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Title: A Short History of Astronomy
Author: Berry, Arthur
Language: English
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Transcriber’s Notes

Obvious typographical errors have been silently corrected. Variations
in hyphenation has been standardised but all other spelling and
punctuation remains unchanged.

Italics are represented thus _italic_, =bold=, superscripts thus y^{en}
and subscripts thus x_{1}.

The footnotes are placed at the end of the book.



[Illustration: The moon. From a photograph taken at the Lick Observatory.
                                    [_Frontispiece._               ]



                        _THE UNIVERSITY SERIES_

                            A Short History

                                  of

                               Astronomy

                         BY ARTHUR BERRY, M.A.

       FELLOW AND ASSISTANT TUTOR OF KING’S COLLEGE, CAMBRIDGE;
                 FELLOW OF UNIVERSITY COLLEGE, LONDON


        _Wagner._ Verzeiht! es ist ein gross Ergetzen
           Sich in den Geist der Zeiten zu versetzen.
           Zu schauen wie vor uns ein weiser Mann gebracht,
           Und wie wir’s dann zuletzt so herrlich weit gebracht.
        _Faust._ O ja, bis an die Sterne weit!
                                                 GOETHE’S _Faust_.


                               NEW YORK
                        CHARLES SCRIBNER’S SONS
                                 1899



PREFACE.


I have tried to give in this book an outline of the history of
astronomy from the earliest historical times to the present day, and to
present it in a form which shall be intelligible to a reader who has no
special knowledge of either astronomy or mathematics, and has only an
ordinary educated person’s power of following scientific reasoning.

In order to accomplish my object within the limits of one small volume
it has been necessary to pay the strictest attention to compression;
this has been effected to some extent by the omission of all but
the scantiest treatment of several branches of the subject which
would figure prominently in a book written on a different plan or
on a different scale. I have deliberately abstained from giving any
connected account of the astronomy of the Egyptians, Chaldaeans,
Chinese, and others to whom the early development of astronomy is
usually attributed. On the one hand, it does not appear to me possible
to form an independent opinion on the subject without a first-hand
knowledge of the documents and inscriptions from which our information
is derived; and on the other, the various Oriental scholars who
have this knowledge still differ so widely from one another in the
interpretations that they give that it appears premature to embody
their results in the dogmatic form of a textbook. It has also seemed
advisable to lighten the book by omitting—except in a very few simple
and important cases—all accounts of astronomical instruments; I do not
remember ever to have derived any pleasure or profit from a written
description of a scientific instrument before seeing the instrument
itself, or one very similar to it, and I have abstained from attempting
to give to my readers what I have never succeeded in obtaining myself.
The aim of the book has also necessitated the omission of a number of
important astronomical discoveries, which find their natural expression
in the technical language of mathematics. I have on this account
only been able to describe in the briefest and most general way the
wonderful and beautiful superstructure which several generations of
mathematicians have erected on the foundations laid by Newton. For the
same reason I have been compelled occasionally to occupy a good deal
of space in stating in ordinary English what might have been expressed
much more briefly, as well as more clearly, by an algebraical formula:
for the benefit of such mathematicians as may happen to read the book
I have added a few mathematical footnotes; otherwise I have tried to
abstain scrupulously from the use of any mathematics beyond simple
arithmetic and a few technical terms which are explained in the text.
A good deal of space has also been saved by the total omission of, or
the briefest possible reference to, a very large number of astronomical
facts which do not bear on any well-established general theory; and for
similar reasons I have generally abstained from noticing speculative
theories which have not yet been established or refuted. In particular,
for these and for other reasons (stated more fully at the beginning
of chapter XIII.), I have dealt in the briefest possible way with the
immense mass of observations which modern astronomy has accumulated;
it would, for example, have been easy to have filled one or more
volumes with an account of observations of sun-spots made during the
last half-century, and of theories based on them, but I have in fact
only given a page or two to the subject.

I have given short biographical sketches of leading astronomers (other
than living ones), whenever the material existed, and have attempted
in this way to make their personalities and surroundings tolerably
vivid; but I have tried to resist the temptation of filling up space
with merely picturesque details having no real bearing on scientific
progress. The trial of Kepler’s mother for witchcraft is probably
quite as interesting as that of Galilei before the Inquisition, but I
have entirely omitted the first and given a good deal of space to the
second, because, while the former appeared to be chiefly of curious
interest, the latter appeared to me to be not merely a striking
incident in the life of a great astronomer, but a part of the history
of astronomical thought. I have also inserted a large number of
dates, as they occupy very little space, and may be found useful by
some readers, while they can be ignored with great ease by others; to
facilitate reference the dates of birth and death (when known) of every
astronomer of note mentioned in the book (other than living ones) have
been put into the Index of Names.

I have not scrupled to give a good deal of space to descriptions of
such obsolete theories as appeared to me to form an integral part of
astronomical progress. One of the reasons why the history of a science
is worth studying is that it sheds light on the processes whereby a
scientific theory is formed in order to account for certain facts, and
then undergoes successive modifications as new facts are gradually
brought to bear on it, and is perhaps finally abandoned when its
discrepancies with facts can no longer be explained or concealed. For
example, no modern astronomer as such need be concerned with the Greek
scheme of epicycles, but the history of its invention, of its gradual
perfection as fresh observations were obtained, of its subsequent
failure to stand more stringent tests, and of its final abandonment
in favour of a more satisfactory theory, is, I think, a valuable and
interesting object-lesson in scientific method. I have at any rate
written this book with that conviction, and have decided very largely
from that point of view what to omit and what to include.

The book makes no claim to be an original contribution to the subject;
it is written largely from second-hand sources, of which, however,
many are not very accessible to the general reader. Particulars of the
authorities which have been used are given in an appendix.

It remains gratefully to acknowledge the help that I have received
in my work. Mr. W. W. Rouse Ball, Tutor of Trinity College, whose
great knowledge of the history of mathematics—a subject very closely
connected with astronomy—has made his criticisms of special value, has
been kind enough to read the proofs, and has thereby saved me from
several errors; he has also given me valuable information with regard
to portraits of astronomers. Miss H. M. Johnson has undertaken the
laborious and tedious task of reading the whole book in manuscript
as well as in proof, and of verifying the cross-references. Miss F.
Hardcastle, of Girton College, has also read the proofs, and verified
most of the numerical calculations, as well as the cross-references. To
both I am indebted for the detection of a large number of obscurities
in expression, as well as of clerical and other errors and of
misprints. Miss Johnson has also saved me much time by making the Index
of Names, and Miss Hardcastle has rendered me a further service of
great value by drawing a considerable number of the diagrams. I am
also indebted to Mr. C. E. Inglis, of this College, for fig. 81; and I
have to thank Mr. W. H. Wesley, of the Royal Astronomical Society, for
various references to the literature of the subject, and in particular
for help in obtaining access to various illustrations.

I am further indebted to the following bodies and individual
astronomers for permission to reproduce photographs and drawings, and
in some cases also for the gift of copies of the originals: the Council
of the Royal Society, the Council of the Royal Astronomical Society,
the Director of the Lick Observatory, the Director of the Instituto
Geographico-Militare of Florence, Professor Barnard, Major Darwin, Dr.
Gill, M. Janssen, M. Loewy, Mr. E. W. Maunder, Mr. H. Pain, Professor
E. C. Pickering, Dr. Schuster, Dr. Max Wolf.

                                                        ARTHUR BERRY.

 KING’S COLLEGE, CAMBRIDGE



CONTENTS.


                                                                    PAGE
  PREFACE                                                              v


  CHAPTER I.

  PRIMITIVE ASTRONOMY, §§ 1-18                                      1-20

  § 1. Scope of astronomy                                              1

  §§ 2-5. First notions: the motion of the sun: the motion
  and _phases_ of the moon: daily motion of the
  stars                                                                1

  § 6. Progress due to early civilised peoples: Egyptians,
  Chinese, Indians, and Chaldaeans                                     3

  § 7. The _celestial sphere_: its scientific value: _apparent
  distance_ between the stars: the measurement of _angles_             4

  §§ 8-9. The rotation of the celestial sphere: the _North_ and
  _South poles_: the _daily motion_: the _celestial
  equator_: _circumpolar stars_                                        7

  §§ 10-11. The _annual motion_ of the sun: _great circles_:
  the _ecliptic_ and its _obliquity_: the _equinoxes_ and
  _equinoctial points_: the _solstices_ and _solstitial
  points_                                                              8

  §§ 12-13. _The constellations_: _the zodiac_, _signs of the zodiac_,
  and _zodiacal constellations_: the _first point of
  Aries_ (♈), and the _first point of Libra_ (♎)                     12

  § 14. The five _planets_: _direct_ and _retrograde_ motions:
  _stationary points_                                                 14

  § 15. The order of nearness of the planets: _occultations_:
  _superior_ and _inferior_ planets                                   15

  § 16. Measurement of time: the day and its division into
  hours: the _lunar month_: the year: the week                        17

  § 17. _Eclipses_: the _saros_                                       19

  § 18. The rise of _Astrology_                                       20


  CHAPTER II.

  GREEK ASTRONOMY (FROM ABOUT 600 B.C. TO ABOUT
  400 A.D.), §§ 19-54                                              21-75

  §§ 19-20. =Astronomy up to the time of Aristotle.= The
  Greek calendar: _full_ and _empty_ months:
  the _octaeteris_: _Meton’s cycle_                                   21

  § 21. The Roman calendar: introduction of the
  _Julian Calendar_                                                   22

  § 22. The _Gregorian Calendar_                                      23

  § 23. Early Greek speculative astronomy: _Thales_
  and _Pythagoras_: the spherical form of the
  earth: the celestial spheres: the _music of
  the spheres_                                                        24

  § 24. _Philolaus_ and other Pythagoreans: early believers
  in the motion of the earth: _Aristarchus_
  and _Seleucus_                                                      25

  § 25. _Plato_: uniform circular and spherical motions               26

  § 26. _Eudoxus_: representation of the celestial
  motions by combinations of spheres: description
  of the constellations. _Callippus_                                  27

  §§ 27-30. _Aristotle_: his spheres: the phases of the moon:
  proofs that the earth is spherical: his
  arguments against the motion of the earth:
  relative distances of the celestial bodies:
  other speculations: estimate of his astronomical
  work                                                                29

  §§ 31-2. =The early Alexandrine school=: its rise: Aristarchus:
  his estimates of the distances of the
  sun and moon. Observations by _Timocharis_
  and _Aristyllus_                                                    34

  §§ 33-4. Development of _spherics_: the _Phenomena_ of
  _Euclid_: the _horizon_, the _zenith_, _poles_ of a
  great circle, _verticals_, _declination circles_, the
  _meridian_, _celestial latitude_ and _longitude_,
  _right ascension_ and _declination_. Sun-dials                      36

  § 35. The division of the surface of the earth into
  zones                                                               37

  § 36. _Eratosthenes_: his measurement of the earth:
  and of the obliquity of the ecliptic                                39

  § 37. =Hipparchus=: his life and chief contributions to
  astronomy. _Apollonius’s_ representation of
  the celestial motions by means of circles.
  General account of the theory of eccentrics
  and epicycles                                                       40

  §§ 38-9. Hipparchus’s representation of the motion of
  the sun, by means of an _eccentric_: _apogee_,
  _perigee_, _line of apses_, _eccentricity_: _equation of
  the centre_: the _epicycle_ and the _deferent_                      41

  § 40. Theory of the moon: _lunation_ or _synodic month_
  and _sidereal month_: motion of the moon’s
  _nodes_ and apses: _draconitic month_ and
  _anomalistic month_                                                 47

  § 41. Observations of planets: eclipse method of connecting
  the distances of the sun and moon:
  estimate of their distances                                         49

  § 42. His star catalogue. Discovery of the _precession
  of the equinoxes_: the _tropical year_ and the
  _sidereal year_                                                     51

  § 43. Eclipses of the sun and moon: _conjunction_
  and _opposition_: _partial_, _total_, and _annular_
  eclipses: _parallax_                                                56

  § 44. Delambre’s estimate of Hipparchus                             61

  § 45. The slow progress of astronomy after the time of Hipparchus:
  _Pliny’s_ proof that the earth is round:
  new measurements of the earth by _Posidonius_                       61

  § 46. =Ptolemy.= The _Almagest_ and the _Optics_: theory of
  _refraction_                                                        62

  § 47. Account of the _Almagest_: Ptolemy’s postulates:
  arguments against the motion of the earth                           63

  § 48. The theory of the moon: _evection_ and _prosneusis_           65

  § 49. The _astrolabe_. Parallax, and distances of the
  sun and moon                                                        67

  § 50. The star catalogue: precession                                68

  § 51. Theory of the planets: the _equant_                           69

  § 52. Estimate of Ptolemy                                           73

  § 53. The decay of ancient astronomy: _Theon_ and _Hypatia_         73

  § 54. Summary and estimate of Greek astronomy                       74


  CHAPTER III.

  THE MIDDLE AGES (FROM ABOUT 600 A.D. TO ABOUT
  1500 A.D.), §§ 55-69                                             76-91

  § 55. The slow development of astronomy during this
  period                                                              76

  § 56. =The East.= The formation of an astronomical school
  at the court of the Caliphs: revival of astrology:
  translations from the Greek by _Honein ben Ishak_,
  _Ishak ben Honein_, _Tabit ben Korra_, and others                   76

  §§ 57-8. The Bagdad observatory. Measurement of the
  earth. Corrections of the astronomical data
  of the Greeks: _trepidation_                                        78

  § 59. _Albategnius_: discovery of the motion of the
  sun’s apogee                                                        79

  § 60. _Abul Wafa_: supposed discovery of the _variation_
  of the moon. _Ibn Yunos_: the _Hakemite
  Tables_                                                             79

  § 61. Development of astronomy in the Mahometan
  dominions in Morocco and Spain: _Arzachel_:
  the _Toletan Tables_                                                80

  § 62. _Nassir Eddin_ and his school: _Ilkhanic Tables_:
  more accurate value of precession                                   81

  § 63. Tartar astronomy: _Ulugh Begh_: his star catalogue            82

  § 64. Estimate of oriental astronomy of this period:
  Arabic numerals: survivals of Arabic names
  of stars and astronomical terms: _nadir_                            82

  § 65. =The West.= General stagnation after the fall of the
  Roman Empire: _Bede_. Revival of learning
  at the court of Charlemagne: _Alcuin_                               83

  § 66. Influence of Mahometan learning: _Gerbert_:
  translations from the Arabic: _Plato of Tivoli_,
  _Athelard of Bath_, _Gherardo of Cremona_.
  _Alfonso X._ and his school: the _Alfonsine
  Tables_ and the _Libros del Saber_                                  84

  § 67. The schoolmen of the thirteenth century,
  _Albertus Magnus_, _Cecco d’Ascoli_, _Roger
  Bacon_. _Sacrobosco’s Sphaera Mundi_                                85

  § 68. _Purbach_ and _Regiomontanus_: influence of the
  original Greek authors: the Nürnberg school:
  _Walther_: employment of printing: conflict
  between the views of Aristotle and of
  Ptolemy: the celestial spheres of the Middle
  Ages: the _firmament_ and the _primum mobile_                       86

  § 69. _Lionardo da Vinci_: _earthshine_. _Fracastor_ and
  _Apian_: observations of comets. _Nonius._
  _Fernel’s_ measurement of the earth                                 90


  CHAPTER IV.

  COPPERNICUS (FROM 1473 A.D. TO 1543 A.D.), §§ 70-92             92-124

  § 70. The Revival of Learning                                       92

  §§ 71-4. Life of _Coppernicus_: growth of his ideas: publication
  of the _Commentariolus_: _Rheticus_ and the
  _Prima Narratio_: publication of the _De Revolutionibus_            93

  § 75. The central idea in the work of Coppernicus:
  relation to earlier writers                                         99

  §§ 76-9. The =De Revolutionibus=. _The first book_: the
  postulates: the principle of _relative motion_,
  with applications to the apparent annual
  motion of the sun, and to the daily motion
  of the celestial sphere                                            100

  § 80. The two motions of the earth: answers to
  objections                                                         105

  § 81. The motion of the planets                                    106

  § 82. The seasons                                                  108

  § 83. End of first book. _The second book_: decrease
  in the obliquity of the ecliptic: the star
  catalogue                                                          110

  § 84. _The third book_: precession                                 110

  § 85. _The third book_: the annual motion of the earth:
  _aphelion_ and _perihelion_. _The fourth book_:
  theory of the moon: distances of the sun
  and moon: eclipses                                                 111

  §§ 86-7. The _fifth_ and _sixth_ books: theory of the planets:
  _synodic_ and _sidereal periods_                                   112

  § 88. Explanation of the stationary points                         118

  §§ 89-90. Detailed theory of the planets: defects of the
  theory                                                             121

  § 91. Coppernicus’s use of epicycles                               122

  § 92. A difficulty in his system                                   123


  CHAPTER V.

  THE RECEPTION OF THE COPPERNICAN THEORY AND THE PROGRESS
  OF OBSERVATION (FROM ABOUT 1543 A.D.
  TO ABOUT 1601 A.D.), §§ 93-112                                 125-144

  §§ 93-4. The first reception of the _De Revolutionibus_:
  _Reinhold_: the _Prussian Tables_                                  125

  § 95. Coppernicanism in England: _Field_, _Recorde_, _Digges_      127

  § 96. Difficulties in the Coppernican system: the need for
  progress in dynamics and for fresh observations                    127

  §§ 97-8. The Cassel Observatory: the Landgrave _William
  IV._, _Rothmann_, and _Bürgi_: the star catalogue:
  Bürgi’s invention of the pendulum clock                            128

  § 99. =Tycho Brahe=: his early life                                130

  § 100. The new star of 1572: travels in Germany                    131

  §§ 101-2. His establishment in Hveen: Uraniborg and
  Stjerneborg: life and work in Hveen                                132

  § 103. The comet of 1577, and others                               135

  § 104. Books on the new star and on the comet of 1577              136

  § 105. Tycho’s system of the world: quarrel with
  _Reymers Bär_                                                      136

  § 106. Last years at Hveen: breach with the King                   138

  § 107. Publication of the _Astronomiae Instauratae
  Mechanica_ and of the star catalogue: invitation
  from the Emperor                                                   139

  § 108. Life at Benatek: co-operation of Kepler: death              140

  § 109. Fate of Tycho’s instruments and observations                141

  § 110. Estimate of Tycho’s work: the accuracy of his
  observations: improvements in the art of
  observing                                                          141

  § 111. Improved values of astronomical constants.
  Theory of the moon: the _variation_ and the
  _annual equation_                                                  143

  § 112. The star catalogue: rejection of trepidation:
  unfinished work on the planets                                     144


  CHAPTER VI.

  GALILEI (FROM 1564 A.D. TO 1642 A.D.), §§ 113-134              145-178

  § 113. Early life                                                  145

  § 114. The pendulum                                                146

  § 115. Diversion from medicine to mathematics: his first
  book                                                               146

  § 116. Professorship at Pisa: experiments on falling
  bodies: protests against the principle of
  authority                                                          147

  § 117. Professorship at Padua: adoption of Coppernican
  views                                                              148

  § 118. =The telescopic discoveries.= Invention of the telescope
  by _Lippersheim_: its application to
  astronomy by _Harriot_, _Simon Marius_, and
  Galilei                                                            149

  § 119. The _Sidereus Nuncius_: observations of the moon            150

  § 120. New stars: resolution of portions of the Milky
  Way                                                                151

  § 121. The discovery of Jupiter’s _satellites_: their importance
  for the Coppernican controversy:
  controversies                                                      151

  § 122. Appointment at the Tuscan court                             153

  § 123. Observations of Saturn. Discovery of the
  phases of Venus                                                    154

  § 124. Observations of _sun-spots_ by Fabricius, Harriot,
  _Scheiner_, and Galilei: the _Macchie Solari_:
  proof that the spots were not planets: observations
  of the _umbra_ and _penumbra_                                      154

  § 125. Quarrel with Scheiner and the Jesuits: theological
  controversies: _Letter to the Grand Duchess
  Christine_                                                         157

  § 126. Visit to Rome. The first condemnation: prohibition
  of Coppernican books                                               159

  § 127. Method for finding longitude. Controversy on
  comets: _Il Saggiatore_                                            160

  § 128. =Dialogue on the Two Chief Systems of the World.=
  Its preparation and publication                                    162

  § 129. The speakers: argument for the Coppernican
  system based on the telescopic discoveries:
  discussion of stellar parallax: the _differential
  method of parallax_                                                163

  § 130. Dynamical arguments in favour of the motion of
  the earth: the _First Law of Motion_. The tides                    165

  § 131. =The trial and condemnation.= The thinly veiled
  Coppernicanism of the _Dialogue_: the remarkable
  preface                                                            168

  § 132. Summons to Rome: trial by the Inquisition:
  condemnation, abjuration, and punishment:
  prohibition of the _Dialogue_                                      169

  § 133. Last years: life at Arcetri: _libration_ of the moon:
  the _Two New Sciences_: _uniform acceleration_, and
  the first law of motion. Blindness and death                       172

  § 134. Estimate of Galilei’s work: his scientific method           176


  CHAPTER VII.

  KEPLER (FROM 1571 A.D. TO 1630 A.D.), §§ 135-151               179-197

  § 135. Early life and theological studies                          179

  § 136. Lectureship on mathematics at Gratz: astronomical
  studies and speculations: the _Mysterium Cosmographicum_           180

  § 137. Religious troubles in Styria: work with Tycho               181

  § 138. Appointment by the Emperor Rudolph as successor
  to Tycho: writings on the new star of 1604 and
  on Optics: theory of refraction and a new form
  of telescope                                                       182

  § 139. Study of the motion of Mars: unsuccessful attempts
  to explain it                                                      183

  §§ 140-1. The _ellipse_: discovery of the first two of _Kepler’s
  Laws_ for the case of Mars: the _Commentaries
  on Mars_                                                           184

  § 142. Suggested extension of Kepler’s Laws to the other
  planets                                                            186

  § 143. Abdication and death of Rudolph: appointment at
  Linz                                                               188

  § 144. The _Harmony of the World_: discovery of _Kepler’s
  Third Law_: the “music of the spheres”                             188

  § 145. _Epitome of the Copernican Astronomy_: its prohibition:
  fanciful correction of the distance of
  the sun: observation of the sun’s _corona_                         191

  § 146. Treatise on _Comets_                                        193

  § 147. Religious troubles at Linz: removal to Ulm                  194

  § 148. The _Rudolphine Tables_                                     194

  § 149. Work Under Wallenstein: death                               195

  § 150. Minor discoveries: speculations on gravity                  195

  § 151. Estimate of Kepler’s work and intellectual character        197


  CHAPTER VIII.

  FROM GALILEI TO NEWTON (FROM ABOUT 1638 A.D. TO
  ABOUT 1687 A.D.), §§ 152-163                                   198-209

  § 152. The general character of astronomical progress
  during the period                                                  198

  § 153. Scheiner’s observations of _faculae_ on the sun. _Hevel_:
  his _Selenographia_ and his writings on comets:
  his star catalogue. _Riccioli’s New Almagest_                      198

  § 154. Planetary observations; _Huygens’s_ discovery of a
  satellite of Saturn and of its ring                                199

  § 155. _Gascoigne’s_ and _Auzout’s_ invention of the _micrometer_:
  _Picard’s_ telescopic “sights”                                     202

  § 156. _Horrocks_: extension of Kepler’s theory to the
  moon: observation of a transit of Venus                            202

  §§ 157-8. Huygens’s rediscovery of the pendulum clock:
  his theory of circular motion                                      203

  § 159. Measurements of the earth by _Snell_, _Norwood_, and
  Picard                                                             204

  § 160. The Paris Observatory: _Domenico Cassini_: his
  discoveries of four new satellites of Saturn: his
  other work                                                         204

  § 161. _Richer’s_ expedition to Cayenne: pendulum observations:
  observations of Mars in opposition: _horizontal
  parallax_: _annual_ or _stellar parallax_                          205

  § 162. _Roemer_ and the velocity of light                          208

  § 163. _Descartes_                                                 208


  CHAPTER IX.

