled{\raisebox{-0.25ex}{1}}
+& 2.76,7265 & 25,7.1 & 1681.414 & 4,.60
+& 11.1 & .07631
+& 10\ 3,7\ 10 & 30\ 4,6\ 39 & 149\ 3,7\ 49 & 83\ 4,0\ 31 \\[1ex]
+\hline
+\multirow{4}{*}{\rotatebox{90}{\parbox{5em}{\centering Major Planets.}}}
+& Jupiter\rule{0pt}{3ex} & \Jupiter
+& 5.20,2800 & 48,3.3 & 4332.580 & 11,.86
+& \phantom{0}8.1 & .04825
+& 1\ 1,8\ 41 & 98\ 5,6\ 17 & 11\ 5,4\ 58 & 160\ \phantom{0},1\ 19
+\\
+& Saturn & \Saturn
+& 9.53,8861 & 88,6.0 & 10759.22 & 29,.46
+& \phantom{0}6.0 & .05607
+& 2\ 2,9\ 40 & 112\ 2,0\ 53 & 90\ \phantom{0},6\ 38 & 14\ 5,2\ 28
+\\
+& Uranus & \Uranus
+& 19.18,329 & 178,1.9 & 30686.82 & 84,.02
+& \phantom{0}4.2 & .04634
+& 0\ 4,6\ 20 & 73\ 1,3\ 54 & 170\ 5,0\ \phantom{0}7 & 29\ 1,7\ 51
+\\
+& Neptune & \Neptune
+& 30.05,508 & 279,1.6 & 60181.11 & 164,.78
+& \phantom{0}3.4 & .00896
+& 1\ 4,7\ \phantom{0}2 & 130\ \phantom{0},6\ 25 & 45\ 5,9\ 43 & 334\ 3,3\ \ 29 \\[1ex]
+\hline \hline
+\end{tabular}
+\bigskip
+
+\setlength{\tabcolsep}{3pt}
+\noindent
+\begin{tabular}{l|l|c|l| D{,}{,}{-1}|D{.}{.}{-1}|c|c| c|c|c|l| D{,}{}{-1}|c|c|c}
+\hline \hline
+  \multicolumn{2}{c|}{\multirow{2}{3em}{\centering \textsc{Name.}}}
+& \multirow{2}{*}{\rotatebox{90}{\textsc{Symbol.}}}
+& \multicolumn{1}{c|}{\multirow{2}{5em}{\centering Apparent Angular Diameter.}}
+& \multicolumn{2}{c|}{\textsc{Mean Diameter}\rule{0pt}{4ex}}
+& \multicolumn{2}{c|}{\textsc{Mass.}}
+& \multicolumn{1}{c|}{\textsc{Volume.}}
+& \multicolumn{2}{c|}{\textsc{Density.}}
+& \multicolumn{1}{c|}{\multirow{2}{5em}{\centering Time of Axial Rotation.}}
+& \multicolumn{1}{c|}{\multirow{2}{4.8em}{\centering Inclination Equator to Orbit.}}
+& \multicolumn{1}{c|}{\multirow{2}{4em}{\centering Oblate\-ness, or Elliptic\-ity.}}
+& \multicolumn{1}{c|}{\multirow{2}{4em}{\centering Gravity at Surface.}}
+& \multicolumn{1}{c }{\multirow{2}{4em}{\centering Albedo.}}
+\\[2ex]
+\cline{5-11}
+  \multicolumn{2}{c|}{} &&
+& \multicolumn{1}{c|}{in Miles.\rule{0pt}{4ex}}
+& \multicolumn{1}{c|}{$\oplus=1$.}
+& \multicolumn{1}{c|}{$\astrosun=1$.}
+& \multicolumn{1}{c|}{$\oplus=1$.}
+& \multicolumn{1}{c|}{$\oplus=1$.}
+& \multicolumn{1}{c|}{$\oplus=1$.}
+& \multicolumn{1}{c|}{Water $=1$.}
+&&&&&
+\\[2ex]
+\hline
+& Sun       & \astrosun\rule{0pt}{3ex}
+& $32'\: 04''$ mean & 866,400 & 109.4 & 1.00000 & 331100 & 1\:310\:000
+& 0.25 & 1.39 & $25^\text{d}\: 7^\text{h}\: 48^\text{m}$
+& 7\text{°}\: 1,5' & ? & \llap{2}7.65
+\\
+& Moon      & \leftmoon
+& $31'\: 07''$ \quad ``   & 2,163 & 0.273 & $\frac{1}{26821000}$ & $\frac{1}{81}$ & 0.020
+& 0.61 & 3.40 & $27\phantom{^d}\: 7\phantom{^d}\: 43$
+& 6\phantom{\text{°}}\: 3,3 & ? & $\frac{1}{6}$ & 0.17 \\[1ex]
+\hline
+\multirow{4}{*}{\rotatebox{90}{\parbox{5em}{\centering Terrestrial Planets.}}}
+& Mercury   & \Mercury\rule{0pt}{3ex}
+& $5''$ to $13''$  & 3,030 & 0.382 & $\frac{1}{2668700}$? & $\frac{1}{8}$? & 0.056
+& 2.23\rlap{?} & 12.44? & \qquad ?
+& ,? & ?  & 0.85 & 0.13
+\\
+& Venus     & \Venus
+& $11''$ to $67''$ &  7,700 & 0.972 & $\frac{1}{425000}$ &  0.78 & 0.920
+& 0.86 & 4.85 & $23^\text{h}\: 21^\text{m}\: 23^\text{s}$
+& ,? & ? & 0.83 & 0.50
+\\
+& The Earth & $\oplus$
+& \qquad $\ldots$   & 7,918 & 1.000 & $\frac{1}{331100}$ & 1.000
+& 1.000 & 1.00 & 5.58 & $23\phantom{^d}\: 56\: \phantom{0}4.09$
+& 23\text{°}\: 2,7'\: 12'' & $\frac{1}{203}$ & 1.00 & 0.20\rlap{?}
+\\
+& Mars      & \Mars
+& $3''.6$ to $24''.5$ &  4,230 & 0.534 & $\frac{1}{3093500}$ & $\frac{1}{9.31}$ & 0.152
+& 0.72 & 4.01 & $24\phantom{^d}\: 37\: 22.67$
+& 24\phantom{\text{°}}\: 5,0 & $\frac{1}{220}$ & 0.38 & 0.26 \\[1ex]
+\hline
+& Ceres     &  \textcircled{\raisebox{-0.25ex}{1}}\rule{0pt}{3ex}
+& \qquad \ldots & \multicolumn{1}{c|}{100?} & \multicolumn{1}{c|}{\ldots}
+& \ldots?  & $\frac{1}{70000}$?
+& $\frac{1}{50000}$? & ? & ? & \qquad ? & ,? & ? & ? \\[1ex]
+\hline
+\multirow{4}{*}{\rotatebox{90}{\parbox{5em}{\centering Major Planets.}}}
+& Jupiter   & \Jupiter\rule{0pt}{3ex}
+& $32''$ to $50''$  & 86,500 & 10.92 & $\frac{1}{1048}$ & 316.0 & 1309
+& 0.24 & 1.33 & $\phantom{0}9^\text{h}\: 55^\text{m}\ ±$
+& 3\text{°}\: 0,5' & $\frac{1}{1710}$ & 2.65 & 0.62
+\\
+& Saturn    & \Saturn
+& $14''$ to $20''$ & 73,000 & 9.17 & $\frac{1}{3490}$ & \phantom{0}94.9 & \phantom{0}760
+& 0.13 & 0.72 & $10^\text{h}\: 14^\text{m}\: 24^\text{s}$
+& 26\phantom{\text{°}}\: 4,9 & $\frac{10}{92}$ & 1.18 & 0.52
+\\
+  & Uranus    & \Uranus
+& $3''.8$ to $4''.1$ & 31,900 & 4.03 & $\frac{1}{22600}$ & \phantom{0}14.7 & \phantom{00}65
+& 0.22 & 1.22 & \qquad ? & ,? & $\frac{1}{11}$ & 0.91 & 0.64
+\\
+& Neptune   & \Neptune
+& $2''.7$ to $2''.9$ & 34,800 & 4.39 & $\frac{1}{19380}$ & \phantom{0}17.1 & \phantom{00}85
+& 0.20 & 1.11 & \qquad ? & ,? & ? & 0.88 & 0.46 \\[1ex]
+\hline \hline
+\end{tabular}
+\renewcommand{\arraystretch}{1}
+\end{sidewaystable}
+
+%% -----File: 541.png---Folio 530-------
+%% -----File: 542.png---Folio 531-------
+\clearpage
+
+\begin{sidewaystable}
+\centering
+\tiny
+\noindent
+\setlength{\tabcolsep}{1pt}
+\nblabel{app:II}
+\begin{tabular}{c| l@{}| l@{~}c| D{.}{.}{-1}| D{,}{}{-1}| *{3}{r@{}r@{\:}}r@{}l |*{3}{r@{}r@{\:}}r@{}l| *{2}{r@{}l@{\ }}r@{}l@{\,} | c| D{,}{}{-1}| r@{}c@{\,} | c | l@{}}
+\multicolumn{34}{c}{\footnotesize TABLE II.---THE SATELLITES OF THE SOLAR SYSTEM.}\\[2ex]
+\hline \hline
+
+& & \rule{0pt}{3ex}
+&& \multicolumn{1}{c|}{\multirow{3}{7em}{\centering Dist.\ in Equa\-torial Radii of Planet}}
+& \multicolumn{1}{c|}{\multirow{3}{4em}{\centering Mean Distance in Miles.}}
+&&&&&&&&
+&&&&&&&&
+& \multicolumn{6}{c|}{\multirow{3}{5em}{\centering Inc.\ of Orbit to Ecliptic.}}
+& \multicolumn{1}{c|}{\multirow{3}{6em}{\centering Inc.\ to Plane of Planet's Orbit.}}
+& \multicolumn{1}{c|}{\multirow{3}{3em}{\centering Eccen\-tricity.}}
+& \multicolumn{2}{c|}{\multirow{3}{2em}{\scalebox{0.9}{\rotatebox{90}{\parbox{3.5em}{\centering Diam'r in Miles}}}}}
+%& \multicolumn{2}{c|}{\multirow{3}{2em}{\scalebox{0.9}{\rotatebox{90}{\parbox{3.5em}{\centering Diam'r in Miles}}}}}
+& \multicolumn{1}{c|}{\multirow{3}{4em}{\centering Mass in Terms of Primary.}}
+&
+\\
+& \multicolumn{1}{c|}{\textsc{Name}.}
+& \multicolumn{2}{c|}{Discovery.} &&
+& \multicolumn{8}{c|}{Sidereal Period.}
+& \multicolumn{8}{c|}{Synodic Period.}
+&&&&&&&&&&&
+& \multicolumn{1}{c }{Remarks.}
+\\
+&&&& &&&&&&&&&
+ &&&&&&&&&&&&&&&&&&~&&~% ~ needed to help LaTeX realise it needs to draw column separator lines in the third row of the header
+\\[1ex]
+\hline
+
+  & Moon \dotfill \rule{0pt}{3ex}
+& \multicolumn{2}{@{\ }c@{\ }|}{\dotfill} & 60.27035 & 23,8\,840
+& 27&$^\text{d}$ &  7&$^\text{h}$ & 43&$^\text{m}$ & 11&$^\text{s}.5$
+& 29&$^\text{d}$ & 12&$^\text{h}$ & 44&$^\text{m}$ & 2&$^\text{s}.7$
+&  5&$°$ & 08&$'$ & 40&$''$
+& \multicolumn{1}{c|}{ --- \qquad --- }
+& 0.05,491 &  2162&& $\frac{1}{81}$
+& \parbox{7.5em}{\hspace{0.5em}Specific gravity\\ 3.44.}
+\\[1ex]
+\hline
+\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITES OF MARS}
+\\[1ex]
+\hline
+1 & Phobos \dotfill \rule{0pt}{3ex}
+& Hall, & 1877 &   2.771 &  ,5\,850
+& &&  7&$^\text{h}$ & 39&$^\text{m}$ & 15&$^\text{s}.1$
+& \multicolumn{8}{c|}{ --- \qquad --- }
+& 26&$°$ & 17&$'$\rlap{.2} &&
+& $28°$ & ,0 & 7&? &  ?
+& \raisebox{0.8ex}{\multirow{2}{8em}{\hspace{0.5em}Orbits sensibly coincident with planet's equator.}}
+\\
+2 & Deimos \dotfill \rule{0pt}{3ex}
+& \quad `` & ``  &   6.921 &  1,4\,650
+& 1&$^\text{d}$ & 6&  & 17&  & 54&.0
+& \multicolumn{8}{c|}{ --- \qquad --- }
+& 25&           & 47&\rlap{.2}           &&
+&  $28°$\rlap{$ \pm$}
+& ,0 & 5&? & ?  &
+\\[1ex]
+\hline
+\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITES OF JUPITER} \\[1ex]
+\hline
+1 & Io \dotfill \rule{0pt}{3ex}
+&  Galileo, & 1610 &   5.933  &  26,1\,000
+&  1&$^\text{d}$ & 18&$^\text{h}$ & 27&$^\text{m}$ & 33&$^\text{s}.5$
+& 1&$^\text{d}$ & 18&$^\text{h}$ & 28&$^\text{m}$ & 35&$^\text{s}.9$
+& 2&$°$ & 08&$'$  & 3&$''$
+& --- \qquad ---   &  ,0 & 2500&   & .00001688
+& \multirow{4}{8em}{\hspace{0.5em}The diameters are Englemann's. The rest of the data are from Damoiseau.}
+\\
+2 & Europa \dotfill \rule{0pt}{3ex}
+& \quad `` & `` &   9.430 & 41,5\,000
+&  3&  & 13&  & 13&  &  42&.1
+&  3&  & 13&  & 17&  &  53&.7
+&  1&  & 38&  &  57&
+& --- \qquad ---  &  ,0 & 2100&   & .00002323   &
+\\
+3 & Ganymede \dotfill \rule{0pt}{3ex}
+& \quad `` & `` &  15.057 &   66,4\,000
+&  7&  & 3&  & 42&  & 33&.4
+&  7&  & 3&  & 59&  & 35&.9
+&  1&  & 59&  & 53&
+& --- \qquad ---  & .0,013 & 3550&   & .00008844   &
+\\
+4 & Callisto \dotfill \rule{0pt}{3ex}
+& \quad `` & `` &  26.486 & 1\,16,7\,000
+& 16&   & 16&   & 32&   & 11&.2
+& 16&   & 18&   &  5&   &  6&.9
+&  1&   & 57&   & 00&
+& --- \qquad ---  & .0,072 & 2960&  & .00004248  &
+\\[1ex]
+\hline
+\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITES OF SATURN} \\[1ex]
+\hline
+1 & Mimas \dotfill \rule{0pt}{3ex}
+& W. Herschel, & 1789 &  3.11 &  11,7\,000
+&  &  & 22&$^\text{h}$ & 37&$^\text{m}$ & 5&$^\text{s}.7$
+& \multicolumn{8}{l|}{Long.\ of Ascend.}
+& 28&$°$   & 10&$'$   &10&$''$
+& \multicolumn{1}{l|}{About $27°$.} &  ,0 &  600&? & ?
+& \multirow{5}{7em}{\hspace{0.5em}The planes of the 5 inner orbits sensibly coincide with the plane of the ring.}
+\\
+2 & Enceladus \dotfill \rule{0pt}{3ex}
+& \quad `` \qquad `` & `` &  3.98 & 15,7\,000
+& 1&$^\text{d}$ & 8&  &  53&  &  6&.9
+& \multicolumn{8}{l|}{Node of orbits on}   &&& ``  &&&
+& \multicolumn{1}{l|}{Inclination of the} &  ,0 &  800&? & ?  &
+\\
+3 & Tethys \dotfill \rule{0pt}{3ex}
+& J. D. Cassini, & 1684 &  4.95 &  18,6\,000
+& 1&  &21&  &18&  & 25&.6
+& \multicolumn{8}{l|}{ecliptic for 1900*,} &&& ``  &&&
+& \multicolumn{1}{l|}{5 inner satellites}  &  ,0  & 1100&? &  ?  &
+\\
+4 & Dione \dotfill \rule{0pt}{3ex}
+& \quad `` \qquad `` & ``  &  6.34 &  23,8\,000
+& 2&  &17&  & 41&  & 9&.3
+& \multicolumn{8}{l|}{$168°\: 10'\: 35''$.}
+&&& ``  &&&
+& \multicolumn{1}{l|}{to plane of celes-}   &   ,0 & 1200&? & ?  &
+\\
+5 & Rhea \dotfill \rule{0pt}{3ex}
+& \quad `` \qquad ``  & 1672  &  8.86 &  33,2\,000
+& 4&  & 12&  & 25&  & 11&.6
+& \multicolumn{8}{l|}{(5 inner satellites}  &&& `` &&&
+& \multicolumn{1}{l|}{tial equator =}  &   ,0 &  1500&?  & ?  &
+\\
+6 & Titan \dotfill \rule{0pt}{3ex}
+& Huyghens, &  1655  &  20.48 &  77,1\,000
+&  13&  & 22&  & 41&   & 23&.2
+& \multicolumn{8}{l|}{and ring.)}
+& 27&  & 38&   & 49&
+&\multicolumn{1}{l|}{$6°\: 57'\: 43''$ (1900)}
+&  0.0,299 & 3500&?  & $\frac{1}{4600}$
+& \multirow{3}{7.5em}{$\left\{ \parbox{6.5em}{\hspace{0.5em}Discovered independently by Lassell.} \right.$}
+\\
+7 & Hyperion \dotfill \rule{0pt}{3ex}
+& G. P. Bond, &  1848  &  25.07 &  93,4\,000
+&  21&  &   6&   & 39&   & 27&.0
+&&&&&&&&
+& 27&  & 4&.8  &&
+& --- \qquad ---  &  0.1,189 &  500&?  &  ?  &
+\\
+8 & Iapetus \dotfill \rule{0pt}{3ex}
+& J. D. Cassini,  &  1671  &  59.58  & 2\,22,5\,000
+&  79&   &  7&   & 54&   &17&.1
+&&&&&&&&
+& 18&  & 31&.5  &&
+& --- \qquad ---  &  0.0,296 & 2000&? & ?  &
+\\[1ex]
+\hline
+\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITES OF URANUS} \\[1ex]
+\hline
+1 & Ariel  \dotfill \rule{0pt}{3ex}
+& Lassell,  &  1851 &   7.52 &  12,0\,000
+&  2&$^\text{d}$ & 12&$^\text{h}$ & 20&$^\text{m}$ & 21&$^\text{s}.1$
+& \multicolumn{8}{l|}{Long.\ of Ascend.}
+&  97&$°$ & 51&$'$ &  &
+\phantom{$'$}\llap{\raisebox{-2ex}[0ex][0ex]{$=-82°\: 09'$}}
+&  & ,0 & 500&?  &  ? &
+\\
+2 & Umbriel  \dotfill \rule{0pt}{3ex}
+& \quad `` & `` & 10.46 &  16,7\,000
+&  4&   &  3&   &  27&  & 37&.2
+& \multicolumn{8}{l|}{Node of orbits on}
+& \multicolumn{2}{r}{``} &\multicolumn{2}{c }{ ``}  &&
+& \multicolumn{1}{l|}{Inc.\ to celestial} &  ,0 & 400&? & ?
+& \hspace{0.5em}Retrograde.
+\\
+3 & Titania \dotfill \rule{0pt}{3ex}
+& W. Herschel, & 1787 &  17.12  &  27,3\,000
+&  8&   &  16&   & 56&   & 29&.5
+& \multicolumn{8}{l|}{plane of ecliptic}
+& \multicolumn{2}{r}{``} &\multicolumn{2}{c }{ ``}  &&
+&\multicolumn{1}{l|}{equator $75°\: 18'$} & ,0 & 1000&? & ? &
+\\
+4 & Oberon \dotfill \rule{0pt}{3ex}
+& \quad `` \qquad ``  & ``  &  22.90 &  36,5\,000
+&  13&   &  11&   &  7&   & 6&.4
+& \multicolumn{8}{l|}{$= 165°\: 32'$ (1900).}
+& \multicolumn{2}{r}{``} &\multicolumn{2}{c }{ ``}  &&
+& \multicolumn{1}{l|}{(1900).} & ,0 & 800&? & ? &
+\\[1ex]
+\hline
+\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITE OF NEPTUNE} \\[1ex]
+\hline
+1 & Nameless \dotfill \rule{0pt}{3ex}
+& Lassell,  &  1846 &  12.93  & 22,5\,000
+&  5&$^\text{d}$ &  21&$^\text{h}$ &  2&$^\text{m}$ & 44&$^\text{s}.2$
+& \multicolumn{8}{m{8em}|}{Long.\ Asc.\ Node, $184°\: 25'$ (1900).}
+& \raisebox{1.2ex}{145}&\raisebox{1.2ex}{$°$} & \raisebox{1.2ex}{12}& \raisebox{1.2ex}{$'$}
+& &\phantom{$'$}\llap{\raisebox{-1.2ex}{$=-34°\: 48'$}}
+& \multicolumn{1}{l|}{$120°\: 05'$ (1900)}
+&  ,0 & 2000&? & ?  & \hspace{0.5em}Retrograde.
+\\[1ex]
+
+\hline
+\hline
+\end{tabular}
+
+\end{sidewaystable}
+\clearpage
+
+%% -----File: 543.png---Folio 532-------
+
+\begin{sidewaystable}
+\footnotesize
+\noindent
+\setlength\tabcolsep{3pt}
+\nblabel{app:III}
+\begin{tabular}{@{}r | l@{}| r@{, }l@{ }r | r@{.}l | r@{.}l | r@{.}l | c | r@{}l@{ }r@{}l@{ }r@{}l | r@{}l@{ }r@{}l@{ }r@{}l | r@{}l@{ }r@{}l@{ }r@{}l@{} }
+\multicolumn{30}{c}{TABLE~III.---PERIODIC COMETS WHICH HAVE BEEN OBSERVED AT MORE THAN ONE PERIHELION PASSAGE.}
+\\
+\multicolumn{30}{c}{FROM THE ANNUAIRE DU BUREAU DES LONGITUDES, 1888.}
+\\[1ex]
+\hline\hline
+\multirow{2}{*}{\begin{sideways}\textsc{\footnotesize No}.\end{sideways}}
+& \multicolumn{1}{c|}{\multirow{2}{*}{\textsc{\footnotesize Name}.}}
+& \multicolumn{3}{c|}{\footnotesize Perihelion\rule{0pt}{3.5ex}}
+& \multicolumn{2}{c|}{\footnotesize Period.}
+& \multicolumn{2}{c|}{\footnotesize Perihelion}
+& \multicolumn{2}{c|}{\footnotesize Aphelion}
+& \multirow{2}{*}{\footnotesize Eccentricity.}
+& \multicolumn{6}{c|}{\footnotesize Longitude of}
+& \multicolumn{6}{c|}{\footnotesize Longitude of}
+& \multicolumn{6}{c}{\multirow{2}{*}{\footnotesize Inclination.}}
+\\[-.5ex]
+ &
+ & \multicolumn{3}{c|}{\footnotesize Passage.}
+ & \multicolumn{2}{c|}{\footnotesize (Years.)}
+ & \multicolumn{2}{c|}{\footnotesize Distance.}
+ & \multicolumn{2}{c|}{\footnotesize Distance.} &
+ & \multicolumn{6}{c|}{\footnotesize Perihelion.}
+ & \multicolumn{6}{c|}{\footnotesize Node.}  &
+\\[1.5ex]
+\hline
+1 & Encke \dotfill\rule{0pt}{3ex}
+& 1885 & Mar.\ &  7 & 3&307 & 0&342309 & 4&096935 & 0.845781
+& 158&$°$ & 32&$'$ & 45&$''$
+& 334&$°$ & 36&$'$ & 55&$''$
+&  12&$°$ & 54&$'$ & 00&$''$
+\\[.5ex]
+2 & Tempel \dotfill
+& 1883 & Nov.\ & 20 & 5&209 & 1&344665 & 4&665563 & 0.552541
+& 306 &&  7 &&  4 &    & 121 &&  2 &&  8 &    & 12 && 45 && 17 &
+\\[.5ex]
+3 & Tempel-Swift \dotfill
+& 1886 & May &  9 & 5&505 & 1&072638 & 5&162744 & 0.655951
+& 43 && 9 && 54 &    & 297 && 0 && 39 &    & 5 && 23 && 37 &
+\\[.5ex]
+4 & Brorsen \dotfill
+& 1879 & Mar.\ & 30 & 5 & 462 & 0 & 589892 & 5 & 612808 & 0.809797
+& 116 && 15 && 3 &    & 101 && 19 && 16 &    & 29 && 23 && 10 &
+\\[.5ex]
+5 & Winnecke \dotfill
+& 1886 & Sept.\ & 4 & 5 & 812 & 0 & 883240 & 5 & 582030 & 0.726775
+& 276 && 4 &&  &    & 101 && 56 &&  &    & 14 && 27 &&  &
+\\[.5ex]
+6 & Tempel \dotfill
+& 1885 & Sept.\ & 25 & 6 & 507 & 2 & 073322 & 4 & 897332 & 0.405128
+& 241 && 21 && 50 &    &  72 && 24 &&  9 &    & 10 && 50 && 27 &
+\\[.5ex]
+7 & Biela, 2d nucleus, \dotfill
+& 1852 & Sept.\ & 23 & 6 & 629 & 0 & 860592 & 6 & 196874 & 0.755119
+& 108 && 58 && 17 &    & 245 && 58 && 29 &    &  12 && 33 && 50 &
+\\[.5ex]
+8 & D'Arrest \dotfill
+& 1884 & Jan.\ & 13 & 6 & 686 & 1 & 326420 & 5 & 771986 & 0.626277
+& 319 && 11 && 11 &    & 146 &&  7 && 21 &    & 15 && 41 && 47 &
+\\[.5ex]
+9 & Faye \dotfill
+& 1881 & Jan.\ & 22 & 7 & 566 & 1 & 738140 & 5 & 970090 & 0.549017
+&  50 && 48 && 47 &    & 209 && 35 && 25 &    &  11 && 19 && 40 &
+\\[.5ex]
+10 & Tuttle \dotfill
+& 1885 & Sept.\ & 11 & 13 & 760 & 1 & 024728 & 10 & 459624 & 0.821544
+& 116 && 28 && 59 &    & 269 && 42 &&  1 &    &  55 && 14 && 23 &
+\\[.5ex]
+11 & Pons-Brooks\dotfill
+& 1884 & Jan.\ & 25 & 71 & 48 & 0 & 77511 & 33 & 67129 & 0.954996
+&  93 && 20 && 48 &    & 254 &&  6 && 15 &    & 74 &&  3 && 20 &
+\\[.5ex]
+12 & Olbers \dotfill
+& 1887 & Oct.\ &  8 & 72 & 63 & 1 & 19961 & 33 & 61592 & 0.931088
+& 149 && 45 && 47 &    &  84 && 29 && 41 &    &  44 && 33 && 53 &
+\\[.5ex]
+13 & Halley \dotfill
+& 1835 & Nov.\ & 15 & 76 & 37 & 0 & 58895 & 35 & 41121 & 0.967281
+& 165 && 48 && 48 &    &  55 && 10 && 15 &    & 162 && 15 &&  7 &
+\\[1ex]
+\hline\hline
+\end{tabular}
+\end{sidewaystable}
+%% -----File: 544.png---Folio 533-------
+
+\clearpage
+
+\begin{center}
+\nblabel{app:IV}
+TABLE~IV.---STELLAR PARALLAX.
+\end{center}
+
+{\footnotesize
+The star-places are only approximate,---sufficient merely for identification. The parallaxes
+are mostly from Houzeau's ``Vade Mecum,'' but a few are included from later authorities.
+
+\bigskip
+\setlength\tabcolsep{3pt}
+\tiny
+\begin{tabular}{@{}r@{ }l@{}|l| r@{\ \ }r@{\quad}|@{ }r@{}r@{\ \ }r| l|l@{\,}|l|r|rl}
+\hline\hline
+
+  \multicolumn{2}{@{}c|}{\strut}
+& \multicolumn{1}{c|} {\multirow{4}{*}{\begin{sideways}Magnitude.\mbox{\ \ }\end{sideways}} }
+& \multicolumn{5}{c|}{\strut}
+& \multicolumn{1}{c|}{\strut}
+& \multicolumn{1}{c|}{\strut}
+& \multicolumn{1}{c|}{\strut}
+& \multicolumn{1}{c|}{\multirow{4}{*} {\begin{sideways}Parallax.\mbox{\quad\ }\end{sideways}}}
+& \multicolumn{2}{c} {\multirow{4}{*}{\begin{sideways}\begin{tabular}{c}Distance.\\ Light-Years.\end{tabular} \end{sideways}}}\\[5ex]
+
+\multicolumn{2}{@{}c|}{\textsc{Name of Star.}}
+& & \multicolumn{5}{c|}{Approx.~Place.}
+& \multicolumn{1}{c|}{Authority. }
+& \multicolumn{1}{c|}{Date. }
+& \multicolumn{1}{c|}{Method. }
+&& \\
+
+&& & \multicolumn{5}{c|}{(1900)} & &&&&\\[1ex]
+
+&& & \multicolumn{2}{c }{$\alpha$}
+   & \multicolumn{3}{c|}{$\delta$} & &&&&\\[1ex]
+
+\hline &&&&&&&&&&&&\\[-1.5ex]
+
+1. & $\beta$~Cassiopeiæ \dotfill & 2
+& 0\rlap{$^\text{h}$} & 4\rlap{$^\text{m}$} & $+$&58\rlap{$°$} & 36\rlap{$'$}
+& Pritchard & 1887 & Photography & $0''.187$ & 17.4\\
+
+2. & Groombridge 34\DPtypo{,}{}
+\dotfill & 8 & 0 & 12 &$+$&43 & 26
+& Auwers & 1867 & Mer.\ Circle & 0.307 & 10.6\\
+
+3. & $\eta$~Cassiopeiæ \dotfill & 4 & 0 & 43 & $+$&57 & 17
+& O. Struve & 1856 & Micrometer & 0.154 & 21.3\\
+
+4. & $\mu$~Cassiopeiæ \dotfill & 5.5 & 1 & 1 & $+$&54 & 26
+& O. Struve &1856 & Micrometer & 0.342 & 9.5\rlap{\rdelim\}{2}{10pt}}\\
+&& & && &&&  Pritchard & 1887 & Photography & 0.036 & 90.6\\
+
+5. & Polaris \dotfill & 2 & 1 & 22 & \DPtypo{}{$+$} &88 & 46
+& Peters & 1846--47 & Mer.\ Circle & 0.073 & 44.7\rlap{\rdelim\}{2}{10pt}}\\
+&& & && &&&  Pritchard & 1887 & Photography & 0.052 & 62.7\\
+
+6. & $\epsilon$~Eridani \dotfill & 4.4 & 3 & 16 &  $-$&43 & 27
+& Elkin & 1882 & Heliometer & 0.143 & 22.8\\
+
+7. & $o_{2}$ Eridani \dotfill & 4.4 & 4 & 11 & $-$& 7 & 46
+& Gill & 1882 & Heliometer & 0.166 & 19.6\rlap{\rdelim\}{2}{10pt}}\\
+&& & && &&&  Hall & 1884 & Micrometer & 0.223 & 14.6\\
+
+8. & $\alpha$~Aurigæ \dotfill & 1  & 5 & 9 & $+$&45 & 54
+& Peters & 1846 & Mer.\ Circle & 0.046 &70.9\rlap{\rdelim\}{2}{10pt}}\\
+
+\multicolumn{2}{c|}{(Capella) }
+ & & && &&&  O. Struve & 1856 & Micrometer & 0.305 & 10.7\\
+
+9. & $\alpha$~Canis Maj. \dotfill & 1 & 6 & 41 & $-$&16 & 35
+& Gyldén & 1864 & Mer.\ Circle & 0.193 & 16.9\rlap{\rdelim\}{3}{10pt}}\\
+\multicolumn{2}{c|}{(Sirius) }
+ & & && &&&  Abbe & 1868 & Mer.\ Circle & 0.273 & 11.9\\
+&& & && &&&  Gill \& Elkin & 1882 & Heliometer & 0.380 & 8.6\\
+
+10. & $\alpha$~Geminorum \dotfill & 1.5& 7 & 28 & $+$&32 & 6
+& Johnson & 1856 & Heliometer& 0.198 & 16.5\\
+\multicolumn{2}{c|}{(Castor) } & & && &&& &&&&\\
+
+11. & $\alpha$~Canis Min. \dotfill & 1 & 7 & 34 & $+$& 5 & 29
+& Auwers & 1873 & Micrometer & 0.123 & 26.5\\
+\multicolumn{2}{c|}{(Procyon) } & & && &&& &&&&\\
+
+12. & $\iota$~Ursæ Maj. \dotfill & 3.5 & 8 & 52 & $+$&48 & 26
+& Peters & 1846 & Mer.\ Circle & 0.133 & 24.5\\
+
+13. & Lalande 21\,185 \dotfill & 7 & 10 & 56 & $+$&36 & 42
+& Winnecke
+& \begin{tabular}{@{}l}1858\rlap{\rdelim\}{2}{10pt}} \\
+                       1872
+  \end{tabular}
+& Micrometer & 0.506 & 6.6\\
+
+14. & Lalande 21\,258 \dotfill & 8.5 & 11 & 1 & $+$&44 & 02
+& Auwers & 1863 & Micrometer & 0.262 & 12.4\rlap{\rdelim\}{2}{10pt}}\\
+&& & && &&&  Krüger & 1864 & Micrometer & 0.260 & 12.6\\
+
+15. & Groombr'ge 1830 \dotfill & 6.5 & 11 & 7 & $+$&38 & 32
+& Peters & 1846 & Micrometer &0.226 & 14.4\rlap{\rdelim\}{5}{10pt}}\\
+&& & && &&&  Wichmann  & 1848 & Micrometer & 0.182 & 17.9\\
+&& & && &&&  O. Struve & 1850 & Micrometer & 0.034 & 95.9\\
+&& & && &&&  Johnson   & 1854 & Heliometer & 0.033 & 98.8\\
+&& & && &&&  Brünnow   & 1873 & Micrometer & 0.095 & 34.3\\
+
+16. & Oeltzen 11\,677 \dotfill & 9.5 & 11 & 15 & $+$&66 & 23
+& Geelmuyden & 1880 & Mer.\ Circle & 0.265 & 12.3\\
+
+17. & $\alpha$~Bootis \dotfill & 1 & 14 & 11 & $+$&19 & 42
+& Peters & 1846 & Mer.\ Circle& 0.127 & 25.7\rlap{\rdelim\}{2}{10pt}}\\
+\multicolumn{2}{c|}{(Arcturus) }
+& & && &&&  Johnson & 1856 & Heliometer & 0.138 & 23.6\\
+
+18. & $\alpha$~Centauri \dotfill & 1 & 14 & 33 & $-$&60 & 25
+& Henderson & 1842 & Mer.\ Circle & 0.913 & 3.6\rlap{\rdelim\}{3}{10pt}}\\
+&& & \multicolumn{5}{c|}{(Probably best.)}
+& Gill \& Elkin & 1882 & Heliometer & 0.750 & 4.35\\
+
+19. & Oeltzen 17\,415 \dotfill & 9 & 17 & 37 & $+$&68 & 28
+& Krüger & 1863 & Micrometer & 0.243 & 13.4\\
+
+20. & $\gamma$~Draconis \dotfill & 2 & 17 & 54 & $+$&51 & 30
+& \begin{tabular}{@{}l@{}l}
+    \hspace*{2pt}\llap{\ldelim\{{2}{6pt}} & Pond \\ & Gyldén
+  \end{tabular}
+& \begin{tabular}{@{}l}1817\rlap{\rdelim\}{2}{10pt}} \\ 1877 \end{tabular}
+& Mural Circle & 0.127 & 25.7\\
+
+21. & 70, p, Ophiuchi \dotfill & 4.5 & 18 & 00 & $+$& 2 & 33
+& Krüger & 1859--63 & Micrometer & 0.162 & 20.1\\
+
+22. & $\alpha$~Lyræ \dotfill & 1 & 18 & 34 & $+$&38 & 41
+& W. Struve & 1840 & Micrometer & 0.262 & 12.4\rlap{\rdelim\}{6}{10pt}}\\
+
+\multicolumn{2}{c|}{(Vega) }
+ & & && &&&  Peters    & 1846 & Mer.\ Circle & 0.103 & 32.4 \\
+&& & && &&&  Johnson   & 1856 & Heliometer   & 0.140 & 23.3 \\
+&& & && &&&  O. Struve & 1859 & Micrometer   & 0.140 & 23.3 \\
+&& & && &&&  Brünnow   & 1873 & Micrometer   & 0.188 & 17.3 \\
+&& & && &&&  Hall      & 1881 & Micrometer   & 0.134 & 24.3 \\
+
+23. & $\alpha$~Draconis \dotfill & 5 & 19 & 33 & $+$&69 & 30
+& Brünnow & 1870--73 & Micrometer & 0.234 & 14.0\\
+
+24. & 61 Cygni \dotfill & 5.5 & 21 & 2 & $+$&38 & 15
+& Bessel & 1838--40 & Heliometer & 0.348 & 9.4\rlap{\rdelim\}{8}{10pt}}\\
+&& & && &&&  Pogson    & 1853     & Heliometer  & 0.384 & 8.5  \\
+&& & && &&&  Johnson   & 1854     & Heliometer  & 0.397 & 8.2  \\
+&& & && &&&  O. Struve & 1852--53 & Micrometer  & 0.505 & 6.5  \\
+&& & && &&&  Auwers    & 1863     & Micrometer  & 0.564 & 5.8  \\
+&& & && &&&  Ball      & 1878     & Micrometer  & 0.465 & 7.0  \\
+&& & && &&&  Hall      & 1880--86 & Micrometer  & 0.270 & 12.1 \\
+&& & && &&&  Pritchard & 1887     & Photography & 0.432 & 7.55 \\
+
+25. & $\epsilon$~Indi \dotfill & 5.5 & 21 & 56 &  \DPtypo{}{$-$}  &57 & 69
+& Gill \& Elkin & 1881--82 & Heliometer & 0.222 & 14.6\\
+
+26. & Lacaille 9352 \dotfill & 7.5 & 22 & 59 &  \DPtypo{}{$-$}  &36 & 26
+& Gill & 1881 & Heliometer & 0.285 & 11.4\\
+
+27. & Bradley 3077 \dotfill & 6 & 23 &  8 &  \DPtypo{}{$+$}  &56 & 37
+& Brünnow & 1873 & Micrometer & 0.069 & 47.3\\
+
+28. & 85 Pegasi \dotfill & 6 & 23 & 57 &  \DPtypo{}{$+$}  &26 & 35
+& Brünnow & 1873 & Micrometer & 0.054 & 60.4\\
+\hline\hline
+\end{tabular}
+}%end group for footnotesize, etc
+%% -----File: 545.png---Folio 534-------
+\clearpage
+
+\begin{sidewaystable}
+\footnotesize
+\nblabel{app:V}
+\begin{tabular} {@{}r | l@{}l@{}| r @{} l @{ } r @{.} l | l @{} r @{} l @{ } r @{} l | r @{.} l | r @{.} l | c | c | l @{} l @{ } r @{} l | p{3.25cm}@{} }
+\multicolumn{23}{c}{TABLE V.---ORBITS OF BINARY STARS\@.  FROM HOUZEAU'S ``VADE MECUM.''}\\[1ex]
+\hline\hline
+& \multicolumn{2}{@{}c@{}|}{\textsc{Name of Star.\rule{0pt}{4ex}}}
+& \multicolumn{4}{c|}{$\alpha$ (1880).}
+& \multicolumn{5}{c|}{$\delta$ (1880).}
+& \multicolumn{2}{c|}{Period.}
+& \multicolumn{2}{c|}{a.}
+& c.
+&Periastron.
+& \multicolumn{4}{c|}{Magnitudes.}
+&Calculator of Orbit.
+\\[2ex]
+\hline
+  1 & $\delta$~Equulei \dotfill\rule{0pt}{3ex}
+&&21&$^{\text{h}}$ & $8^{\text{m}}$&6    &$+$& 9&$°$&32&$'$
+& \multicolumn{2}{l}{$10^{\text{y}} \text{ to } 12^{\text{y}}$}\vline&$0''$&30    &?    &  ?    &   &4.5,& 5&
+&\parbox[t]{9em}{\parindent=0em\hangindent=1em No satisfactory orbit yet computed.}
+\\
+ 2 & 42 Comæ Ber. \dotfill   && 13 &   & 4&2    & $+$&18 &    & 10 &   & 25&71  & 0&657 & 0.480 & 1869.92 &   & 6,  & 6 &    & Dubiago.
+\\
+ 3 & $\zeta$~Herculis \dotfill    && 16 &   & 38&8    & $+$&31&    & 49 &   & 34&411 & 1&284 & 0.463 & 1864.79 &   & 3,  & 5&.5   & Doberck.
+\\
+ 4 & $\eta$~Coronæ Bor. \dotfill    && 15 &   & 18&2    & $+$&30&    & 43 &   & 41&576 & 0&827   & 0.263 & 1850.26 &   & 6,  & 6&.5   & Duner.
+\\
+ 5 & Sirius \dotfill    && 6 &   & 39&7    & $-$&16 &    & 32 &   & 44&0       & 8&53    & 0.591 & 1889.44    & $-$&1,  & 9&   & Pritchard.
+\\
+ 6 & $\mu_{2}$ Herculis \dotfill    && 17 &   & 41&8    & $+$&27&    & 48 &   & 54&25      & 1&46    & 0.302 & 1877.13    &   & 9.5, & 10&.5   & Doberck.