  UNIVERSAL GRAVITATION (FROM 1643 A.D. TO 1727 A.D.),
  §§ 164-195                                                     210-246

  § 164. Division of _Newton’s_ life into three periods              210

  § 165. Early life, 1643 to 1665                                    210

  § 166. Great productive period, 1665-87                            211

  § 167. Chief divisions of his work: astronomy, optics, pure
  mathematics                                                        211

  § 168. Optical discoveries: the _reflecting telescopes_ of
  _Gregory_ and Newton: the _spectrum_                               211

  § 169. Newton’s description of his discoveries in 1665-6           212

  § 170. The beginning of his work on gravitation: the
  falling apple: previous contributions to the
  subject by Kepler, _Borelli_, and Huygens                          213

  § 171. The problem of circular motion: _acceleration_              214

  § 172. The _law of the inverse square_ obtained from Kepler’s
  Third Law for the planetary orbits, treated as
  circles                                                            215

  § 173. Extension of the earth’s gravity as far as the moon:
  imperfection of the theory                                         217

  § 174. _Hooke’s_ and _Wren’s_ speculations on the planetary
  motions and on gravity. Newton’s second calculation
  of the motion of the moon: agreement
  with observation                                                   221

  § 175-6. Solution of the problem of elliptic motion:
  _Halley’s_ visit to Newton                                         221

  § 177. Presentation to the Royal Society of the tract _De
  Motu_: publication of the _Principia_                              222

  § 178. The =Principia=: its divisions                              223

  §§ 179-80. The _Laws of Motion_: the First Law: acceleration
  in its general form: _mass_ and _force_:
  the Third Law                                                      223

  § 181. Law of _universal gravitation_ enunciated                   227

  § 182. The attraction of a sphere                                  228

  § 183. The general problem of accounting for the
  motions of the solar system by means of
  gravitation and the Laws of Motion:
  _perturbations_                                                    229

  § 184. Newton’s lunar theory                                       230

  § 185. Measurement of the mass of a planet by means
  of its attraction of its satellites                                231

  § 186. Motion of the sun: _centre of gravity_ of the solar
  system: relativity of motion                                       231

  § 187. The non-spherical form of the earth, and of Jupiter         233

  § 188. Explanation of precession                                   234

  § 189. The tides: the mass of the moon deduced from
  tidal observations                                                 235

  § 190. The motions of comets: _parabolic_ orbits                   237

  § 191. Reception of the _Principia_                                239

  § 192. Third period of Newton’s life, 1687-1727: Parliamentary
  career: improvement of the lunar
  theory: appointments at the Mint and removal
  to London: publication of the _Optics_ and of the
  second and third editions of the _Principia_, edited
  by _Cotes_ and Pemberton: death                                    240

  § 193. Estimates of Newton’s work by Leibniz, by Lagrange,
  and by himself                                                     241

  § 194. Comparison of his astronomical work with that of
  his predecessors: “explanation” and “description”:
  conception of the material universe
  as made up of bodies attracting one another
  according to certain laws                                          242

  § 195. Newton’s scientific method: “_Hypotheses non fingo_”        245


  CHAPTER X.

  OBSERVATIONAL ASTRONOMY IN THE EIGHTEENTH
  CENTURY, §§ 196-227                                            247-286

  § 196. _Gravitational astronomy_: its development due
  almost entirely to Continental astronomers: use
  of _analysis_: English observational astronomy                     247

  §§ 197-8. _Flamsteed_: foundation of the Greenwich Observatory:
  his star catalogue                                                 249

  § 199. =Halley=: catalogue of Southern stars                       253

  § 200. Halley’s comet                                              253

  § 201. _Secular acceleration of the moon’s mean motion_            254

  § 202. Transits of Venus                                           254

  § 203. _Proper motions_ of the fixed stars                         255

  §§ 204-5. Lunar and planetary tables: career at Greenwich:
  minor work                                                         255

  § 206. =Bradley=: career                                           257

  §§ 207-11. Discovery and explanation of _aberration_: the
  _constant of aberration_                                           258

  § 212. Failure to detect parallax                                  265

  §§ 213-5. Discovery of _nutation_: _Machin_                        265

  §§ 216-7. Tables of Jupiter’s satellites by Bradley and by
  _Wargentin_: determination of longitudes,
  and other work                                                     269

  § 218. His observations: _reduction_                               271

  § 219. The density of the earth: _Maskelyne_: the _Cavendish
  experiment_                                                        273

  § 220. The _Cassini-Maraldi_ school in France                      275

  § 221. Measurements of the earth: the Lapland and
  Peruvian arcs: _Maupertuis_                                        275

  §§ 222-4. _Lacaille_: his career: expedition to the Cape:
  star catalogues, and other work                                    279

  §§ 225-6. _Tobias Mayer_: his observations: lunar tables:
  the longitude prize                                                282

  § 227. The transits of Venus in 1761 and 1769: distance
  of the sun                                                         284


  CHAPTER XI.

  GRAVITATIONAL ASTRONOMY IN THE EIGHTEENTH CENTURY,
  §§ 228-250                                                     287-322

  § 228. Newton’s problem: the _problem of three bodies_:
  methods of approximation: _lunar theory_ and
  _planetary theory_                                                 287

  § 229. The progress of Newtonian principles in France:
  popularisation by Voltaire. The five great
  mathematical astronomers: the pre-eminence of
  France                                                             290

  § 230. _Euler_: his career: St. Petersburg and Berlin:
  extent of his writings                                             291

  § 231. _Clairaut_: figure of the earth: return of Halley’s
  comet                                                              293

  § 232. _D’Alembert_: his dynamics: precession and nutation:
  his versatility: rivalry with Clairaut                             295

  §§ 233-4. The lunar theories and lunar tables of Euler,
  Clairaut, and D’Alembert: advance on Newton’s
  lunar theory                                                       297

  § 235. Planetary theory: Clairaut’s determination of the
  masses of the moon and of Venus: _Lalande_                         299

  § 236. Euler’s planetary theory: method of the _variation
  of elements_ or _parameters_                                       301

  § 237. _Lagrange_: his career: Berlin and Paris: the
  _Mécanique Analytique_                                             304

  § 238. _Laplace_: his career: the _Mécanique Céleste_ and the
  _Système du Monde_: political appointments and
  distinctions                                                       306

  § 239. Advance made by Lagrange and Laplace on the
  work of their immediate predecessors                               308

  § 240. Explanation of the moon’s secular acceleration by
  Laplace                                                            308

  § 241. Laplace’s lunar theory: tables of _Bürg_ and _Burckhardt_   309

  § 242. _Periodic_ and _secular_ inequalities                       310

  § 243. Explanation of the mutual perturbation of Jupiter
  and Saturn: _long inequalities_                                    312

  §§ 244-5. Theorems on the _stability_ of the solar system:
  the _eccentricity fund_ and the _inclination fund_                 313

  § 246. The magnitudes of some of the secular inequalities          318

  § 247. Periodical inequalities: solar and planetary tables
  _Mécanique Céleste_                                                318

  § 248. Minor problems of gravitational astronomy: the
  satellites: Saturn’s ring: precession and nutation:
  figure of the earth: tides: comets: masses
  of planets and satellites                                          318

  § 249. The solution of Newton’s problem by the astronomers
  of the eighteenth century                                          319

  § 250. The _nebular hypothesis_: its speculative character         320


  CHAPTER XII.

  HERSCHEL (FROM 1738 A.D. TO 1822 A.D.), §§ 251-271             323-353

  §§ 251-2. William Herschel’s early career: Bath: his
  first telescope                                                    323

  §§ 253-4. The discovery of the planet Uranus, and its
  consequences: Herschel’s removal to Slough                         325

  § 255. Telescope-making: marriage: the forty-foot telescope:
  discoveries of satellites of Saturn and of
  Uranus                                                             327

  § 256. Life and work at Slough: last years: _Caroline
  Herschel_                                                          328

  § 257. Herschel’s astronomical programme: the study of
  the fixed stars                                                    330

  § 258. The distribution of the stars in space: _star-gauging_:
  the “grindstone” theory of the universe: defects of the
  fundamental assumption: its partial withdrawal. Employment of
  brightness as a test of nearness: measurement of brightness:
  “space-penetrating” power of a telescope                           332

  § 259. _Nebulae_ and _star clusters_: Herschel’s great catalogues  336

  § 260. Relation of nebulae to star clusters: the “island
  universe” theory of nebulae: the “shining fluid”
  theory: distribution of nebulae                                    337

  § 261. Condensation of nebulae into clusters and stars             339

  § 262. The irresolvability of the Milky Way                        340

  § 263. _Double stars_: their proposed employment for finding
  parallax: catalogues: probable connection
  between members of a pair                                          341

  § 264. Discoveries of the revolution of double stars:
  _binary stars_: their uselessness for parallax                     343

  § 265. The motion of the sun in space: the various
  positions suggested for the _apex_                                 344

  § 266. Variable stars: _Mira_ and _Algol_: catalogues of
  comparative brightness: method of _sequences_:
  variability of α _Herculis_                                        346

  § 267. Herschel’s work on the solar system: new satellites:
  observations of Saturn, Jupiter, Venus, and Mars                   348

  § 268. Observations of the sun: _Wilson_: theory of the
  structure of the sun                                               350

  § 269. Suggested variability of the sun                            351

  § 270. Other researches                                            352

  § 271. Comparison of Herschel with his contemporaries:
  _Schroeter_                                                        352


  CHAPTER XIII.

  THE NINETEENTH CENTURY, §§ 272-320                             354-409

  § 272. The three chief divisions of astronomy, _observational_,
  _gravitational_, and _descriptive_                                 354

  § 273. The great growth of descriptive astronomy in the
  nineteenth century                                                 355

  § 274. =Observational Astronomy.= Instrumental advances:
  the introduction of photography                                    357

  § 275. The method of _least squares_: _Legendre_ and _Gauss_       357

  § 276. Other work by Gauss: the _Theoria Motus_: rediscovery
  of the minor planet Ceres                                          358

  § 277. _Bessel_: his improvement in methods of reduction:
  his table of refraction: the _Fundamenta
  Nova_ and _Tabulae Regiomontanae_                                  359

  § 278. The parallax of 61 _Cygni_: its distance                    360

  § 279. _Henderson’s_ parallax of α _Centauri_ and _Struve’s_
  of _Vega_: later parallax determinations                           362

  § 280. Star catalogues: the photographic chart                     362

  §§ 281-4. The distance of the sun: transits of Venus:
  observations of Mars and of the minor planets
  in opposition: _diurnal method_: gravitational
  methods, lunar and planetary: methods
  based on the velocity of light: summary of results                 363

  § 285. Variation in latitude: rigidity of the earth                367

  § 286. =Gravitational Astronomy.= Lunar theory: _Damoiseau_,
  _Poisson_, _Pontécoulant_, _Lubbock_, _Hansen_,
  _Delaunay_, Professor _Newcomb_, _Adams_, Dr.
  _Hill_                                                             367

  § 287. Secular acceleration of the moon’s mean motion:
  Adams’s correction of Laplace: Delaunay’s
  explanation by means of _tidal friction_                           369

  § 288. Planetary theory: _Leverrier_, _Gyldén_, M. _Poincaré_      370

  § 289. The discovery of Neptune by Leverrier and Dr.
  _Galle_: Adams’s work                                              371

  § 290. Lunar and planetary tables: outstanding discrepancies
  between theory and observation                                     372

  § 291. Cometary orbits: return of Halley’s comet in
  1835: Encke’s and other periodic comets                            372

  § 292. Theory of tides: analysis of tidal observations
  by Lubbock, _Whewell_, Lord _Kelvin_, and
  Professor _Darwin_: bodily tides in the earth
  and its rigidity                                                   373

  § 293. The stability of the solar system                           374

  § 294. =Descriptive Astronomy.= Discovery of the _minor
  planets_ or _asteroids_: their number, distribution,
  and size                                                           376

  § 295. Discoveries of satellites of Neptune, Saturn,
  Uranus, Mars, and Jupiter, and of the _crape
  ring_ of Saturn                                                    380

  § 296. The surface of the moon: _rills_: the lunar atmosphere      382

  § 297. The surfaces of Mars, Jupiter, and Saturn: the
  _canals_ on Mars: _Maxwell’s_ theory of Saturn’s
  rings: the rotation of Mercury and of Venus                        383

  § 298. The surface of the sun: _Schwabe’s_ discovery of
  the periodicity of sun-spots: connection between
  sun-spots and terrestrial magnetism:
  _Carrington’s_ observations of the motion and
  distribution of spots: Wilson’s theory of spots                    385

  §§ 299-300. _Spectrum analysis_: Newton, _Wollaston_, _Fraunhofer_,
  _Kirchhoff_: the chemistry of the sun                              386

  § 301. Eclipses of the sun: the _corona_, _chromosphere_,
  and _prominences_: spectroscopic methods of
  observation                                                        389

  § 302. Spectroscopic method of determining motion to
  or from the observer: _Doppler’s principle_:
  application to the sun                                             391

  § 303. The constitution of the sun                                 392

  §§ 304-5. Observations of comets: _nucleus_: theory of the
  formation of their tails: their spectra: relation
  between comets and _meteors_                                       393

  §§ 306-8. Sidereal astronomy: career of _John Herschel_: his
  catalogues of nebulae and of double stars:
  the expedition to the Cape: measurement of
  the sun’s heat by Herschel and by _Pouillet_                       396

  § 309. Double stars: observations by Struve and
  others: orbits of binary stars                                     398

  § 310. Lord _Rosse’s_ telescopes: his observations of
  nebulae: revival of the “island universe”
  theory                                                             400

  § 311. Application of the spectroscope to nebulae:
  distinction between nebulae and clusters                           401

  § 312. Spectroscopic classification of stars by _Secchi_:
  chemistry of stars: stars with bright-line
  spectra                                                            401

  §§ 313-4. Motion of stars in the line of sight. Discovery
  of binary stars by the spectroscope: eclipse
  theory of variable stars                                           402

  § 315. Observations of variable stars                              403

  § 316. Stellar _photometry_: _Pogson’s_ light ratio: the
  Oxford, Harvard, and Potsdam photometries                          403

  § 317. Structure of the sidereal system: relations of
  stars and nebulae                                                  405

  §§ 318-20. Laplace’s nebular hypothesis in the light of
  later discoveries: the sun’s heat: _Helmholtz’s_
  shrinkage theory. Influence of tidal friction on
  the development of the solar system: Professor
  Darwin’s theory of the birth of the moon.
  Summary                                                            406

  LIST OF AUTHORITIES AND OF BOOKS FOR STUDENTS                      411

  INDEX OF NAMES                                                     417

  GENERAL INDEX                                                      425



LIST OF ILLUSTRATIONS.


  FIG.                                                              PAGE
     The moon                                        _Frontispiece_

  1. The celestial sphere                                              5

  2. The daily paths of circumpolar stars             _To face p._     8

  3. The circles of the celestial sphere                               9

  4. The equator and the ecliptic                                     11

  5. The Great Bear                                   _To face p._    12

  6. The apparent path of Jupiter                                     16

  7. The apparent path of Mercury                                     17

  8-11. The phases of the moon                                    30, 31

  12. The curvature of the earth                                      32

  13. The method of Aristarchus for comparing the distances of
  the sun and moon                                                    34

  14. The equator and the ecliptic                                    36

  15. The equator, the horizon, and the meridian                      38

  16. The measurement of the earth                                    39

  17. The eccentric                                                   44

  18. The position of the sun’s apogee                                45

  19. The epicycle and the deferent                                   47

  20. The eclipse method of connecting the distances of the sun
  and moon                                                            50

  21. The increase of the longitude of a star                         52

  22. The movement of the equator                                     53

  23, 24. The precession of the equinoxes                         53, 54

  25. The earth’s shadow                                              57

  26. The ecliptic and the moon’s path                                57

  27. The sun and moon                                                58

  28. Partial eclipse of the moon                                     58

  29. Total eclipse of the moon                                       58

  30. Annular eclipse of the sun                                      59

  31. Parallax                                                        60

  32. Refraction by the atmosphere                                    63

  33. Parallax                                                        68

  34. Jupiter’s epicycle and deferent                                 70

  35. The equant                                                      71

  36. The celestial spheres                                           89

  PORTRAIT OF COPPERNICUS                             _To face p._    94

  37. Relative motion                                                102

  38. The relative motion of the sun and moon                        103

  39. The daily rotation of the earth                                104

  40. The solar system according to Coppernicus                      107

  41, 42. Coppernican explanation of the seasons                108, 109

  43. The orbits of Venus and of the earth                           113

  44. The synodic and sidereal periods of Venus                      114

  45. The epicycle of Jupiter                                        116

  46. The relative sizes of the orbits of the earth and of a superior
  planet                                                             117

  47. The stationary points of Mercury                               119

  48. The stationary points of Jupiter                               120

  49. The alteration in a planet’s apparent position due to an
  alteration in the earth’s distance from the sun                    122

  50. Stellar parallax                                               124

  51. Uraniborg                                                      133

  52. Tycho’s system of the world                                    137

  PORTRAIT OF TYCHO BRAHE                             _To face p._   139

  53. One of Galilei’s drawings of the moon                „         150

  54. Jupiter and its satellites as seen on January 7, 1610          152

  55. Sun-spots                                       _To face p._   154

  56. Galilei’s proof that sun-spots are not planets                 156

  57. The differential method of parallax                            165

  PORTRAIT OF GALILEI                                  _To face p._  171

  58. The daily libration of the moon                                173

  PORTRAIT OF KEPLER                                   _To face p._  183

  59. An ellipse                                                     185

  60. Kepler’s second law                                            186

  61. Diagram used by Kepler to establish his laws of planetary
  motion                                                             187

  62. The “music of the spheres” according to Kepler                 190

  63. Kepler’s idea of gravity                                       196

  64. Saturn’s ring, as drawn by Huygens              _To face p._   200

  65. Saturn, with the ring seen edge-wise                 „         200

  66. The phases of Saturn’s ring                                    201

  67. Early drawings of Saturn                        _To face p._   202

  68. Mars in opposition                                             206

  69. The parallax of a planet                                       206

  70. Motion in a circle                                             214

  71. The moon as a projectile                                       220

  72. The spheroidal form of the earth                               234

  73. An elongated ellipse and a parabola                            238

  PORTRAIT OF NEWTON                                  _To face p._   240

  PORTRAIT OF BRADLEY                                      „         258

  74, 75. The aberration of light                               262, 263

  76. The aberrational ellipse                                       264

  77. Precession and nutation                                        268

  78. The varying curvature of the earth                             277

  79. Tobias Mayer’s map of the moon                  _To face p._   282

  80. The path of Halley’s comet                                     294

  81. A varying ellipse                                              303

  PORTRAIT OF LAGRANGE                                _To face p._   305

  PORTRAIT OF LAPLACE                                      „         307

  PORTRAIT OF WILLIAM HERSCHEL                             „         327

  82. Herschel’s forty-foot telescope                                329

  83. Section of the sidereal system                                 333

  84. Illustrating the effect of the sun’s motion in space           345

  85. _61 Cygni_ and the two neighbouring stars used by Bessel       360

  86. The parallax of _61 Cygni_                                     361

  87. The path of Halley’s comet                                     373

  88. Photographic trail of a minor planet            _To face p._   377

  89. Paths of minor planets                                         378

  90. Comparative sizes of three minor planets and the moon          379

  91. Saturn and its system                                          380

  92. Mars and its satellites                                        381

  93. Jupiter and its satellites                                     382

  94. The Apennines and the adjoining regions }       _To face p._   383
  of the moon                                 }

  95. Saturn and its rings                                  „        384

  96. A group of sun-spots                                  „        385

  97. Fraunhofer’s map of the solar spectrum                „        387

  98. The total solar eclipse of 1886                       „        390

  99. The great comet of 1882                               „        393

  100. The nebula about η _Argus_                           „        397

  101. The orbit of ξ _Ursae_                                        399

  102. Spiral nebulae                                 _To face p._   400

  103. The spectrum of β _Aurigae_                          „        403

  104. The Milky Way near the cluster in Perseus            „        405



A SHORT HISTORY OF ASTRONOMY.



CHAPTER I.

PRIMITIVE ASTRONOMY.

           “The never-wearied Sun, the Moon exactly round,
   And all those Stars with which the brows of ample heaven are crowned,
   Orion, all the Pleiades, and those seven Atlas got,
   The close beamed Hyades, the Bear, surnam’d the Chariot,
   That turns about heaven’s axle tree, holds ope a constant eye
   Upon Orion, and of all the cressets in the sky
   His golden forehead never bows to th’ Ocean empery.”
                                    _The Iliad_ (Chapman’s translation).


1. Astronomy is the science which treats of the sun, the moon, the
stars, and other objects such as comets which are seen in the sky. It
deals to some extent also with the earth, but only in so far as it has
properties in common with the heavenly bodies. In early times astronomy
was concerned almost entirely with the observed motions of the heavenly
bodies. At a later stage astronomers were able to discover the
distances and sizes of many of the heavenly bodies, and to weigh some
of them; and more recently they have acquired a considerable amount of
knowledge as to their nature and the material of which they are made.

2. We know nothing of the beginnings of astronomy, and can only
conjecture how certain of the simpler facts of the science—particularly
those with a direct influence on human life and comfort—gradually
became familiar to early mankind, very much as they are familiar to
modern savages.

With these facts it is convenient to begin, taking them in the order in
which they most readily present themselves to any ordinary observer.

3. The sun is daily seen to rise in the eastern part of the sky, to
travel across the sky, to reach its highest position in the south in
the middle of the day, then to sink, and finally to set in the western
part of the sky. But its daily path across the sky is not always the
same: the points of the horizon at which it rises and sets, its height
in the sky at midday, and the time from sunrise to sunset, all go
through a series of changes, which are accompanied by changes in the
weather, in vegetation, etc.; and we are thus able to recognise the
existence of the seasons, and their recurrence after a certain interval
of time which is known as a year.

4. But while the sun always appears as a bright circular disc, the
next most conspicuous of the heavenly bodies, the moon, undergoes
changes of form which readily strike the observer, and are at once
seen to take place in a regular order and at about the same intervals
of time. A little more care, however, is necessary in order to observe
the connection between the form of the moon and her position in the
sky with respect to the sun. Thus when the moon is first visible soon
after sunset near the place where the sun has set, her form is a thin
crescent (cf. fig. 11 on p. 31), the hollow side being turned away
from the sun, and she sets soon after the sun. Next night the moon is
farther from the sun, the crescent is thicker, and she sets later; and
so on, until after rather less than a week from the first appearance of
the crescent, she appears as a semicircular disc, with the flat side
turned away from the sun. The semicircle enlarges, and after another
week has grown into a complete disc; the moon is now nearly in the
opposite direction to the sun, and therefore rises about at sunset
and sets about at sunrise. She then begins to approach the sun on the
other side, rising before it and setting in the daytime; her size
again diminishes, until after another week she is again semicircular,
the flat side being still turned away from the sun, but being now
turned towards the west instead of towards the east. The semicircle
then becomes a gradually diminishing crescent, and the time of rising
approaches the time of sunrise, until the moon becomes altogether
invisible. After two or three nights the new moon reappears, and the
whole series of changes is repeated. The different forms thus assumed
by the moon are now known as her =phases=; the time occupied by this
series of changes, the month, would naturally suggest itself as a
convenient measure of time; and the day, month, and year would thus
form the basis of a rough system of time-measurement.

5. From a few observations of the stars it could also clearly be seen
that they too, like the sun and moon, changed their positions in the
sky, those towards the east being seen to rise, and those towards
the west to sink and finally set, while others moved across the sky
from east to west, and those in a certain northern part of the sky,
though also in motion, were never seen either to rise or set. Although
anything like a complete classification of the stars belongs to a
more advanced stage of the subject, a few star groups could easily be
recognised, and their position in the sky could be used as a rough
means of measuring time at night, just as the position of the sun to
indicate the time of day.

6. To these rudimentary notions important additions were made when
rather more careful and prolonged observations became possible, and
some little thought was devoted to their interpretation.

Several peoples who reached a high stage of civilisation at an early
period claim to have made important progress in astronomy. Greek
traditions assign considerable astronomical knowledge to Egyptian
priests who lived some thousands of years B.C., and some of the
peculiarities of the pyramids which were built at some such period
are at any rate plausibly interpreted as evidence of pretty accurate
astronomical observations; Chinese records describe observations
supposed to have been made in the 25th century B.C.; some of the Indian
sacred books refer to astronomical knowledge acquired several centuries
before this time; and the first observations of the Chaldaean priests
of Babylon have been attributed to times not much later.