+\\
+ 7 & $\zeta$~Cancri \dotfill  && 8 &   & 5&3    & $+$&18&    & 01 &   & 60&327       & 0&853   & 0.391 & 1866.02    &   & 5.5, & 6&.2   & Seeliger.
+\\
+ 8 & $\xi$~Ursæ Maj. \dotfill   && 11 &   & 11&8    & $+$&32&    & 13 &   & 60&80      & 2&580   & 0.416 & 1875.26    &   & 4,  & 5 &   & Pritchard.
+\\
+ 9 & $\alpha$~Centauri \dotfill   && 14 &   & 32&0    & $-$&60&    & 22 &   & 77&42      & 17&50    & 0.526 & 1875.97    &   & 1,  & 4 &   & Elkin.
+\\
+10 & 70, p, Ophiuchi \dotfill   && 17 &   & 59&4    & $+$& 2 &    & 32 &   & 94&44      & 4&790   & 0.467 & 1808.90    &   & 4.5, & 6 &   & Pritchard.
+\\
+11 & $\gamma$~Coronæ Bor. \dotfill    && 15 &   & 37&7    & $+$&26 &    & 40 &   & 95&50      & 0&70    & 0.350 & 1843.70    &   & 4,  & 7 &   & Doberck.
+\\
+12 & $\omega$~Leonis \dotfill   && 9 &   & 22&0    & $+$& 9 &    & 35 &    & 110&82      & 0&890   & 0.536 & 1841.81    &   & 6,  & 7 &   & Doberck.
+\\
+13 & $\xi$~Bootis \dotfill    && 14 &   & 45&8    & $+$&19 &    & 36 &    & 127&35      & 4&860   & 0.708 & 1770.69    &   & 4.5, & 6&.5   & Doberck.
+\\
+14 & 4 Aquarii \dotfill   && 20 &   &45&0    & $-$& 6 &    & 5 &    & 129&84      & 0&717   & 0.461 & 1751.96    &   & 6,  & 7 &   & Doberck.
+\\
+15 & $\gamma$~Virginis \dotfill   && 12 &   & 35&6    & $-$& 0 &    & 47 &    & 185&01      & 3&97    & 0.896 & 1836.68    &   & 3,  & 3&.2   & Thiele.
+\\
+16 & $\eta$~Cassiopeiæ \dotfill   && 0 &   & 41&8    & $+$&57 &    & 11 &    & 195&24      & 8&639   & 0.624 & 1901.95    &   & 4,  & 7&.3   & Gruber.
+\\
+17 & 36 Andromedæ \dotfill    && 0 &   & 48&0    & $+$&23 &    & 53 &    & 349&10      & 1&54    & 0.634 & 1798.80    &   & 6,  & 7 &   & Doberck.
+\\
+18 & $\gamma$~Leonis \dotfill   && 10 &   & 31&3    & $+$&20 &    & 27 &    & 407&04      & 1&98    & 0.733 & 1741.00    &   & 2,  & 3&.5   & Doberck.
+\\
+19 & $\sigma$~Coronæ Bor. \dotfill    && 16 &   & 10&2    & $+$&34 &    & 10 &    & 845&86      & 5&88    & 0.752 & 1826.93    &   & 5.5, & 6&.5   & Doberck.
+\\
+20 & $\alpha$~Geminorum \dotfill    && 7 &   & 27&0    & $ +$&32 &    & 2 &    &\ 996&85      & 7&538   & 0.344 & 1750.33    &   & 2.5, & 3 &   & Thiele.
+\\[1ex]
+\hline\hline
+\end{tabular}
+\end{sidewaystable}
+%% -----File: 546.png---Folio 535-------
+\clearpage
+
+\begin{center}
+\nblabel{app:VI}
+TABLE VI.---THE PRINCIPAL VARIABLE STARS.
+\end{center}
+
+{\small
+A selection from S. C. Chandler's catalogue of 225 variables (Astronomical Journal, Sept.\
+1888), containing such as are visible to the naked eye, have a range of variation exceeding half
+a magnitude, and can be seen in the United States.
+
+\bigskip
+\tiny
+\noindent
+\renewcommand{\arraystretch}{1.4}
+\setlength\tabcolsep{4pt}
+\begin{tabular}{r|l| *{2}{r@{}c@{\,}r@{}l|} l@{ }c@{}r@{}l| D{!}{}{-1}| @{}l }
+\hline\hline
+
+& \multicolumn{1}{c|}{\multirow{2}{5em}{\centering \textsc{Name}.\rule{0pt}{4ex}}}
+& \multicolumn{8}{c|}{Place, 1900.\rule{0pt}{3ex}}
+& \multicolumn{4}{m{5em}|}{\multirow{2}{4.5em}{\centering \rule{0pt}{3ex}Range of Variation.}}
+& \multicolumn{1}{c|}{\multirow{2}{7em}{\centering Period (days).\rule{0pt}{4ex}}}
+& \multicolumn{1}{c}{\multirow{2}{*}{Remarks.\rule{0pt}{4ex}}}
+\\[1ex]
+\cline{3-10}
+
+  \rotatebox{90}{\rlap{\textsc{\:No}.}}
+&& \multicolumn{4}{c|}{$\alpha$\rule{0pt}{3ex}}
+ & \multicolumn{4}{c|}{$\delta$} &&&&&
+\\[1ex]
+\hline
+
+1 & T Ceti\dotfill \rule{0pt}{3ex}
+& 0&${}^\text{h}$& 16${}^\text{m}$.&7  & $-20$&°& 37&$'$
+& 5.1 & to & 7 && 65!\ ± &\hspace*{0.6em}Irregular.
+\\
+2 & R Andromedæ\dotfill
+& 0 && 18.8 && $+38$ &&  1 && 5.6 && 13 && 411!.2 &
+\\
+3 & R Sculptoris\dotfill
+& 1 && 22.4 && $-33$ &&  4 && 5.8 && 7&.8 & 207!
+& \multirow{3}{*}{$\left\{\parbox{7em}{\textit{Mira}. Variations in length of period.}\right.$}
+\\
+4 & $o$ Ceti\dotfill
+& 2 && 14.3 && $-\phantom{0}3$ && 26 && 1.7 && 9&.5 & 331!.3363 &
+\\
+5 & $\rho$~Persei\dotfill
+& 2 && 58.7 && $+38$ && 27 && 3.4 && 4&.2 &  33!  &
+\\
+6 & $\beta$~Persei\dotfill
+&  3 &&  1.6 && $+40$ && 34 && 2.3 && 3&.5
+& 2^\text{d}\, 23^\text{h}\, 48!^\text{m}\, 55^\text{s}.43
+& \smash{$\Bigl\{$\parbox{7em}{\textit{Algol}. Period now shortening.}}
+\\
+7 & $\lambda$~Tauri\dotfill
+&  3 && 55.1 && $+12$ && 12 && 3.4 && 4&.2
+& 3^\text{d}\, 22^\text{h}\, 52!^\text{m}\, 12^\text{s}
+&
+\\
+8 & $\epsilon$~Aurigæ\dotfill
+&  4 && 54.8 && $+43$ && 41 && 3   && 4&.5
+&\multicolumn{1}{c|}{\hspace*{0.6em}Irregular}
+&\smash{$\Bigl\{$\parbox{7em}{Algol type, but irregular.}}
+\\
+9 & $\alpha$~Orionis\dotfill
+&  5 && 49.7 && $+\phantom{0}7$ && 23 && 1 && 1&.6 & 196!\ ?
+& \hspace*{0.6em}Irregular.
+\\
+10 & $\eta$~Geminorum\dotfill
+&  6 &&  8.8 && $+22$ && 32 && 3.2 && 4&.2 & 229!.1  &
+\\
+11 & $\zeta$~Geminorum\dotfill
+&  6 && 58.2 && $+20$ && 43 && 3.7 && 4&.5
+& 10^\text{d}\, \phantom{0}3^\text{h}\, 41!^\text{m}\, 30^\text{s} &
+\\
+12 & R Canis Maj.\dotfill
+&  7 && 14.9 && $-16$ && 12 && 5.9 && 6&.7
+& 1^\text{d}\, \phantom{0}3^\text{h}\, 15!^\text{m}\, 55^\text{s}
+& \hspace*{0.6em}Algol type.
+\\
+13 & R Leonis Min.\dotfill
+&  9 && 39.6 && $+34$ && 58 && 6   && 13 && 373!.5 & \hspace*{0.6em}Period short'ing.
+\\
+14 & R Leonis\dotfill
+&  9 && 42.2 && $+11$ && 54 && 5.2 && 10 && 312!.87 &
+\\
+15 & U Hydræ\dotfill
+& 10 && 32.6 && $-12$ && 52 && 4.5 &&  6&.3 & 194!.65 &
+\\
+16 & R Ursæ Maj.\dotfill
+& 10 && 37.6 && $+69$ && 18 && 6.0 && 13&.2 & 305!.4
+& \hspace*{0.6em}Period short'ing.
+\\
+17 & R Hydræ\dotfill
+& 13 && 24.2 && $-22$ && 46 && 3.5 &&  9&.7 & 496!.91
+& \hspace*{0.6em}Period short'ing.
+\\
+18 & S Virginis\dotfill
+& 13 && 27.8 && $-\phantom{0}6$ && 41 && 5.7 && 12&.5 & 376!.0 &
+\\
+19 & R Bootis\dotfill
+& 14 && 32.8 && $+27$ && 10 && 5.9 && 12&.2 &  223!.9 &
+\\
+20 & $\delta$~Libræ\dotfill
+& 14 && 55.6 && $-\phantom{0}8$ && 7 && 5.0 && 6&.2
+& 2^\text{d}\, \phantom{0}7^\text{h}\, 51!^\text{m}\, 22^\text{s}.8 & \hspace*{0.6em}Algol type.
+\\
+21 & S Coronæ\dotfill
+& 15 && 17.3 && $+31$ && 44 && 6.0 && 12&.5 & 360!.57 &
+\\
+22 & R Coronæ\dotfill
+& 15 && 44.4 && $+28$ && 28 && 5.8 && 13&
+& \multicolumn{1}{c|}{\quad Irregular} &
+\\
+23 & R Serpentis\dotfill
+& 15 && 46.1 && $+15$ && 26 && 5.6 && 13&   & 357!.6 &
+\\
+24 & $\alpha$~Herculis\dotfill
+& 17 && 10.1 && $+14$ && 30 && 3.1 &&  3&.9
+& \multicolumn{2}{c}{Two or three months, but very irreg.}
+\\
+25 & U Ophiuchi\dotfill
+& 17 && 11.5 && $+\phantom{0}1$ && 19 && 6.0 && 6&.7
+& 20^\text{h}\, \phantom{0}7!^\text{m}\, 41^\text{s}.6
+& \smash{$\Bigl\{$\parbox{7em}{Shortest period known.}}
+\\
+26 & X Sagittarii\dotfill
+& 17 && 41.3 && $-27$ && 48 && 4   &&  6&   & 7!.01185 &
+\\
+27 & W Sagittarii\dotfill
+& 17 && 58.6 && $-29$ && 35 && 5   &&  6&.5 & 7!.59445 &
+\\
+28 & Y Sagittarii\dotfill
+& 18 && 15.5 && $-18$ && 54 && 5.8 &&  6&.6 & 5!.76900 &
+\\
+29 & R Scuti\dotfill
+& 18 && 42.1 && $-\phantom{0}5$ && 49 && 4.7 && 9&   & 71!.10
+& \multirow{3}{8em}{$\Biggl\{$\parbox{7em}{Secondary minimum about midway.}}
+\\
+30 & $\beta$~Lyræ\dotfill
+& 18 && 46.4 && $+33$ && 15 && 3.4 &&  4&.5
+& 12^\text{d}\ 21^\text{h}\, 46!^\text{m}\, 58^\text{s}.3 &
+\\
+31 & R Cygni\dotfill
+& 19 && 34.1 && $+49$ && 58 && 5.9 && 13&   & 425!.7 &
+\\
+32 & $\chi$~Cygni\dotfill
+& 19 && 46.7 && $+32$ && 40 && 4.0 && 13&.5 & 406!.045
+& \hspace*{0.6em}Period length'ing
+\\
+33 & $\eta$~Aquilæ\dotfill
+& 19 && 47.4 && $+\phantom{0}0$ && 45 && 3.5 && 4&.7
+& 7^\text{d}\, \phantom{0}4^\text{h}\, 14!^\text{m}\, \phantom{0}0^\text{s}.0 &
+\\
+34 & S Sagittæ\dotfill
+& 19 && 51.4 && $+16$ && 22 && 5.6 && 6&.4
+& 8^\text{d}\, \phantom{0}9^\text{h}\, 11!^\text{m} &
+\\
+35 & X Cygni\dotfill
+& 20 && 39.5 && $+35$ && 13 && 6.4 && 7&.7
+& 15^\text{d}\, 14^\text{h}\, 24!^\text{m}\,
+& \smash{$\Bigl\{$\parbox{7em}{Minimum not constant.}}
+\\
+36 & T Vulpeculæ\dotfill
+& 20 && 47.2 && $+27$ && 52 && 5.5 && 6&.5
+& 4^\text{d}\, 10^\text{h}\, 29!^\text{m}
+\\
+37 & T Cephei\dotfill
+& 21 &&  8.2 && $+68$ && \phantom{0}5  &&  5.6 && 9&.9 & 383!.20 &
+\\
+38 & $\mu$~Cephei\dotfill
+& 21 && 40.4 && $+58$ && 19 && 4   &&  5&   & 432!\ ? &
+\\
+39 & $\delta$~Cephei\dotfill
+& 22 && 25.4 && $+57$ && 54 && 3.7 &&  4&.9
+& 5^\text{d}\, \phantom{0}8^\text{h}\, 47!^\text{m}\, 39^\text{s}.97 &
+\\
+40 & $\beta$~Pegasi\dotfill
+& 22 && 58.9 && $+27$ && 32 && 2.2 &&  2&.7
+& \multicolumn{1}{c|}{\quad Irregular} &
+\\
+41 & R Aquarii\dotfill
+& 23 && 38.6 && $-15$ && 50 && 5.8 && 11&   & 387!.16 &
+\\
+42 & R Cassiopeiæ\dotfill
+& 23 && 53.3 && $+50$ && 50 && 4.8 && 12&   & 429!.00
+\\[0.5ex]\hline\hline
+\end{tabular}
+} %end group for font size changes etc
+\chelabel{APPENDIX}
+%% -----File: 547.png---Folio 536-------
+%% -----File: 548.png---Folio 537-------
+
+\clearpage
+\pdfbookmark[1]{Index}{Index}
+\chapter*{INDEX.}
+\chslabel{INDEX}
+\markboth{INDEX.}{}
+\setlength\headheight{20.5pt}
+\renewcommand\headrule{\vspace{-6pt}}
+\fancyhead[C]{\begin{tabular}{c}\footnotesize\leftmark\\[-0.5ex]
+\tiny[All references, unless expressly stated to the contrary, are to \textit{articles} and not to \textit{pages}.]\end{tabular}}
+\fancyhead[R]{\begin{tabular}{r}\thepage\\[-0.5ex]
+\tiny~\end{tabular}}
+
+\thispagestyle{empty}
+\nbrule\bigskip
+
+{
+{\centering\scriptsize
+[All references, unless expressly stated to the contrary, are to \textit{articles} and not to \textit{pages}.]
+\par}
+
+\setlength\columnseprule{0.5pt}%
+\setlength\columnsep{1.5em}%
+\begin{multicols}{2}
+\sloppy
+\footnotesize
+\setlength\parindent{0pt}%
+\setlength\parskip{0pt plus 0.1pt}%
+
+\idxsection*{A.}
+
+\idxb{Aberration} of light, annual, \idxart{99}, \idxart{224}--\idxart{226};
+used to determine the solar parallax, \idxart{692};
+diurnal, \idxart{226*};
+spherical and chromatic, \idxart{39}.
+
+\idxc{Aboul Wefa}, discoverer of the lunar variation, \idxart{457}.
+
+\idxb{Absolute} scale of stellar magnitude, \idxart{819}.
+
+\idxb{Acceleration} of Encke's comet, \idxart{710};
+of Winnecke's comet, \idxart{711};
+of the sun's equator, \idxart{283}--\idxart{285};
+secular, of moon's mean motion, \idxart{459}--\idxart{461};
+secular, of moon's mean motion as affected by meteors, \idxart{778}.
+
+\idxb{Achromatic} object-glasses for telescopes, \idxart{41}.
+
+\idxb{Actinometer} of Violle, \idxart{341}.
+
+\idxc{Adams, J.~C.}, the discovery of Neptune, \idxart{654};
+investigation of the orbit of the Leonids, \idxart{785}.
+
+\idxb{Adjustments} of the transit instrument, \idxart{60}.
+
+\idxb{Aerolites}.    See \idxsee{Meteorites}.
+
+\idxb{Age}, relative, of the planets, \idxart{913}, \idxart{915};
+of the solar system, \idxart{922};
+of the sun, \idxart{359}.
+
+\idxb{Air-currents }at high elevations, \idxart{773}, \textit{note}.
+
+\idxc{Airy, G.~B.}, density of the earth, \idxart{169}.
+
+\idxb{Albedo} defined and determined, \idxart{546};
+of Jupiter, \idxart{644};
+of Mars, \idxart{583};
+of Mercury, \idxart{558};
+of the Moon, \idxart{259};
+of Neptune, \idxart{660};
+of Saturn, \idxart{636};
+of Uranus, \idxart{648};
+of Venus, \idxart{572}.
+
+\idxb{Algol}, or $\beta$~Persei, \idxart{848}.
+
+\idxb{Almagest} of Ptolemy, \idxart{500}, \idxart{700}, \idxart{795}.
+
+\idxb{Almucantar} defined, \idxart{12}.
+
+\idxb{Altitude} defined, \idxart{21};
+parallels of, \idxart{12};
+of pole equals latitude, \idxart{30};
+of sun, how measured with sextant, \idxart{77}.
+
+\idxb{Altitude} and azimuth instrument, \idxart{71}.
+
+\idxb{Amplitude} defined, \idxart{22}.
+
+\idxb{Andromeda}, the nebula in, \idxart{886};
+temporary star in the nebula of, \idxart{845}.
+
+\idxb{Andromedes}, the, \idxart{780}, \idxart{784}, \idxart{786}.
+
+\idxb{Angle}, position, of a double star, \idxart{868};
+of the vertical, \idxart{156}.
+
+\idxb{Angular} and linear dimensions, \idxart{5};
+velocity under central force, its law, \idxart{408}, \idxart{409}.
+
+\idxb{Annual} equation of the moon's motion, \idxart{458};
+motion of the sun, \idxart{172}, \idxart{173}.
+
+\idxb{Annular} eclipse, \idxart{382};
+nebula in Lyra, \idxart{888}.
+
+\idxb{Anomalistic} month, the, \idxart{397}, \textit{note};
+revolution of the moon, \idxart{250};
+year defined, \idxart{216}.
+
+\idxb{Anomaly} defined, mean and true, \idxart{189}.
+
+\idxb{Apertures}, limiting, in photometry, \idxart{825}.
+
+\idxb{Apex} of the sun's way, \idxart{805}.
+
+\idxb{Apparition}, perpetual, circle of, \idxart{33}.
+
+\idxb{Apsides}, line of, defined, \idxart{183};
+its revolution in case of the earth's orbit, \idxart{199};
+its revolution in case of the moon's orbit, \idxart{454};
+its revolution in case of the planets' orbits, \idxart{527}.
+
+\idxb{Arc} of meridian, how measured, \idxart{147}.
+
+\idxb{Areal} or areolar velocity, law of, under
+central force, \idxart{402}--\idxart{406}.
+
+\idxb{Areas}, equable description of, in earth's orbit, \idxart{186}, \idxart{187}.
+
+\idxc{Argelander}, his \textit{Durchmusterung} and zones, \idxart{795}, \idxart{833};
+his star magnitudes, \idxart{817}, \idxart{833}.
+
+\idxb{Argus}, $\eta$, \idxart{841}.
+
+\idxb{Ariel}, the inner satellite of Uranus, \idxart{650}.
+
+\idxb{Aries}, first of, \idxart{17}.
+
+\idxc{Aristarchus}, method of determining the sun's distance, \idxart{666}, \idxart{670}.
+
+\idxb{Artificial} horizon, the, \idxart{78}.
+
+\idxb{Ashes} of meteors, \idxart{775}.
+
+\idxb{Aspects} of planets defined by diagram, \idxart{494}.
+
+\idxb[Asteroid]{Asteroids}, the, or minor planets, \idxart{592}--\idxart{601}\DPtypo{,}{;}
+theories as to their origin, \idxart{600}.
+
+\idxb{Astræa}, the fifth asteroid, discovered by Heneke, \idxart{593}.
+%% -----File: 549.png---Folio 538-------
+
+\idxb{Astro-Physics} defined, \idxart{2}.
+
+\idxb{Atlases} of the stars, \idxart{793}.
+
+\idxb{Atmosphere} of the moon, \idxart{255}--\idxart{257};
+of Venus, \idxart{573};
+height of the earth's, \idxart{98}.
+
+\idxb{Attraction}, intensity of the solar, on the earth, \idxart{436};
+within a hollow sphere,\idxart{169};
+of universal gravitation, \idxart{161}, \idxart{162}.
+
+\idxb{Axis} of the earth, its direction, \idxart{14};
+of the earth, disturbed by precession, \idxart{206};
+of the sun, its direction, \idxart{282}.
+
+\idxb{Azimuth} defined, \idxart{22};
+determination of, \idxart{127};
+method of reckoning, \idxart{22};
+of transit instrument, its adjustment, \idxart{60}.
+
+\idxsection* {B.}
+
+\idxc{Baily}, determination of the density of
+the earth, \idxart{166}.
+
+\idxb{Balance}, common, used in determining
+the density of the earth, \idxart{170};
+torsion, used in determining the density of the
+earth, \idxart{165}.
+
+\idxb{Barometer}, changes of, affecting atmospheric
+refraction, \idxart{91};
+effect on height of the tides, \idxart{480}.
+
+\idxb{Barometric} error of a clock and its compensation,
+52.
+
+\idxb{Beginning} of the day, \idxart{123};
+of the year, \idxart{222}.
+
+\idxc{Benzenberg}, experiments on the deviation
+of falling bodies, \idxart{138}.
+
+\idxc{Bessel}, the parallax of 61 Cygni, \idxart{809}, \idxart{811};
+formation of comets' tails, \idxart{728};
+his ``zones,'' \idxart{795}.
+
+\idxc{Biela's} comet, \idxart{744}--\idxart{746}.
+
+\idxb{Bielids}, the, \idxart{746}, \idxart{780}, \idxart{784}, \idxart{786}.
+
+\idxb{Bielid} meteorite, Mazapil, \idxart{784}.
+
+\idxb[Binary Stars]{Binary stars}, \idxart{872}--\idxart{875};
+number known at present, \idxart{872};
+their masses, \idxart{877}, \idxart{878};
+their ``mass-brightness,'' \idxart{879};
+their orbits, \idxart{875}--\idxart{877}.
+
+\idxb{Bissextile} year, explanation of term, \idxart{219}.
+
+\idxb{Black Drop}, the, at a transit of Venus,
+681.
+
+\idxc{Bode's} law, \idxart{488}, \idxart{489}.
+
+\idxb{Bolides}, or detonating meteors, \idxart{768}.
+
+\idxb{Bolometer}, the, of Langley, \idxart{343}.
+
+\idxc{Bond, G. P.}, first photograph of a double
+star, \idxart{868}.
+
+\idxc{Bond, W. C.}, discovery of Hyperion, \idxart{643};
+of Saturn's dusky ring, \idxart{637}.
+
+\idxc{Boyle}, law of, \idxart{360}, \textit{note}.
+
+\idxc{Brahe, Tycho}.  See \idxsee{Tycho}.
+
+\idxc{Bredichin}, his theory of comets' tails,
+731, \idxart{732}.
+
+\idxb{Brightness} of comets, \idxart{699};
+of planets in various positions, Mercury, \idxart{551};
+Venus, \idxart{563}, \idxart{568};
+Mars, \idxart{579};
+Asteroids, \idxart{596}, \idxart{599};
+Jupiter, \idxart{610};
+Saturn, \idxart{632};
+Uranus, \idxart{647};
+Neptune, \idxart{660};
+of an object in the telescope, \idxart{38};
+of shooting stars, \idxart{773};
+of stars, causes of the difference in this respect, \idxart{836};
+of stars, its measurement, \idxart{823}--\idxart{831}.
+
+\idxsection* {C.}
+
+\idxb{Calendar}, the, \idxart{217}--\idxart{223}.
+
+\idxb{Callisto}, the outer satellite of Jupiter,
+621, \idxart{627}.
+
+\idxb{Calories} of different magnitude, \idxart{338}, \textit{note.}
+
+\idxb{Candle} power, its mechanical equivalent, \idxart{776};
+power of sunlight, \idxart{332}, \idxart{333}.
+
+\idxb{Candle} standard, \idxart{333}, \textit{note.}
+
+\idxb{Capture} theory of comets, \idxart{740}.
+
+\idxb{Cardinal} points defined, \idxart{20}.
+
+\idxc{Carlini}, earth's density, \idxart{168}.
+
+\idxc{Carrington}, law of the sun's rotation,
+283, \idxart{284}.
+
+\idxc{Cassegrainian} telescope, \idxart{48}.
+
+\idxc{Cassini, J. D.}, discovery of the division
+in  Saturn's  ring, \idxart{637};
+discovery of four satellites of Saturn, \idxart{643}.
+
+\idxb{Catalogues} of stars, \idxart{795}.
+
+\idxc{Cavendish}, the torsion balance, \idxart{165}.
+
+\idxb{Celestial} latitude and longitude, \idxart{178}, \idxart{179};
+sphere, conceptions of it, \idxart{4}.
+
+\idxb{Cenis, Mt.}, determination of the earth's
+density, \idxart{168}.
+
+\idxb{Central} force, motion under it, \idxart{400}--\idxart{410};
+force, its measure in case of circular motion, \idxart{411}.
+
+\idxb{Central} suns, \idxart{807}, \idxart{903}.
+
+\idxb{Centrifugal} force of the earth's rotation,
+154.
+
+\idxb{Ceres}, discovery of, \idxart{592}.
+
+\idxc{Chandler, S. C.}, catalogue of variable
+stars, \idxart{852}, Appendix, \hyperref[app:VI]{Table VI.}
+
+\idxb{Changes} on the moon's surface, \idxart{268};
+in the nebulæ, \idxart{892}.
+
+\idxb{Characteristics} of different meteoric
+swarms, \idxart{783}.
+
+\idxb{Charts} of the stars, \idxart{798}.
+
+\idxb{Chemical} elements recognized in comets, \idxart{724}, \idxart{725};
+elements recognized in stars, \idxart{856};
+elements recognized in the sun, \idxart{315}--\idxart{317}.
+
+\idxb{Chromatic} aberration of a lens, \idxart{39}.
+
+\idxb{Chromosphere}, the, \idxart{291}, \idxart{322}, \idxart{363}.
+
+\idxb{Chronograph}, the, \idxart{56}.
+
+\idxb{Chronometer}, the, \idxart{54};
+longitude by, \idxart{121} [A].
+
+\idxb{Circle}, the meridian, \idxart{63}.
+
+\idxb{Circles} of perpetual apparition and occultation, \idxart{33}.
+
+\idxb{Circular} motion, central force in, \idxart{411}.
+
+\idxc{Clairaut's} equation concerning the
+ellipticity of the earth, \idxart{155}.
+%% -----File: 550.png---Folio 539-------
+
+\idxc{Clarke, Col}., dimensions of the earth,
+145 and \hyperref[app:spheroid]{Appendix}.
+
+\idxb{Classification} of stellar spectra, \idxart{857}, \idxart{858}.
+
+\idxc{Clerke, Miss A. M.}, her history of astronomy,
+Preface, \idxart{570}, \idxart{626}, \textit{note}, \idxart{723},
+746, \idxart{900}.
+
+\idxb{Clocks}, general remarks on, \idxart{50}.
+
+\idxb{Clock-breaks} (electric), \idxart{57}.
+
+\idxb{Clock-error}, or correction, and rate, \idxart{53};
+or correction determined by transit instrument, \idxart{59}.
+
+\idxb{Clusters} of stars, \idxart{883}--\idxart{885}.
+
+\idxc{Coggia's} Comet, \idxart{730}.
+
+\idxb{Collimating} eye-piece, \idxart{67}.
+
+\idxb{Collimation} of transit instrument, \idxart{60}.
+
+\idxb{Collimator}, the, used with transit instrument, \idxart{60};
+of a spectroscope, \idxart{311}.
+
+\idxb{Collision} theory of variable stars, \idxart{850}.
+
+\idxb{Colors} of stars in photometry, \idxart{830};
+of double stars, \idxart{867}.
+
+\idxb{Colures} defined, \idxart{25}.
+
+\idxb{Comet}, Biela's, \idxart{744}; Donati's, \idxart{727}, \idxart{730}, \idxart{747};
+Encke's, \idxart{710}, \idxart{743};
+great, of 1882, \idxart{748}--\idxart{752};
+Halley's, \idxart{742};
+Winnecke's, \idxart{711}.
+
+\idxb{Comets}, acceleration of Encke's and Winnecke's,
+710, \idxart{711};
+brightness of, \idxart{699}, \idxart{723};
+capture theory of, \idxart{740};
+chemical elements in, \idxart{724}, \idxart{725};
+constituent parts, \idxart{713};
+contraction of head when near the sun, \idxart{715};
+danger from, \idxart{753}, \idxart{754};
+density of, \idxart{720};
+designation of, \idxart{697};
+dimensions of, \idxart{714}, \idxart{717};
+ejection theory, \idxart{741};
+fall upon earth or sun, probable effect, \idxart{754};
+groups of, with similar orbits, \idxart{705};
+their light, \idxart{721};
+their masses, \idxart{718}, \idxart{719};
+and meteors, their connection, \idxart{785}--\idxart{787};
+nature of, \idxart{737};
+their orbits, \idxart{700}--\idxart{709};
+origin of, \idxart{738}--\idxart{741};
+perihelia, distribution of, \idxart{706};
+physical characteristics, \idxart{712};
+planetary families of, \idxart{739};
+their spectra, \idxart{724}, \idxart{726};
+superstitions regarding them, \idxart{695};
+their tails or trains, \idxart{713}, \idxart{717}, \idxart{728}--\idxart{736};
+variations in brightness, \idxart{723};
+visitors in the solar system, \idxart{709}.
+
+\idxb{Comparison} of starlight with sunlight,
+334, \idxart{832}.
+
+\idxb{Compensation} pendulums, \idxart{51}.
+
+\idxb{Compensation} of pendulum for barometric
+changes, \idxart{52}.
+
+\idxb{Components} of the disturbing force, \idxart{445}.
+
+\idxb{Co-ordinates}, astronomical, \idxart{20}.
+
+\idxc{Common, A. A.}, photographs of nebulæ,
+893.
+
+\idxb{Conics}, the. \idxart{422}, \idxart{423}.
+
+\idxb{Connection} between comets and meteors,
+785--\idxart{787}.
+
+\idxb{Constant} of aberration, the, \idxart{225};
+the solar, \idxart{338}--\idxart{340}.
+
+\idxb{Constancy}, secular, of the mean distances
+and periods of the planets, \idxart{526}.
+
+\idxb{Constellations}, list of, \idxart{792};
+their origin, \idxart{791}.
+
+\idxb{Contact} observations, transit of Venus,
+679--\idxart{682}.
+
+\idxb{Contraction} theory of solar heat, \idxart{356}.
+
+\idxb{Conversion} of R. A. and Decl.\ to latitude
+and longitude, \idxart{180}.
+
+\idxc{Copernicus}, his system, \idxart{503};
+``Triumphans,'' \idxart{809}.
+
+\idxc{Cornu}, determination of the earth's density, \idxart{166};
+photometric observation of
+eclipses of Jupiter's satellites, \idxart{630}.
+
+\idxb{Corona}, the solar, \idxart{291}, \idxart{327}--\idxart{331}, \idxart{364}.
+
+\idxb{Cosmogony}, \idxart{905}--\idxart{917}.
+
+\idxb{Cotidal} lines, \idxart{475}.
+
+\idxb{Craters} on the moon, \idxart{265}--\idxart{267}.
+
+\idxc{Crew, H.}, spectroscopic observations of
+the sun's rotation, \idxart{285}, \textit{note}.
+
+\idxb{Crust} of meteorites, \idxart{761}.
+
+\idxb{Curvature} of comet's tails, \idxart{729}.
+
+\idxb{Curvilinear} motion the effect of force, \idxart{401}.
+
+\idxb{Cycle}, the metonic, \idxart{218}.
+
+\idxb{Cyclones} as proofs of the earth's rotation,
+143.
+
+\idxsection* {D.}
+
+\idxc{Dalton}, his law of gaseous mixtures, \idxart{360},
+\textit{note}.
+
+\idxb{Danger} from comets, \idxart{753}, \idxart{754}.
+
+\idxb{Darkening} of the sun's limb, \idxart{337}.
+
+\idxc{Darwin, G. H.}, rigidity of the earth, \idxart{171};
+tidal evolution, \idxart{484}, \idxart{916}.
+
+\idxc{Dawes}, diameter  of  the  spurious  discs
+of stars, \idxart{43};
+nucleoli in sun spots, \idxart{293}.
+
+\idxb{Day}, the civil and the astronomical, \idxart{117};
+effect of tidal friction upon its length, \idxart{461};
+changes in its length, \idxart{144};
+where it begins, \idxart{123}.
+
+\idxb{Declination} defined, \idxart{23};
+parallels of, \idxart{23};
+determined with the meridian circle, \idxart{128}.
+
+\idxb{Degree} of the meridian, how measured,
+135, \idxart{147}.
+
+\idxb{Deimos}, the outer satellite of Mars, \idxart{590},
+591.
+
+\idxc{Delisle}, method of determining the solar
+parallax, \idxart{682}.
+
+\idxc{Denning}, drawings of Jupiter's red spot,
+618.
+
+\idxb{Density} of comets, \idxart{720};
+of the earth, determinations of it, \idxart{164}--\idxart{170};
+of the moon, \idxart{246};
+of a planet, how determined, \idxart{540};
+of the sun, \idxart{279}.
+
+\idxb{Detonating} meteors, or ``Bolides,'' \idxart{768}.
+%% -----File: 551.png---Folio 540-------
+
+\idxb{Development} of sun spots, \idxart{297}.
+
+\idxb{Dhurmsala} meteorite, ice-coated, \idxart{765}.
+
+\idxb{Diameter} (apparent) as related to distance, \idxart{6};
+of a planet, how measured, \idxart{534}.
+
+\idxb{Differential} method of determining a body's place, \idxart{129};
+method of determining stellar parallax, \idxart{811}.
+
+\idxb{Diffraction} of an object-glass, \idxart{43}.
+
+\idxb{Dione}, fourth satellite of Saturn, \idxart{643}, \textit{note}.
+
+\idxb{Dip} of the horizon, \idxart{81}.
+
+\idxb{Disc}, spurious, of stars in telescope, \idxart{43}.
+
+\idxb{Discovery} of comets, \idxart{698}.
+
+\idxb{Dissipation} of energy, \idxart{925}.
+
+\idxb{Distance} of the moon, \idxart{239};
+of the nebulæ, \idxart{896};
+and parallax, relation between, \idxart{84};
+of a planet in astronomical units, how determined, \idxart{515}--\idxart{518};
+of the stars, \idxart{808}--\idxart{815};
+of the sun, \idxart{274}, \idxart{275}, also \chapref{CHAPTERXVI}{Chap.~XVI.}
+
+\idxb{Distinctness} of telescopic image, its conditions, \idxart{39}.
+
+\idxb{Distribution} of the nebulæ, \idxart{895};
+of the stars, \idxart{899};
+of the sun spots, \idxart{301}.
+
+\idxb{Disturbing} force, the, \idxart{439}--\idxart{444};
+force, diagram of, \idxart{441};
+force, its resolution into components, \idxart{445}.
+
+\idxb{Diurnal} aberration, \idxart{226*};
+inequality of the tides, \idxart{471};
+parallax, \idxart{82}, \idxart{86};
+phenomena in various latitudes, \idxart{191}.
+
+\idxb{Divisions} of astronomy, \idxart{2}.
+
+\idxc{Doerfel} proves that a comet moves in a parabola, \idxart{700}.
+
+\idxc{Donati's} comet, \idxart{727}, \idxart{730}, \idxart{747}.
+
+\idxc{Doppler's} principle, \idxart{321}, \textit{note}.
+
+\idxb[Double Stars]{Double} stars, \idxart{866}--\idxart{879};
+their colors, \idxart{867};
+criterion for distinguishing between these optically and physically double, \idxart{870};
+method of measuring them, \idxart{868};
+optically and physically double, \idxart{869};
+having orbital motion, see \idxsee{Binary Stars}.
+
+\idxc{Draper, H.}, oxygen in the sun, \idxart{316};
+photograph of the nebula in Orion, \idxart{893};
+photography of stellar spectra, \idxart{859};
+memorial, the, \idxart{859}.
+
+\idxb{Duration}, future, of the sun, \idxart{358};
+of sun spots, \idxart{300}.
+
+\idxsection*{E.}
+
+\idxb{Earth}, the, her annual motion proved by aberration and stellar parallax, \idxart{174};
+approximate dimensions, how measured, \idxart{134}, \idxart{135};
+constitution of its interior, \idxart{171};
+its dimensions, \hyperref[app:spheroid]{Appendix} and \idxart{145};
+its dimensions determined geodetically, \idxart{147}--\idxart{149};
+form of, from pendulum experiments, \idxart{152}--\idxart{155};
+growth of, by accession of meteoric matter, \idxart{777};
+mass compared with that of the sun, \idxart{278};
+its mass and density, \idxart{159}--\idxart{170};
+its orbit, form of, determined, \idxart{182};
+principal facts relating to it, \idxart{132};
+proofs of its rotation, \idxart{138}--\idxart{143}.
+
+\idxb{Earth-shine} on the moon, \idxart{254}.
+
+\idxb{Eccentricity} of the earth's orbit, discovered by Hipparchus, \idxart{184};
+of the earth's orbit, how determined, \idxart{185};
+of the earth's orbit, secular change of, \idxart{198};
+of an ellipse defined, \idxart{183}, \idxart{506}.
+
+\idxb{Eclipses}, duration of lunar, \idxart{373};
+duration of solar, \idxart{385};
+number in a year, \idxart{391}--\idxart{393};
+recurrence of, the saros, \idxart{395};
+of Jupiter's satellites, \idxart{627}--\idxart{630};
+of the moon, \idxart{370}--\idxart{378};
+of the sun, \idxart{379}--\idxart{390};
+total, of the sun, as showing the solar atmosphere and corona, \idxart{319}, \idxart{323}.
+
+\idxb{Ecliptic}, the, defined, \idxart{175};
+obliquity of, \idxart{176};
+limits, lunar, \idxart{374}, \idxart{375};
+limits, solar, \idxart{386}.
+
+\idxb{Effective} temperature of the sun, \idxart{351}.
+
+\idxb{Ejection} theory of comets and meteors, \idxart{741}.
+
+\idxb{Elbowed} equatorial, the, \idxart{74}.
+
+\idxb{Electrical} registration of observations, \idxart{56}.
+
+\idxb{Electro-dynamic} theory of gravitation, \idxart{602}.
+
+\idxb{Elements}, chemical, not truly elementary, \idxart{318};
+chemical, recognized in comets, \idxart{724}, \idxart{725};
+chemical, recognized in stars, \idxart{856};
+chemical, recognized in sun, \idxart{316}, \idxart{317};
+of a planet's orbit, \idxart{505}--\idxart{508}.
+
+\idxc{Elkin}, stellar parallaxes, \idxart{808}, \idxart{814}, \idxart{815}, and Appendix, \hyperref[app:IV]{Table~IV.}
+
+\idxb{Ellipse} defined, \idxart{183};
+described as a conic, \idxart{422}, \idxart{423}.
+
+\idxb{Elliptic} comets, their number, \idxart{702};
+their orbits, \idxart{703};
+recognition of, \idxart{704}.
+
+\idxb{Ellipticity} or oblateness of a planet defined, \idxart{150}.
+
+\idxb{Elongation} of moon or planet defined, \idxart{230}.
+
+\idxb{Enceladus}, the second satellite of Saturn, \idxart{643}, \textit{note}.
+
+\idxc{Encke's} comet, \idxart{710}, \idxart{743}.
+
+\idxc{Encke}, his reduction of the transits of Venus, \idxart{667}.
+
+\idxb{Energy}, the dissipation of, \idxart{925};
+and work of solar radiation, \idxart{345}.
+
+\idxb{Enlargement}, apparent, of bodies near horizon, \idxart{4}, \textit{note}, \idxart{88}, \idxart{93}.
+
+\idxb{Envelopes} in the head of a comet, \idxart{713}, \idxart{727}.
+
+\idxb{Epoch} of a planet's orbit defined, \idxart{508}.
+
+\idxb{Epsilon} Lyræ, \idxart{653}, \idxart{866}, \idxart{882}.
+
+\idxb{Equal} altitudes, determination of time, \idxart{115}.
+%% -----File: File: 552.png---Folio 541-------
+
+\idxb{Equation}, annual, of moon's motion, \idxart{458};
+of the centre, \idxart{189};
+of the equinoxes, \idxart{213};
+of light, by means of Jupiter's satellites, \idxart{628}--\idxart{630};
+of time explained, \idxart{201}--\idxart{204};
+expressing the relation between the light of different stellar magnitudes, \idxart{820}.
+
+\idxb[Equator]{Equator}, the celestial, \idxart{16}.
+
+\idxb{Equatorial} acceleration of the sun's rotation, \idxart{283}--\idxart{285};
+coudé, Paris, \idxart{74};
+parallax, \idxart{85};
+telescope, \idxart{72};
+telescope used to determine the place of a heavenly body, \idxart{129}.