On the other hand, the earliest recorded astronomical observation the
authenticity of which may be accepted without scruple belongs only to
the 8th century B.C.

For the purposes of this book it is not worth while to make any attempt
to disentangle from the mass of doubtful tradition and conjectural
interpretation of inscriptions, bearing on this early astronomy, the
few facts which lie embedded therein; and we may proceed at once to
give some account of the astronomical knowledge, other than that
already dealt with, which is discovered in the possession of the
earliest really historical astronomers—the Greeks—at the beginning of
their scientific history, leaving it an open question what portions of
it were derived from Egyptians, Chaldaeans, their own ancestors, or
other sources.

7. If an observer looks at the stars on any clear night he sees an
apparently innumerable[1] host of them, which seem to lie on a portion
of a spherical surface, of which he is the centre. This spherical
surface is commonly spoken of as the sky, and is known to astronomy
as the =celestial sphere=. The visible part of this sphere is bounded
by the earth, so that only half can be seen at once; but only the
slightest effort of the imagination is required to think of the other
half as lying below the earth, and containing other stars, as well as
the sun. This sphere appears to the observer to be very large, though
he is incapable of forming any precise estimate of its size.[2]

Most of us at the present day have been taught in childhood that
the stars are at different distances, and that this sphere has in
consequence no real existence. The early peoples had no knowledge of
this, and for them the celestial sphere really existed, and was often
thought to be a solid sphere of crystal.

[Illustration: FIG. 1.—The celestial sphere.]

Moreover modern astronomers, as well as ancient, find it convenient
for very many purposes to make use of this sphere, though it has no
material existence, as a means of representing the directions in which
the heavenly bodies are seen and their motions. For all that direct
observation can tell us about the position of such an object as a star
is its _direction_; its distance can only be ascertained by indirect
methods, if at all. If we draw a sphere, and suppose the observer’s eye
placed at its centre O (fig. 1), and then draw a straight line from O
to a star S, meeting the surface of the sphere in the point _s_; then
the star appears exactly in the same position as if it were at _s_, nor
would its apparent position be changed if it were placed at any other
point, such as S′ or S″, on this same line. When we speak, therefore,
of a star as being at a point _s_ on the celestial sphere, all that we
mean is that it is in the same direction as the point _s_, or, in other
words, that it is situated somewhere on the straight line through O and
S. The advantages of this method of representing the position of a star
become evident when we wish to compare the positions of several stars.
The difference of direction of two stars is the angle between the lines
drawn from the eye to the stars; _e.g._, if the stars are R, S, it is
the angle R O S. Similarly the difference of direction of another pair
of stars, P, Q, is the angle P O Q. The two stars P and Q appear nearer
together than do R and S, or farther apart, according as the angle P
O Q is less or greater than the angle R O S. But if we represent the
stars by the corresponding points _p_, _q_, _r_, _s_ on the celestial
sphere, then (by an obvious property of the sphere) the angle P O Q
(which is the same as _p_ O _q_) is less or greater than the angle R O
S (or _r_ O _s_) according as the arc joining _p q_ on the sphere is
less or greater than the arc joining _r s_, and in the same proportion;
if, for example, the angle R O S is twice as great as the angle P O
Q, so also is the arc _p q_ twice as great as the arc _r s_. We may
therefore, in all questions relating only to the directions of the
stars, replace the angle between the directions of two stars by the arc
joining the corresponding points on the celestial sphere, or, in other
words, by the distance between these points on the celestial sphere.
But such arcs on a sphere are easier both to estimate by eye and to
treat geometrically than angles, and the use of the celestial sphere
is therefore of great value, apart from its historical origin. It is
important to note that this =apparent distance= of two stars, _i.e._
their distance from one another on the celestial sphere, is an entirely
different thing from their actual distance from one another in space.
In the figure, for example, Q is actually much nearer to S than it is
to P, but the apparent distance measured by the arc _q s_ is several
times greater than _q p_. The apparent distance of two points on the
celestial sphere is measured numerically by the angle between the lines
joining the eye to the two points, expressed in =degrees=, =minutes=,
and =seconds=.[3]

We might of course agree to regard the celestial sphere as of a
particular size, and then express the distance between two points on
it in miles, feet, or inches; but it is practically very inconvenient
to do so. To say, as some people occasionally do, that the distance
between two stars is so many feet is meaningless, unless the supposed
size of the celestial sphere is given at the same time.

It has already been pointed out that the observer is always at the
centre of the celestial sphere; this remains true even if he moves to
another place. A sphere has, however, only one centre, and therefore
if the sphere remains fixed the observer cannot move about and yet
always remain at the centre. The old astronomers met this difficulty
by supposing that the celestial sphere was so large that any possible
motion of the observer would be insignificant in comparison with
the radius of the sphere and could be neglected. It is often more
convenient—when we are using the sphere as a mere geometrical device
for representing the position of the stars—to regard the sphere as
moving with the observer, so that he always remains at the centre.

8. Although the stars all appear to move across the sky (§ 5), and
their rates of motion differ, yet the distance between any two stars
remains unchanged, and they were consequently regarded as being
attached to the celestial sphere. Moreover a little careful observation
would have shown that the motions of the stars in different parts of
the sky, though at first sight very different, were just such as would
have been produced by the celestial sphere—with the stars attached
to it—turning about an axis passing through the centre and through a
point in the northern sky close to the familiar pole-star. This point
is called the =pole=. As, however, a straight line drawn through the
centre of a sphere meets it in two points, the axis of the celestial
sphere meets it again in a second point, opposite the first, lying in
a part of the celestial sphere which is permanently below the horizon.
This second point is also called a pole; and if the two poles have to
be distinguished, the one mentioned first is called the =north pole=,
and the other the =south pole=. The direction of the rotation of the
celestial sphere about its axis is such that stars near the north pole
are seen to move round it in circles in the direction opposite to
that in which the hands of a clock move; the motion is uniform, and a
complete revolution is performed in four minutes less than twenty-four
hours; so that the position of any star in the sky at twelve o’clock
to-night is the same as its position at four minutes to twelve
to-morrow night.

The moon, like the stars, shares this motion of the celestial sphere
and so also does the sun, though this is more difficult to recognise
owing to the fact that the sun and stars are not seen together.

As other motions of the celestial bodies have to be dealt with, the
general motion just described may be conveniently referred to as the
=daily motion= or =daily rotation= of the celestial sphere.

9. A further study of the daily motion would lead to the recognition of
certain important circles of the celestial sphere.

Each star describes in its daily motion a circle, the size of
which depends on its distance from the poles. Fig. 2 shews the
paths described by a number of stars near the pole, recorded
photographically, during part of a night. The pole-star describes so
small a circle that its motion can only with difficulty be detected
with the naked eye, stars a little farther off the pole describe larger
circles, and so on, until we come to stars half-way between the two
poles, which describe the largest circle which can be drawn on the
celestial sphere. The circle on which these stars lie and which is
described by any one of them daily is called the =equator=. By looking
at a diagram such as fig. 3, or, better still, by looking at an actual
globe, it can easily be seen that half the equator (E Q W) lies above
and half (the dotted part, W R E) below the horizon, and that in
consequence a star, such as _s_, lying on the equator, is in its daily
motion as long a time above the horizon as below. If a star, such as
S, lies on the north side of the equator, _i.e._ on the side on which
the north pole P lies, more than half of its daily path lies above the
horizon and less than half (as shewn by the dotted line) lies below;
and if a star is near enough to the north pole (more precisely, if it
is nearer to the north pole than the nearest point, K, of the horizon),
as σ, it never sets, but remains continually above the horizon. Such a
star is called a (=northern=) =circumpolar= star. On the other hand,
less than half of the daily path of a star on the south side of the
equator, as S′, is above the horizon, and a star, such as σ′, the
distance of which from the north pole is greater than the distance of
the farthest point, H, of the horizon, or which is nearer than H to the
south pole, remains continually below the horizon.

[Illustration:
 FIG. 2.—The paths of circumpolar stars, shewing their movement during
 seven hours. From a photograph by Mr. H. Pain. The thickest line is
 the path of the pole star.                        [_To face p. 8._  ]

10. A slight familiarity with the stars is enough to shew any one
that the same stars are not always visible at the same time of night.
Rather more careful observation, carried out for a considerable time,
is necessary in order to see that the aspect of the sky changes in a
regular way from night to night, and that after the lapse of a year the
same stars become again visible at the same time. The explanation of
these changes as due to the motion of the sun on the celestial sphere
is more difficult, and the unknown discoverer of this fact certainly
made one of the most important steps in early astronomy.

[Illustration: FIG. 3.—The circles of the celestial sphere.]

If an observer notices soon after sunset a star somewhere in the
west, and looks for it again a few evenings later at about the same
time, he finds it lower down and nearer to the sun; a few evenings
later still it is invisible, while its place has now been taken by
some other star which was at first farther east in the sky. This star
can in turn be observed to approach the sun evening by evening. Or
if the stars visible after sunset low down in the east are noticed
a few days later, they are found to be higher up in the sky, and
their place is taken by other stars at first too low down to be seen.
Such observations of stars rising or setting about sunrise or sunset
shewed to early observers that the stars were gradually changing their
position with respect to the sun, or that the sun was changing its
position with respect to the stars.

The changes just described, coupled with the fact that the stars do not
change their positions with respect to one another, shew that the stars
as a whole perform their daily revolution rather more rapidly than the
sun, and at such a rate that they gain on it one complete revolution
in the course of the year. This can be expressed otherwise in the
form that the stars are all moving westward on the celestial sphere,
relatively to the sun, so that stars on the east are continually
approaching and those on the west continually receding from the sun.
But, again, the same facts can be expressed with equal accuracy and
greater simplicity if we regard the stars as fixed on the celestial
sphere, and the sun as moving on it from west to east among them (that
is, in the direction _opposite_ to that of the daily motion), and at
such a rate as to complete a circuit of the celestial sphere and to
return to the same position after a year.

This =annual motion= of the sun is, however, readily seen not to be
merely a motion from west to east, for if so the sun would always rise
and set at the same points of the horizon, as a star does, and its
midday height in the sky and the time from sunrise to sunset would
always be the same. We have already seen that if a star lies on the
equator half of its daily path is above the horizon, if the star is
north of the equator more than half, and if south of the equator less
than half; and what is true of a star is true for the same reason of
any body sharing the daily motion of the celestial sphere. During the
summer months therefore (March to September), when the day is longer
than the night, and more than half of the sun’s daily path is above the
horizon, the sun must be north of the equator, and during the winter
months (September to March) the sun must be south of the equator. The
change in the sun’s distance from the pole is also evident from the
fact that in the winter months the sun is on the whole lower down in
the sky than in summer, and that in particular its midday height is
less.

11. The sun’s path on the celestial sphere is therefore oblique to the
equator, lying partly on one side of it and partly on the other. A good
deal of careful observation of the kind we have been describing must,
however, have been necessary before it was ascertained that the sun’s
annual path on the celestial sphere (see fig. 4) is a =great circle=
(that is, a circle having its centre at the centre of the sphere).
This great circle is now called the =ecliptic= (because eclipses take
place only when the moon is in or near it), and the angle at which it
cuts the equator is called the =obliquity= of the ecliptic. The Chinese
claim to have measured the obliquity in 1100 B.C., and to have found
the remarkably accurate value 23° 52′ (cf. chapter II., § 35). The
truth of this statement may reasonably be doubted, but on the other
hand the statement of some late Greek writers that either Pythagoras
or Anaximander (6th century B.C.) was the _first_ to discover the
obliquity of the ecliptic is almost certainly wrong. It must have been
known with reasonable accuracy to both Chaldaeans and Egyptians long
before.

[Illustration: FIG. 4.—The equator and the ecliptic.]

When the sun crosses the equator the day is equal to the night, and the
times when this occurs are consequently known as the =equinoxes=, the
=vernal equinox= occurring when the sun crosses the equator from south
to north (about March 21st), and the =autumnal equinox= when it crosses
back (about September 23rd). The points on the celestial sphere where
the sun crosses the equator (A, C in fig. 4), _i.e._ where ecliptic
and equator cross one another, are called the =equinoctial points=,
occasionally also the equinoxes.

After the vernal equinox the sun in its path along the ecliptic
recedes from the equator towards the north, until it reaches, about
three months afterwards, its greatest distance from the equator, and
then approaches the equator again. The time when the sun is at its
greatest distance from the equator on the north side is called the
=summer solstice=, because then the northward motion of the sun is
arrested and it temporarily appears to stand still. Similarly the sun
is at its greatest distance from the equator towards the south at the
=winter solstice=. The points on the ecliptic (B, D in fig. 4) where
the sun is at the solstices are called the =solstitial points=, and are
half-way between the equinoctial points.

12. The earliest observers probably noticed particular groups of stars
remarkable for their form or for the presence of bright stars among
them, and occupied their fancy by tracing resemblances between them
and familiar objects, etc. We have thus at a very early period a rough
attempt at dividing the stars into groups called =constellations= and
at naming the latter.

In some cases the stars regarded as belonging to a constellation form a
well-marked group on the sky, sufficiently separated from other stars
to be conveniently classed together, although the resemblance which
the group bears to the object after which it is named is often very
slight. The seven bright stars of the Great Bear, for example, form
a group which any observer would very soon notice and naturally make
into a constellation, but the resemblance to a bear of these and the
fainter stars of the constellation is sufficiently remote (see fig.
5), and as a matter of fact this part of the Bear has also been called
a Waggon and is in America familiarly known as the Dipper; another
constellation has sometimes been called the Lyre and sometimes also the
Vulture. In very many cases the choice of stars seems to have been made
in such an arbitrary manner, as to suggest that some fanciful figure
was first imagined and that stars were then selected so as to represent
it in some rough sort of way. In fact, as Sir John Herschel remarks,
“The constellations seem to have been purposely named and delineated
to cause as much confusion and inconvenience as possible. Innumerable
snakes twine through long and contorted areas of the heavens where no
memory can follow them; bears, lions, and fishes, large and small,
confuse all nomenclature.” (_Outlines of Astronomy_, § 301.)

[Illustration: FIG. 5.—The Great Bear. From Bayer’s _Uranometria_
(1603).                                          [_To face p. 12._]

The constellations as we now have them are, with the exception of a
certain number (chiefly in the southern skies) which have been added
in modern times, substantially those which existed in early Greek
astronomy; and such information as we possess of the Chaldaean and
Egyptian constellations shews resemblances indicating that the Greeks
borrowed some of them. The names, as far as they are not those of
animals or common objects (Bear, Serpent, Lyre, etc.), are largely
taken from characters in the Greek mythology (Hercules, Perseus, Orion,
etc.). The constellation Berenice’s Hair, named after an Egyptian
queen of the 3rd century B.C., is one of the few which commemorate a
historical personage.[4]

13. Among the constellations which first received names were those
through which the sun passes in its annual circuit of the celestial
sphere, that is those through which the ecliptic passes. The moon’s
monthly path is also a great circle, never differing very much from
the ecliptic, and the paths of the planets (§ 14) are such that they
also are never far from the ecliptic. Consequently the sun, the moon,
and the five planets were always to be found within a region of the
sky extending about 8° on each side of the ecliptic. This strip of the
celestial sphere was called the =zodiac=, because the constellations
in it were (with one exception) named after living things (Greek ζῷον,
an animal); it was divided into twelve equal parts, the =signs of the
zodiac=, through one of which the sun passed every month, so that the
position of the sun at any time could be roughly described by stating
in what “sign” it was. The stars in each “sign” were formed into a
constellation, the “sign” and the constellation each receiving the
same name. Thus arose twelve =zodiacal constellations=, the names of
which have come down to us with unimportant changes from early Greek
times.[5] Owing, however, to an alteration of the position of the
equator, and consequently of the equinoctial points, the sign Aries,
which was defined by Hipparchus in the second century B.C. (see chapter
II., § 42) as beginning at the vernal equinoctial point, no longer
contains the constellation Aries, but the preceding one, Pisces: and
there is a corresponding change throughout the zodiac. The more precise
numerical methods of modern astronomy have, however, rendered the signs
of the zodiac almost obsolete: but the =first point of Aries= (♈), and
the =first point of Libra= (♎), are still the recognised names for the
equinoctial points.

In some cases individual stars also received special names, or were
called after the part of the constellation in which they were situated,
_e.g._ Sirius, the Eye of the Bull, the Heart of the Lion, etc.; but
the majority of the present names of single stars are of Arabic origin
(chapter III., § 64).

14. We have seen that the stars, as a whole, retain invariable
positions on the celestial sphere,[6] whereas the sun and moon change
their positions. It was, however, discovered in prehistoric times that
five bodies, at first sight barely distinguishable from the other
stars, also changed their places. These five—Mercury, Venus, Mars,
Jupiter, and Saturn—with the sun and moon, were called =planets=,[7] or
wanderers, as distinguished from the =fixed stars=. Mercury is never
seen except occasionally near the horizon just after sunset or before
sunrise, and in a climate like ours requires a good deal of looking
for; and it is rather remarkable that no record of its discovery should
exist. Venus is conspicuous as the Evening Star or as the Morning Star.
The discovery of the identity of the Evening and Morning Stars is
attributed to Pythagoras (6th century B.C.), but must almost certainly
have been made earlier, though the Homeric poems contain references to
both, without any indication of their identity. Jupiter is at times as
conspicuous as Venus at her brightest, while Mars and Saturn, when well
situated, rank with the brightest of the fixed stars.

The paths of the planets on the celestial sphere are, as we have seen
(§ 13), never very far from the ecliptic; but whereas the sun and moon
move continuously along their paths from west to east, the motion of
a planet is sometimes from west to east, or =direct=, and sometimes
from east to west, or =retrograde=. If we begin to watch a planet when
it is moving eastwards among the stars, we find that after a time
the motion becomes slower and slower, until the planet hardly seems
to move at all, and then begins to move with gradually increasing
speed in the opposite direction; after a time this westward motion
becomes slower and then ceases, and the planet then begins to move
eastwards again, at first slowly and then faster, until it returns to
its original condition, and the changes are repeated. When the planet
is just reversing its motion it is said to be =stationary=, and its
position then is called a =stationary point=. The time during which a
planet’s motion is retrograde is, however, always considerably less
than that during which it is direct; Jupiter’s motion, for example, is
direct for about 39 weeks and retrograde for 17, while Mercury’s direct
motion lasts 13 or 14 weeks and the retrograde motion only about 3
weeks (see figs. 6, 7). On the whole the planets advance from west to
east and describe circuits round the celestial sphere in periods which
are different for each planet. The explanation of these irregularities
in the planetary motions was long one of the great difficulties of
astronomy.

[Illustration:
 FIG. 6.—The apparent path of Jupiter from Oct. 28, 1897, to Sept. 3,
 1898. The dates printed in the diagram shew the positions of Jupiter.]

15. The idea that some of the heavenly bodies are nearer to the
earth than others must have been suggested by eclipses (§ 17) and
=occultations=, _i.e._ passages of the moon over a planet or fixed
star. In this way the moon would be recognised as nearer than any
of the other celestial bodies. No direct means being available for
determining the distances, rapidity of motion was employed as a test
of probable nearness. Now Saturn returns to the same place among the
stars in about 29-1∕2 years, Jupiter in 12 years, Mars in 2 years, the
sun in one year, Venus in 225 days, Mercury in 88 days, and the moon in
27 days; and this order was usually taken to be the order of distance,
Saturn being the most distant, the moon the nearest. The stars being
seen above us it was natural to think of the most distant celestial
bodies as being the highest, and accordingly Saturn, Jupiter, and Mars
being beyond the sun were called =superior planets=, as distinguished
from the two =inferior planets= Venus and Mercury. This division
corresponds also to a difference in the observed motions, as Venus and
Mercury seem to accompany the sun in its annual journey, being never
more than about 47 and 29° respectively distant from it, on either
side; while the other planets are not thus restricted in their motions.

[Illustration:
 FIG. 7.—The apparent path of Mercury from Aug. 1 to Oct. 3, 1898. The
 dates printed in capital letters shew the positions of the sun; the
 other dates shew those of Mercury.]

16. One of the purposes to which applications of astronomical knowledge
was first applied was to the measurement of time. As the alternate
appearance and disappearance of the sun, bringing with it light and
heat, is the most obvious of astronomical facts, so the day is the
simplest unit of time.[8] Some of the early civilised nations divided
the time from sunrise to sunset and also the night each into 12 equal
hours. According to this arrangement a day-hour was in summer longer
than a night-hour and in winter shorter, and the length of an hour
varied during the year. At Babylon, for example, where this arrangement
existed, the length of a day-hour was at midsummer about half as long
again as in midwinter, and in London it would be about twice as long.
It was therefore a great improvement when the Greeks, in comparatively
late times, divided the whole day into 24 equal hours. Other early
nations divided the same period into 12 double hours, and others again
into 60 hours.

The next most obvious unit of time is the =lunar month=, or period
during which the moon goes through her phases. A third independent unit
is the year. Although the year is for ordinary life much more important
than the month, yet as it is much longer and any one time of year is
harder to recognise than a particular phase of the moon, the length of
the year is more difficult to determine, and the earliest known systems
of time-measurement were accordingly based on the month, not on the
year. The month was found to be nearly equal to 29-1∕2 days, and as a
period consisting of an exact number of days was obviously convenient
for most ordinary purposes, months of 29 or 30 days were used, and
subsequently the calendar was brought into closer accord with the moon
by the use of months containing alternately 29 and 30 days (cf. chapter
II., § 19).

Both Chaldaeans and Egyptians appear to have known that the year
consisted of about 365-1∕4 days; and the latter, for whom the
importance of the year was emphasised by the rising and falling of the
Nile, were probably the first nation to use the year in preference to
the month as a measure of time. They chose a year of 365 days.

The origin of the week is quite different from that of the month or
year, and rests on certain astrological ideas about the planets. To
each hour of the day one of the seven planets (sun and moon included)
was assigned as a “ruler,” and each day named after the planet which
ruled its first hour. The planets being taken in the order already
given (§ 15), Saturn ruled the first hour of the first day, and
therefore also the 8th, 15th, and 22nd hours of the first day, the 5th,
12th, and 19th of the second day, and so on; Jupiter ruled the 2nd,
9th, 16th, and 23rd hours of the first day, and subsequently the 1st
hour of the 6th day. In this way the first hours of successive days
fell respectively to Saturn, the Sun, the Moon, Mars, Mercury, Jupiter,
and Venus. The first three are easily recognised in our Saturday,
Sunday, and Monday; in the other days the names of the Roman gods have
been replaced by their supposed Teutonic equivalents—Mercury by Wodan,
Mars by Thues, Jupiter by Thor, Venus by Freia.[9]

17. =Eclipses= of the sun and moon must from very early times have
excited great interest, mingled with superstitious terror, and the hope
of acquiring some knowledge of them was probably an important stimulus
to early astronomical work. That eclipses of the sun only take place
at new moon, and those of the moon only at full moon, must have been
noticed after very little observation; that eclipses of the sun are
caused by the passage of the moon in front of it must have been only a
little less obvious; but the discovery that eclipses of the moon are
caused by the earth’s shadow was probably made much later. In fact even
in the time of Anaxagoras (5th century B.C.) the idea was so unfamiliar
to the Athenian public as to be regarded as blasphemous.

One of the most remarkable of the Chaldaean contributions to astronomy
was the discovery (made at any rate several centuries B.C.) of the
recurrence of eclipses after a period, known as the =saros=, consisting
of 6,585 days (or eighteen of our years and ten or eleven days,
according as five or four leap-years are included). It is probable that
the discovery was made, not by calculations based on knowledge of the
motions of the sun and moon, but by mere study of the dates on which
eclipses were recorded to have taken place. As, however, an eclipse of
the sun (unlike an eclipse of the moon) is only visible over a small
part of the surface of the earth, and eclipses of the sun occurring
at intervals of eighteen years are not generally visible at the same
place, it is not at all easy to see how the Chaldaeans could have
established their cycle for this case, nor is it in fact clear that the
saros was supposed to apply to solar as well as to lunar eclipses. The
saros may be illustrated in modern times by the eclipses of the sun
which took place on July 18th, 1860, on July 29th, 1878, and on August
9th, 1896; but the first was visible in Southern Europe, the second in
North America, and the third in Northern Europe and Asia.