+
+\idxb{Equinoctial}, the, see \idxsee{Equator}, celestial;
+points, or equinoxes, \idxart{17}.
+
+\idxb{Equinoxes}, the, equation of, \idxart{213};
+precession of, \idxart{205}--\idxart{212}.
+
+\idxc{Eratosthenes}, his measure of the earth, \idxart{136}.
+
+\idxb{Erecting} eye-piece for telescope, \idxart{45}.
+
+\idxc{Ericsson}, his solar engine, \idxart{345};
+experiment upon radiation of molten iron, \idxart{350}.
+
+\idxb{Eruptive} prominences, \idxart{325}.
+
+\idxb{Escapement} of clock, \idxart{50}.
+
+\idxb{Establishment} of a port (harbor) defined, \idxart{463}.
+
+\idxb{Europa}, the second satellite of Jupiter, \idxart{621}.
+
+\idxb{Evection}, the, \idxart{456}.
+
+\idxb{Evolution}, tidal, \idxart{484}, \idxart{916}.
+
+\idxb{Eye-pieces}, telescopic, \idxart{44}.
+
+\idxb{Extinctions}, the method of, in photometry, \idxart{825}.
+
+
+\idxsection*{F.}
+
+\idxb{Faculæ}, solar, \idxart{292}.
+
+\idxb{Fall} of a planet to the sun, time required, \idxart{413}, 3;
+of a comet on the sun, probable effect, \idxart{754}.
+
+\idxb{Falling} bodies, eastward deviation, \idxart{138}.
+
+\idxb{Families} (planetary) of comets, \idxart{739}.
+
+\idxc{Faye, H. A.}, his modification of the nebular hypothesis, \idxart{915};
+theory of sun spots, \idxart{304}.
+
+\idxb{Flattening}, apparent, of the celestial sphere, 4, \textit{note}.
+
+\idxb{Force}, evidenced not by motion, but by \textit{change} of motion, \idxart{400};
+projectile, term carelessly used, \idxart{401};
+central, motion under it, \idxart{400}--\idxart{410};
+repulsive, action on comets, \idxart{728}--\idxart{733}.
+
+\idxb{Form} of the earth, \idxart{145}--\idxart{155}.
+
+\idxb{Formation} of comets' tails, \idxart{728}.
+
+\idxc{Foucault}, the gyroscope, showing earth's rotation, \idxart{142};
+his pendulum experiment, showing earth's rotation, \idxart{139}--\idxart{141};
+measures velocity of light, \idxart{690}.
+
+\idxb{Fourteen} hundred and seventy-four line of the spectrum of the corona, \idxart{329}.
+
+\idxc{Fraunhofer} lines in the solar spectrum, \idxart{315}, \idxart{855};
+observations on stellar spectra, \idxart{855}.
+
+\idxb{Free} wave, velocity of, \idxart{473}.
+
+\idxb{Frequency}, relative, of solar and lunar eclipses, \idxart{394}.
+
+
+\idxsection*{G.}
+
+\idxb{Galaxy}, the, \idxart{898}.
+
+\idxc{Galileo}, discovery of Jupiter's satellites, \idxart{621};
+discovery of Saturn's rings, \idxart{637};
+discovery of phases of Venus, \idxart{567};
+use of pendulum in time-keeping, \idxart{50}.
+
+\idxc{Galle}, optical discovery of Neptune, \idxart{654}.
+
+\idxb{Ganymede}, the third satellite of Jupiter, \idxart{621}.
+
+\idxb{Gas} contracting by loss of heat, Lane's law, \idxart{357}.
+
+\idxc{Gauss}, computes the orbit of Ceres, \idxart{592};
+determination of the elements of an orbit, \idxart{519};
+peculiar form of achromatic object-glass, \idxart{41}.
+
+\idxc{Gay Lussac}, law of gaseous expansion, \idxart{360}, \textit{note}.
+
+\idxb{Geocentric} latitude, \idxart{156};
+place of a heavenly body, \idxart{511}.
+
+\idxb{Geodetic} determination of the earth's dimensions, \idxart{147}, \idxart{149}.
+
+\idxb{Genesis} of the solar system, \idxart{908}--\idxart{915};
+of star clusters and nebulæ, \idxart{924}.
+
+\idxb{Georgium Sidus}, the original name for Uranus, \idxart{645}.
+
+\idxc{Gill}, solar parallax from observations of Mars, \idxart{676};
+stellar parallaxes, \idxart{808}.
+Appendix, \hyperref[app:IV]{Table~IV.}
+
+\idxb{Globe}, celestial, rectification of, \idxart{33}, \textit{note}.
+
+\idxb{Gnomon}, determination of latitude with it, \idxart{107};
+determination of the obliquity of the ecliptic, \idxart{176}.
+
+\idxb{Golden} number, the, \idxart{218}.
+
+\idxb{Gradual} changes in the light of the stars, \idxart{839}.
+
+\idxb{Graduation} errors of a circle, \idxart{69}.
+
+\idxb{Grating} diffraction, \idxart{311}, \textit{note}.
+
+\idxb{Gravitation}, electro-dynamic, theory of, \idxart{602};
+law stated, \idxart{161};
+nature unknown, \idxart{161};
+law extending to the stars, \idxart{872}, \textit{note}, \idxart{873}, \idxart{901}, \textit{note};
+Newton's verification of the law by means of the moon's motion, \idxart{419}, \idxart{420}.
+
+\idxb{Gravitational} astronomy defined, \idxart{2};
+methods of determining the solar parallax, \idxart{687}--\idxart{689}.
+
+\idxb{Gravity}, increase of, below the earth's surface, \idxart{169};
+variation of, between equator and pole, \idxart{152}.
+%% -----File: File: 553.png---Folio 542-------
+
+\idxc{Gregorian} calendar, the, and its adoption in England, \idxart{220}, \idxart{221};
+telescope, \idxart{48}.
+
+\idxb{Groups}, cometary, \idxart{705};
+of stars having common motion, \idxart{803}.
+
+\idxb{Growth} of the earth by meteoric matter, \idxart{777}.
+
+\idxb{Gyroscope}, Foucault's proof of earth's rotation, \idxart{142};
+illustrating the precession of the equinoxes, \idxart{210}, \idxart{211}.
+
+\idxsection*{H.}
+
+\idxc{Hall, A.}, discovery of the satellites of Mars, \idxart{590};
+on the question whether it is certain that gravitation extends through the stellar universe, \idxart{901}, \textit{note}.
+
+\idxc{Halley}, his comet, \idxart{742};
+his computation of cometary orbits, \idxart{700};
+his method of determining the sun's parallax, \idxart{679}, \idxart{680};
+the moon's secular acceleration, \idxart{459};
+proper motions of stars, \idxart{800}.
+
+\idxc{Hansen}, correction of the solar parallax, \idxart{667};
+opinion on the form of the moon, \idxart{258}.
+
+\idxc{Harding} discovers Juno, \idxart{593}.
+
+\idxc{Harkness}, observations on the light of meteors, \idxart{776};
+observation of the corona spectrum, \idxart{329}.
+
+\idxb{Harmonic} law, Kepler's, \idxart{412}--\idxart{117}.
+
+\idxb{Harton} colliery, density of the earth, \idxart{169}.
+
+\idxb{Harvard} photometry, the, \idxart{827}, \idxart{828}.
+
+\idxb{Harvest} and hunter's moons explained, \idxart{237}.
+
+\idxb{Heat} and light of meteors explained, \idxart{765};
+of the moon, \idxart{260};
+of the sun, \idxart{338}--\idxart{358};
+received by the earth from meteors, \idxart{355}, \idxart{779};
+from the stars, \idxart{834}.
+
+\idxb{Height} of lunar mountains, \idxart{270}.
+
+\idxc{Heis}, enumeration of naked-eye stars, \idxart{818}.
+
+\idxb{Heliocentric} place of a planet, \idxart{512}.
+
+\idxb{Heliometer}, the, \idxart{677};
+used in determining solar parallax, \idxart{676}, \idxart{683};
+used in determining stellar parallax, \idxart{811}, \idxart{815}.
+
+\idxb{Helioscopes}, or solar eye-pieces, \idxart{286}, \idxart{287}.
+
+\idxb{Helium}, an unidentified metal in the solar chromosphere, \idxart{323}.
+
+\idxc{Helmholtz}, contraction theory of solar heat, \idxart{356}.
+
+\idxc{Hencke}, discovers Astræa, the fifth asteroid, \idxart{593}.
+
+\idxc{Henderson}, measures the parallax of $\alpha$~Centauri, \idxart{809}, \idxart{810}.
+
+\idxc{Henry Brothers}, astronomical photography, \idxart{798}.
+
+\idxc{Henry, Prof.\ J.}, heat of sun spots, \idxart{310};
+at sun's limb, \idxart{348}.
+
+\idxc{Herschel, Sir John}, astrometry, \idxart{819};
+illustration of the planetary system, \idxart{664}.
+
+\idxc{Herschel, Sir W.}, discovery of the sun's motion in space, \idxart{804};
+discovery of two satellites of Saturn, \idxart{643};
+discovery of Uranus, \idxart{645};
+discovery of two satellites of Uranus, \idxart{650};
+star-gauges, \idxart{899};
+theory of sun spots, \idxart{302};
+his reflecting telescope, \idxart{48}.
+
+\idxc{Hevelius}, his view of cometary orbits, \idxart{700}.
+
+\idxc{Hipparchus}, discovers eccentricity of earth's orbit, \idxart{184};
+discovers precession, \idxart{205};
+his value of the solar parallax, \idxart{671};
+the first star-catalogue, \idxart{795}.
+
+\idxc{Holden, E. S.}, on changes in nebulæ, \idxart{892}.
+
+\idxb{Horizon}, apparent enlargement of bodies near it, \idxart{4}, \textit{note}, \idxart{88}, \idxart{93};
+artificial, \idxart{78};
+rational and apparent defined, \idxart{10};
+dip of, \idxart{81};
+visible, defined, \idxart{11}.
+
+\idxb{Horizontal} parallax, \idxart{83}, \idxart{84};
+point of the meridian circle, \idxart{67}.
+
+\idxb{Hour-angle} defined, \idxart{24}.
+
+\idxb{Hour-circle} defined, \idxart{18}.
+
+\idxc{Huggins, W.}, attempts to photograph the solar corona without an eclipse, \idxart{328};
+attempted observation of stellar heat, \idxart{834};
+observations of stellar spectra, \idxart{856};
+photography of stellar spectra, \idxart{859};
+spectroscopic observations of T coronæ, \idxart{844};
+star-motions in line of sight, \idxart{802}.
+
+\idxc{Humboldt, A. von}, classification of the planets, \idxart{549}.
+
+\idxc{Hunt, Sterry}, carbonic acid brought to earth by comets, \idxart{735}.
+
+\idxc{Huyghens}, discovery of Saturn's rings, \idxart{637};
+discovery of Saturn's satellite, Titan, \idxart{643};
+invention of the pendulum clock, \idxart{50};
+his long telescope, \idxart{40}.
+
+\idxb{Hydrogen} in the solar chromosphere and prominences, \idxart{323}--\idxart{325};
+bright lines of its spectrum in the nebulæ, \idxart{890};
+bright lines of its spectrum in temporary stars, \idxart{844};
+bright lines of its spectrum in variable stars, \idxart{857}.
+
+\idxb{Hyperbola}, the, described as a conic, \idxart{422}.
+
+\idxb{Hyperbolic} comets, \idxart{702}.
+
+\idxb{Hyperion}, the seventh and last discovered satellite of Saturn, \idxart{643}, \idxart{644}.
+
+\idxb{Hypothesis}, nebular. See \idxsee{Nebular} hypothesis.
+
+
+\idxsection*{I.}
+
+\idxb{Iapetus}, the outermost satellite of Saturn, \idxart{643}.
+
+\idxc{Ibn Jounis}, use of pendulum in observation, \idxart{50}.
+%% -----File: File: 554.png---Folio 543-------
+
+\idxb{Ice}, amount melted by solar radiation, \idxart{344}--\idxart{346}, \idxart{364*}.
+
+\idxb{Illumination} of the moon's disc during a lunar eclipse, \idxart{376}.
+
+\idxb{Image}, telescopic, conditions of distinctness, \idxart{39}.
+
+\idxb{Inequality}, diurnal, of the tides, \idxart{471}.
+
+\idxb{Inferences} deducible from Kepler's laws, \idxart{418}.
+
+\idxb{Inferior} planet, motion of, \idxart{497}.
+
+\idxb{Infinity}, velocity from, \idxart{429}.
+
+\idxb{Influences} of the moon on the earth, \idxart{262}.
+
+\idxb{Intra-Mercurial} planets, \idxart{602}--\idxart{606};
+planets,  supposed observations of, during solar eclipse, \idxart{605}.
+
+\idxb{Interior} of the earth, its constitution,  \idxart{171}.
+
+\idxb{Invariable} plane of the solar system, \idxart{531}.
+
+\idxb{Invariability} of the earth's rotation, \idxart{144}.
+
+\idxb{Io}, the first satellite of Jupiter, \idxart{621}.
+
+\idxb{Iron} meteorites, \idxart{758};
+in the sun, \idxart{315}.
+
+\idxb{Irradiation} in micrometric measures,  \idxart{256}, \idxart{534}.
+
+
+\idxsection*{J.}
+
+\idxc{Janssen}, discovery of the spectroscopic  method of observing the solar prominences,  \idxart{323};
+solar photography, \idxart{289}.
+
+\idxb{Jets} issuing from the nucleus of a comet,  \idxart{713}, \idxart{727}.
+
+\idxc{Jolly}, observations of the earth's density,  \idxart{170}.
+
+\idxb{Julian} calendar, \idxart{219}.
+
+\idxb{Juno}, discovered by Harding, \idxart{593}.
+
+\idxb{Jupiter}, the planet, \idxart{609}--\idxart{631};
+brightness as seen from $\alpha$~Centauri, \idxart{881};
+a semi-sun,  \idxart{619}.
+
+
+\idxsection*{K.}
+
+\idxc{Kant}, proposes the nebular hypothesis,  \idxart{908}.
+
+\idxc{Kepler}, his belief as to cometary orbits, \idxart{700};
+his three laws of planetary motion,  \idxart{412}--\idxart{418};
+his ``problem,'' \idxart{188};
+his ``regular solid'' theory of the planetary distances, \idxart{592}, \textit{note}.
+
+\idxc{Kirchoff}, his fundamental principles of spectrum analysis, \idxart{314}.
+
+\idxsection*{L.}
+
+\idxc{Langley, S. P.}, his Bolometer, \idxart{343};
+the color of the sun, \idxart{337};
+observations on lunar heat, \idxart{260}, \idxart{261};
+on solar heat, \idxart{348};
+sun-spot drawings, \idxart{292};
+heat in sun spots, \idxart{293}.
+
+\idxc{Lane's} law, rise of temperature consequent on the contraction of a gaseous mass, \idxart{357}.
+
+\idxc{La Place}, his equations relating to the eccentricities and inclinations of the  planetary orbits, \idxart{532};
+explanation of  the moon's secular acceleration, \idxart{459},  \idxart{460};
+the invariable plane of the solar system, \idxart{531};
+the nebular hypothesis, \idxart{901}--\idxart{911}.
+
+\idxc{Lassell}, discovery of the two inner satellites of Uranus, \idxart{650};
+independent discovery of Hyperion, \idxart{643}.
+
+\idxb{Latitude} (astronomical) of a place on the earth's surface, \idxart{30}, \idxart{100}, \idxart{156};
+astronomical,  geodetic, and geocentric, distinguished, \idxart{156};
+determination of, methods used, \idxart{101}--\idxart{107};
+at sea, its determination, \idxart{103};
+possible variations of it, \idxart{108};
+station errors, \idxart{158};
+celestial, defined, \idxart{178}, \idxart{179};
+and longitude (celestial), conversion into $\alpha$ and $\delta$, \idxart{180}.
+
+\idxb{Law} of angular velocity under central force, \idxart{408};
+Bode's, \idxart{488}, \idxart{489};
+of Boyle or Mariotte, density of a gas, \idxart{360}, \textit{note};
+of Dalton, mixture of gases, \idxart{360}, \textit{note};
+of earth's orbital motion, \idxart{186}, \idxart{187};
+of  equal areas, \idxart{186}, \idxart{402}--\idxart{406}, \idxart{412};
+of Gay Lussac, gaseous expansion, \idxart{360}, \textit{note};
+of gravitation, \idxart{161}, \idxart{162}, \idxart{419}, \idxart{872};
+Lane's, of temperature in gaseous contraction, \idxart{357};
+of linear velocity in angular motion, \idxart{407}.
+
+\idxb{Laws} of Kepler, \idxart{412}--\idxart{418};
+motion under a central force, \idxart{400}--\idxart{411}.
+
+\idxb{Leap} year, rule for, \idxart{220}.
+
+\idxb{Length}, of the day, possible changes in it, \idxart{144};
+of the year, its invariability, \idxart{526}, \idxart{778}.
+
+\idxb{Leonids}, the, \idxart{780}, \idxart{786}.
+
+\idxc{Lescarbault}, supposed discovery of Vulcan, \idxart{603}.
+
+\idxb{Level} adjustment of the transit instrument, \idxart{60}.
+
+\idxc{Leverrier}, discovery of Neptune, \idxart{653}, \idxart{654};
+on an intra-Mercurial planet, \idxart{603};
+motion of the perihelion of Mercury's orbit, \idxart{602};
+method of determining the solar parallax by planetary perturbations, \idxart{689}.
+
+\idxc{Lexell's} comet, approach to Jupiter, \idxart{718};
+recognition of Uranus as a planet, \idxart{645}.
+
+\idxb{Librations} of the moon, \idxart{249}, \idxart{250}, \idxart{251}.
+
+\idxb{Light} of comets, \idxart{722};
+of the moon, \idxart{259};
+emitted by certain stars, \idxart{835};
+received by the earth from certain stars, \idxart{832};
+of the sun, \idxart{332}--\idxart{337};
+total, of the stars, \idxart{833};
+equation of, from Jupiter's satellites, \idxart{628}--\idxart{630};
+mechanical equivalent
+%% -----File: File: 555.png---Folio 544-------
+of, Thomsen, \idxart{776};
+time occupied by, in coming from the sun, \idxart{275}, \idxart{629};
+velocity of, \idxart{225}, \textit{note}, \idxart{668}, \idxart{690}.
+
+\idxb{Light-curves} of variable stars, \idxart{848}.
+
+\idxb{Light-gathering} power of telescopes, \idxart{38}.
+
+\idxb{Light-ratio}, the, in scale of star-magnitudes, \idxart{819}.
+
+\idxb{Light-year}, the, the unit of stellar distance, \idxart{814}.
+
+\idxb{Limb} of the sun, darkening of, \idxart{337};
+of the sun, diminution of heat, \idxart{348}.
+
+\idxb{Limiting} apertures in stellar photometry, \idxart{825}.
+
+\idxb{Linear} and angular dimensions, their relation, \idxart{5};
+velocity under central force, its law, \idxart{407}, \idxart{409}.
+
+\idxb{Linné}, lunar crater supposed to have changed, \idxart{269}.
+
+\idxc{Listing}, dimensions of the earth, \idxart{145}.
+
+\idxb{Local} and standard time, \idxart{122}.
+
+\idxc{Lockyer, J. N.}, discovery of the spectroscopic method of observing the solar prominences, \idxart{323};
+his ``collision theory'' of variable stars, \idxart{850};
+views as to the compound nature of the so-called chemical ``elements,'' \idxart{318};
+origin of the Fraunhofer lines, \idxart{320};
+theory of sun spots, \idxart{306};
+meteoric theory of nebulæ, \idxart{894};
+meteoric hypothesis, \idxart{926}.
+
+\idxc{Loewy}, peculiar method of determining the refraction, \idxart{95}.
+
+\idxb{Longitude}, arcs of, to determine the earth's dimensions, \idxart{151};
+(terrestrial), determination of, \idxart{118}--\idxart{121};
+(celestial), \idxart{178}--\idxart{180};
+of perihelion, \idxart{505}, \idxart{506}.
+
+\idxb{Luminosity} of bodies at low temperatures, \idxart{737}, \textit{note.}
+
+\idxb{Lunar} distances, \idxart{120}, B;
+eclipses, \idxart{370}--\idxart{378};
+influences on the earth, \idxart{262};
+methods of determining the longitude, \idxart{120};
+perturbations, \idxart{448}--\idxart{461};
+perturbations used to determine the solar parallax, \idxart{687}.
+
+\idxb{Lyræ}, $\alpha$, see \idxsee{Vega};
+$\beta$, variable star, \idxart{847};
+$\epsilon$, quadruple star, \idxart{653}, \idxart{866}, \idxart{882}.
+
+
+\idxsection*{M.}
+
+\idxc{Mädler}, speculations as to a central sun, \idxart{807}.
+
+\idxb{Magnifying} power of a telescope, \idxart{37};
+power, highest available, \idxart{43}.
+
+\idxb{Magnitudes} of stars, \idxart{816}--\idxart{822}.
+
+\idxb{Magnitude} of smallest star visible in a given telescope, \idxart{822}.
+
+\idxb{Magnesium} in the nebulæ, \idxart{890}, \idxart{894}.
+
+\idxb{Maintenance} of the solar heat, \idxart{353}--\idxart{356}.
+
+\idxb{Mars}, the planet, \idxart{578}--\idxart{591};
+observed for solar parallax, \idxart{673}--\idxart{677}.
+
+\idxb{Maskelyne}, his mountain method of determining the earth's density, \idxart{164}.
+
+\idxb{Mass} and weight, distinction between them, \idxart{159}, \idxart{160};
+of comets, \idxart{718}, \idxart{719};
+of the earth compared with the sun, \idxart{278};
+of the earth in terms of the sun as determining the solar parallax, \idxart{689};
+of the moon, its determination, \idxart{243};
+of a planet, how determined, \idxart{536}--\idxart{539};
+of the sun, compared with the earth, \idxart{278};
+probable, of shooting stars, \idxart{776}.
+
+\idxb{Mass-brightness} of binary stars, \idxart{879}.
+
+\idxb{Masses} of binary stars, \idxart{877}, \idxart{878}.
+
+\idxc{Mayer, R.}, meteoric theory of the solar heat, \idxart{353}.
+
+\idxb{Maxwell, Clerk}, meteoric theory of Saturn's rings, \idxart{641}.
+
+\idxb{Mazapil}, the meteorite of, \idxart{784}.
+
+\idxb{Mechanical} equivalent of light, \idxart{776}.
+
+\idxb{Mercury}, the planet, \idxart{551}--\idxart{562}.
+
+\idxb{Mercury's} orbit, motion of its perihelion, \idxart{602}.
+
+\idxb{Meridian}, the celestial, defined, \idxart{19};
+arc of, how measured, \idxart{147}.
+
+\idxb[Meridian Circle]{Meridian-circle}, the, \idxart{63};
+used to determine the place of a heavenly body, \idxart{128}.
+
+\idxb{Meridian} photometer, the, \idxart{828}.
+
+\idxb{Meteors} and shooting stars, \idxart{755}--\idxart{787};
+ashes of, \idxart{775};
+and comets, their connection, \idxart{785}--\idxart{787};
+daily number of, \idxart{771};
+detonating, \idxart{768};
+effect on the earth's orbital motion, \idxart{778};
+effect upon the moon's motion, \idxart{778};
+effect upon the transparency of space, \idxart{779};
+explanation of their light and heat, \idxart{765};
+heat from them, \idxart{355}, \idxart{779};
+magnitude of, \idxart{762};
+method of observing them, \idxart{764};
+their trains, \idxart{766}.
+
+\idxb{Meteoric} growth of the earth, \idxart{777};
+showers, \idxart{780}--\idxart{786};
+shower of the Bielids, 1872, 1886, \idxart{746};
+shower of the Leonids, 1833, 1866--67, \idxart{781};
+swarms and rings, \idxart{783};
+swarms, special characteristics, \idxart{783};
+theory of Saturn's rings, \idxart{641};
+theory of the sun's heat, \idxart{353}--\idxart{355};
+theory of sun spots, \idxart{306}.
+
+\nblabel{idx:Meteorite}% Typo in Uranolith index entry refers to Meteorite rather than Meteorites
+\idxb[Meteorites]{Meteorites}, or uranoliths, or aerolites, \idxart{755}--\idxart{769};
+chemical elements in them, \idxart{760};
+crust, \idxart{761};
+fall of, \idxart{756};
+which have fallen in the United States, \idxart{759};
+iron, list of, \idxart{758};
+number of, \idxart{769};
+question of their origin, \idxart{767};
+their paths, \idxart{763}.
+
+\idxb{Metonic} cycle, the, \idxart{218}.
+%% -----File: File: 556.png---Folio 545-------
+
+\idxc{Michelson}, determination of the velocity of light, \idxart{225}, \textit{note}, \idxart{668}, \idxart{690}.
+
+\idxb{Micrometer}, the filar, \idxart{73}, \idxart{534}, \idxart{867}.
+
+\idxb{Microscope}, the reading, \idxart{64}.
+
+\idxb{Midnight} sun, the, \idxart{191}.
+
+\idxb{Milky Way}, or galaxy, \idxart{898}.
+
+\idxb{Mimas}, the innermost satellite of Saturn, \idxart{643} and \textit{note}.
+
+\idxb{Minor} planets, or asteroids, \idxart{592}--\idxart{601}.
+
+\idxb{Mira}, Omicron Ceti, \idxart{846}.
+
+\idxb{Missing} stars, \idxart{840}.
+
+\idxb{Mohammedan} calendar, \idxart{217}.
+
+\idxc{Monck}, the mass-brightness of binary stars, \idxart{879}.
+
+\idxb{Monocentric} eye-piece for telescope, \idxart{45}.
+
+\idxb{Moon}, the, \idxart{227}--\idxart{272};
+her atmosphere, \idxart{255}--\idxart{258};
+the, regarded as a clock, \idxart{120};
+distance of, etc., \idxart{240};
+her heat and temperature, \idxart{260}, \idxart{261};
+her light as compared with sunlight, \idxart{259};
+influences on the earth, \idxart{262};
+mass of, determined, \idxart{243}, \idxart{244};
+her motion (apparent), \idxart{228};
+her motion relative to the sun, \idxart{241};
+her mountains, measurement of their elevation, \idxart{270};
+her orbit with reference to the earth, \idxart{238};
+her parallax determined, \idxart{239};
+perturbations, \idxart{448}--\idxart{461};
+her rotation and librations, \idxart{248}--\idxart{251};
+her phases, \idxart{253};
+her surface character, \idxart{263}--\idxart{270};
+culminations for longitude, \idxart{120}, A.
+
+\idxb{Month}, the anomalistic, \idxart{397}, \textit{note};
+the nodical, \idxart{397}, \textit{note};
+the sidereal, \idxart{229}, \idxart{232};
+the synodic, \idxart{229}, \idxart{232};
+length of, increased by perturbation, \idxart{453};
+slightly shortened by the secular acceleration, \idxart{459}.
+
+\idxb{Motion}, direct and retrograde, of the planets, \idxart{494};
+of the solar system in space, \idxart{804}--\idxart{807};
+in line of sight, effect on spectrum, \idxart{321};
+of stars in line of sight, spectroscopically observed, \idxart{802}.
+
+\idxb{Motions}, proper, of the stars, \idxart{800}--\idxart{803}.
+
+\idxb{Mountains}, lunar, their height, \idxart{270}.
+
+\idxb{Mountain} method of determining the earth's density, \idxart{164}.
+
+\idxb{Multiple} stars, \idxart{882}.
+
+\idxb{Mural} circle, the, \idxart{70}.
+
+
+\idxsection*{N.}
+
+\idxb{Nadir}, the, defined, \idxart{9};
+point of meridian circle, \idxart{67}.
+
+\idxb{Names} of the constellations, \idxart{792};
+of Jupiter's satellites, \idxart{621};
+of satellites of Mars, \idxart{590};
+of the planets, \idxart{487}, \idxart{489};
+of Saturn's satellites, \idxart{643}, \textit{note};
+of the satellites of Uranus, \idxart{650};
+of stars, \idxart{794}.
+
+\idxb{Neap} tide defined, \idxart{463}.
+
+\idxb{Nebula}, the great, in Andromeda, \idxart{886};
+the annular, in Lyra, \idxart{888};
+of Orion, the, \idxart{886}, \idxart{892}, \idxart{893}.
+
+\idxb{Nebulæ}, the, \idxart{886}--\idxart{897};
+changes in, \idxart{892};
+their distance, \idxart{896};
+Lockyer's meteoric theory, \idxart{894};
+their nature, \idxart{894};
+their number and distribution, \idxart{895};
+photographs of, \idxart{893};
+planetary, \idxart{888};
+spiral, \idxart{888};
+their spectra, and chemical elements in them, \idxart{890}, \idxart{891}.
+
+\idxb[Nebular]{Nebular} hypothesis, the, \idxart{908}--\idxart{915};
+modifications of the original theory, \idxart{912}, \idxart{913}.
+
+\idxb{Negative} eye-pieces for the telescope, \idxart{44};
+shadow of the moon, \idxart{381};
+star magnitudes, \idxart{821}.
+
+\idxb{Neptune}, the planet, \idxart{653}--\idxart{661};
+anomalous retrograde rotation, in relation to the nebular hypothesis, \idxart{914};
+appearance of sun and solar system from it, \idxart{658};
+(actual) discovery by Galle, \idxart{654};
+theoretical discovery by Leverrier and Adams, \idxart{653}, \idxart{654};
+its discovery ``no accident,'' \idxart{655};
+the computed elements erroneous, \idxart{655};
+its satellite, \idxart{661};
+spectrum of, \idxart{660}.
+
+\idxc{Newcomb, S.}, conclusions as to sun's age and duration, \idxart{358}, \idxart{359};
+observations on meteors, \idxart{776};
+on the moon's secular acceleration, \idxart{461};
+on the structure of the heavens, \idxart{900};
+his value of the solar parallax, \idxart{667};
+velocity of light, \idxart{225}, \textit{note}, \idxart{668}, \idxart{690}.
+
+\idxc{Newton}, Prof. H.~A., daily number of meteors, \idxart{771};
+investigation of meteoric orbits, \idxart{767}, \idxart{785};
+theory of the constitution of a comet, \idxart{737}.
+
+\idxc{Newton, Sir Isaac}, discovery of gravitation, \idxart{161}, \idxart{419};
+verification of the idea of gravitation by means of the moon's motion, \idxart{419}, \idxart{420};
+discovery that planetary orbits must be conics, \idxart{421};
+computation of a cometary orbit, \idxart{700};
+his reflecting telescope, \idxart{48}.
+
+\idxb{Nitrogen}, suspected in the nebulæ, \idxart{890}.
+
+\idxb{Node} of an orbit defined, \idxart{233}, \idxart{506}.
+
+\idxb{Nodes} of moon's orbit, their regression, \idxart{455};
+of the planetary orbits, their motion, \idxart{527}.
+
+\idxb{Nodical} month, the, \idxart{249}, \idxart{397}, \textit{note}.
+
+\idxc{Nordenskiold}, meteoric ashes, \idxart{775}.
+
+\idxb{Nucleus} of a comet, \idxart{713}, \idxart{716}.
+
+\idxb{Number} of eclipses in a year, \idxart{391};
+in a saros, \idxart{398};
+of meteors and meteorites, \idxart{759}, \idxart{769}, \idxart{771};
+and designation of variable stars, \idxart{852}.
+
+\idxb{Nutation} of the earth's axis, \idxart{214}, \idxart{215}.
+%% -----File: File: 557.png---Folio 546-------
+
+\idxsection*{O.}
+
+\idxb{Oberon}, the outer satellite of Uranus, \idxart{650}.
+
+\idxb{Object-glass}, achromatic, \idxart{41};
+residual or secondary spectrum of, \idxart{42};
+designed for photography, \idxart{42}.
+
+\idxb{Oblateness}, or ellipticity of a spheroid, \idxart{150}.
+
+\idxb{Oblique} sphere, the, \idxart{33}.
+
+\idxb{Obliquity} of the ecliptic defined and measured, \idxart{176};
+of the ecliptic, secular change of, \idxart{197}.
+
+\idxb{Occultation}, circle of perpetual, \idxart{33};
+of stars, \idxart{399};
+of stars used for longitude determination, \idxart{120}~C;
+of stars proving absence of lunar atmosphere, \idxart{256}.
+
+\idxc{Olbers}, discovers Pallas and Vesta, \idxart{593}.
+
+\idxc{Olmsted, D.}, his researches on meteors, \idxart{785}.
+
+\idxc{Oppolzer}, effect of meteors on the moon's motion, \idxart{778};
+orbit of Tempel's Comet, \idxart{786};
+motion of Winnecke's Comet, \idxart{711}, \textit{note}.
+
+\idxb{Orbit} of the earth, its form determined, \idxart{182};
+of the earth, effect of meteors upon it, \idxart{778};
+of the earth, perturbations, \idxart{197};
+of the moon, \idxart{238};
+of the moon, its perturbations, \idxart{448}--\idxart{461};
+of a planet, determined graphically, \idxart{428}, \idxart{431}, \idxart{432};
+planetary, its elements, \idxart{505}--\idxart{510};
+planetary, its elements, determination of, \idxart{519}.
+
+\idxb{Orbits} of binary stars, \idxart{873};
+of comets, \idxart{700}--\idxart{709};
+of planets, diagram, \idxart{489};
+of sun and stars in the stellar system, \idxart{904}.
+
+\idxb{Origin} of comets, \idxart{738}--\idxart{741};
+of meteorites or aerolites, \idxart{767}.
+
+\idxb{Orthogonal} component of the disturbing force, \idxart{445}, \idxart{455}.
+
+\idxsection*{P.}
+
+\idxb{Pallas}, discovered by Olbers, \idxart{593}.
+
+\idxc{Palisa}, discoverer of sixty-five asteroids, \idxart{593}.
+
+\idxb{Parabola}, the, described as a conic, \idxart{422}, \idxart{423}.
+
+\idxb{Parabolic} comets, their number, \idxart{702};
+velocity, the, \idxart{429}.
+
+\idxb[Parallax of the Sun]{Parallax} (diurnal), defined and discussed, \idxart{82}, \idxart{83};
+of the moon, determined, \idxart{239};
+of the sun, classification of methods, \idxart{669};
+of the sun, gravitational methods, \idxart{687}--\idxart{689};
+of the sun, history of investigations, \idxart{666}--\idxart{668};
+of the sun, method of Aristarchus, \idxart{670};
+of the sun, method of Hipparchus, \idxart{671};
+of the sun by observations on Mars, \idxart{673}, \idxart{676};
+of the sun by transits of Venus, \idxart{678}, \idxart{686};
+of the sun by the velocity of light, \idxart{690}--\idxart{692};
+of the sun, Ptolemy's value, \idxart{671};
+of the stars (annual), \idxart{808}--\idxart{814};
+of $\alpha$~Centauri, Henderson, \idxart{809}, \idxart{810};
+of 61 Cygni, Bessel, \idxart{809}--\idxart{811};
+of $\alpha$~Lyræ, negative, Pond, \idxart{809};
+stellar, absolute method, \idxart{810};
+stellar, differential method, \idxart{811};
+stellar, table of, Appendix, \hyperref[app:IV]{Table IV.}
+
+\idxb{Parallactic} inequality of the moon, \idxart{687};
+orbit of a star, \idxart{808}.
+
+\idxb{Parallel} sphere, the, \idxart{32}.
+
+\idxb{Parallels} of declination, \idxart{23}.
+
+\idxc{Peirce, B.}, heat from meteors, \idxart{355};
+on the mass of comets, \idxart{719};
+theory of sun spots, \idxart{306}.
+
+\idxb{Pendulum}, compensation, \idxart{51};
+use in clocks, \idxart{50};
+used in determining form of the earth, \idxart{152}--\idxart{155};
+free, of Foucault, showing earth's rotation, \idxart{139}--\idxart{141}.
+
+\idxb{Penumbra} of the earth's shadow, \idxart{368};
+of the moon's shadow, \idxart{383};
+of a sun spot, \idxart{295}.
+
+\idxb{Perigee} and apogee defined, \idxart{238}.
+
+\idxb{Perihelia} of comets, their distribution, \idxart{706}.
+
+\idxb{Perihelion} of earth's orbit defined, \idxart{183};
+its motion, \idxart{199};
+of Mercury's orbit, its motion, \idxart{602}.
+
+\idxb{Period}, sidereal and synodic, of the moon, \idxart{229}--\idxart{232};
+sidereal and synodic, of a planet, defined, \idxart{490};
+sidereal, of a planet, determined, \idxart{513}, \idxart{514}.
+
+\idxb{Periodic} comets, \idxart{703}, \idxart{704}, \idxart{738}--\idxart{740};
+table of comets of short period, Appendix, \hyperref[app:III]{Table III.}
+
+\idxb{Periodicity} of sun spots, \idxart{307}--\idxart{309}.
+
+\idxb{Persei}, $\beta$, or Algol, \idxart{848}.
+
+\idxb{Perseids}, the, meteoric swarm, \idxart{780}, \idxart{782}, \idxart{783}.
+
+\idxb{Personal} equation, \idxart{114}, \idxart{120}~A, \idxart{121}~B.
+
+\idxb{Perturbations}, lunar, \idxart{448}--\idxart{461};
+planetary, \idxart{521}--\idxart{523};
+of Mars and Venus by the earth as determining the sun's parallax, \idxart{689}.
+
+\idxc{Peters, C.~H.~F.}, discovers fifty-two asteroids, \idxart{593}.
+
+\idxc{Piazzi}, discovery of Ceres, \idxart{592}.
+
+\idxc{Picard}, measure of earth's diameter, \idxart{136}.
+
+\idxc{Pickering, E.~C.}, his meridian photometer, \idxart{828};
+photography of stellar spectra, \idxart{860}, \idxart{862};
+photometric observations of the eclipses of Jupiter's satellites, \idxart{630};
+the Harvard photometry, \idxart{827}.
+
+\idxb{Phases} of Mercury, Venus, and Mars, \idxart{559}, \idxart{567}, \idxart{582};
+of the moon, \idxart{253}.
+
+\idxb{Phobos}, the inner satellite of Mars, \idxart{589}.
+
+\idxb{Photographs} of the moon, \idxart{272};
+of the nebulæ, \idxart{893};
+of the solar corona, \idxart{328};
+of the sun's surface and spots, \idxart{289}.
+%% -----File: File: 558.png---Folio 547-------
+
+\idxb{Photographic} object-glasses, \idxart{42};
+observations of eclipses of Jupiter's satellites, \idxart{630}, and \textit{note};
+observations of transit of Venus, \idxart{684}--\idxart{686}.
+
+\idxb{Photography} as a means of photometry, \idxart{829};
+solar, \idxart{289};
+spectroscopic, motion in line of sight, \idxart{802};
+applied to star-charting, \idxart{798};
+in determination of stellar parallax, \idxart{812};
+of stellar spectra, \idxart{859}--\idxart{863}.
+
+\idxb{Photometer}, the meridian, \idxart{828};
+polarization, \idxart{827};
+the wedge, \idxart{826}.
+
+\idxb{Photometry}, Harvard, the, \idxart{827}, \idxart{828};
+by means of photography, \idxart{829};
+by the spectroscope, \idxart{831};
+of sunlight, \idxart{332}--\idxart{335};
+stellar, \idxart{823}--\idxart{831}.
+
+\idxb{Photosphere of the} sun, its nature, \idxart{291}, \idxart{292}, \idxart{361}.
+
+\idxb{Photo-tachymetrical} determination of the sun's parallax, \idxart{690}--\idxart{692}.
+
+\idxb{Physical} characteristics of comets, \idxart{712};
+method of determining sun's parallax, \idxart{690}--\idxart{692}.
+
+\idxb{Planet}, intra-Mercurial, \idxart{602}--\idxart{606};
+trans-Neptunian, \idxart{662}.
+
+\idxb{Planets} attending certain stars, \idxart{880}, \idxart{881};
+distances and periods, \idxart{489};
+enumerated, \idxart{486}, \idxart{487};
+relative age, according to nebular hypothesis, \idxart{913}, \idxart{915};
+orbits, diagram of, \idxart{489};
+orbits, elements of, \idxart{505}, \idxart{510}.
+
+\idxb{Planetoid}. See \idxsee{Asteroid}, \idxart{591}.
+
+\idxb{Planetary} data, tables of, Appendix, \hyperref[app:I]{Table I.};
+data, accuracy of, \idxart{653};
+nebulæ, \idxart{888};
+system, facts suggesting the theory of its origin, \idxart{907};
+system, Sir J. Herschel's illustration of its dimensions, \idxart{664}.
+
+\idxb{Pleiades}, the, \idxart{884}.
+
+\idxc{Pogson}, his absolute scale of star-magnitudes, \idxart{819}.
+
+\idxb{Pole} of the earth, \idxart{28};
+(celestial), defined, \idxart{14};
+its altitude equal to the latitude, \idxart{30}, \idxart{100};
+its place affected by precession, \idxart{206}, \idxart{207}.
+
+\idxb{Pole-star}, ancient, $\gamma$~Draconis, \idxart{207};
+its position and recognition, \idxart{15}.
+
+\idxb{Polar} distance, defined, \idxart{23};
+point of meridian circle, \idxart{66}.
+
+\idxb{Position-angle} of a double star, \idxart{868}.
+
+\idxb{Position} of a heavenly body, how determined, \idxart{128}, \idxart{129}.
+
+\idxb{Positive} eye-pieces for telescopes, \idxart{44}.
+
+\idxc{Pouillet}, his pyrheliometer, \idxart{340}.
+
+\idxb{Power}, magnifying, of telescope, \idxart{37}.
+
+\idxc{Poynting}, determination of the earth's density, \idxart{170}.