18. To the Chaldaeans may be assigned also the doubtful honour of
having been among the first to develop =astrology=, the false science
which has professed to ascertain the influence of the stars on human
affairs, to predict by celestial observations wars, famines, and
pestilences, and to discover the fate of individuals from the positions
of the stars at their birth. A belief in some form of astrology has
always prevailed in oriental countries; it flourished at times among
the Greeks and the Romans; it formed an important part of the thought
of the Middle Ages, and is not even quite extinct among ourselves
at the present day.[10] It should, however, be remembered that if
the history of astrology is a painful one, owing to the numerous
illustrations which it affords of human credulity and knavery, the
belief in it has undoubtedly been a powerful stimulus to genuine
astronomical study (cf. chapter III., § 56, and chapter V., §§ 99, 100)



CHAPTER II.

GREEK ASTRONOMY.

 “The astronomer discovers that geometry, a pure abstraction of the
 human mind, is the measure of planetary motion.”
                                                             EMERSON.


19. In the earlier period of Greek history one of the chief functions
expected of astronomers was the proper regulation of the calendar.
The Greeks, like earlier nations, began with a calendar based on the
moon. In the time of Hesiod a year consisting of 12 months of 30
days was in common use; at a later date a year made up of 6 =full=
months of 30 days and 6 =empty= months of 29 days was introduced.
To Solon is attributed the merit of having introduced at Athens,
about 594 B.C., the practice of adding to every alternate year a
“full” month. Thus a period of two years would contain 13 months of
30 days and 12 of 29 days, or 738 days in all, distributed among 25
months, giving, for the average length of the year and month, 369
days and about 29-1∕2 days respectively. This arrangement was further
improved by the introduction, probably during the 5th century B.C.,
of the =octaeteris=, or eight-year cycle, in three of the years of
which an additional “full” month was introduced, while the remaining
years consisted as before of 6 “full” and 6 “empty” months. By this
arrangement the average length of the year was reduced to 365-1∕4
days, that of the month remaining nearly unchanged. As, however, the
Greeks laid some stress on beginning the month when the new moon was
first visible, it was necessary to make from time to time arbitrary
alterations in the calendar, and considerable confusion resulted, of
which Aristophanes makes the Moon complain in his play _The Clouds_,
acted in 423 B.C.:

                          “Yet you will not mark your days
    As she bids you, but confuse them, jumbling them all sorts of ways.
    And, she says, the Gods in chorus shower reproaches on her head,
    When, in bitter disappointment, they go supperless to bed.
    Not obtaining festal banquets, duly on the festal day.”

20. A little later, the astronomer _Meton_ (born about 460 B.C.)
made the discovery that the length of 19 years is very nearly equal
to that of 235 lunar months (the difference being in fact less than
a day), and he devised accordingly an arrangement of 12 years of 12
months and 7 of 13 months, 125 of the months in the whole cycle being
“full” and the others “empty.” Nearly a century later _Callippus_
made a slight improvement, by substituting in every fourth period of
19 years a “full” month for one of the “empty” ones. Whether =Meton’s
cycle=, as it is called, was introduced for the civil calendar or not
is uncertain, but if not it was used as a standard by reference to
which the actual calendar was from time to time adjusted. The use of
this cycle seems to have soon spread to other parts of Greece, and it
is the basis of the present ecclesiastical rule for fixing Easter.
The difficulty of ensuring satisfactory correspondence between the
civil calendar and the actual motions of the sun and moon led to the
practice of publishing from time to time tables (παραπήγματα)
not unlike our modern almanacks, giving for a series of years the dates
of the phases of the moon, and the rising and setting of some of the
fixed stars, together with predictions of the weather. Owing to the
same cause the early writers on agriculture (_e.g._ Hesiod) fixed the
dates for agricultural operations, not by the calendar, but by the
times of the rising and setting of constellations, _i.e._ the times
when they first became visible before sunrise or were last visible
immediately after sunset—a practice which was continued long after the
establishment of a fairly satisfactory calendar, and was apparently by
no means extinct in the time of Galen (2nd century A.D.).

21. The Roman calendar was in early times even more confused than the
Greek. There appears to have been at one time a year of either 304 or
354 days; tradition assigned to Numa the introduction of a cycle of
four years, which brought the calendar into fair agreement with the
sun, but made the average length of the month considerably too short.
Instead, however, of introducing further refinements the Romans cut the
knot by entrusting to the ecclesiastical authorities the adjustment of
the calendar from time to time, so as to make it agree with the sun
and moon. According to one account, the first day of each month was
proclaimed by a crier. Owing either to ignorance, or, as was alleged,
to political and commercial favouritism, the priests allowed the
calendar to fall into a state of great confusion, so that, as Voltaire
remarked, “les généraux romains triomphaient toujours, mais ils ne
savaient pas quel jour ils triomphaient.”

A satisfactory reform of the calendar was finally effected by Julius
Caesar during the short period of his supremacy at Rome, under the
advice of an Alexandrine astronomer _Sosigenes_. The error in the
calendar had mounted up to such an extent, that it was found necessary,
in order to correct it, to interpolate three additional months in a
single year (46 B.C.), bringing the total number of days in that year
up to 445. For the future the year was to be independent of the moon;
the ordinary year was to consist of 365 days, an extra day being added
to February every fourth year (our leap-year), so that the average
length of the year would be 365-1∕4 days.

The new system began with the year 45 B.C., and soon spread, under the
name of the =Julian Calendar=, over the civilised world.

22. To avoid returning to the subject, it may be convenient to deal
here with the only later reform of any importance.

The difference between the average length of the year as fixed by
Julius Caesar and the true year is so small as only to amount to about
one day in 128 years. By the latter half of the 16th century the date
of the vernal equinox was therefore about ten days earlier than it was
at the time of the Council of Nice (A.D. 325), at which rules for the
observance of Easter had been fixed. Pope Gregory XIII. introduced
therefore, in 1582, a slight change;, ten days were omitted from that
year, and it was arranged to omit for the future three leap-years in
four centuries (viz. in 1700, 1800, 1900, 2100, etc., the years 1600,
2000, 2400, etc., remaining leap-years). The =Gregorian Calendar=, or
=New Style=, as it was commonly called, was not adopted in England till
1752, when 11 days had to be omitted; and has not yet been adopted in
Russia and Greece, the dates there being now 12 days behind those of
Western Europe.

23. While their oriental predecessors had confined themselves chiefly
to astronomical observations, the earlier Greek philosophers appear to
have made next to no observations of importance, and to have been far
more interested in inquiring into causes of phenomena. _Thales_, the
founder of the Ionian school, was credited by later writers with the
introduction of Egyptian astronomy into Greece, at about the end of the
7th century B.C.; but both Thales and the majority of his immediate
successors appear to have added little or nothing to astronomy,
except some rather vague speculations as to the form of the earth and
its relation to the rest of the world. On the other hand, some real
progress seems to have been made by _Pythagoras_[11] and his followers.
Pythagoras taught that the earth, in common with the heavenly bodies,
is a sphere, and that it rests without requiring support in the middle
of the universe. Whether he had any real evidence in support of these
views is doubtful, but it is at any rate a reasonable conjecture that
he knew the moon to be bright because the sun shines on it, and the
phases to be caused by the greater or less amount of the illuminated
half turned towards us; and the curved form of the boundary between
the bright and dark portions of the moon was correctly interpreted by
him as evidence that the moon was spherical, and not a flat disc, as
it appears at first sight. Analogy would then probably suggest that
the earth also was spherical. However this may be, the belief in the
spherical form of the earth never disappeared from Greek thought, and
was in later times an established part of Greek systems, whence it has
been handed down, almost unchanged, to modern times. This belief is
thus 2,000 years older than the belief in the rotation of the earth
and its revolution round the sun (chapter IV.), doctrines which we
are sometimes inclined to couple with it as the foundations of modern
astronomy.

In Pythagoras occurs also, perhaps for the first time, an idea
which had an extremely important influence on ancient and mediaeval
astronomy. Not only were the stars supposed to be attached to a crystal
sphere, which revolved daily on an axis through the earth, but each
of the seven planets (the sun and moon being included) moved on a
sphere of its own. The distances of these spheres from the earth were
fixed in accordance with certain speculative notions of Pythagoras
as to numbers and music; hence the spheres as they revolved produced
harmonious sounds which specially gifted persons might at times hear:
this is the origin of the idea of the =music of the spheres= which
recurs continually in mediaeval speculation and is found occasionally
in modern literature. At a later stage these spheres of Pythagoras
were developed into a scientific representation of the motions of the
celestial bodies, which remained the basis of astronomy till the time
of Kepler (chapter VII.).

24. The Pythagorean _Philolaus_, who lived about a century later than
his master, introduced for the first time the idea of the motion of
the earth: he appears to have regarded the earth, as well as the sun,
moon, and five planets, as revolving round some central fire, the earth
rotating on its own axis as it revolved, apparently in order to ensure
that the central fire should always remain invisible to the inhabitants
of the known parts of the earth. That the scheme was a purely fanciful
one, and entirely different from the modern doctrine of the motion of
the earth, with which later writers confused it, is sufficiently shewn
by the invention as part of the scheme of a purely imaginary body, the
counter-earth (ἀντιχθών), which brought the number of moving
bodies up to ten, a sacred Pythagorean number. The suggestion of such
an important idea as that of the motion of the earth, an idea so
repugnant to uninstructed common sense, although presented in such a
crude form, without any of the evidence required to win general assent,
was, however, undoubtedly a valuable contribution to astronomical
thought. It is well worth notice that Coppernicus in the great book
which is the foundation of modern astronomy (chapter IV., § 75)
especially quotes Philolaus and other Pythagoreans as authorities for
his doctrine of the motion of the earth.

Three other Pythagoreans, belonging to the end of the 6th century
and to the 5th century B.C., _Hicetas_ of Syracuse, _Heraclitus_,
and _Ecphantus_, are explicitly mentioned by later writers as having
believed in the rotation of the earth.

An obscure passage in one of Plato’s dialogues (the _Timaeus_) has
been interpreted by many ancient and modern commentators as implying a
belief in the rotation of the earth, and Plutarch also tells us, partly
on the authority of Theophrastus, that Plato in old age adopted the
belief that the centre of the universe was not occupied by the earth
but by some better body.[12]

Almost the only scientific Greek astronomer who believed in the motion
of the earth was _Aristarchus_ of Samos, who lived in the first half
of the 3rd century B.C., and is best known by his measurements of the
distances of the sun and moon (§ 32). He held that the sun and fixed
stars were motionless, the sun being in the centre of the sphere on
which the latter lay, and that the earth not only rotated on its axis,
but also described an orbit round the sun. _Seleucus_ of Seleucia, who
belonged to the middle of the 2nd century B.C., also held a similar
opinion. Unfortunately we know nothing of the grounds of this belief in
either case, and their views appear to have found little favour among
their contemporaries or successors.

It may also be mentioned in this connection that Aristotle (§ 27)
clearly realised that the apparent daily motion of the stars could be
explained by a motion either of the stars or of the earth, but that he
rejected the latter explanation.

25. _Plato_ (about 428-347 B.C.) devoted no dialogue especially to
astronomy, but made a good many references to the subject in various
places. He condemned any careful study of the actual celestial motions
as degrading rather than elevating, and apparently regarded the subject
as worthy of attention chiefly on account of its connection with
geometry, and because the actual celestial motions suggested ideal
motions of greater beauty and interest. This view of astronomy he
contrasts with the popular conception, according to which the subject
was useful chiefly for giving to the agriculturist, the navigator, and
others a knowledge of times and seasons.[13] At the end of the same
dialogue he gives a short account of the celestial bodies, according
to which the sun, moon, planets, and fixed stars revolve on eight
concentric and closely fitting wheels or circles round an axis passing
through the earth. Beginning with the body nearest to the earth, the
order is Moon, Sun, Mercury, Venus, Mars, Jupiter, Saturn, stars. The
Sun, Mercury, and Venus are said to perform their revolutions in the
same time, while the other planets move more slowly, statements which
shew that Plato was at any rate aware that the motions of Venus and
Mercury are different from those of the other planets. He also states
that the moon shines by reflected light received from the sun.

Plato is said to have suggested to his pupils as a worthy problem
the explanation of the celestial motions by means of a combination
of uniform circular or spherical motions. Anything like an accurate
theory of the celestial motions, agreeing with actual observation, such
as Hipparchus and Ptolemy afterwards constructed with fair success,
would hardly seem to be in accordance with Plato’s ideas of the true
astronomy, but he may well have wished to see established some simple
and harmonious geometrical scheme which would not be altogether at
variance with known facts.

26. Acting to some extent on this idea of Plato’s, _Eudoxus_ of
Cnidus (about 409-356 B.C.) attempted to explain the most obvious
peculiarities of the celestial motions by means of a combination of
uniform circular motions. He may be regarded as representative of the
transition from speculative to scientific Greek astronomy. As in
the schemes of several of his predecessors, the fixed stars lie on
a sphere which revolves daily about an axis through the earth; the
motion of each of the other bodies is produced by a combination of
other spheres, the centre of each sphere lying on the surface of the
preceding one. For the sun and moon three spheres were in each case
necessary: one to produce the daily motion, shared by all the celestial
bodies; one to produce the annual or monthly motion in the opposite
direction along the ecliptic; and a third, with its axis inclined to
the axis of the preceding, to produce the smaller motion to and from
the ecliptic. Eudoxus evidently was well aware that the moon’s path
is not coincident with the ecliptic, and even that its path is not
always the same, but changes continuously, so that the third sphere
was in this case necessary; on the other hand, he could not possibly
have been acquainted with the minute deviations of the sun from the
ecliptic with which modern astronomy deals. Either therefore he used
erroneous observations, or, as is more probable, the sun’s third sphere
was introduced to explain a purely imaginary motion conjectured to
exist by “analogy” with the known motion of the moon. For each of the
five planets four spheres were necessary, the additional one serving
to produce the variations in the speed of the motion and the reversal
of the direction of motion along the ecliptic (chapter I., § 14, and
below, § 51). Thus the celestial motions were to some extent explained
by means of a system of 27 spheres, 1 for the stars, 6 for the sun
and moon, 20 for the planets. There is no clear evidence that Eudoxus
made any serious attempt to arrange either the size or the time of
revolution of the spheres so as to produce any precise agreement with
the observed motions of the celestial bodies, though he knew with
considerable accuracy the time required by each planet to return to
the same position with respect to the sun; in other words, his scheme
represented the celestial motions qualitatively but not quantitatively.
On the other hand, there is no reason to suppose that Eudoxus regarded
his spheres (with the possible exception of the sphere of the fixed
stars) as material; his known devotion to mathematics renders it
probable that in his eyes (as in those of most of the scientific
Greek astronomers who succeeded him) the spheres were mere geometrical
figures, useful as a means of resolving highly complicated motions
into simpler elements. Eudoxus was also the first Greek recorded to
have had an observatory, which was at Cnidus, but we have few details
as to the instruments used or as to the observations made. We owe,
however, to him the first systematic description of the constellations
(see below, § 42), though it was probably based, to a large extent, on
rough observations borrowed from his Greek predecessors or from the
Egyptians. He was also an accomplished mathematician, and skilled in
various other branches of learning.

Shortly afterwards Callippus (§ 20) further developed Eudoxus’s scheme
of revolving spheres by adding, for reasons not known to us, two
spheres each for the sun and moon and one each for Venus, Mercury, and
Mars, thus bringing the total number up to 34.

27. We have a tolerably full account of the astronomical views of
_Aristotle_ (384-322 B.C.), both by means of incidental references, and
by two treatises—the _Meteorologica_ and the _De Coelo_—though another
book of his, dealing specially with the subject, has unfortunately been
lost. He adopted the planetary scheme of Eudoxus and Callippus, but
imagined on “metaphysical grounds” that the spheres would have certain
disturbing effects on one another, and to counteract these found it
necessary to add 22 fresh spheres, making 56 in all. At the same time
he treated the spheres as material bodies, thus converting an ingenious
and beautiful geometrical scheme into a confused mechanism.[14]
Aristotle’s spheres were, however, not adopted by the leading Greek
astronomers who succeeded him, the systems of Hipparchus and Ptolemy
being geometrical schemes based on ideas more like those of Eudoxus.

[Illustration: FIG. 8.—The phases of the moon.]

[Illustration: FIG. 9.—The phases of the moon.]

28. Aristotle, in common with other philosophers of his time, believed
the heavens and the heavenly bodies to be spherical. In the case of the
moon he supports this belief by the argument attributed to Pythagoras
(§ 23), namely that the observed appearances of the moon in its
several phases are those which would be assumed by a spherical body
of which one half only is illuminated by the sun. Thus the visible
portion of the moon is bounded by two planes passing nearly through its
centre, perpendicular respectively to the lines joining the centre of
the moon to those of the sun and earth. In the accompanying diagram,
which represents a section through the centres of the sun (S), earth
(E), and moon (M), A B C D representing on a much enlarged scale a
section of the moon itself, the portion D A B which is turned away from
the sun is dark, while the portion A D C, being turned away from the
observer on the earth, is in any case invisible to him. The part of the
moon which appears bright is therefore that of which B C is a section,
or the portion represented by F B G C in fig. 9 (which represents the
complete moon), which consequently appears to the eye as bounded by
a semicircle F C G, and a portion F B G of an oval curve (actually
an ellipse). The breadth of this bright surface clearly varies with
the relative positions of sun, moon, and earth; so that in the course
of a month, during which the moon assumes successively the positions
relative to sun and earth represented by 1, 2, 3, 4, 5, 6, 7, 8 in
fig. 10, its appearances are those represented by the corresponding
numbers in fig. 11, the moon thus passing through the familiar phases
of crescent, half full, gibbous, full moon, and gibbous, half full,
crescent again.[15]

[Illustration: FIG. 10.—The phases of the moon.]

Aristotle then argues that as one heavenly body is spherical, the
others must be so also, and supports this conclusion by another
argument, equally inconclusive to us, that a spherical form is
appropriate to bodies moving as the heavenly bodies appear to do.

[Illustration: FIG. 11.—The phases of the moon.]

29. His proofs that the earth is spherical are more interesting. After
discussing and rejecting various other suggested forms, he points out
that an eclipse of the moon is caused by the shadow of the earth cast
by the sun, and argues from the circular form of the boundary of the
shadow as seen on the face of the moon during the progress of the
eclipse, or in a partial eclipse, that the earth must be spherical;
for otherwise it would cast a shadow of a different shape. A second
reason for the spherical form of the earth is that when we move
north and south the stars change their positions with respect to the
horizon, while some even disappear and fresh ones take their place.
This shows that the direction of the stars has changed as compared
with the observer’s horizon; hence, the actual direction of the stars
being imperceptibly affected by any motion of the observer on the
earth, the horizons at two places, north and south of one another,
are in different directions, and the earth is therefore curved. For
example, if a star is visible to an observer at A (fig. 12), while
to an observer at B it is at the same time invisible, _i.e._ hidden
by the earth, the surface of the earth at A must be in a different
direction from that at B. Aristotle quotes further, in confirmation of
the roundness of the earth, that travellers from the far East and the
far West (practically India and Morocco) alike reported the presence of
elephants, whence it may be inferred that the two regions in question
are not very far apart. He also makes use of some rather obscure
arguments of an _a priori_ character.

[Illustration: FIG. 12.—The curvature of the earth.]

There can be but little doubt that the readiness with which Aristotle,
as well as other Greeks, admitted the spherical form of the earth and
of the heavenly bodies, was due to the affection which the Greeks
always seem to have had for the circle and sphere as being “perfect,”
_i.e._ perfectly symmetrical figures.

30. Aristotle argues against the possibility of the revolution of the
earth round the sun, on the ground that this motion, if it existed,
ought to produce a corresponding apparent motion of the stars. We
have here the first appearance of one of the most serious of the many
objections ever brought against the belief in the motion of the earth,
an objection really only finally disposed of during the present
century by the discovery that such a motion of the stars can be seen in
a few cases, though owing to the almost inconceivably great distance
of the stars the motion is imperceptible except by extremely refined
methods of observation (cf. chapter XIII., §§ 278, 279). The question
of the distances of the several celestial bodies is also discussed,
and Aristotle arrives at the conclusion that the planets are farther
off than the sun and moon, supporting his view by his observation of
an occultation of Mars by the moon (_i.e._ a passage of the moon in
front of Mars), and by the fact that similar observations had been
made in the case of other planets by Egyptians and Babylonians. It is,
however, difficult to see why he placed the planets beyond the sun,
as he must have known that the intense brilliancy of the sun renders
planets invisible in its neighbourhood, and that no occultations of
planets by the sun could really have been seen even if they had been
reported to have taken place. He quotes also, as an opinion of “the
mathematicians,” that the stars must be at least nine times as far off
as the sun.

There are also in Aristotle’s writings a number of astronomical
speculations, founded on no solid evidence and of little value; thus
among other questions he discusses the nature of comets, of the Milky
Way, and of the stars, why the stars twinkle, and the causes which
produce the various celestial motions.

In astronomy, as in other subjects, Aristotle appears to have collected
and systematised the best knowledge of the time; but his original
contributions are not only not comparable with his contributions to
the mental and moral sciences, but are inferior in value to his work
in other natural sciences, _e.g._ Natural History. Unfortunately the
Greek astronomy of his time, still in an undeveloped state, was as it
were crystallised in his writings, and his great authority was invoked,
centuries afterwards, by comparatively unintelligent or ignorant
disciples in support of doctrines which were plausible enough in his
time, but which subsequent research was shewing to be untenable. The
advice which he gives to his readers at the beginning of his exposition
of the planetary motions, to compare his views with those which they
arrived at themselves or met with elsewhere, might with advantage have
been noted and followed by many of the so-called Aristotelians of the
Middle Ages and of the Renaissance.[16]

31. After the time of Aristotle the centre of Greek scientific thought
moved to Alexandria. Founded by Alexander the Great (who was for a
time a pupil of Aristotle) in 332 B.C., Alexandria was the capital
of Egypt during the reigns of the successive Ptolemies. These kings,
especially the second of them, surnamed Philadelphos, were patrons of
learning; they founded the famous Museum, which contained a magnificent
library as well as an observatory, and Alexandria soon became the home
of a distinguished body of mathematicians and astronomers. During the
next five centuries the only astronomers of importance, with the great
exception of Hipparchus (§ 37), were Alexandrines.

[Illustration: FIG. 13.—The method of Aristarchus for comparing the
distances of the sun and moon.]

32. Among the earlier members of the Alexandrine school were
_Aristarchus_ of Samos, _Aristyllus_, and _Timocharis_, three nearly
contemporary astronomers belonging to the first half of the 3rd
century B.C. The views of Aristarchus on the motion of the earth have
already been mentioned (§ 24). A treatise of his _On the Magnitudes
and Distances of the Sun and Moon_ is still extant: he there gives an
extremely ingenious method for ascertaining the comparative distances
of the sun and moon. If, in the figure, E, S, and M denote respectively
the centres of the earth, sun, and moon, the moon evidently appears
to an observer at E half full when the angle E M S is a right angle.
If when this is the case the angular distance between the centres of
the sun and moon, _i.e._ the angle M E S, is measured, two angles
of the triangle M E S are known; its shape is therefore completely
determined, and the ratio of its sides E M, E S can be calculated
without much difficulty. In fact, it being known (by a well-known
result in elementary geometry) that the angles at E and S are together
equal to a right angle, the angle at S is obtained by subtracting
the angle S E M from a right angle. Aristarchus made the angle at S
about 3°, and hence calculated that the distance of the sun was from
18 to 20 times that of the moon, whereas, in fact, the sun is about
400 times as distant as the moon. The enormous error is due to the
difficulty of determining with sufficient accuracy the moment when
the moon is half full: the boundary separating the bright and dark
parts of the moon’s face is in reality (owing to the irregularities on
the surface of the moon) an ill-defined and broken line (cf. fig. 53
and the frontispiece), so that the observation on which Aristarchus
based his work could not have been made with any accuracy even with
our modern instruments, much less with those available in his time.
Aristarchus further estimated the apparent sizes of the sun and moon
to be about equal (as is shewn, for example, at an eclipse of the sun,
when the moon sometimes rather more than hides the surface of the
sun and sometimes does not quite cover it), and inferred correctly
that the real diameters of the sun and moon were in proportion to
their distances. By a method based on eclipse observations which was
afterwards developed by Hipparchus (§ 41), 1∕3 that of the earth, a
result very near to the truth; and the same method supplied data from
which the distance of the moon could at once have been expressed in
terms of the radius of the earth, but his work was spoilt at this point
by a grossly inaccurate estimate of the apparent size of the moon (2°
instead of 1∕2°), and his conclusions seem to contradict one another.
He appears also to have believed the distance of the fixed stars to
be immeasurably great as compared with that of the sun. Both his
speculative opinions and his actual results mark therefore a decided
advance in astronomy.