+
+\idxb{Practical} astronomy defined, \idxart{2}.
+
+\idxb{Precession} of the equinoxes, \idxart{205}--\idxart{212}.
+
+\idxb{Prime} vertical defined, \idxart{19};
+vertical instrument, \idxart{62}, \idxart{106}.
+
+\idxb{Priming} and lagging of the tides, \idxart{470}.
+
+\idxc{Pritchard, Prof. C.}, determination of stellar parallax by means of photography, \idxart{812};
+stellar photometry, \idxart{826};
+Uranometria Oxoniensis, \idxart{826}.
+
+\idxc{Pritchett, C.~W.}, discovery of the great red spot on Jupiter, \idxart{618}.
+
+\idxb{Problem} of three bodies, \idxart{437}--\idxart{461};
+of two bodies, \idxart{424}--\idxart{433}.
+
+\idxb{Problems} illustrating Kepler's third law, \idxart{413}.
+
+\idxc{Proctor, R.~A.}, on the origin of comets, \idxart{741};
+determination of the rotation period of Mars, \idxart{584}.
+
+\idxb{Projectiles}, deviation caused by earth's rotation, \idxart{143};
+their path near the earth, \idxart{435}.
+
+\idxb{Projectile} force, careless use of the term, \idxart{401}.
+
+\idxb{Prominences}, or protuberances, the solar, \idxart{291}, \idxart{323}--\idxart{326}, \idxart{363};
+quiescent and eruptive, \idxart{325}, \idxart{326}.
+
+\idxb{Proper} motions of the stars, \idxart{800}, \idxart{803}.
+
+\idxb{Proximity} of a star, indications of it, \idxart{813}.
+
+\idxb{Ptolemaic} system, the, \idxart{500}, \idxart{502}.
+
+\idxc{Ptolemy}, his almagest, \idxart{500}, \idxart{700}, \idxart{795}.
+
+\idxb{Pyrheliometer} of Pouillet, \idxart{340}.
+
+
+\idxsection*{Q.}
+
+\idxb{Quantity} of the solar radiation in calories, \idxart{338}--\idxart{340};
+of sunlight in candle power, \idxart{332}, \idxart{333}.
+
+\idxb{Quiescent} prominences, \idxart{325}.
+
+
+\idxsection*{R.}
+
+\idxb{Radial} compound of the disturbing force, \idxart{446}.
+
+\idxb{Radian}, the, defined as angular unit, \idxart{5}, \textit{note}.
+
+\idxb{Radiant}, the, in meteoric showers, \idxart{780}.
+
+\idxb{Radius} of curvature of a meridian, \idxart{149}.
+
+\idxc{Ranyard}, peculiar theory of the repulsive force operative in comets' tails, \idxart{733}.
+
+\idxb{Rate} of a clock defined, \idxart{53}.
+
+\idxb{Reading} microscope, the, \idxart{64}.
+
+\idxb{Recognition} of elliptic comets, difficulties, \idxart{704}.
+
+\idxb{Red} spot of Jupiter, the, \idxart{618}.
+
+\idxb{Reduction} of mean star places to apparent and \textit{vice versa}, \idxart{797}.
+
+\idxb{Reflecting} telescope, various forms, \idxart{47}, \idxart{48};
+telescopes, large instruments, \idxart{48}.
+%% -----File: File: 559.png---Folio 548-------
+
+\idxb{Refraction}, atmospheric, its law, \idxart{89}, \idxart{90};
+determination of its amount, \idxart{94}, \idxart{95};
+effect of temperature and barometric pressure, \idxart{91};
+effect upon form and size of discs of sun and moon near the horizon, \idxart{93};
+effect upon time of sunrise and sunset, \idxart{92}.
+
+\idxb{Refracting} telescope (simple), \idxart{36};
+telescope, achromatic, \idxart{41}.
+
+\idxb{Refractors} and reflectors compared, \idxart{49}.
+
+\idxc{Reich}, determination of the density of the earth, \idxart{166};
+experiments upon falling bodies, \idxart{138}.
+
+\idxb{Relative} motion, law of, \idxart{492};
+sizes of the planets, diagram, \idxart{550}.
+
+\idxb{Repulsive} force acting on comets, \idxart{728}, \idxart{731}, \idxart{732}, \idxart{734}.
+
+\idxb{Retardation} of earth's rotation by the tides, \idxart{461}, \idxart{483}.
+
+\idxb{Reticle} used in telescope for pointing, \idxart{46}.
+
+\idxb{Retrograde} revolution of the satellites of Uranus and Neptune, \idxart{652}, \idxart{661}, \idxart{914}.
+
+\idxb{Reversing} layer of the solar atmosphere, \idxart{291}, \idxart{319}, \idxart{320}, \idxart{362}.
+
+\idxb{Reversal} of the spectrum, \idxart{314}.
+
+\idxb{Rhea}, the fifth satellite of Saturn, \idxart{643}, \textit{note}.
+
+\idxb{Rigidity} of the earth, \idxart{171}, \idxart{482}.
+
+\idxb{Right} ascension defined, \idxart{25}, \idxart{27};
+ascension determined by transit instrument, \idxart{59}, \idxart{128}, \idxart{129};
+sphere, the, \idxart{31}.
+
+\idxb{Rings} of Saturn, \idxart{637}--\idxart{642}.
+
+\idxc{Rosse, Lord}, observations of lunar heat, \idxart{260}, \idxart{261};
+his great telescope, \idxart{48};
+spiral nebulæ, \idxart{888}.
+
+\idxb{Rotation}, distinguished from revolution, \idxart{248}, \idxart{248*};
+of the earth, affected by the tides, \idxart{461}, \idxart{483};
+of the earth, proofs of, \idxart{138}--\idxart{143};
+of planets, how determined, \idxart{543};
+period of Jupiter, \idxart{615};
+period of Mars, \idxart{584};
+period of the moon, \idxart{248}, \idxart{252};
+period of Saturn, \idxart{635};
+of the sun, \idxart{281}, \idxart{283};
+period of Venus, \idxart{570};
+periods, see also Appendix, \hyperref[app:I]{Table I.}
+
+
+\idxsection*{S.}
+
+\idxb{Saros}, the, \idxart{395}--\idxart{398};
+number of eclipses in a saros, \idxart{398}.
+
+\idxb{Satellites} of Jupiter, \idxart{621}--\idxart{631};
+of Mars, \idxart{590}, \idxart{591};
+of Neptune, \idxart{661};
+of Saturn, \idxart{643}, \idxart{644};
+of Uranus, \idxart{650}--\idxart{652};
+general table of, Appendix, \hyperref[app:II]{Table II.}
+
+\idxb{Satellite} orbits, generally circular, \idxart{548}.
+
+\idxb{Saturn}, the planet, \idxart{632}--\idxart{644}.
+
+\idxb{Schehallien}, determination of the earth's mass, \idxart{164}.
+
+\idxc{Schiaparelli}, connection between comets and meteors, \idxart{786};
+his map of Mars, \idxart{588}.
+
+\idxc{Schröter}, the rotation of Mercury, \idxart{559};
+the rotation of Venus, \idxart{570}.
+
+\idxc{Schwabe}, discovery of the periodicity of sun spots, \idxart{307}.
+
+\idxb{Scintillation} of the stars, \idxart{864}.
+
+\idxb{Sea}, ship's place at, \idxart{103}, \idxart{120}~B, \idxart{121}~A, \idxart{124}--\idxart{126}.
+
+\idxb{Seasons}, the, explained, \idxart{190}, \idxart{192}, \idxart{193};
+difference between northern and southern hemispheres, \idxart{194}, \idxart{195}.
+
+\idxc{Secchi}, theories of sun spots, \idxart{303}, \idxart{305};
+observations on stellar spectra, \idxart{856}, \idxart{857}.
+
+\idxc{Seidel}, his photometer, \idxart{827}.
+
+\idxb{Secular} acceleration of the moon's mean motion, \idxart{459}--\idxart{461};
+changes in the earth's orbit, \idxart{196}--\idxart{200};
+perturbations in the planetary system, \idxart{525}--\idxart{529}.
+
+\idxb{Semi-diameter}, augmentation of the moon's, \idxart{88};
+correction for, in sextant observations, \idxart{88}.
+
+\idxb{Semi-major} axis of a planet's orbit, defined and discussed, \idxart{505}, \idxart{506};
+axis of the planets' orbits, invariable, \idxart{526};
+axis as depending on planet's velocity, \idxart{428}--\idxart{430}.
+
+\idxb{Separating} power of a telescope, Dawes, \idxart{43}.
+
+\idxb{Sequences}, method of, in stellar photometry, \idxart{824}.
+
+\idxb{Sextant}, the, described, \idxart{76};
+the, used to determine latitude, \idxart{103};
+the, used in finding a ship's place at sea, \idxart{103}, \idxart{116}, \idxart{125}, \idxart{126};
+the, used in determining time, \idxart{115}, \idxart{116}.
+
+\idxb{Shadow} of the earth, its dimensions, \idxart{367};
+of the moon, \idxart{379}, \idxart{380};
+of the moon, its velocity over the earth, \idxart{384}.
+
+\idxb{``Sheath''} of the comet of 1882, \idxart{752}.
+
+\idxb{Ship's} place at sea, determination of, \idxart{103}, \idxart{120}~B, \idxart{121}~A, \idxart{124}--\idxart{126}.
+
+\idxb[Shooting Stars]{Shooting Stars}, \idxart{770}, \idxart{787};
+ashes of, \idxart{775};
+brightness of, \idxart{773};
+comparative numbers in morning and evening, \idxart{772};
+daily number of, \idxart{771};
+elevation of, \idxart{774};
+mass of, \idxart{776};
+materials of, \idxart{775};
+path of, \idxart{774};
+showers of, \idxart{780}--\idxart{786};
+spectra of, \idxart{775};
+velocity of, \idxart{774}.
+
+\idxb{Short-period} comets, \idxart{703};
+comets, table of, Appendix, \hyperref[app:III]{Table III.}
+
+\idxb{Showers}, meteoric, \idxart{780}--\idxart{786}.
+
+\idxb{Sidereal} day defined, \idxart{26}, \idxart{110};
+month, \idxart{229};
+time, \idxart{26}, \idxart{110};
+year, its length, \idxart{216}.
+
+\idxb{Signals} used in determining difference of longitude, \idxart{119}.
+%% -----File: File: 560.png---Folio 549-------
+
+\idxb{Signs} of the zodiac, \idxart{177}.
+
+\idxb{Single-altitude} method of determining local time, \idxart{116}.
+
+\idxb{Sirius} and its companion, \idxart{875};
+its light compared with the sun's, \idxart{334}, \idxart{832}, \idxart{835};
+its mass, \idxart{877}.
+
+\idxb{Sky}, apparent distance of, \idxart{6}.
+
+\idxb{Slitless} spectroscope, the, \idxart{860}--\idxart{862}.
+
+\idxb{Solar} constant, the, \idxart{338}--\idxart{340};
+eclipses, \idxart{319}, \idxart{323}, \idxart{327}, \idxart{329}, \idxart{387}--\idxart{390}, \idxart{393}, \idxart{398};
+eclipses, their rarity, \idxart{398};
+engine of Ericsson and Mouchot, \idxart{345};
+eye-pieces, \idxart{286}, \idxart{287};
+parallax, see \idxsee{Parallax of the Sun};
+system, age of, \idxart{922};
+time, apparent and mean, defined, \idxart{111}, \idxart{112}.
+
+\idxb{Solstice} defined, \idxart{176}.
+
+\idxc{Sosigenes}, devises the Julian calendar, \idxart{219}.
+
+\idxb{Spectra} of comets, \idxart{724}--\idxart{726};
+of meteors, \idxart{775};
+of nebulæ, \idxart{890}, \idxart{891};
+of stars, \idxart{855}--\idxart{863}.
+
+\idxb{Spectroscope}, principles of its construction, \idxart{311}--\idxart{313};
+how it shows the solar prominences, \idxart{324};
+slitless, \idxart{860}--\idxart{862}.
+
+\idxb{Spectroscopic} measurement of motions in the line of vision, \idxart{321}, \idxart{802}.
+
+\idxb{Spectrum} of the chromosphere and prominences, \idxart{323};
+of the corona, \idxart{329};
+solar (photosphere), \idxart{312};
+solar, compared with iron, \idxart{315};
+of sun spots, \idxart{321};
+analysis, fundamental principles, \idxart{314};
+photometry, \idxart{831}.
+
+\idxb{Sphere}, the celestial, conceptions of it, \idxart{4};
+the oblique, \idxart{33};
+the parallel, \idxart{32};
+the right, \idxart{31}.
+
+\idxb{Spheres}, attraction of, \idxart{162}.
+
+\idxb{Spheroid}, terrestrial, its dimensions, \idxart{145}, Appendix, page~\pageref{app:spheroid}.
+
+\idxb{Spherical} aberration of a lens, \idxart{39};
+astronomy, defined, \idxart{3};
+shell, its attraction, \idxart{169}.
+
+\idxb{Spider} lines in a reticle, \idxart{46}.
+
+\idxb{Spring} tide defined, etc., \idxart{463}.
+
+\idxb{Spurious} disc of stars in the telescope, \idxart{43}.
+
+\idxb{Stability} of the planetary system, \idxart{530}--\idxart{533}.
+
+\idxb{Standard} and local time, \idxart{122}.
+
+\idxb{Stars}, binary, see \idxsee{Binary Stars};
+causes of the difference in their brightness, \idxart{836};
+colors of, \idxart{830};
+dark, \idxart{836};
+designations and names, \idxart{794};
+their real diameters, \idxart{837};
+distribution of, \idxart{899};
+double, see \idxsee{Double Stars};
+gradual changes in their light, \idxart{839};
+heat from them, \idxart{834};
+light compared with sunlight, \idxart{334}, \idxart{832}, \idxart{835};
+magnitudes of, \idxart{816}--\idxart{822};
+missing, \idxart{840};
+nature, as being suns, \idxart{789};
+number of, \idxart{790};
+parallax and distance, \idxart{808}--\idxart{814};
+photography of, \idxart{798};
+photometric observations of, \idxart{823}--\idxart{831};
+proper motions of, \idxart{800}--\idxart{803};
+proximity of, its indications, \idxart{813};
+seen by day with telescope, \idxart{38};
+shooting, see \idxsee{Shooting Stars};
+temporary, \idxart{842}--\idxart{845};
+triple and multiple, \idxart{882};
+twinkling of, or scintillation, \idxart{864};
+variable, see \idxsee{Variable Stars}.
+
+\idxb{Star-atlases}, \idxart{793}.
+
+\idxb{Star-catalogues}, \idxart{795}.
+
+\idxb{Star-charts}, \idxart{798}.
+
+\idxb{Star-clusters}, \idxart{883}, \idxart{884}.
+
+\idxb{Star-gauges}, \idxart{899}.
+
+\idxb{Star-motions}, \idxart{799}--\idxart{803}.
+
+\idxb{Star-places}, how affected by aberration, etc., \idxart{226};
+their determination, \idxart{796};
+mean and apparent, \idxart{797}.
+
+\idxb{Statical} theory of the tides, \idxart{469}.
+
+\idxb{Station} errors, \idxart{158}.
+
+\idxb{Stellar} spectra, \idxart{855}, \idxart{856};
+classification of, \idxart{857}, \idxart{858};
+photography of, \idxart{859}--\idxart{863};
+system, the hypothetical, \idxart{901}--\idxart{904}.
+
+\idxc{Stone, E.~J.}, attempted observation of stellar heat, \idxart{834}.
+
+\idxb{Stripe}, central, in comets' tails, \idxart{730}.
+
+\idxb{Structure} of the heavens, \idxart{900}--\idxart{904}.
+
+\idxc{Struve, von, F.~G.~W.}, on distribution of stars, \idxart{899}.
+
+\idxc{Struve, von, Ludwig}, investigation of sun's motion in space, \idxart{806}.
+
+\idxc{Struve, von, Otto}, Saturn's rings, \idxart{637}, \idxart{642}.
+
+\idxc{Sumner, Capt.}, his method of determining a ship's place at sea, \idxart{125}, \idxart{126}.
+
+\idxb{Sun}, the, \idxart{273}--\idxart{364};
+age and duration of, \idxart{359}, \idxart{922};
+apparent annual motion of, \idxart{172}, \idxart{173};
+attraction on the earth, its intensity, \idxart{436};
+candle power of sunlight, \idxart{332}, \idxart{333};
+chemical elements in it, \idxart{315}--\idxart{317};
+diameter, surface, and volume, \idxart{276}, \idxart{277};
+distance and parallax, \idxart{274}, \idxart{275}, \idxart{665}--\idxart{693};
+gravity at its surface, \idxart{280};
+heat emission, \idxart{338}--\idxart{357};
+light, \idxart{332}--\idxart{336};
+mass and density, \idxart{278}, \idxart{279};
+its motion in space, \idxart{804}--\idxart{805};
+physical constitution, \idxart{360}--\idxart{364};
+its temperature, \idxart{349}--\idxart{351};
+the central, \idxart{807}.
+
+\idxb{Sun spots}, their development and changes, \idxart{297}, \idxart{298};
+distribution on sun's surface, \idxart{301};
+general description, \idxart{293}, \idxart{294};
+influence on terrestrial conditions, \idxart{309}, \idxart{310};
+periodicity of, \idxart{307}--\idxart{309};
+their spectrum, \idxart{321};
+theories as to their cause and nature, \idxart{302}--\idxart{306}.
+
+\idxb{Sun's way}, apex of, \idxart{804}--\idxart{806}.
+
+\idxb{Sunrise} and sunset affected by refraction, \idxart{92}.
+%% -----File: File: 561.png---Folio 550-------
+
+\idxb{Superior} planet, motions of, \idxart{496}.
+
+\idxb{Surface} errors in lenses and mirrors, \idxart{49};
+of the moon, \idxart{263}--\idxart{270}.
+
+\idxb{Swarms}, meteoric, \idxart{783}.
+
+\idxc{Swedenborg}, a proposer of the nebular hypothesis, \idxart{908}.
+
+\idxb{System}, planetary, facts suggesting a theory of its origin, \idxart{907};
+numerical data, Appendix, \hyperref[app:I]{Table I.};
+stellar, \idxart{901}--\idxart{904}.
+
+\idxb{Synodic} month, or revolution, of moon, \idxart{229}--\idxart{231};
+period, general definition of, \idxart{490}.
+
+\idxb{Syzygy}, defined, \idxart{230}.
+
+
+\idxsection*{T.}
+
+\idxb{Tables}, \textit{Appendix}. Greek alphabet, page~\pageref{app:greek};
+miscellaneous symbols, page~\pageref{app:misc};
+dimensions of the earth, page~\pageref{app:spheroid};
+time constants, page~\pageref{app:time};
+Table I., elements of solar system, page~\pageref{app:I};
+Table II., satellites of the system, page~\pageref{app:II};
+Table III., short-period comets, page~\pageref{app:III};
+Table IV., parallaxes of stars, page~\pageref{app:IV};
+Table V., orbits of binary stars, page~\pageref{app:V};
+Table VI., the variable stars, page~\pageref{app:VI}.
+
+\idxb{Tables}, \textit{in body of the book}. The constellations, \idxart{792};
+approximate distances and periods of the planets, \idxart{489};
+distance of sun corresponding to certain values of the parallax, \idxart{668};
+distribution of stars with reference to the galaxy, \idxart{899};
+iron meteors seen to fall, \idxart{758};
+naked-eye stars north of celestial equator, \idxart{818};
+orbits and masses of certain binary stars, \idxart{877};
+parallaxes of first-magnitude stars, Elkin, \idxart{815};
+proper motions of certain stars, \idxart{800};
+signs of the zodiac, \idxart{177};
+telescopic aperture required to show stars of given magnitude, \idxart{822};
+temporary stars, \idxart{842};
+total light from stars of different magnitude, \idxart{833};
+velocity of free wave at various depths, \idxart{473}.
+
+\idxb{Tails} or trains of comets, \idxart{713}, \idxart{717}, \idxart{728}--\idxart{736}.
+
+\idxc{Talcott, Capt.}, his zenith telescope, \idxart{105}.
+
+\idxb{Tangential} component of the disturbing force, \idxart{447}.
+
+\idxb{Telegraph} used in determination of longitude, \idxart{121}~B.
+
+\idxb{Telescope}, the, achromatic, \idxart{41};
+distinctness of image, \idxart{39};
+equatorial, \idxart{72};
+eye-pieces, \idxart{44};
+the general theory, \idxart{35};
+invention of, \idxart{35};
+light-gathering power, \idxart{38};
+long, of Huyghens, \idxart{40};
+magnifying power, \idxart{37};
+object-glass, various forms, \idxart{41}, \idxart{42};
+reflecting, various forms, \idxart{47}, \idxart{48};
+refracting, simple, \idxart{36};
+relation of its aperture to the ``magnitude'' of the smallest star visible with it, \idxart{822};
+separating or dividing power, \idxart{43}.
+
+\idxb{Telespectroscope}, \idxart{313}.