Timocharis and Aristyllus were the first to ascertain and to record
the positions of the chief stars, by means of numerical measurements
of their distances from fixed positions on the sky; they may thus
be regarded as the authors of the first real star catalogue, earlier
astronomers having only attempted to fix the position of the stars
by more or less vague verbal descriptions. They also made a number
of valuable observations of the planets, the sun, etc., of which
succeeding astronomers, notably Hipparchus and Ptolemy, were able to
make good use.

[Illustration: FIG. 14.—The equator and the ecliptic.]

33. Among the important contributions of the Greeks to astronomy must
be placed the development, chiefly from the mathematical point of
view, of the consequences of the rotation of the celestial sphere and
of some of the simpler motions of the celestial bodies, a development
the individual steps of which it is difficult to trace. We have,
however, a series of minor treatises or textbooks, written for the
most part during the Alexandrine period, dealing with this branch of
the subject (known generally as =Spherics=, or the Doctrine of the
Sphere), of which the _Phenomena_ of the famous geometer _Euclid_
(about 300 B.C.) is a good example. In addition to the points and
circles of the sphere already mentioned (chapter I., §§ 8-11), we now
find explicitly recognised the =horizon=, or the great circle in which
a horizontal plane through the observer meets the celestial sphere, and
its =pole=,[17] the =zenith=,[18] or point on the celestial sphere
vertically above the observer; the =verticals=, or great circles
through the zenith, meeting the horizon at right angles; and the
=declination circles=, which pass through the north and south poles and
cut the equator at right angles. Another important great circle was the
=meridian=, passing through the zenith and the poles. The well-known
Milky Way had been noticed, and was regarded as forming another great
circle. There are also traces of the two chief methods in common use at
the present day of indicating the position of a star on the celestial
sphere, namely, by reference either to the equator or to the ecliptic.
If through a star S we draw on the sphere a portion of a great circle
S N, cutting the ecliptic ♈ N at right angles in N, and another great
circle (a declination circle) cutting the equator at M, and if ♈
be the first point of Aries (§ 13), where the ecliptic crosses the
equator, then the position of the star is completely defined _either_
by the lengths of the arcs ♈ N, N S, which are called the =celestial
longitude= and =latitude= respectively, _or_ by the arcs ♈ M, M S,
called respectively the =right ascension= and =declination=.[19] For
some purposes it is more convenient to find the position of the star
by the first method, _i.e._ by reference to the ecliptic; for other
purposes in the second way, by making use of the equator.

34. One of the applications of Spherics was to the construction of
sun-dials, which were supposed to have been originally introduced
into Greece from Babylon, but which were much improved by the Greeks,
and extensively used both in Greek and in mediaeval times. The proper
graduation of sun-dials placed in various positions, horizontal,
vertical, and oblique, required considerable mathematical skill. Much
attention was also given to the time of the rising and setting of the
various constellations, and to similar questions.

35. The discovery of the spherical form of the earth led to a
scientific treatment of the differences between the seasons in
different parts of the earth, and to a corresponding division of
the earth into zones. We have already seen that the height of the
pole above the horizon varies in different places, and that it was
recognised that, if a traveller were to go far enough north, he would
find the pole to coincide with the zenith, whereas by going south
he would reach a region (not very far beyond the limits of actual
Greek travel) where the pole would be on the horizon and the equator
consequently pass through the zenith; in regions still farther south
the north pole would be permanently invisible, and the south pole would
appear above the horizon.

[Illustration: FIG. 15.—The equator, the horizon, and the meridian.]

Further, if in the figure H E K W represents the horizon, meeting the
equator Q E R W in the east and west points E W, and the meridian H Q Z
P K in the south and north points H and K, Z being the zenith and P the
pole, then it is easily seen that Q Z is equal to P K, the height of
the pole above the horizon. Any celestial body, therefore, the distance
of which from the equator towards the north (declination) is less than
P K, will cross the meridian to the south of the zenith, whereas if
its declination be greater than P K, it will cross to the north of the
zenith. Now the greatest distance of the sun from the equator is equal
to the angle between the ecliptic and the equator, or about 23-1∕2°,
Consequently at places at which the height of the pole is less than
23-1∕2° the sun will, during part of the year, cast shadows at midday
towards the south. This was known actually to be the case not very far
south of Alexandria. It was similarly recognised that on the other side
of the equator there must be a region in which the sun ordinarily cast
shadows towards the south, but occasionally towards the north. These
two regions are the torrid zones of modern geographers.

Again, if the distance of the sun from the equator is 23-1∕2°, its
distance from the pole is 66-1∕2°; therefore in regions so far north
that the height P K of the north pole is more than 66-1∕2°, the sun
passes in summer into the region of the circumpolar stars which never
set (chapter I., § 9), and therefore during a portion of the summer
the sun remains continuously above the horizon. Similarly in the same
regions the sun is in winter so near the south pole that for a time it
remains continuously below the horizon. Regions in which this occurs
(our Arctic regions) were unknown to Greek travellers, but their
existence was clearly indicated by the astronomers.

[Illustration: FIG. 16.—The measurement of the earth.]

36. To _Eratosthenes_ (276 B.C. to 195 or 196 B.C.), another member of
the Alexandrine school, we owe one of the first scientific estimates
of the size of the earth. He found that at the summer solstice the
angular distance of the sun from the zenith at Alexandria was at midday
1∕50th of a complete circumference, or about 7°, whereas at Syene
in Upper Egypt the sun was known to be vertical at the same time.
From this he inferred, assuming Syene to be due south of Alexandria,
that the distance from Syene to Alexandria was also 1∕50th of the
circumference of the earth. Thus if in the figure S denotes the sun, A
and B Alexandria and Syene respectively, C the centre of the earth, and
A Z the direction of the zenith at Alexandria, Eratosthenes estimated
the angle S A Z, which, owing to the great distance of S, is sensibly
equal to the angle S C A, to be 7°, and hence inferred that the arc
A B was to the circumference of the earth in the proportion of 7° to
360° or 1 to 50. The distance between Alexandria and Syene being known
to be 5,000 _stadia_, Eratosthenes thus arrived at 250,000 stadia as
an estimate of the circumference of the earth, a number altered into
252,000 in order to give an exact number of stadia (700) for each
degree on the earth. It is evident that the data employed were rough,
though the principle of the method is perfectly sound; it is, however,
difficult to estimate the correctness of the result on account of
the uncertainty as to the value of the _stadium_ used. If, as seems
probable, it was the common Olympic stadium, the result is about 20 per
cent. too great, but according to another interpretation[20] the result
is less than 1 per cent. in error (cf. chapter X., § 221).

Another measurement due to Eratosthenes was that of the obliquity of
the ecliptic, which he estimated at 22∕83 of a right angle, or 23° 51′,
the error in which is only about 7′.

37. An immense advance in astronomy was made by _Hipparchus_, whom all
competent critics have agreed to rank far above any other astronomer of
the ancient world, and who must stand side by side with the greatest
astronomers of all time. Unfortunately only one unimportant book of his
has been preserved, and our knowledge of his work is derived almost
entirely from the writings of his great admirer and disciple Ptolemy,
who lived nearly three centuries later (§§ 46 _seqq._). We have also
scarcely any information about his life. He was born either at Nicaea
in Bithynia or in Rhodes, in which island he erected an observatory
and did most of his work. There is no evidence that he belonged to the
Alexandrine school, though he probably visited Alexandria and may have
made some observations there. Ptolemy mentions observations made by
him in 146 B.C., 126 B.C., and at many intermediate dates, as well as
a rather doubtful one of 161 B.C. The period of his greatest activity
must therefore have been about the middle of the 2nd century B.C.

Apart from individual astronomical discoveries, his chief services
to astronomy may be put under four heads. He invented or greatly
developed a special branch of mathematics,[21] which enabled processes
of numerical calculation to be applied to geometrical figures,
whether in a plane or on a sphere. He made an extensive series of
observations, taken with all the accuracy that his instruments would
permit. He systematically and critically made use of old observations
for comparison with later ones so as to discover astronomical
changes too slow to be detected within a single lifetime. Finally,
he systematically employed a particular geometrical scheme (that
of eccentrics, and to a less extent that of epicycles) for the
representation of the motions of the sun and moon.

38. The merit of suggesting that the motions of the heavenly bodies
could be represented more simply by combinations of uniform _circular_
motions than by the revolving _spheres_ of Eudoxus and his school (§
26) is generally attributed to the great Alexandrine mathematician
_Apollonius_ of Perga, who lived in the latter half of the 3rd century
B.C., but there is no clear evidence that he worked out a system in any
detail.

On account of the important part that this idea played in astronomy for
nearly 2,000 years, it may be worth while to examine in some detail
Hipparchus’s theory of the sun, the simplest and most successful
application of the idea.

We have already seen (chapter I., § 10) that, in addition to the
daily motion (from east to west) which it shares with the rest of the
celestial bodies, and of which we need here take no further account,
the sun has also an annual motion on the celestial sphere in the
reverse direction (from west to east) in a path oblique to the equator,
which was early recognised as a great circle, called the ecliptic.
It must be remembered further that the celestial sphere, on which
the sun appears to lie, is a mere geometrical fiction introduced for
convenience; all that direct observation gives is the change in the
sun’s direction, and therefore the sun may consistently be supposed
to move in such a way as to vary its distance from the earth in any
arbitrary manner, provided only that the alterations in the apparent
size of the sun, caused by the variations in its distance, agree with
those observed, or that at any rate the differences are not great
enough to be perceptible. It was, moreover, known (probably long before
the time of Hipparchus) that the sun’s apparent motion in the ecliptic
is not quite uniform, the motion at some times of the year being
slightly more rapid than at others.

Supposing that we had such a complete set of observations of the motion
of the sun, that we knew its position from day to day, how should we
set to work to record and describe its motion? For practical purposes
nothing could be more satisfactory than the method adopted in our
almanacks, of giving from day to day the position of the sun; after
observations extending over a few years it would not be difficult
to verify that the motion of the sun is (after allowing for the
irregularities of our calendar) from year to year the same, and to
predict in this way the place of the sun from day to day in future
years.

But it is clear that such a description would not only be long, but
would be felt as unsatisfactory by any one who approached the question
from the point of view of intellectual curiosity or scientific
interest. Such a person would feel that these detailed facts ought to
be capable of being exhibited as consequences of some simpler general
statement.

A modern astronomer would effect this by expressing the motion of the
sun by means of an algebraical formula, _i.e._ he would represent the
velocity of the sun or its distance from some fixed point in its path
by some symbolic expression representing a quantity undergoing changes
with the time in a certain definite way, and enabling an expert to
compute with ease the required position of the sun at any assigned
instant.[22]

The Greeks, however, had not the requisite algebraical knowledge for
such a method of representation, and Hipparchus, like his predecessors,
made use of a geometrical representation of the required variations in
the sun’s motion in the ecliptic, a method of representation which is
in some respects more intelligible and vivid than the use of algebra,
but which becomes unmanageable in complicated cases. It runs moreover
the risk of being taken for a mechanism. The circle, being the simplest
curve known, would naturally be thought of, and as any motion other
than a uniform motion would itself require a special representation,
the idea of Apollonius, adopted by Hipparchus, was to devise a proper
combination of uniform circular motions.

39. The simplest device that was found to be satisfactory in the case
of the sun was the use of the =eccentric=, _i.e._ a circle the centre
of which (C) does not coincide with the position of the observer on
the earth (E). If in fig. 17 a point, S, describes the eccentric
circle A F G B uniformly, so that it always passes over equal arcs of
the circle in equal times and the angle A C S increases uniformly,
then it is evident that the angle A E S, or the apparent distance of
S from A, does not increase uniformly. When S is near the point A,
which is farthest from the earth and hence called the =apogee=, it
appears on account of its greater distance from the observer to move
more slowly than when near F or G; and it appears to move fastest when
near B, the point nearest to E, hence called the =perigee=. Thus the
motion of S varies in the same sort of way as the motion of the sun as
actually observed. Before, however, the eccentric could be considered
as satisfactory, it was necessary to show that it was possible to
choose the direction of the line B E C A (the =line of apses=) which
determines the positions of the sun when moving fastest and when moving
most slowly, and the magnitude of the ratio of E C to the radius C A of
the circle (the =eccentricity=), so as to make the calculated positions
of the sun in various parts of its path differ from the observed
positions at the corresponding times of year by quantities so small
that they might fairly be attributed to errors of observation.

[Illustration: FIG. 17.—The eccentric.]

This problem was much more difficult than might at first sight appear,
on account of the great difficulty experienced in Greek times and long
afterwards in getting satisfactory observations of the sun. As the
sun and stars are not visible at the same time, it is not possible
to measure directly the distance of the sun from neighbouring stars
and so to fix its place on the celestial sphere. But it is possible,
by measuring the length of the shadow cast by a rod at midday, to
ascertain with fair accuracy the height of the sun above the horizon,
and hence to deduce its distance from the equator, or the declination
(figs. 3, 14). This one quantity does not suffice to fix the sun’s
position, but if also the sun’s right ascension (§ 33), or its distance
east and west from the stars, can be accurately ascertained, its place
on the celestial sphere is completely determined. The methods available
for determining this second quantity were, however, very imperfect. One
method was to note the time between the passage of the sun across some
fixed position in the sky (_e.g._ the meridian), and the passage of a
star across the same place, and thus to ascertain the angular distance
between them (the celestial sphere being known to turn through 15° in
an hour), a method which with modern clocks is extremely accurate,
but with the rough water-clocks or sand-glasses of former times was
very uncertain. In another method the moon was used as a connecting
link between sun and stars, her position relative to the latter being
observed by night, and with respect to the former by day; but owing
to the rapid motion of the moon in the interval between the two
observations, this method also was not susceptible of much accuracy.

[Illustration: FIG. 18.—The position of the sun’s apogee.]

In the case of the particular problem of the determination of the line
of apses, Hipparchus made use of another method, and his skill is shewn
in a striking manner by his recognition that both the eccentricity and
position of the apse line could be determined from a knowledge of the
lengths of two of the seasons of the year, _i.e._ of the intervals into
which the year is divided by the solstices and the equinoxes (§ 11). By
means of his own observations, and of others made by his predecessors,
he ascertained the length of the spring (from the vernal equinox to the
summer solstice) to be 94 days, and that of the summer (summer solstice
to autumnal equinox) to be 92-1∕2 days, the length of the year being
365-1∕4 days. As the sun moves in each season through the same angular
distance, a right angle, and as the spring and summer make together
more than half the year, and the spring is longer than the summer, it
follows that the sun must, on the whole, be moving more slowly during
the spring than in any other season, and that it must therefore pass
through the apogee in the spring. If, therefore, in fig. 18, we draw
two perpendicular lines Q E S, P E R to represent the directions of
the sun at the solstices and equinoxes, P corresponding to the vernal
equinox and R to the autumnal equinox, the apogee must lie at some
point A between P and Q. So much can be seen without any mathematics:
the actual calculation of the position of A and of the eccentricity is
a matter of some complexity. The angle P E A was found to be about 65°,
so that the sun would pass through its apogee about the beginning of
June; and the eccentricity was estimated at 1∕24.

The motion being thus represented geometrically, it became merely a
matter of not very difficult calculation to construct a table from
which the position of the sun for any day in the year could be easily
deduced. This was done by computing the so-called =equation of the
centre=, the angle C S E of fig. 17, which is the excess of the actual
longitude of the sun over the longitude which it would have had if
moving uniformly.

Owing to the imperfection of the observations used (Hipparchus
estimated that the times of the equinoxes and solstices could only be
relied upon to within about half a day), the actual results obtained
were not, according to modern ideas, very accurate, but the theory
represented the sun’s motion with an accuracy about as great as that of
the observations. It is worth noticing that with the same theory, but
with an improved value of the eccentricity, the motion of the sun can
be represented so accurately that the error never exceeds about 1′, a
quantity insensible to the naked eye.

The theory of Hipparchus represents the variations in the distance
of the sun with much less accuracy, and whereas in fact the angular
diameter of the sun varies by about 1∕30th part of itself, or by about
1′ in the course of the year, this variation according to Hipparchus
should be about twice as great. But this error would also have been
quite imperceptible with his instruments.

[Illustration: FIG. 19.—The epicycle and the deferent.]

Hipparchus saw that the motion of the sun could equally well be
represented by the other device suggested by Apollonius, the
=epicycle=. The body the motion of which is to be represented is
supposed to move uniformly round the circumference of one circle,
called the epicycle, the centre of which in turn moves on another
circle called the =deferent=. It is in fact evident that if a circle
equal to the eccentric, but with its centre at E (fig. 19), be taken as
the deferent, and if S′ be taken on this so that E S′ is parallel to C
S, then S′ S is parallel and equal to E C; and that therefore the sun
S, moving uniformly on the eccentric, may equally well be regarded as
lying on a circle of radius S′ S, the centre S′ of which moves on the
deferent. The two constructions lead in fact in this particular problem
to exactly the same result, and Hipparchus chose the eccentric as being
the simpler.

40. The motion of the moon being much more complicated than that of
the sun has always presented difficulties to astronomers,[23] and
Hipparchus required for it a more elaborate construction. Some further
description of the moon’s motion is, however, necessary before
discussing his theory.

We have already spoken (chapter I., § 16) of the lunar month as the
period during which the moon returns to the same position with respect
to the sun; more precisely this period (about 29-1∕2 days) is spoken
of as a =lunation= or =synodic month=: as, however, the sun moves
eastward on the celestial sphere like the moon but more slowly, the
moon returns to the same position with respect to the _stars_ in a
somewhat shorter time; this period (about 27 days 8 hours) is known as
the =sidereal month=. Again, the moon’s path on the celestial sphere is
slightly inclined to the ecliptic, and may be regarded approximately
as a great circle cutting the ecliptic in two =nodes=, at an angle
which Hipparchus was probably the first to fix definitely at about 5°.
Moreover, the moon’s path is always changing in such a way that, the
inclination to the ecliptic remaining nearly constant (but cf. chapter
V., § 111), the nodes move slowly backwards (from east to west) along
the ecliptic, performing a complete revolution in about 19 years.
It is therefore convenient to give a special name, the =draconitic
month=,[24] to the period (about 27 days 5 hours) during which the moon
returns to the same position with respect to the nodes.

Again, the motion of the moon, like that of the sun, is not uniform,
the variations being greater than in the case of the sun. Hipparchus
appears to have been the first to discover that the part of the moon’s
path in which the motion is most rapid is not always in the same
position on the celestial sphere, but moves continuously; or, in other
words, that the line of apses (§ 39) of the moon’s path moves. The
motion is an advance, and a complete circuit is described in about nine
years. Hence arises a fourth kind of month, the =anomalistic month=,
which is the period in which the moon returns to apogee or perigee.

To Hipparchus is due the credit of fixing with greater exactitude than
before the lengths of each of these months. In order to determine them
with accuracy he recognised the importance of comparing observations of
the moon taken at as great a distance of time as possible, and saw that
the most satisfactory results could be obtained by using Chaldaean and
other eclipse observations, which, as eclipses only take place near the
moon’s nodes, were simultaneous records of the position of the moon,
the nodes, and the sun.

To represent this complicated set of motions, Hipparchus used, as
in the case of the sun, an eccentric, the centre of which described
a circle round the earth in about nine years (corresponding to the
motion of the apses), the plane of the eccentric being inclined to the
ecliptic at an angle of 5°, and sliding back, so as to represent the
motion of the nodes already described.

The result cannot, however, have been as satisfactory as in the case of
the sun. The variation in the rate at which the moon moves is not only
greater than in the case of the sun, but follows a less simple law,
and cannot be adequately represented by means of a single eccentric;
so that though Hipparchus’ work would have represented the motion of
the moon in certain parts of her orbit with fair accuracy, there must
necessarily have been elsewhere discrepancies between the calculated
and observed places. There is some indication that Hipparchus was aware
of these, but was not able to reconstruct his theory so as to account
for them.

41. In the case of the planets Hipparchus found so small a supply of
satisfactory observations by his predecessors, that he made no attempt
to construct a system of epicycles or eccentrics to represent their
motion, but collected fresh observations for the use of his successors.
He also made use of these observations to determine with more accuracy
than before the average times of revolution of the several planets.

[Illustration: FIG. 20.—The eclipse method of connecting the distances
of the sun and moon.]

He also made a satisfactory estimate of the size and distance of the
moon, by an eclipse method, the leading idea of which was due to
Aristarchus (§ 32); by observing the angular diameter of the earth’s
shadow (Q R) at the distance of the moon at the time of an eclipse, and
comparing it with the known angular diameters of the sun and moon, he
obtained, by a simple calculation,[25] a relation between the
distances of the sun and moon, which gives either when the other is
known. Hipparchus knew that the sun was very much more distant than
the moon, and appears to have tried more than one distance, that of
Aristarchus among them, and the result obtained in each case shewed
that the distance of the moon was nearly 59 times the radius of the
earth. Combining the estimates of Hipparchus and Aristarchus, we
find the distance of the sun to be about 1,200 times the radius of
the earth—a number which remained substantially unchanged for many
centuries (chapter VIII., § 161).

42. The appearance in 134 B.C. of a new star in the Scorpion is said
to have suggested to Hipparchus the construction of a new catalogue of
the stars. He included 1,080 stars, and not only gave the (celestial)
latitude and longitude of each star, but divided them according to
their brightness into six magnitudes. The constellations to which he
refers are nearly identical with those of Eudoxus (§ 26), and the
list has undergone few alterations up to the present day, except for
the addition of a number of southern constellations, invisible in the
civilised countries of the ancient world. Hipparchus recorded also a
number of cases in which three or more stars appeared to be in line
with one another, or, more exactly, lay on the same great circle, his
object being to enable subsequent observers to detect more easily
possible changes in the positions of the stars. The catalogue remained,
with slight alterations, the standard one for nearly sixteen centuries
(cf. chapter III., § 63).

The construction of this catalogue led to a notable discovery, the best
known probably of all those which Hipparchus made. In comparing his
observations of certain stars with those of Timocharis and Aristyllus
(§ 33), made about a century and a half earlier, Hipparchus found that
their distances from the equinoctial points had changed. Thus, in the
case of the bright star Spica, the distance from the equinoctial points
(measured eastwards) had increased by about 2° in 150 years, or at
the rate of 48″ per annum. Further inquiry showed that, though the
roughness of the observations produced considerable variations in the
case of different stars, there was evidence of a general increase in
the longitude of the stars (measured from west to east), unaccompanied
by any change of latitude, the amount of the change being estimated
by Hipparchus as at least 36″ annually, and possibly more. The
agreement between the motions of different stars was enough to justify
him in concluding that the change could be accounted for, not as a
motion of individual stars, but rather as a change in the position
of the equinoctial points, from which longitudes were measured. Now
these points are the intersection of the equator and the ecliptic:
consequently one or another of these two circles must have changed.
But the fact that the latitudes of the stars had undergone no change
shewed that the ecliptic must have retained its position and that the
change had been caused by a motion of the equator. Again, Hipparchus
measured the obliquity of the ecliptic as several of his predecessors
had done, and the results indicated no appreciable change. Hipparchus
accordingly inferred that the equator was, as it were, slowly sliding
backwards (_i.e._ from east to west), keeping a constant inclination to
the ecliptic.

[Illustration: FIG. 21.—The increase of the longitude of a star.]

The argument may be made clearer by figures. In fig. 21 let ♈ M
denote the ecliptic, ♈ N the equator, S a star as seen by Timocharis,
S M a great circle drawn perpendicular to the ecliptic. Then S M is
the latitude, ♈ M the longitude. Let S′ denote the star as seen by
Hipparchus; then he found, that S′ M was equal to the former S M, but
that ♈ M′ was greater than the former ♈ M, or that M′ was slightly to
the east of M. This change M M′ being nearly the same for all stars,
it was simpler to attribute it to an equal motion in the opposite
direction of the point ♈, say from ♈ to ♈′ (fig. 22), _i.e._ by a
motion of the equator from ♈ N to ♈′ N′, its inclination N′ ♈′ M
remaining equal to its former amount N ♈ M. The general effect of this
change is shewn in a different way in fig. 23, where ♈ ♈′ ♎ ♎′ being
the ecliptic, A B C D represents the equator as it appeared in the
time of Timocharis, A′ B′ C′ D′ (printed in red) the same in the time
of Hipparchus, ♈, ♎ being the earlier positions of the two equinoctial
points, and ♈′, ♎′ the later positions.

[Illustration: FIG. 22.—The movement of the equator.]

[Illustration: FIG. 33.—The precession of the equinoxes.]

The annual motion ♈ ♈′ was, as has been stated, estimated by Hipparchus
as being at least 36″ (equivalent to one degree in a century), and
probably more. Its true value is considerably more, namely about 50″.

[Illustration: FIG. 24.—The precession of the equinoxes.]

An important consequence of the motion of the equator thus discovered
is that the sun in its annual journey round the ecliptic, after
starting from the equinoctial point, returns to the new position of the
equinoctial point a little before returning to its original position
with respect to the stars, and the successive equinoxes occur slightly
earlier than they otherwise would. From this fact is derived the name
=precession of the equinoxes=, or more shortly, =precession=, which is
applied to the motion that we have been considering. Hence it becomes
necessary to recognise, as Hipparchus did, two different kinds of year,
the =tropical year= or period required by the sun to return to the same
position with respect to the equinoctial points, and the =sidereal
year= or period of return to the same position with respect to the
stars. If ♈ ♈′ denote the motion of the equinoctial point during a
tropical year, then the sun after starting from the equinoctial point
at ♈ arrives—at the end of a tropical year—at the new equinoctial point
at ♈′; but the sidereal year is only complete when the sun has further
described the arc ♈′ ♈ and returned to its original starting-point ♈.
Hence, taking the modern estimate 50″ of the arc ♈ ♈′, the sun, in
the sidereal year, describes an arc of 360°, in the tropical year an
arc less by 50″, or 359° 59′ 10″; the lengths of the two years are
therefore in this proportion, and the amount by which the sidereal year
exceeds the tropical year bears to either the same ratio as 50″ to
360° (or 1,296,000″), and is therefore (365-1∕4 × 50)∕1296000 days of
about 20 minutes.

Another way of expressing the amount of the precession is to say
that the equinoctial point will describe the complete circuit of the
ecliptic and return to the same position after about 26,000 years.

The length of each kind of year was also fixed by Hipparchus with
considerable accuracy. That of the tropical year was obtained by
comparing the times of solstices and equinoxes observed by earlier
astronomers with those observed by himself. He found, for example, by
comparison of the date of the summer solstice of 280 B.C., observed
by Aristarchus of Samos, with that of the year 135 B.C., that the
current estimate of 365-1∕4 days for the length of the year had to
be diminished by 1∕300th of a day or about five minutes, an estimate
confirmed roughly by other cases. It is interesting to note as an
illustration of his scientific method that he discusses with some
care the possible error of the observations, and concludes that the
time of a solstice may be erroneous to the extent of about 3∕4 day,
while that of an equinox may be expected to be within 1∕4 day of
the truth. In the illustration given, this would indicate a possible
error of 1-1∕2 days in a period of 145 years, or about 15 minutes in
a year. Actually his estimate of the length of the year is about six
minutes too great, and the error is thus much less than that which he
indicated as possible. In the course of this work he considered also
the possibility of a change in the length of the year, and arrived at
the conclusion that, although his observations were not precise enough
to show definitely the invariability of the year, there was no evidence
to suppose that it had changed.

The length of the tropical year being thus evaluated at 365 days 5
hours 55 minutes, and the difference between the two kinds of year
being given by the observations of precession, the sidereal year was
ascertained to exceed 365-1∕4 days by about 10 minutes, a result
agreeing almost exactly with modern estimates. That the addition of two
erroneous quantities, the length of the tropical year and the amount
of the precession, gave such an accurate result was not, as at first
sight appears, a mere accident. The chief source of error in each
case being the erroneous times of the several equinoxes and solstices
employed, the errors in them would tend to produce errors of opposite
kinds in the tropical year and in precession, so that they would
in part compensate one another. This estimate of the length of the
sidereal year was probably also to some extent verified by Hipparchus
by comparing eclipse observations made at different epochs.

43. The great improvements which Hipparchus effected in the theories of
the sun and moon naturally enabled him to deal more successfully than
any of his predecessors with a problem which in all ages has been of
the greatest interest, the prediction of eclipses of the sun and moon.

That eclipses of the moon were caused by the passage of the moon
through the shadow of the earth thrown by the sun, or, in other words,
by the interposition of the earth between the sun and moon, and
eclipses of the sun by the passage of the moon between the sun and the
observer, was perfectly well known to Greek astronomers in the time of
Aristotle (§ 29), and probably much earlier (chapter I., § 17), though
the knowledge was probably confined to comparatively few people and
superstitious terrors were long associated with eclipses.

The chief difficulty in dealing with eclipses depends on the fact that
the moon’s path does not coincide with the ecliptic. If the moon’s path
on the celestial sphere were identical with the ecliptic, then, once
every month, at new moon, the moon (M) would pass exactly between the
earth and the sun, and the latter would be eclipsed, and once every
month also, at full moon, the moon (M′) would be in the opposite
direction to the sun as seen from the earth, and would consequently be
obscured by the shadow of the earth.

[Illustration: FIG. 25.—The earth’s shadow.]

[Illustration: FIG. 26.—The ecliptic and the moon’s path.]

As, however, the moon’s path is inclined to the ecliptic (§ 40), the
latitudes of the sun and moon may differ by as much as 5°, either when
they are in =conjunction=, _i.e._ when they have the same longitudes,
or when they are in =opposition=, _i.e._ when their longitudes differ
by 180°, and they will then in either case be too far apart for an
eclipse to occur. Whether then at any full or new moon an eclipse will
occur or not, will depend primarily on the latitude of the moon at the
time, and hence upon her position with respect to the nodes of her
orbit (§ 40). If conjunction takes place when the sun and moon happen
to be near one of the nodes (N), as at S M in fig. 26, the sun and moon
will be so close together that an eclipse will occur; but if it occurs
at a considerable distance from a node, as at S′ M′, their centres are
so far apart that no eclipse takes place.

Now the apparent diameter of either sun or moon is, as we have seen (§
32), about 1∕2°; consequently when their discs just touch, as in fig.
27, the distance between their centres is also about 1∕2°. If then
at conjunction the distance between their centres is less than this
amount, an eclipse of the sun will take place; if not, there will be
no eclipse. It is an easy calculation to determine (in fig. 26) the
length of the side N S or N M of the triangle N M S, when S M has this
value, and hence to determine the greatest distance from the node at
which conjunction can take place if an eclipse is to occur. An eclipse
of the moon can be treated in the same way, except that we there have
to deal with the moon and the shadow of the earth at the distance of
the moon. The apparent size of the shadow is, however, considerably
greater than the apparent size of the moon, and an eclipse of the moon
takes place if the distance between the centre of the moon and the
centre of the shadow is less than about 1°. As before, it is easy to
compute the distance of the moon or of the centre of the shadow from
the node when opposition occurs, if an eclipse just takes place. As,
however, the apparent sizes of both sun and moon, and consequently also
that of the earth’s shadow, vary according to the distances of the sun
and moon, a variation of which Hipparchus had no accurate knowledge,
the calculation becomes really a good deal more complicated than at
first sight appears, and was only dealt with imperfectly by him.

[Illustration: FIG. 27.—The sun and moon.]

[Illustration: FIG. 28.—Partial eclipse of the moon.]

[Illustration: FIG. 29.—Total eclipse of the moon.]

Eclipses of the moon are divided into =partial= or =total=, the former
occurring when the moon and the earth’s shadow only overlap partially
(as in fig. 28), the latter when the moon’s disc is completely
immersed in the shadow (fig. 29). In the same way an eclipse of the sun
may be partial or total; but as the sun’s disc may be at times slightly
larger than that of the moon, it sometimes happens also that the whole
disc of the sun is hidden by the moon, except a narrow ring round the
edge (as in fig. 30): such an eclipse is called =annular=. As the
earth’s shadow at the distance of the moon is always larger than the
moon’s disc, annular eclipses of the moon cannot occur.

[Illustration: FIG. 30.—Annular eclipse of the sun.]

Thus eclipses take place if, and only if, the distance of the moon from
a node at the time of conjunction or opposition lies within certain
limits approximately known; and the problem of predicting eclipses
could be roughly solved by such knowledge of the motion of the moon
and of the nodes as Hipparchus possessed. Moreover, the length of
the synodic and draconitic months (§ 40) being once ascertained, it
became merely a matter of arithmetic to compute one or more periods
after which eclipses would recur nearly in the same manner. For if any
period of time contains an exact number of each kind of month, and if
at any time an eclipse occurs, then after the lapse of the period,
conjunction (or opposition) again takes place, and the moon is at the
same distance as before from the node and the eclipse recurs very much
as before. The saros, for example (chapter I., § 17), contained very
nearly 223 synodic or 242 draconitic months, differing from either by
less than an hour. Hipparchus saw that this period was not completely
reliable as a means of predicting eclipses, and showed how to allow for
the irregularities in the moon’s and sun’s motion (§§ 39, 40) which
were ignored by it, but was unable to deal fully with the difficulties
arising from the variations in the apparent diameters of the sun or
moon.

An important complication, however, arises in the case of eclipses
of the sun, which had been noticed by earlier writers, but which
Hipparchus was the first to deal with. Since an eclipse of the moon is
an actual darkening of the moon, it is visible to anybody, wherever
situated, who can see the moon at all; for example, to possible
inhabitants of other planets, just as we on the earth can see precisely
similar eclipses of Jupiter’s moons. An eclipse of the sun is, however,
merely the screening off of the sun’s light from a particular observer,
and the sun may therefore be eclipsed to one observer while to another
elsewhere it is visible as usual. Hence in computing an eclipse of the
sun it is necessary to take into account the position of the observer
on the earth. The simplest way of doing this is to make allowance for
the difference of direction of the moon as seen by an observer at the
place in question, and by an observer in some standard position on the
earth, preferably an ideal observer at the centre of the earth. If, in
fig. 31, M denote the moon, C the centre of the earth, A a point on the
earth between C and M (at which therefore the moon is overhead), and B
any other point on the earth, then observers at C (or A) and B see the
moon in slightly different directions, C M, B M, the difference between
which is an angle known as the =parallax=, which is equal to the angle
B M C and depends on the distance of the moon, the size of the earth,
and the position of the observer at B. In the case of the sun, owing to
its great distance, even as estimated by the Greeks, the parallax was
in all cases too small to be taken into account, but in the case of the
moon the parallax might be as much as 1° and could not be neglected.

[Illustration: FIG. 31.—Parallax.]

If then the path of the moon, as seen from the centre of the earth,
were known, then the path of the moon as seen from any particular
station on the earth could be deduced by allowing for parallax, and the
conditions of an eclipse of the sun visible there could be computed
accordingly.

From the time of Hipparchus onwards lunar eclipses could easily be
predicted to within an hour or two by any ordinary astronomer; solar
eclipses probably with less accuracy; and in both cases the prediction
of the extent of the eclipse, _i.e._ of what portion of the sun or moon
would be obscured, probably left very much to be desired.

44. The great services rendered to astronomy by Hipparchus can hardly
be better expressed than in the words of the great French historian of
astronomy, Delambre, who is in general no lenient critic of the work of
his predecessors:—

 “When we consider all that Hipparchus invented or perfected, and
 reflect upon the number of his works and the mass of calculations
 which they imply, we must regard him as one of the most astonishing
 men of antiquity, and as the greatest of all in the sciences which are
 not purely speculative, and which require a combination of geometrical
 knowledge with a knowledge of phenomena, to be observed only by
 diligent attention and refined instruments.”[26]

45. For nearly three centuries after the death of Hipparchus, the
history of astronomy is almost a blank. Several textbooks written
during this period are extant, shewing the gradual popularisation of
his great discoveries. Among the few things of interest in these books
may be noticed a statement that the stars are not necessarily on the
surface of a sphere, but may be at different distances from us, which,
however, there are no means of estimating; a conjecture that the sun
and stars are so far off that the earth would be a mere point seen
from the sun and invisible from the stars; and a re-statement of an
old opinion traditionally attributed to the Egyptians (whether of the
Alexandrine period or earlier is uncertain), that Venus and Mercury
revolve round the sun. It seems also that in this period some attempts
were made to explain the planetary motions by means of epicycles, but
whether these attempts marked any advance on what had been done by
Apollonius and Hipparchus is uncertain.

It is interesting also to find in _Pliny_ (A.D. 23-79) the well-known
modern argument for the spherical form of the earth, that when a
ship sails away the masts, etc., remain visible after the hull has
disappeared from view.

A new measurement of the circumference of the earth by _Posidonius_
(born about the end of Hipparchus’s life) may also be noticed; he
adopted a method similar to that of Eratosthenes (§ 36), and arrived at
two different results. The later estimate, to which he seems to have
attached most weight, was 180,000 stadia, a result which was about as
much below the truth as that of Eratosthenes was above it.

46. The last great name in Greek astronomy is that of Claudius
Ptolemaeus, commonly known as _Ptolemy_, of whose life nothing is known
except that he lived in Alexandria about the middle of the 2nd century
A.D. His reputation rests chiefly on his great astronomical treatise,
known as the _Almagest_,[27] which is the source from which by far
the greater part of our knowledge of Greek astronomy is derived, and
which may be fairly regarded as the astronomical Bible of the Middle
Ages. Several other minor astronomical and astrological treatises are
attributed to him, some of which are probably not genuine, and he was
also the author of an important work on geography, and possibly of a
treatise on _Optics_, which is, however, not certainly authentic and
maybe of Arabian origin. The _Optics_ discusses, among other topics,
the =refraction= or bending of light, by the atmosphere on the earth:
it is pointed out that the light of a star or other heavenly body S,
on entering our atmosphere (at A) and on penetrating to the lower
and denser portions of it, must be gradually bent or =refracted=,
the result being that the star appears to the observer at B nearer
to the zenith Z than it actually is, _i.e._ the light appears to
come from S′ instead of from S; it is shewn further that this effect
must be greater for bodies near the horizon than for those near the
zenith, the light from the former travelling through a greater extent
of atmosphere; and these results are shewn to account for certain
observed deviations in the daily paths of the stars, by which they
appear unduly raised up when near the horizon. Refraction also explains
the well-known flattened appearance of the sun or moon when rising or
setting, the lower edge being raised by refraction more than the upper,
so that a contraction of the vertical diameter results, the horizontal
contraction being much less.[28]

[Illustration: FIG. 32.—Refraction by the atmosphere.]

47. The _Almagest_ is avowedly based largely on the work of earlier
astronomers, and in particular on that of Hipparchus, for whom Ptolemy
continually expresses the greatest admiration and respect. Many of its
contents have therefore already been dealt with by anticipation, and
need not be discussed again in detail. The book plays, however, such
an important part in astronomical history, that it may be worth while
to give a short outline of its contents, in addition to dealing more
fully with the parts in which Ptolemy made important advances.

The _Almagest_ consists altogether of 13 books. The first two deal with
the simpler observed facts, such as the daily motion of the celestial
sphere, and the general motions of the sun, moon, and planets, and
also with a number of topics connected with the celestial sphere and
its motion, such as the length of the day and the times of rising and
setting of the stars in different zones of the earth; there are also
given the solutions of some important mathematical problems,[29] and
a mathematical table[30] of considerable accuracy and extent. But the
most interesting parts of these introductory books deal with what may
be called the postulates of Ptolemy’s astronomy (Book I., chap. ii.).
The first of these is that the earth is spherical; Ptolemy discusses
and rejects various alternative views, and gives several of the usual
positive arguments for a spherical form, omitting, however, one of the
strongest, the eclipse argument found in Aristotle (§ 29), possibly as
being too recondite and difficult, and adding the argument based on the
increase in the area of the earth visible when the observer ascends to
a height. In his geography he accepts the estimate given by Posidonius
that the circumference of the earth is 180,000 stadia. The other
postulates which he enunciates and for which he argues are, that the
heavens are spherical and revolve like a sphere; that the earth is in
the centre of the heavens, and is merely a point in comparison with the
distance of the fixed stars, and that it has no motion. The position
of these postulates in the treatise and Ptolemy’s general method of
procedure suggest that he was treating them, not so much as important
results to be established by the best possible evidence, but rather as
assumptions, more probable than any others with which the author was
acquainted, on which to base mathematical calculations which should
explain observed phenomena.[31] His attitude is thus essentially
different from that either of the early Greeks, such as Pythagoras, or
of the controversialists of the 16th and early 17th centuries, such
as Galilei (chapter VI.), for whom the truth or falsity of postulates
analogous to those of Ptolemy was of the very essence of astronomy and
was among the final objects of inquiry. The arguments which Ptolemy
produces in support of his postulates, arguments which were probably
the commonplaces of the astronomical writing of his time, appear to us,
except in the case of the shape of the earth, loose and of no great
value. The other postulates were, in fact, scarcely, capable of either
proof or disproof with the evidence which Ptolemy had at command. His
argument in favour of the immobility of the earth is interesting, as
it shews his clear perception that the more obvious appearances can be
explained equally well by a motion of the stars or by a motion of the
earth; he concludes, however, that it is easier to attribute motion to
bodies like the stars which seem to be of the nature of fire than to
the solid earth, and points out also the difficulty of conceiving the
earth to have a rapid motion of which we are entirely unconscious. He
does not, however, discuss seriously the possibility that the earth or
even Venus and Mercury may revolve round the sun.

The third book of the _Almagest_ deals with the length of the year
and theory of the sun, but adds nothing of importance to the work of
Hipparchus.

48. The fourth book of the _Almagest_, which treats of the length of
the month and of the theory of the moon, contains one of Ptolemy’s most
important discoveries. We have seen that, apart from the motion of
the moon’s orbit as a whole, and the revolution of the line of apses,
the chief irregularity or inequality was the so-called equation of
the centre (§§ 39, 40), represented fairly accurately by means of an
eccentric, and depending only on the position of the moon with respect
to its apogee. Ptolemy, however, discovered, what Hipparchus only
suspected, that there was a further inequality in the moon’s motion—to
which the name =evection= was afterwards given—and that this depended
partly on its position with respect to the sun. Ptolemy compared the
observed positions of the moon with those calculated by Hipparchus
in various positions relative to the sun and apogee, and found that,
although there was a satisfactory agreement at new and full moon, there
was a considerable error when the moon was half-full, provided it was
also not very near perigee or apogee. Hipparchus based his theory of
the moon chiefly on observations of eclipses, _i.e._ on observations
taken necessarily at full or new moon (§ 43), and Ptolemy’s discovery
is due to the fact that he checked Hipparchus’s theory by observations
taken at other times. To represent this new inequality, it was found
necessary to use an epicycle and a deferent, the latter being itself a
moving eccentric circle, the centre of which revolved round the earth.
To account, to some extent, for certain remaining discrepancies between
theory and observation, which occurred neither at new and full moon,
nor at the =quadratures= (half-moon), Ptolemy introduced further a
certain small to-and-fro oscillation of the epicycle, an oscillation to
which he gave the name of =prosneusis=.[32] Ptolemy thus succeeded in
fitting his theory on to his observations so well that the error seldom
exceeded 10′, a small quantity in the astronomy of the time, and on the
basis of this construction he calculated tables from which the position
of the moon at any required time could be easily deduced.

One of the inherent weaknesses of the system of epicycles occurred
in this theory in an aggravated form. It has already been noticed in
connection with the theory of the sun (§ 39), that the eccentric or
epicycle produced an erroneous variation in the distance of the sun,
which was, however, imperceptible in Greek times. Ptolemy’s system,
however, represented the moon as being sometimes nearly twice as far
off as at others, and consequently the apparent diameter ought at some
times to have been not much more than half as great as at others—a
conclusion obviously inconsistent with observation. It seems probable
that Ptolemy noticed this difficulty, but was unable to deal with it;
it is at any rate a significant fact that when he is dealing with
eclipses, for which the apparent diameters of the sun and moon are of
importance, he entirely rejects the estimates that might have been
obtained from his lunar theory and appeals to direct observation (cf.
also § 51, note).

49. The fifth book of the _Almagest_ contains an account of the
construction and use of Ptolemy’s chief astronomical instrument, a
combination of graduated circles known as the =astrolabe=.[33]

Then follows a detailed discussion of the moon’s parallax (§ 43), and
of the distances of the sun and moon. Ptolemy obtains the distance of
the moon by a parallax method which is substantially identical with
that still in use. If we know the direction of the line C M (fig. 33)
joining the centres of the earth and moon, or the direction of the moon
as seen by an observer at A; and also the direction of the line B M,
that is the direction of the moon as seen by an observer at B, then the
angles of the triangle C B M are known, and the ratio of the sides C B,
C M is known. Ptolemy obtained the two directions required by means
of observations of the moon, and hence found that C M was 59 times C
B, or that the distance of the moon was equal to 59 times the radius
of the earth. He then uses Hipparchus’s eclipse method to deduce the
distance of the sun from that of the moon thus ascertained, and finds
the distance of the sun to be 1,210 times the radius of the earth. This
number, which is substantially the same as that obtained by Hipparchus
(§ 41), is, however, only about 1∕20 of the true number, as indicated
by modern work (chapter XIII., § 284).

[Illustration: FIG. 33.—Parallax.]

The sixth book is devoted to eclipses, and contains no substantial
additions to the work of Hipparchus.

50. The seventh and eighth books contain a catalogue of stars, and
a discussion of precession (§ 42). The catalogue, which contains
1,028 stars (three of which are duplicates), appears to be nearly
identical with that of Hipparchus, It contains none of the stars
which were visible to Ptolemy at Alexandria, but not to Hipparchus at
Rhodes. Moreover, Ptolemy professes to deduce from a comparison of
his observations with those of Hipparchus and others the (erroneous)
value 36″ for the precession, which Hipparchus had given as the least
possible value, and which Ptolemy regards as his final estimate. But
an examination of the positions assigned to the stars in Ptolemy’s
catalogue agrees better with their actual positions in the time of
Hipparchus, _corrected for precession at the supposed rate of 36″
annually_, than with their actual positions in Ptolemy’s time. It is
therefore probable that the catalogue as a whole does not represent
genuine observations made by Ptolemy, but is substantially the
catalogue of Hipparchus corrected for precession and only occasionally
modified by new observations by Ptolemy or others.