+
+\idxb{Temperature}, cause of the annual change, \idxart{192}, \idxart{193};
+of the moon, \idxart{261};
+of the sun, \idxart{349}--\idxart{351}.
+
+\idxb{Temporary} stars, \idxart{842}--\idxart{845}.
+
+\idxb{``Terminator,''} the, on the moon's surface, its form, \idxart{253}.
+
+\idxb{Tethys}, the third satellite of Saturn, \idxart{643}, \textit{note}.
+
+\idxc{Thomsen} of Copenhagen, the mechanical equivalent of light, \idxart{776}.
+
+\idxc{Thomson, Sir W.}, on the temperature of meteors, \idxart{765};
+rigidity of the earth, \idxart{171}, \idxart{482}.
+
+\idxb{Three} bodies, the problem of, \idxart{437}, \idxart{438}.
+
+\idxb{Tidal} evolution, \idxart{484}, \idxart{916};
+friction, effect on the earth's rotation, \idxart{461}, \idxart{483};
+wave, its origin and course, \idxart{476}.
+
+\idxb{Tides}, the, definition of terms relating to them, \idxart{463};
+priming and lagging of, \idxart{470};
+statical theory of, \idxart{469};
+wave theory of, \idxart{472}.
+
+\idxb{Tide-raising} force, the, \idxart{464}--\idxart{467}.
+
+\idxb{Time}, defined as an hour-angle, \idxart{109};
+its determination by the sextant, \idxart{115}, \idxart{116};
+its determination by the transit instrument, \idxart{113};
+equation of, explained and discussed, \idxart{201}--\idxart{204};
+sidereal, defined, \idxart{26}, \idxart{110};
+solar, apparent, \idxart{111};
+solar, mean, \idxart{112};
+standard and local, \idxart{122}.
+
+\idxc{Tisserand} on peculiarities of satellite orbits, \idxart{548}.
+
+\idxb{Titan}, the sixth and great satellite of Saturn, \idxart{643}.
+
+\idxb{Titania}, the third satellite of Uranus, \idxart{650}.
+
+\idxc{Todd, Prof. D.~P.}, search for trans-Neptunian planet, \idxart{662}.
+
+\idxb{Torsion} balance, determination of the earth's density, \idxart{165}.
+
+\idxb{Trade} winds, proving rotation of the earth, \idxart{143}.
+
+\idxb{Trains} of meteors, \idxart{766}, \idxart{773}.
+
+\idxb{Transits} of moon across meridian, the interval between them, \idxart{235};
+of Mercury, \idxart{561}, \idxart{562};
+of Venus, law of recurrence, \idxart{575}--\idxart{577};
+of Venus, used for determination of solar parallax, \idxart{678}--\idxart{686}.
+
+\idxb{Transit} circle, see \idxsee{Meridian Circle}, \idxart{63};
+instrument, \idxart{59}--\idxart{61};
+instrument used in determining time, \idxart{113}.
+%% -----File: File: 562.png---Folio 551-------
+
+\idxb{Trans-Neptunian} planet, hypothetical, \idxart{662}.
+
+\idxb{Transparency} of space as affected by meteors, \idxart{779}.
+
+\idxb{Triple} and multiple stars, \idxart{882}.
+
+\idxb{Tropics}, defined, \idxart{176}.
+
+\idxb{Tropical} year, its definition and its length, \idxart{216}.
+
+\idxb{Twilight}, theory and duration of, \idxart{96}, \idxart{97}, \idxart{130}.
+
+\idxb{Twinkling}, or scintillation, of the stars, \idxart{864}.
+
+\idxb{Two} bodies, problem of, \idxart{424}--\idxart{433}.
+
+\idxc[Tycho]{Tycho Brahe} discovers the lunar variation, \idxart{457};
+observations of comet of 1577, \idxart{700};
+temporary star in Cassiopeia, \idxart{843};
+his planetary system, \idxart{504}.
+
+
+\idxsection*{U.}
+
+\idxb{Umbriel}, the second satellite of Uranus, \idxart{650}.
+
+\idxb{Unit} of stellar distances, the light-year, \idxart{814}.
+
+\idxb{Uranolith.}  See \DPtypo{\idxsee{Meteorite}}{\idxsee{Meteorites}}.
+
+\idxb{Uranometria Nova:} Argelander, \idxart{817};
+Oxoniensis, \idxart{826}.
+
+\idxb{Uranus} and Neptune, their anomalous rotation in relation to the nebular hypothesis, \idxart{914};
+the planet, \idxart{645}--\idxart{652}.
+
+\idxb{Utility} of astronomy, \idxart{2}.
+
+
+\idxsection*{V.}
+
+\idxc{Van der Kolk's} theorem, \idxart{434}.
+
+\idxb{Vanishing} point of a system of parallel lines, \idxart{7}.
+
+\idxb{Variable} nebulæ, \idxart{889}.
+
+\idxb[Variable Stars]{Variable stars}, \idxart{838}--\idxart{854};
+classification of, \idxart{838};
+explanation of their variation, \idxart{849}--\idxart{851};
+methods of observation, \idxart{854};
+their number and designation, \idxart{852};
+their range of variation, \idxart{853}.
+
+\idxb{Variation}, the lunar, \idxart{457}.
+
+\idxb[Vega]{Vega}, or $\alpha$~Lyræ, its light compared with the sun's, \idxart{334}, \idxart{832}, \idxart{835};
+its spectrum, \idxart{859}.
+
+\idxb{Velocity} of air currents at high elevations, \idxart{773}, \textit{note};
+areal, linear and angular, law of, \idxart{407}--\idxart{409};
+of earth in her orbit, \idxart{225}, \textit{note}, \idxart{278};
+of light, \idxart{225}, \textit{note,} \idxart{690}--\idxart{692};
+of the moon's shadow, \idxart{384};
+parabolic, or velocity from infinity, \idxart{429};
+of planet at any point in its orbit, \idxart{434};
+of stellar motions, \idxart{801}.
+
+\idxb{Venus}, the planet, \idxart{563}--\idxart{577};
+her atmosphere and its effect upon observations of a transit, \idxart{681};
+transits of, used to determine solar parallax,  \idxart{678}--\idxart{686}.
+
+\idxb{Vertical}, angle of the, \idxart{156};
+circles defined, \idxart{12}.
+
+\idxc{Vogel}, his classification of stellar spectra, \idxart{858};
+star motions in line of sight, \idxart{802}, \idxart{863}.
+
+\idxb{Vulcan}, hypothetical intra-Mercurial
+planet, \idxart{603}, \idxart{604}.
+
+
+\idxsection*{W.}
+
+\idxb{Waste} of solar energy, \idxart{347}.
+
+\idxb{Water}, absence of, on the moon, \idxart{258};
+presence of, in atmosphere of planets, \idxart{560}, \idxart{573}, \idxart{589}.
+
+\idxc{Watson, J.~C.} discovers and endows twenty-two asteroids, \idxart{593}, \idxart{601}.
+
+\idxb{Wave-length} of a ray of light, affected by motion in the line of vision---Doppler's principle, \idxart{321}, \textit{note.}
+
+\idxb{Wave-theory} of the tides, \idxart{472}.
+
+\idxb{Weather}, moon's influence on it, \idxart{262}.
+
+\idxb{Wedge} photometer, the, \idxart{826}.
+
+\idxb{Weight}, loss of, between equator and pole, \idxart{152}--\idxart{155};
+and mass, distinction between them, \idxart{159}, \idxart{160}.
+
+\idxc{Wilsing}, determination of the earth's density, \idxart{167}.
+
+\idxc{Winnecke's} comet, acceleration of, \idxart{711}.
+
+\idxc{Wolf}, periodicity of the sun spots, \idxart{307}.
+
+\idxc{Worms}, formula for the eastward deviation of a falling body, \idxart{138}.
+
+
+\idxsection*{Y.}
+
+\idxb{Year}, bissextile, or leap, \idxart{219};
+beginning of, \idxart{222};
+of confusion, \idxart{219};
+eclipse, \idxart{391};
+sidereal, tropical, and anomalistic, \idxart{216}, also Appendix, page~\pageref{app:time};
+of light, unit of stellar distance, \idxart{814}.
+
+
+\idxsection*{Z.}
+
+\idxb{Zenith}, the astronomical and geocentric, \idxart{8}.
+
+\idxb{Zenith} distance defined, \idxart{21};
+telescope, for determination of latitude, \idxart{105}.
+
+\idxc{Zenker}, theory of a comet's constitution, \idxart{733}.
+
+\idxb{Zero} points of a meridian circle, \idxart{66}, \idxart{67}.
+
+\idxb{Zodiac}, the, and its signs, \idxart{177};
+signs of, as affected by precession, \idxart{208}.
+
+\idxb{Zodiacal} light, the, \idxart{607}, \idxart{608}.
+
+\idxc{Zöllner}, albedo of the planets, \idxart{558}, \idxart{572}, \idxart{583}, \idxart{614}, \idxart{636}, \idxart{648}, \idxart{660}, also Appendix, page~\pageref{app:I};
+his photometer, \idxart{827};
+on the repulsive force acting upon comets, \idxart{732}.
+
+\end{multicols}
+} %end group for font size change etc
+\chelabel{INDEX}
+
+\clearpage
+%% -----File: 563.png---Folio 552-------
+%% -----File: 564.png---Folio 553-------
+
+\markboth{}{}
+\pagestyle{empty}
+\renewcommand\headrule{}
+
+\begin{center}
+\scriptsize
+\nbenlargepage
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+\multicolumn{3}{c}{\Large \textsf{Latin Text-Books.}}  \\
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+\multicolumn{2}{l}{\textsc{Allen \& Greenough}: \textbf{Latin Grammar} \dotfill} & \$1.20  \\
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+& \textbf{Preparatory Course of Latin Prose} \dotfill & 1.40  \\
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+\textsc{Allen} \dotfill & \textbf{New Latin Method} \dotfill & .90  \\
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+& \textbf{Latin Lexicon} \dotfill & .90  \\
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+\textsc{Stickney} \dotfill & \textbf{Cicero de Natura Deorum} \dotfill & 1.40  \\
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+& \textbf{Latin for Sight Reading} \dotfill & 1.00  \\
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+\textsc{White (J. W.)} \dotfill & \textbf{Schmidt's Rhythmic and Metric} \dotfill & 2.50  \\
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+\textsc{White (J. T.)} \dotfill & \textbf{Junior Students' Latin-English Lexicon} (mor.) \dotfill & 1.75  \\
+& \textbf{English-Latin Lexicon} (sheep) \dotfill & 1.50  \\
+& \textbf{Latin-English and English-Latin Lexicon} (sheep) \dotfill & 3.00  \\
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+\textsc{Whiton} \dotfill & \textbf{Auxilia Vergiliana}; or, First Steps in Latin Prosody \dotfill & .15  \\
+& \textbf{Six Weeks' Preparation for Reading Cæsar} \dotfill & .40
+\end{tabular}
+
+\textit{Copies sent to Teachers for Examination, with a view to Introduction,
+on receipt of Introduction Price.}
+
+\nbrule
+
+\medskip
+\textbf{\normalsize GINN \& COMPANY, Publishers,}\\
+\textsc{Boston, New York, and Chicago.}
+\end{center}
+%% -----File: 565.png---Folio 554-------
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+\begin{center}
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+\multicolumn{3}{c}{\textsf{\Large Greek Text-Books.}}  \\
+\multicolumn{3}{r}{\textit{Intro.~Price.}}  \\
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+\textbf{Allen}: & Medea of Euripides \dotfill & \$1.00  \\
+
+\textbf{Flagg}:
+& Hellenic Orations of Demosthenes \dotfill & 1.00  \\
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+\textbf{Goodwin}:
+& Greek Grammar \dotfill & 1.50  \\
+& Greek Reader \dotfill & 1.50  \\
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+\textbf{Keep}: & Essential Uses of the Moods \dotfill & .25  \\
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+\textbf{Parsons}: & Cebes' Tablet \dotfill & .75  \\
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+                                  Cloth & .60 \end{tabular}}  \\
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+\textbf{Sidgwick}: & Greek Prose Composition \dotfill & 1.50  \\
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+& First Lessons in Greek \dotfill & 1.20  \\
+& Schmidt's Rhythmic and Metric \dotfill & 2.50  \\
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+\multicolumn{2}{l}{\textbf{Beckwith}: Euripides' Bacchantes.}  \\
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+\textbf{D'Ooge}: & Sophocles' Antigone.  \\
+& Text and Notes, Paper, .95; Cloth, \$1.25; Text only, & .20.  \\
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+\textbf{Dyer}: & Plato's Apology and Crito.  \\
+& Text and Notes, Paper, .95; Cloth, \$1.25; Text only, & .20.  \\
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+\textbf{Fowler}: & Thucydides, Book V.  \\
+& Text and Notes, Paper, .95; Cloth, \$1.25; Text only, & .20.  \\
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+\multicolumn{2}{l}{\textbf{Humphreys}: Aristophanes' Clouds.}  \\
+& Text and Notes, Paper, .95; Cloth, \$1.25; Text only, & .20.  \\
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+\textbf{Manatt}: & Xenephon's Hellenica, Books I.--IV.  \\
+& Text and Notes, Paper, \$1.20; Cloth, \$1.50; Text only, & .20.  \\
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+\textbf{Morris}: & Thucydides, Book I.  \\
+& Text and Notes, Paper, \$1.20; Cloth, \$1.50; Text only, & .20.  \\
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+\multicolumn{2}{l}{\textbf{Seymour}: Homer's Iliad, Books I.--III.}  \\
+& Text and Notes, Paper, .95: Cloth, \$1.25; Text only, & .20.  \\
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+\textbf{Smith}: & Thucydides, Book VII.  \\
+& Text and Notes, Paper, .95; Cloth, \$1.25; Text only, & .20. \\
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+\end{tabular}
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+\\
+\end{tabular}
+\begin{tabular}{p{0.16\textwidth}@{\ }p{0.6\textwidth}@{\,}r}
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+\multicolumn{3}{c}{\textsf{\Large Sanskrit.}}  \\
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+\textbf{Arrowsmith}: & Kaegi's Rigveda, (\textit{translation}) \dotfill & \$1.50  \\
+
+\textbf{Elwell}: & Nine Jatakas (\textit{Pali}) \dotfill & .60  \\
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+\textbf{Lanman}: & Sanskrit Reader \dotfill & 1.80  \\
+
+\textbf{Perry}: & Sanskrit Primer \dotfill & 1.50  \\
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+\textbf{Whitney}: & Sanskrit Grammar \dotfill & 2.50  \\
+
+\end{tabular}
+
+
+\textit{Copies sent to Teachers for Examination, with a view to Introduction,
+on receipt of Introduction Price.}
+
+\nbrule
+
+\medskip
+\textbf{\normalsize GINN \& COMPANY, Publishers,}\\
+\textsc{Boston, New York, and Chicago.}
+\end{center}
+%% -----File: 566.png---Folio 555-------
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+\clearpage
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+\begin{center}
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+\multicolumn{3}{r}{\tiny\textit{Introd.}}\\[-0.8ex]
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+\textbf{Byerly} \dotfill & Differential Calculus \dotfill & \$2.00  \\
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+\textbf{Ginn} \dotfill & Addition Manual \dotfill & .15  \\
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+\textbf{Hardy} \dotfill & Quaternions \dotfill & 2.00  \\
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+\textbf{Taylor} \dotfill & Elements of the Calculus \dotfill & 1.80 \\
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+& Shorter Course in Algebra \dotfill & 1.00  \\
+& Elements of Algebra \dotfill & 1.12  \\
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+& Plane Geometry \dotfill & .75  \\
+& Plane and Solid Geometry \dotfill & 1.25  \\
+& Plane and Solid Geometry, and Trigonometry \dotfill & 1.40  \\
+& Plane Trigonometry and Tables. \textit{Paper} \dotfill & .60  \\
+& Pl.\ and Sph.\ Trig., Surv., and Navigation \dotfill & 1.12  \\
+& Pl.\ and Sph.\ Trig., Surv., and Tables \dotfill & 1.25  \\
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+\multicolumn{2}{l}{\textbf{Wentworth \& Hill}: Practical Arithmetic \dotfill} & 1.00  \\
+& Abridged Practical Arithmetic \dotfill & .75  \\
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+& \quad Part II. \textit{Examination Manual} \dotfill & .35  \\
+& \quad Answers (to both Parts) \dotfill & .25  \\
+& Exercises in Geometry \dotfill & .70  \\
+& Five-place Log.\ and Trig.\ Tables (7 \textit{Tables}) \dotfill & .50  \\
+& Five-place Log.\ and Trig.\ Tables (\textit{Comp.~Ed.}) \dotfill & 1.00  \\
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+\multicolumn{2}{l}{\textbf{Wentworth \& Reed}: First Steps in Number, \textit{Pupils' Edition} \dotfill} & .30  \\
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+& \multicolumn{1}{r}{Parts I., II., and III. (separate), each} & .30  \\
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+\textbf{Wheeler} \dotfill & Plane and Spherical Trig.\ and Tables \dotfill & 1.00
+
+\end{tabular}
+
+\medskip
+\nbrule
+
+\textit{Copies sent to Teachers for Examination, with a view to Introduction,
+on receipt of Introduction Price.}
+
+\nbrule
+
+\medskip
+\textbf{\normalsize GINN \& COMPANY, Publishers,}\\
+\textsc{\hfill Boston. \hfill New York. \hfill Chicago. \hfill\null}
+\nbVS
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+
+%% -----File: 567.png---Folio 556-------
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+\begin{center}
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+\textbf{Everett}: & Vibratory Motion and Sound \dotfill & \$2.00  \\
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+& Introduction to Physical Science \dotfill & 1.00  \\
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+\textbf{Hale}: & Little Flower-People \dotfill & .40  \\
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+\textbf{Hill}: & Questions on Stewart's Physics \dotfill & .35  \\
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+\multicolumn{2}{l@{\,}}{\textbf{Journal of Morphology} \dotfill (per vol.)} & 6.00  \\
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+\textbf{Knight}: & Primer of Botany \dotfill & .30  \\
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+\textbf{Williams}: & Introduction to Chemical Science \dotfill & .80  \\[3ex]
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+\textbf{Ladd}: & Lotze's Outlines of Metaphysic \dotfill & .80  \\
+& Lotze's Outlines of Philosophy of Religion \dotfill & .80  \\
+& Lotze's Outlines of Practical Philosophy \dotfill & .80  \\
+& Lotze's Outlines of Psychology \dotfill & .80  \\
+& Lotze's Outlines of Æsthetics \dotfill & .80  \\
+& Lotze's Outlines of Logic \dotfill & .80  \\
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+\multicolumn{3}{c}{\textsf{HISTORY.}}  \\[.5ex]
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+\textbf{Allen}: & Readers' Guide to English History \dotfill & .25  \\
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+\textbf{Andrade}: & Historia do Brazil \dotfill & .75  \\
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+\multicolumn{2}{l@{}}{\textbf{Fiske-Irving}: Washington and His Country \dotfill} & 1.00  \\
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+\multicolumn{2}{l@{\,}}{\textbf{Journal of Archaeology} \dotfill (per vol.)} & 5.00  \\
+
+\textbf{Judson}: & Cæsar's Army \dotfill & 1.00  \\
+
+\textbf{Montgomery}: & Leading Facts of English History \dotfill & 1.00  \\
+& English History Reader \dotfill & .60  \\
+
+\textbf{Moore}: & Pilgrims and Puritans \dotfill & .60  \\
+
+\textbf{Myers}: & Mediæval and Modern History \dotfill & 1.50  \\
+& Ancient History \dotfill & 1.40
+
+\end{tabular}
+
+\medskip
+\nbrule
+
+\textit{Copies sent to Teachers for Examination, with a view to Introduction,
+on receipt of Introduction Price.}
+
+\nbrule
+
+\medskip
+\textbf{\normalsize GINN \& COMPANY, Publishers.}\\
+\textsc{\hfill Boston. \hfill New York. \hfill Chicago. \hfill\null}
+\nbVS
+\end{center}
+
+\clearpage
+
+\pdfbookmark[0]{Project Gutenberg License}{Project Gutenberg License}
+
+\pagestyle{fancy}
+\fancyhead[C]{\footnotesize LICENSING}
+\fancyhead[R]{\footnotesize\thepage}
+\markboth{LICENSING}{}
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+\begin{verbatim}
+End of the Project Gutenberg EBook of A Textbook of General Astronomy, by 
+Charles A. Young
+
+*** END OF THIS PROJECT GUTENBERG EBOOK A TEXTBOOK OF GENERAL ASTRONOMY ***
+
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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% End of the Project Gutenberg EBook of A Textbook of General Astronomy, by 
+% Charles A. Young                                                        %
+%                                                                         %
+% *** END OF THIS PROJECT GUTENBERG EBOOK A TEXTBOOK OF GENERAL ASTRONOMY ***
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diff --git a/LICENSE.txt b/LICENSE.txt
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+++ 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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+++ b/README.md
@@ -0,0 +1,2 @@
+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #37275 (https://www.gutenberg.org/ebooks/37275)