51. The last five books deal with the theory of the planets, the
most important of Ptolemy’s original contributions to astronomy. The
problem of giving a satisfactory explanation of the motions of the
planets was, on account of their far greater irregularity, a much more
difficult one than the corresponding problem for the sun or moon. The
motions of the latter are so nearly uniform that their irregularities
may usually be regarded as of the nature of small corrections, and
for many purposes may be ignored. The planets, however, as we have
seen (chapter I., § 14), do not even always move from west to east,
but stop at intervals, move in the reverse direction for a time, stop
again, and then move again in the original direction. It was probably
recognised in early times, at latest by Eudoxus (§ 26), that in the
case of three of the planets, Mars, Jupiter, and Saturn, these motions
could be represented roughly by supposing each planet to oscillate to
and fro on each side of a fictitious planet, moving uniformly round the
celestial sphere in or near the ecliptic, and that Venus and Mercury
could similarly be regarded as oscillating to and fro on each side of
the sun. These rough motions could easily be interpreted by means of
revolving spheres or of epicycles, as was done by Eudoxus and probably
again with more precision by Apollonius. In the case of Jupiter,
for example, we may regard the planet as moving on an epicycle, the
centre of which, _j_, describes uniformly a deferent, the centre of
which is the earth. The planet will then as seen from the earth appear
alternately to the east (as at J_{1}) and to the west (as at J_{2}) of
the fictitious planet _j_; and the extent of the oscillation on each
side, and the interval between successive appearances in the extreme
positions (J_{1}, J_{2}) on either side, can be made right by choosing
appropriately the size and rapidity of motion of the epicycle. It is
moreover evident that with this arrangement the apparent motion of
Jupiter will vary considerably, as the two motions—that on the epicycle
and that of the centre of the epicycle on the deferent—are sometimes
in the same direction, so as to increase one another’s effect, and at
other times in opposite directions. Thus, when Jupiter is most distant
from the earth, that is at J_{3}, the motion is most rapid, at J_{1}
and J_{2} the motion as seen from the earth is nearly the same as that
of _j_; while at J_{4} the two motions are in opposite directions,
and the size and motion of the epicycle having been chosen in the way
indicated above, it is found in fact that the motion of the planet in
the epicycle is the greater of the two motions, and that therefore the
planet when in this position appears to be moving from east to west
(from left to right in the figure), as is actually the case. As then
at J_{1} and J_{2} the planet appears to be moving from west to east,
and at J_{4} in the opposite direction, and sudden changes of motion
do not occur in astronomy, there must be a position between J_{1} and
J_{4}, and another between J_{4} and J_{2}, at which the planet is
just reversing its direction of motion, and therefore appears for the
instant at rest. We thus arrive at an explanation of the stationary
points (chapter I., § 14). An exactly similar scheme explains roughly
the motion of Mercury and Venus, except that the centre of the epicycle
must always be in the direction of the sun.

[Illustration: FIG. 34.—Jupiter’s epicycle and deferent.]

Hipparchus, as we have seen (§ 41), found the current representations
of the planetary motions inaccurate, and collected a number of fresh
observations. These, with fresh observations of his own, Ptolemy now
employed in order to construct an improved planetary system.

As in the case of the moon, he used as deferent an eccentric circle
(centre C), but instead of making the centre _j_ of the epicycle move
uniformly in the deferent, he introduced a new point called an =equant=
(E′), situated at the same distance from the centre of the deferent as
the earth but on the opposite side, and regulated the motion of _j_ by
the condition that the apparent motion _as seen from the equant_ should
be uniform; in other words, the angle A E′ _j_ was made to increase
uniformly. In the case of Mercury (the motions of which have been found
troublesome by astronomers of all periods), the relation of the equant
to the centre of the epicycle was different, and the latter was made to
move in a small circle. The deviations of the planets from the ecliptic
(chapter I., §§ 13, 14) were accounted for by tilting up the planes of
the several deferents and epicycles so that they were inclined to the
ecliptic at various small angles.

[Illustration: FIG. 35.—The equant.]

By means of a system of this kind, worked out with great care, and
evidently at the cost of enormous labour, Ptolemy was able to represent
with very fair exactitude the motions of the planets, as given by the
observations in his possession.

It has been pointed out by modern critics, as well as by some mediaeval
writers, that the use of the equant (which played also a small part in
Ptolemy’s lunar theory) was a violation of the principle of employing
only uniform circular motions, on which the systems of Hipparchus and
Ptolemy were supposed to be based, and that Ptolemy himself appeared
unconscious of his inconsistency. It may, however, fairly be doubted
whether Hipparchus or Ptolemy ever had an abstract belief in the
exclusive virtue of such motions, except as a convenient and easily
intelligible way of representing certain more complicated motions,
and it is difficult to conceive that Hipparchus would have scrupled
any more than his great follower, in using an equant to represent
an irregular motion, if he had found that the motion was thereby
represented with accuracy. The criticism appears to me in fact to be
an anachronism. The earlier Greeks, whose astronomy was speculative
rather than scientific, and again many astronomers of the Middle Ages,
felt that it was on _a priori_ grounds necessary to represent the
“perfection” of the heavenly motions by the most “perfect” or regular
of geometrical schemes; so that it is highly probable that Pythagoras
or Plato, or even Aristotle, would have objected, and certain that the
astronomers of the 14th and 15th centuries ought to have objected (as
some of them actually did), to this innovation of Ptolemy’s. But there
seems no good reason for attributing this _a priori_ attitude to the
later scientific Greek astronomers (cf. also §§ 38, 47).[34]

It will be noticed that nothing has been said as to the actual
distances of the planets, and in fact the apparent motions are
unaffected by any alteration in the scale on which deferent
and epicycle are constructed, provided that both are altered
proportionally. Ptolemy expressly states that he had no means of
estimating numerically the distances of the planets, or even of knowing
the order of the distance of the several planets. He followed tradition
in accepting conjecturally rapidity of motion as a test of nearness,
and placed Mars, Jupiter, Saturn (which perform the circuit of the
celestial sphere in about 2, 12, and 29 years respectively) beyond the
sun in that order. As Venus and Mercury accompany the sun, and may
therefore be regarded as on the average performing their revolutions in
a year, the test to some extent failed in their case, but Ptolemy again
accepted the opinion of the “ancient mathematicians” (_i.e._ probably
the Chaldaeans) that Mercury and Venus lie between the sun and moon,
Mercury being the nearer to us. (Cf. chapter I., § 15.)

52. There has been much difference of opinion among astronomers as to
the merits of Ptolemy. Throughout the Middle Ages his authority was
regarded as almost final on astronomical matters, except where it was
outweighed by the even greater authority assigned to Aristotle. Modern
criticism has made clear, a fact which indeed he never conceals, that
his work is to a large extent based on that of Hipparchus; and that
his observations, if not actually fictitious, were at any rate in most
cases poor. On the other hand his work shews clearly that he was an
accomplished and original mathematician.[35] The most important of his
positive contributions to astronomy were the discovery of evection
and his planetary theory, but we ought probably to rank above these,
important as they are, the services which he rendered by preserving
and developing the great ideas of Hipparchus—ideas which the other
astronomers of the time were probably incapable of appreciating, and
which might easily have been lost to us if they had not been embodied
in the _Almagest_.

53. The history of Greek astronomy practically ceases with Ptolemy.
The practice of observation died out so completely that only eight
observations are known to have been made during the eight and a half
centuries which separate him from Albategnius (chapter III., § 59). The
only Greek writers after Ptolemy’s time are compilers and commentators,
such as _Theon_ (_fl._ A.D. 365), to none of whom original ideas of
any importance can be attributed. The murder of his daughter Hypatia
(A.D. 415), herself also a writer on astronomy, marks an epoch in the
decay of the Alexandrine school; and the end came in A.D. 640, when
Alexandria was captured by the Arabs.[36]

54. It remains to attempt to estimate briefly the value of the
contributions to astronomy made by the Greeks and of their method of
investigation. It is obviously unreasonable to expect to find a brief
formula which will characterise the scientific attitude of a series
of astronomers whose lives extend over a period of eight centuries;
and it is futile to explain the inferiority of Greek astronomy to our
own on some such ground as that they had not discovered the method of
induction, that they were not careful enough to obtain facts, or even
that their ideas were not clear. In habits of thought and scientific
aims the contrast between Pythagoras and Hipparchus is probably greater
than that between Hipparchus on the one hand and Coppernicus or even
Newton on the other, while it is not unfair to say that the fanciful
ideas which pervade the work of even so great a discoverer as Kepler
(chapter VII., §§ 144, 151) place his scientific method in some
respects behind that of his great Greek predecessor.

The Greeks inherited from their predecessors a number of observations,
many of them executed with considerable accuracy, which were nearly
sufficient for the requirements of practical life, but in the matter of
astronomical theory and speculation, in which their best thinkers were
very much more interested than in the detailed facts, they received
virtually a blank sheet on which they had to write (at first with
indifferent success) their speculative ideas. A considerable interval
of time was obviously necessary to bridge over the gulf separating such
data as the eclipse observations of the Chaldaeans from such ideas as
the harmonical spheres of Pythagoras; and the necessary theoretical
structure could not be erected without the use of mathematical methods
which had gradually to be invented. That the Greeks, particularly
in early times, paid little attention to making observations, is
true enough, but it may fairly be doubted whether the collection of
fresh material for observations would really have carried astronomy
much beyond the point reached by the Chaldaean observers. When once
speculative ideas, made definite by the aid of geometry, had been
sufficiently developed to be capable of comparison with observation,
rapid progress was made. The Greek astronomers of the scientific
period, such as Aristarchus, Eratosthenes, and above all Hipparchus,
appear moreover to have followed in their researches the method
which has always been fruitful in physical science—namely, to frame
provisional hypotheses, to deduce their mathematical consequences, and
to compare these with the results of observation. There are few better
illustrations of genuine scientific caution than the way in which
Hipparchus, having tested the planetary theories handed down to him
and having discovered their insufficiency, deliberately abstained from
building up a new theory on data which he knew to be insufficient, and
patiently collected fresh material, never to be used by himself, that
some future astronomer might thereby be able to arrive at an improved
theory.

Of positive additions to our astronomical knowledge made by the Greeks
the most striking in some ways is the discovery of the approximately
spherical form of the earth, a result which later work has only
slightly modified. But their explanation of the chief motions of
the solar system and their resolution of them into a comparatively
small number of simpler motions was, in reality, a far more important
contribution, though the Greek epicyclic scheme has been so remodelled,
that at first sight it is difficult to recognise the relation between
it and our modern views. The subsequent history will, however, show
how completely each stage in the progress of astronomical science has
depended on those that preceded.

When we study the great conflict in the time of Coppernicus between the
ancient and modern ideas, our sympathies naturally go out towards those
who supported the latter, which are now known to be more accurate,
and we are apt to forget that those who then spoke in the name of
the ancient astronomy and quoted Ptolemy were indeed believers in
the doctrines which they had derived from the Greeks, but that their
methods of thought, their frequent refusal to face facts, and their
appeals to authority, were all entirely foreign to the spirit of the
great men whose disciples they believed themselves to be.



CHAPTER III.

THE MIDDLE AGES.

    “The lamp burns low, and through the casement bars
    Grey morning glimmers feebly.”
                                              BROWNING’S _Paracelsus_.


55. About fourteen centuries elapsed between the publication of the
_Almagest_ and the death of Coppernicus (1543), a date which is in
astronomy a convenient landmark on the boundary between the Middle Ages
and the modern world. In this period, nearly twice as long as that
which separated Thales from Ptolemy, almost four times as long as that
which has now elapsed since the death of Coppernicus, no astronomical
discovery of first-rate importance was made. There were some important
advances in mathematics, and the art of observation was improved; but
theoretical astronomy made scarcely any progress, and in some respects
even went backward, the current doctrines, if in some points slightly
more correct than those of Ptolemy, being less intelligently held.

In the Western World we have already seen that there was little to
record for nearly five centuries after Ptolemy. After that time ensued
an almost total blank, and several more centuries elapsed before there
was any appreciable revival of the interest once felt in astronomy.

56. Meanwhile a remarkable development of science had taken place in
the East during the 7th century. The descendants of the wild Arabs
who had carried the banner of Mahomet over so large a part of the
Roman empire, as well as over lands lying farther east, soon began to
feel the influence of the civilisation of the peoples whom they had
subjugated, and Bagdad, which in the 8th century became the capital of
the Caliphs, rapidly developed into a centre of literary and scientific
activity. Al Mansur, who reigned from A.D. 754 to 775, was noted as a
patron of science, and collected round him learned men both from India
and the West. In particular we are told of the arrival at his court
in 772 of a scholar from India bearing with him an Indian treatise on
astronomy,[37] which was translated into Arabic by order of the Caliph,
and remained the standard treatise for nearly half a century. From Al
Mansur’s time onwards a body of scholars, in the first instance chiefly
Syrian Christians, were at work at the court of the Caliphs translating
Greek writings, often through the medium of Syriac, into Arabic. The
first translations made were of the medical treatises of Hippocrates
and Galen; the Aristotelian ideas contained in the latter appear to
have stimulated interest in the writings of Aristotle himself, and thus
to have enlarged the range of subjects regarded as worthy of study.
Astronomy soon followed medicine, and became the favourite science of
the Arabians, partly no doubt out of genuine scientific interest, but
probably still more for the sake of its practical applications. Certain
Mahometan ceremonial observances required a knowledge of the direction
of Mecca, and though many worshippers, living anywhere between the
Indus and the Straits of Gibraltar, must have satisfied themselves
with rough-and-ready solutions of this problem, the assistance which
astronomy could give in fixing the true direction was welcome in
larger centres of population. The Mahometan calendar, a lunar one,
also required some attention in order that fasts and feasts should
be kept at the proper times. Moreover the belief in the possibility
of predicting the future by means of the stars, which had flourished
among the Chaldaeans (chapter I., § 18), but which remained to a great
extent in abeyance among the Greeks, now revived rapidly on a congenial
oriental soil, and the Caliphs were probably quite as much interested
in seeing that the learned men of their courts were proficient in
astrology as in astronomy proper.

The first translation of the _Almagest_ was made by order of Al
Mansur’s successor Harun al Rasid (A.D. 765 or 766-A.D. 809), the
hero of the _Arabian Nights_. It seems, however, to have been found
difficult to translate; fresh attempts were made by _Honein ben
Ishak_ (?-873) and by his son _Ishak ben Honein_ (?-910 or 911), and
a final version by _Tabit ben Korra_ (836-901) appeared towards the
end of the 9th century. Ishak ben Honein translated also a number
of other astronomical and mathematical books, so that by the end of
the 9th century, after which translations almost ceased, most of the
more important Greek books on these subjects, as well as many minor
treatises, had been translated. To this activity we owe our knowledge
of several books of which the Greek originals have perished.

57. During the period in which the Caliphs lived at Damascus an
observatory was erected there, and another on a more magnificent scale
was built at Bagdad in 829 by the Caliph Al Mamun. The instruments
used were superior both in size and in workmanship to those of the
Greeks, though substantially of the same type. The Arab astronomers
introduced moreover the excellent practice of making regular and as
far as possible nearly continuous observations of the chief heavenly
bodies, as well as the custom of noting the positions of known stars
at the beginning and end of an eclipse, so as to have afterwards an
exact record of the times of their occurrence. So much importance was
attached to correct observations that we are told that those of special
interest were recorded in formal documents signed on oath by a mixed
body of astronomers and lawyers.

Al Mamun ordered Ptolemy’s estimate of the size of the earth to be
verified by his astronomers. Two separate measurements of a portion
of a meridian were made, which, however, agreed so closely with one
another and with the erroneous estimate of Ptolemy that they can hardly
have been independent and careful measurements, but rather rough
verifications of Ptolemy’s figures.

58. The careful observations of the Arabs soon shewed the defects in
the Greek astronomical tables, and new tables were from time to time
issued, based on much the same principles as those in the _Almagest_,
but with changes in such numerical data as the relative sizes of the
various circles, the positions of the apogees, and the inclinations of
the planes, etc.

To Tabit ben Korra, mentioned above as the translator of the
_Almagest_, belongs the doubtful honour of the discovery of a supposed
variation in the amount of the precession (chapter II., §§ 42, 50). To
account for this he devised a complicated mechanism which produced a
certain alteration in the position of the ecliptic, thus introducing
a purely imaginary complication, known as the =trepidation=, which
confused and obscured most of the astronomical tables issued during the
next five or six centuries.

59. A far greater astronomer than any of those mentioned in the
preceding articles was the Arab prince called from his birthplace Al
Battani, and better known by the Latinised name _Albategnius_, who
carried on observations from 878 to 918 and died in 929. He tested
many of Ptolemy’s results by fresh observations, and obtained more
accurate values of the obliquity of the ecliptic (chapter I., § 11) and
of precession. He wrote also a treatise on astronomy which contained
improved tables of the sun and moon, and included his most notable
discovery—namely, that the direction of the point in the sun’s orbit
at which it is farthest from the earth (the apogee), or, in other
words, the direction of the centre of the eccentric representing the
sun’s motion (chapter II., § 39), was not the same as that given in
the _Almagest_; from which change, too great to be attributed to mere
errors of observation or calculation, it might fairly be inferred that
the apogee was slowly moving, a result which, however, he did not
explicitly state. Albategnius was also a good mathematician, and the
author of some notable improvements in methods of calculation.[38]

60. The last of the Bagdad astronomers was _Abul Wafa_ (939 or
940-998), the author of a voluminous treatise on astronomy also known
as the _Almagest_, which contained some new ideas and was written on
a different plan from Ptolemy’s book, of which it has sometimes been
supposed to be a translation. In discussing the theory of the moon
Abul Wafa found that, after allowing for the equation of the centre
and for the evection, there remained a further irregularity in the
moon’s motion which was imperceptible at conjunction, opposition, and
quadrature, but appreciable at the intermediate points. It is possible
that Abul Wafa here detected an inequality rediscovered by Tycho Brahe
(chapter _V._, § 111) and known as the =variation=, but it is equally
likely that he was merely restating Ptolemy’s prosneusis (chapter
II., § 48).[39] In either case Abul Wafa’s discovery appears to have
been entirely ignored by his successors and to have borne no fruit.
He also carried further some of the mathematical improvements of his
predecessors.

Another nearly contemporary astronomer, commonly known as _Ibn Yunos_
(?-1008), worked at Cairo under the patronage of the Mahometan rulers
of Egypt. He published a set of astronomical and mathematical tables,
the _Hakemite Tables_, which remained the standard ones for about
two centuries, and he embodied in the same book a number of his
own observations as well as an extensive series by earlier Arabian
astronomers.

61. About this time astronomy, in common with other branches of
knowledge, had made some progress in the Mahometan dominions in Spain
and the opposite coast of Africa. A great library and an academy were
founded at Cordova about 970, and centres of education and learning
were established in rapid succession at Cordova, Toledo, Seville, and
Morocco.

The most important work produced by the astronomers of these places
was the volume of astronomical tables published under the direction
of _Arzachel_ in 1080, and known as the _Toletan Tables_, because
calculated for an observer at Toledo, where Arzachel probably lived.
To the same school are due some improvements in instruments and
in methods of calculation, and several writings were published in
criticism of Ptolemy, without, however, suggesting any improvements on
his ideas.

Gradually, however, the Spanish Christians began to drive back their
Mahometan neighbours. Cordova and Seville were captured in 1236 and
1248 respectively, and with their fall Arab astronomy disappeared from
history.

62. Before we pass on to consider the progress of astronomy in Europe,
two more astronomical schools of the East deserve mention, both
of which illustrate an extraordinarily rapid growth of scientific
interests among barbarous peoples. Hulagu Khan, a grandson of the
Mongol conqueror Genghis Khan, captured Bagdad in 1258 and ended the
rule of the Caliphs there. Some years before this he had received
into favour, partly as a political adviser, the astronomer _Nassir
Eddin_ (born in 1201 at Tus in Khorassan), and subsequently provided
funds for the establishment of a magnificent observatory at Meraga,
near the north-west frontier of modern Persia. Here a number of
astronomers worked under the general superintendence of Nassir Eddin.
The instruments they used were remarkable for their size and careful
construction, and were probably better than any used in Europe in the
time of Coppernicus, being surpassed first by those of Tycho Brahe
(chapter V.).

Nassir Eddin and his assistants translated or commented on nearly all
the more important available Greek writings on astronomy and allied
subjects, including Euclid’s _Elements_, several books by Archimedes,
and the _Almagest_. Nassir Eddin also wrote an abstract of astronomy,
marked by some little originality, and a treatise on geometry. He does
not appear to have accepted the authority of Ptolemy without question,
and objected in particular to the use of the equant (chapter II., §
51), which he replaced by a new combination of spheres. Many of these
treatises had for a long time a great reputation in the East, and
became in their turn the subject-matter of commentary.

But the great work of the Meraga astronomers, which occupied them 12
years, was the issue of a revised set of astronomical tables, based
on the Hakemite Tables of Ibn Yunos (§ 60), and called in honour of
their patron the _Ilkhanic Tables_. They contained not only the usual
tables for computing the motions of the planets, etc., but also a star
catalogue, based to some extent on new observations.

An important result of the observations of fixed stars made at Meraga
was that the precession (chapter II., § 42) was fixed at 51″, or
within about 1″ of its true value. Nassir Eddin also discussed the
supposed trepidation (§ 58), but seems to have been a little doubtful
of its reality. He died in 1273, soon after his patron, and with him
the Meraga School came to an end as rapidly as it was formed.

63. Nearly two centuries later _Ulugh Begh_ (born in 1394), a grandson
of the savage Tartar Tamerlane, developed a great personal interest
in astronomy, and built about 1420 an observatory at Samarcand (in
the present Russian Turkestan), where he worked with assistants. He
published fresh tables of the planets, etc., but his most important
work was a star catalogue, embracing nearly the same stars as that of
Ptolemy, but observed afresh. This was probably the first substantially
independent catalogue made since Hipparchus. The places of the stars
were given with unusual precision, the minutes as well as the degrees
of celestial longitude and latitude being recorded; and although a
comparison with modern observation shews that there were usually errors
of several minutes, it is probable that the instruments used were
extremely good. Ulugh Begh was murdered by his son in 1449, and with
him Tartar astronomy ceased.

64. No great original idea can be attributed to any of the Arab and
other astronomers whose work we have sketched. They had, however, a
remarkable aptitude for absorbing foreign ideas, and carrying them
slightly further. They were patient and accurate observers, and
skilful calculators. We owe to them a long series of observations,
and the invention or introduction of several important improvements
in mathematical methods.[40] Among the most important of their
services to mathematics, and hence to astronomy, must be counted the
introduction, from India, of our present system of writing numbers,
by which the value of a numeral is altered by its position, and fresh
symbols are not wanted, as in the clumsy Greek and Roman systems,
for higher numbers. An immense simplification was thereby introduced
into arithmetical work.[41] More important than the actual original
contributions of the Arabs to astronomy was the service that they
performed in keeping alive interest in the science and preserving the
discoveries of their Greek predecessors.

Some curious relics of the time when the Arabs were the great masters
in astronomy have been preserved in astronomical language. Thus we have
derived from them, usually in very corrupt forms, the current names
of many individual stars, _e.g._ Aldebaran, Altair, Betelgeux, Rigel,
Vega (the constellations being mostly known by Latin translations of
the Greek names), and some common astronomical terms such as zenith
and =nadir= (the invisible point on the celestial sphere opposite the
zenith); while at least one such word, almanack, has passed into common
language.

65. In Europe the period of confusion following the breakup of the
Roman empire and preceding the definite formation of feudal Europe
is almost a blank as regards astronomy, or indeed any other natural
science. The best intellects that were not absorbed in practical life
were occupied with theology. A few men, such as the Venerable Bede
(672-735), living for the most part in secluded monasteries, were
noted for their learning, which included in general some portions of
mathematics and astronomy; none were noted for their additions to
scientific knowledge. Some advance was made by Charlemagne (742-814),
who, in addition to introducing something like order into his extensive
dominions, made energetic attempts to develop education and learning.
In 782 he summoned to his court our learned countryman _Alcuin_
(735-804) to give instruction in astronomy, arithmetic, and rhetoric,
as well as in other subjects, and invited other scholars to join him,
forming thus a kind of Academy of which Alcuin was the head.

Charlemagne not only founded a higher school at his own court, but was
also successful in urging the ecclesiastical authorities in all parts
of his dominions to do the same. In these schools were taught the seven
liberal arts, divided into the so-called trivium (grammar, rhetoric,
and dialectic) and quadrivium, which included astronomy in addition to
arithmetic, geometry, and music.

66. In the 10th century the fame of the Arab learning began slowly
to spread through Spain into other parts of Europe, and the immense
learning of _Gerbert_, the most famous scholar of the century, who
occupied the papal chair as Sylvester II. from 999 to 1003, was
attributed in large part to the time which he spent in Spain, either in
or near the Moorish dominions. He was an ardent student, indefatigable
in collecting and reading rare books, and was especially interested in
mathematics and astronomy. His skill in making astrolabes (chapter II.,
§ 49) and other instruments was such that he was popularly supposed to
have acquired his powers by selling his soul to the Evil One. Other
scholars shewed a similar interest in Arabic learning, but it was not
till the lapse of another century that the Mahometan influence became
important.

At the beginning of the 12th century began a series of translations
from Arabic into Latin of scientific and philosophic treatises,
partly original works of the Arabs, partly Arabic translations of the
Greek books. One of the most active of the translators was _Plato of
Tivoli_, who studied Arabic in Spain about it 1116, and translated
Albategnius’s _Astronomy_ (§ 59), as well as other astronomical books.
At about the same time Euclid’s _Elements_, among other books, was
translated by _Athelard of Bath_. _Gherardo of Cremona_ (1114-1187) was
even more industrious, and is said to have made translations of about
70 scientific treatises, including the _Almagest_, and the _Toletan
Tables_ of Arzachel (§ 61). The beginning of the 13th century was
marked by the foundation of several Universities, and at that of Naples
(founded in 1224) the Emperor Frederick II., who had come into contact
with the Mahometan learning in Sicily, gathered together a number of
scholars whom he directed to make a fresh series of translations from
the Arabic.

Aristotle’s writings on logic had been preserved in Latin translations
from classical times, and were already much esteemed by the scholars of
the 11th and 12th centuries. His other writings were first met with in
Arabic versions, and were translated into Latin during the end of the
12th and during the 13th centuries; in one or two cases translations
were also made from the original Greek. The influence of Aristotle over
mediæval thought, already considerable, soon became almost supreme, and
his works were by many scholars regarded with a reverence equal to or
greater than that felt for the Christian Fathers.

Western knowledge of Arab astronomy was very much increased by the
activity of _Alfonso X._ of Leon and Castile (1223-1284), who collected
at Toledo, a recent conquest from the Arabs, a body of scholars, Jews
and Christians, who calculated under his general superintendence a set
of new astronomical tables to supersede the _Toletan Tables_. These
_Alfonsine Tables_ were published in 1252, on the day of Alfonso’s
accession, and spread rapidly through Europe. They embodied no new
ideas, but several numerical data, notably the length of the year, were
given with greater accuracy than before. To Alfonso is due also the
publication of the _Libros del Saber_, a voluminous encyclopædia of the
astronomical knowledge of the time, which, though compiled largely from
Arab sources, was not, as has sometimes been thought, a mere collection
of translations. One of the curiosities in this book is a diagram
representing Mercury’s orbit as an ellipse, the earth being in the
_centre_ (cf. chapter VII., § 140), this being probably the first trace
of the idea of representing the celestial motions by means of curves
other than circles.

67. To the 13th century belong also several of the great scholars, such
as _Albertus Magnus_, _Roger Bacon_, and _Cecco d’ Ascoli_ (from whom
Dante learnt), who took all knowledge for their province. Roger Bacon,
who was born in Somersetshire about 1214 and died about 1294, wrote
three principal books, called respectively the _Opus Majus_, _Opus
Minus_, and _Opus Tertium_, which contained not only treatises on most
existing branches of knowledge, but also some extremely interesting
discussions of their relative importance and of the right method for
the advancement of learning. He inveighs warmly against excessive
adherence to authority, especially to that of Aristotle, whose books
he wishes burnt, and speaks strongly of the importance of experiment
and of mathematical reasoning in scientific inquiries. He evidently
had a good knowledge of optics and has been supposed to have been
acquainted with the telescope, a supposition which we can hardly regard
as confirmed by his story that the invention was known to Caesar, who
when about to invade Britain surveyed the new country from the opposite
shores of Gaul with a telescope!

Another famous book of this period was written by the Yorkshireman John
Halifax or Holywood, better known by his Latinised name _Sacrobosco_,
who was for some time a well-known teacher of mathematics at Paris,
where he died about 1256. His _Sphaera Mundi_ as an elementary treatise
on the easier parts of current astronomy, dealing in fact with little
but the more obvious results of the daily motion of the celestial
sphere. It enjoyed immense popularity for three or four centuries, and
was frequently re-edited, translated, and commented on: it was one of
the very first astronomical books ever printed; 25 editions appeared
between 1472 and the end of the century, and 40 more by the middle of
the 17th century.

68. The European writers of the Middle Ages whom we have hitherto
mentioned, with the exception of Alfonso and his assistants, had
contented themselves with collecting and rearranging such portions
of the astronomical knowledge of the Greeks and Arabs as they could
master; there were no serious attempts at making progress, and no
observations of importance were made. A new school, however, grew up
in Germany during the 15th century which succeeded in making some
additions to knowledge, not in themselves of first-rate importance,
but significant of the greater independence that was beginning to
inspire scientific work. _George Purbach_, born in 1423, became in
1450 professor of astronomy and mathematics at the University of
Vienna, which had soon after its foundation (1365) become a centre for
these subjects. He there began an _Epitome of Astronomy_ based on the
_Almagest_, and also a Latin version of Ptolemy’s planetary theory,
intended partly as a supplement to Sacrobosco’s textbook, from which
this part of the subject had been omitted, but in part also as a
treatise of a higher order; but he was hindered in both undertakings
by the badness of the only available versions of the _Almagest_—Latin
translations which had been made not directly from the Greek, but
through the medium at any rate of Arabic and very possibly of Syriac
as well (cf. § 56), and which consequently swarmed with mistakes. He
was assisted in this work by his more famous pupil John Müller of
Königsberg (in Franconia), hence known as _Regiomontanus_, who was
attracted to Vienna at the age of 16 (1452) by Purbach’s reputation.
The two astronomers made some observations, and were strengthened
in their conviction of the necessity of astronomical reforms by the
serious inaccuracies which they discovered in the _Alfonsine Tables_,
now two centuries old; an eclipse of the moon, for example, occurring
an hour late and Mars being seen 2° from its calculated place. Purbach
and Regiomontanus were invited to Rome by one of the Cardinals, largely
with a view to studying a copy of the _Almagest_ contained among the
Greek manuscripts which since the fall of Constantinople (1453) had
come into Italy in considerable numbers, and they were on the point of
starting when the elder man suddenly died (1461).

Regiomontanus, who decided on going notwithstanding Purbach’s death,
was altogether seven years in Italy; he there acquired a good knowledge
of Greek, which he had already begun to study in Vienna, and was thus
able to read the _Almagest_ and other treatises in the original; he
completed Purbach’s _Epitome of Astronomy_, made some observations,
lectured, wrote a mathematical treatise[42] of considerable merit,
and finally returned to Vienna in 1468 with originals or copies of
several important Greek manuscripts. He was for a short time professor
there, but then accepted an invitation from the King of Hungary to
arrange a valuable collection of Greek manuscripts. The king, however,
soon turned his attention from Greek to fighting, and Regiomontanus
moved once more, settling this time in Nürnberg, then one of the most
flourishing cities in Germany, a special attraction of which was that
one of the early printing presses was established there. The Nürnberg
citizens received Regiomontanus with great honour, and one rich man
in particular, _Bernard Walther_ (1430-1504), not only supplied him
with funds, but, though an older man, became his pupil and worked with
him. The skilled artisans of Nürnberg were employed in constructing
astronomical instruments of an accuracy hitherto unknown in Europe,
though probably still inferior to those of Nassir Eddin and Ulugh
Begh (§§ 62, 63). A number of observations were made, among the most
interesting being those of the comet of 1472, the first comet which
appears to have been regarded as a subject for scientific study rather
than for superstitious terror. Regiomontanus recognised at once the
importance for his work of the new invention of printing, and, finding
probably that the existing presses were unable to meet the special
requirements of astronomy, started a printing press of his own. Here he
brought out in 1472 or 1473 an edition of Purbach’s book on planetary
theory, which soon became popular and was frequently reprinted. This
book indicates clearly the discrepancy already being felt between the
views of Aristotle and those of Ptolemy. Aristotle’s original view was
that sun, moon, the five planets, and the fixed stars were attached
respectively to eight spheres, one inside the other; and that the outer
one, which contained the fixed stars, by its revolution was the primary
cause of the apparent daily motion of all the celestial bodies. The
discovery of precession required on the part of those who carried on
the Aristotelian tradition the addition of another sphere. According
to this scheme, which was probably due to some of the translators
or commentators at Bagdad (§ 56), the fixed stars were on a sphere,
often called the =firmament=, and outside this was a ninth sphere,
known as the =primum mobile=, which moved all the others; another
sphere was added by Tabit ben Korra to account for trepidation (§
58), and accepted by Alfonso and his school; an eleventh sphere was
added towards the end of the Middle Ages to account for the supposed
changes in the obliquity of the ecliptic. A few writers invented a
larger number. Outside these spheres mediaeval thought usually placed
the Empyrean or Heaven. The accompanying diagram illustrates the whole
arrangement.

[Illustration: FIG. 36.—The celestial spheres. From Apian’s
_Cosmographia_.]

These spheres, which were almost entirely fanciful and in no serious
way even professed to account for the details of the celestial motions,
are of course quite different from the circles known as deferents
and epicycles, which Hipparchus and Ptolemy used. These were mere
geometrical abstractions, which enabled the planetary motions to
be represented with tolerable accuracy. Each planet moved freely in
space, its motion being represented or described (not _controlled_) by
a particular geometrical arrangement of circles. Purbach suggested a
compromise by hollowing out Aristotle’s crystal spheres till there was
room for Ptolemy’s epicycles inside!

From the new Nürnberg press were issued also a succession of almanacks
which, like those of to-day, gave the public useful information about
moveable feasts, the phases of the moon, eclipses, etc.; and, in
addition, a volume of less popular _Ephemerides_, with astronomical
information of a fuller and more exact character for a period of about
30 years. This contained, among other things, astronomical data for
finding latitude and longitude at sea, for which Regiomontanus had
invented a new method.[43]

The superiority of these tables over any others available was such that
they were used on several of the great voyages of discovery of this
period, probably by Columbus himself on his first voyage to America.

In 1475 Regiomontanus was invited to Rome by the Pope to assist in a
reform of the calendar, but died there the next year at the early age
of forty.

Walther carried on his friend’s work and took a number of good
observations; he was the first to make any successful attempt to allow
for the atmospheric refraction of which Ptolemy had probably had some
knowledge (chapter II., § 46); to him is due also the practice of
obtaining the position of the sun by comparison with Venus instead of
with the moon (chapter II., § 39), the much slower motion of the planet
rendering greater accuracy possible.

After Walther’s death other observers of less merit carried on the
work, and a Nürnberg astronomical school of some kind lasted into the
17th century.

69. A few minor discoveries in astronomy belong to this or to a
slightly later period and may conveniently be dealt with here.

_Leonardo da Vinci_ (1452-1519), who was not only a great painter and
sculptor, but also an anatomist, engineer, mechanician, physicist, and
mathematician, was the first to explain correctly the dim illumination
seen over the rest of the surface of the moon when the bright part is
only a thin crescent. He pointed out that when the moon was nearly new
the half of the earth which was then illuminated by the sun was turned
nearly directly towards the moon, and that the moon was in consequence
illuminated slightly by this =earthshine=, just as we are by moonshine.
The explanation is interesting in itself, and was also of some value
as shewing an analogy between the earth and moon which tended to break
down the supposed barrier between terrestrial and celestial bodies
(chapter VI., § 119).

_Jerome Fracastor_ (1483-1543) and _Peter Apian_ (1495-1552), two
voluminous writers on astronomy, made observations of comets of some
interest, both noticing that a comet’s tail continually points away
from the sun, as the comet changes its position, a fact which has been
used in modern times to throw some light on the structure of comets
(chapter XIII., § 304).

_Peter Nonius_ (1492-1577) deserves mention on account of the knowledge
of twilight which he possessed; several problems as to the duration of
twilight, its variation in different latitudes, etc., were correctly
solved by him; but otherwise his numerous books are of no great
interest.[44]

A new determination of the size of the earth, the first since the time
of the Caliph Al Mamun (§ 57), was made about 1528 by the French doctor
_John Fernel_ (1497-1558), who arrived at a result the error in which
(less than 1 per cent.) was far less than could reasonably have been
expected from the rough methods employed.

The life of Regiomontanus overlapped that of Coppernicus by three
years; the four writers last named were nearly his contemporaries; and
we may therefore be said to have come to the end of the comparatively
stationary period dealt with in this chapter.



CHAPTER IV.

COPPERNICUS.

“But in this our age, one rare witte (seeing the continuall errors
that from time to time more and more continually have been discovered,
besides the infinite absurdities in their Theoricks, which they have
been forced to admit that would not confesse any Mobilitie in the
ball of the Earth) hath by long studye, paynfull practise, and rare
invention delivered a new Theorick or Model of the world, shewing
that the Earth resteth not in the Center of the whole world or globe
of elements, which encircled and enclosed in the Moone’s orbit, and
together with the whole globe of mortality is carried yearly round
about the Sunne, which like a king in the middest of all, rayneth and
giveth laws of motion to all the rest, sphaerically dispersing his
glorious beames of light through all this sacred coelestiall Temple.”
                                                   THOMAS DIGGES, 1590.


70. The growing interest in astronomy shewn by the work of such men as
Regiomontanus was one of the early results in the region of science
of the great movement of thought to different aspects of which are
given the names of Revival of Learning, Renaissance, and Reformation,
The movement may be regarded primarily as a general quickening of
intelligence and of interest in matters of thought and knowledge. The
invention of printing early in the 15th century, the stimulus to the
study of the Greek authors, due in part to the scholars who were driven
westwards after the capture of Constantinople by the Turks (1453), and
the discovery of America by Columbus in 1492, all helped on a movement
the beginning of which has to be looked for much earlier.

Every stimulus to the intelligence naturally brings with it a tendency
towards inquiry into opinions received through tradition and based
on some great authority. The effective discovery and the study of
Greek philosophers other than Aristotle naturally did much to shake
the supreme authority of that great philosopher, just as the Reformers
shook the authority of the Church by pointing out what they considered
to be inconsistencies between its doctrines and those of the Bible.
At first there was little avowed opposition to the principle that
truth was to be derived from some authority, rather than to be sought
independently by the light of reason; the new scholars replaced the
authority of Aristotle by that of Plato or of Greek and Roman antiquity
in general, and the religious Reformers replaced the Church by the
Bible. Naturally, however, the conflict between authorities produced
in some minds scepticism as to the principle of authority itself; when
freedom of judgment had to be exercised to the extent of deciding
between authorities, it was but a step further—a step, it is true, that
comparatively few took—to use the individual judgment on the matter at
issue itself.

In astronomy the conflict between authorities had already arisen,
partly in connection with certain divergencies between Ptolemy and
Aristotle, partly in connection with the various astronomical tables
which, though on substantially the same lines, differed in minor
points. The time was therefore ripe for some fundamental criticism of
the traditional astronomy, and for its reconstruction on a new basis.

Such a fundamental change was planned and worked out by the great
astronomer whose work has next to be considered.

71. _Nicholas_ Coppernic or _Coppernicus_[45] was born on February
19th, 1473, in a house still pointed out in the little trading town of
Thorn on the Vistula. Thorn now lies just within the eastern frontier
of the present kingdom of Prussia; in the time of Coppernicus it lay
in a region over which the King of Poland had some sort of suzerainty,
the precise nature of which was a continual subject of quarrel between
him, the citizens, and the order of Teutonic knights, who claimed a
good deal of the neighbouring country. The astronomer’s father (whose
name was most commonly written Koppernigk) was a merchant who came
to Thorn from Cracow, then the capital of Poland, in 1462. Whether
Coppernicus should be counted as a Pole or as a German is an intricate
question, over which his biographers have fought at great length and
with some acrimony, but which is not worth further discussion here.

Nicholas, after the death of his father in 1483, was under the care
of his uncle, Lucas Watzelrode, afterwards bishop of the neighbouring
diocese of Ermland, and was destined by him from a very early date for
an ecclesiastical career. He attended the school at Thorn, and at the
age of 17 entered the University of Cracow. Here he seems to have first
acquired (or shewn) a decided taste for astronomy and mathematics,
subjects in which he probably received help from Albert Brudzewski,
who had a great reputation as a learned and stimulating teacher; the
lecture lists of the University show that the comparatively modern
treatises of Purbach and Regiomontanus (chapter III., § 68) were the
standard textbooks used. Coppernicus had no intention of graduating at
Cracow, and probably left after three years (1494). During the next
year or two he lived partly at home, partly at his uncle’s palace at
Heilsberg, and spent some of the time in an unsuccessful candidature
for a canonry at Frauenburg, the cathedral city of his uncle’s diocese.

The next nine or ten years of his life (from 1496 to 1505 or 1506)
were devoted to studying in Italy, his stay there being broken only
by a short visit to Frauenburg in 1501. He worked chiefly at Bologna
and Padua, but graduated at Ferrara, and also spent some time at
Rome, where his astronomical knowledge evidently made a favourable
impression. Although he was supposed to be in Italy primarily with a
view to studying law and medicine, it is evident that much of his best
work was being put into mathematics and astronomy, while he also paid a
good deal of attention to Greek.

[Illustration: COPPERNICUS.]

During his absence he was appointed (about 1497) to a canonry at
Frauenburg, and at some uncertain date be also received a sinecure
ecclesiastical appointment at Breslau.

72. On returning to Frauenburg from Italy Coppernicus almost
immediately obtained fresh leave of absence, and joined his uncle
at Heilsberg, ostensibly as his medical adviser and really as his
companion.

It was probably during the quiet years spent at Heilsberg that he first
put into shape his new ideas about astronomy, and wrote the first draft
of his book. He kept the manuscript by him, revising and rewriting
from time to time, partly from a desire to make his work as perfect as
possible, partly from complete indifference to reputation, coupled with
dislike of the controversy to which the publication of his book would
almost certainly give rise. In 1509 he published at Cracow his first
book, a Latin translation of a set of Greek letters by Theophylactus,
interesting as being probably the first translation from the Greek ever
published in Poland or the adjacent districts. In 1512, on the death of
his uncle, he finally settled in Frauenburg, in a set of rooms which he
occupied, with short intervals, for the next 31 years. Once fairly in
residence, he took his share in conducting the business of the Chapter:
he acted, for example, more than once as their representative in
various quarrels with the King of Poland and the Teutonic knights; in
1523 he was general administrator of the diocese for a few months after
the death of the bishop; and for two periods, amounting altogether
to six years (1516-1519 and 1520-1521), he lived at the castle of
Allenstein, administering some of the outlying property of the Chapter.
In 1521 he was commissioned to draw up a statement of the grievances
of the Chapter against the Teutonic knights for presentation to the
Prussian Estates, and in the following year wrote a memorandum on the
debased and confused state of the coinage in the district, a paper
which was also laid before the Estates, and was afterwards rewritten
in Latin at the special request of the bishop. He also gave a certain
amount of medical advice to his friends as well as to the poor of
Frauenburg, though he never practised regularly as a physician; but
notwithstanding these various occupations it is probable that a very
large part of his time during the last 30 years of his life was devoted
to astronomy.

73. We are so accustomed to associate the revival of astronomy, as
of other branches of natural science, with increased care in the
collection of observed facts, and to think of Coppernicus as the chief
agent in the revival, that it is worth while here to emphasise the
fact that he was in no sense a great observer. His instruments, which
were mostly of his own construction, were far inferior to those of
Nassir Eddin and of Ulugh Begh (chapter III., §§ 62, 63), and not even
as good as those which he could have procured if he had wished from
the workshops of Nürnberg; his observations were not at all numerous
(only 27, which occur in his book, and a dozen or two besides being
known), and he appears to have made no serious attempt to secure great
accuracy. His determination of the position of one star, which was
extensively used by him as a standard of reference and was therefore of
special importance, was in error to the extent of nearly 40′ (more than
the apparent breadth of the sun or moon), an error which Hipparchus
would have considered very serious. His pupil Rheticus (§ 74) reports
an interesting discussion between his master and himself, in which the
pupil urged the importance of making observations with all imaginable
accuracy; Coppernicus answered that minute accuracy was not to be
looked for at that time, and that a rough agreement between theory and
observation was all that he could hope to attain. Coppernicus moreover
points out in more than one place that the high latitude of Frauenburg
and the thickness of the air were so detrimental to good observation
that, for example, though he had occasionally been able to see the
planet Mercury, he had never been able to observe it properly.

Although he published nothing of importance till towards the end of his
life, his reputation as an astronomer and mathematician appears to have
been established among experts from the date of his leaving Italy, and
to have steadily increased as time went on.

In 1515 he was consulted by a committee appointed by the Lateran
Council to consider the reform of the calendar, which had now fallen
into some confusion (chapter II., § 22), but he declined to give any
advice on the ground that the motions of the sun and moon were as yet
too imperfectly known for a satisfactory reform to be possible. A few
years later (1524) he wrote an open letter, intended for publication,
to one of his Cracow friends, in reply to a tract on precession, in
which, after the manner of the time, he used strong language about the
errors of his opponent.[46]

It was meanwhile gradually becoming known that he held the novel
doctrine that the earth was in motion and the sun and stars at
rest, a doctrine which was sufficiently startling to attract notice
outside astronomical circles. About 1531 he had the distinction of
being ridiculed on the stage at some popular performance in the
neighbourhood; and it is interesting to note (especially in view of the
famous persecution of Galilei at Rome a century later) that Luther in
his _Table Talk_ frankly described Coppernicus as a fool for holding
such opinions, which were obviously contrary to the Bible, and that
Melanchthon, perhaps the most learned of the Reformers, added to a
somewhat similar criticism a broad hint that such opinions should not
be tolerated. Coppernicus appears to have taken no notice of these or
similar attacks, and still continued to publish nothing. No observation
made later than 1529 occurs in his great book, which seems to have been
nearly in its final form by that date; and to about this time belongs
an extremely interesting paper, known as the _Commentariolus_, which
contains a short account of his system of the world, with some of
the evidence for it, but without any calculations. It was apparently
written to be shewn or lent to friends, and was not published; the
manuscript disappeared after the death of the author and was only
rediscovered in 1878. The _Commentariolus_ was probably the basis of a
lecture on the ideas of Coppernicus given in 1533 by one of the Roman
astronomers at the request of Pope Clement VII. Three years later
Cardinal Schomberg wrote to ask Coppernicus for further information as
to his views, the letter showing that the chief features were already
pretty accurately known.

74. Similar requests must have been made by others, but his final
decision to publish his ideas seems to have been due to the arrival at
Frauenburg in 1539 of the enthusiastic young astronomer generally known
as _Rheticus_.[47] Born in 1514, he studied astronomy under Schoner at
Nürnberg, and was appointed in 1536 to one of the chairs of mathematics
created by the influence of Melanchthon at Wittenberg, at that time the
chief Protestant University.

Having heard, probably through the _Commentariolus_, of Coppernicus
and his doctrines, he was so much interested in them that he decided
to visit the great astronomer at Frauenburg. Coppernicus received him
with extreme kindness, and the visit, which was originally intended to
last a few days or weeks, extended over nearly two years. Rheticus set
to work to study Coppernicus’s manuscript, and wrote within a few weeks
of his arrival an extremely interesting and valuable account of it,
known as the First Narrative (_Prima Narratio_), in the form of an open
letter to his old master Schoner, a letter which was printed in the
following spring and was the first easily accessible account of the new
doctrines.[48]

When Rheticus returned to Wittenberg, towards the end of 1541, he took
with him a copy of a purely mathematical section of the great book,
and had it printed as a textbook of the subject (Trigonometry); it had
probably been already settled that he was to superintend the printing
of the complete book itself. Coppernicus, who was now an old man and
would naturally feel that his end was approaching, sent the manuscript
to his friend Giese, Bishop of Kulm, to do what he pleased with.
Giese sent it at once to Rheticus, who made arrangements for having
it printed at Nürnberg. Unfortunately Rheticus was not able to see it
all through the press, and the work had to be entrusted to Osiander, a
Lutheran preacher interested in astronomy. Osiander appears to have
been much alarmed at the thought of the disturbance which the heretical
ideas of Coppernicus would cause, and added a prefatory note of his
own (which he omitted to sign), praising the book in a vulgar way, and
declaring (what was quite contrary to the views of the author) that the
fundamental principles laid down in it were merely abstract hypotheses
convenient for purposes of calculation; he also gave the book the
title _De Revolutionibus Orbium Celestium_ (On the Revolutions of the
Celestial Spheres), the last two words of which were probably his own
addition. The printing was finished in the winter 1542-3, and the
author received a copy of his book on the day of his death (May 24th,
1543), when his memory and mental vigour had already gone.

75. The central idea with which the name of Coppernicus is associated,
and which makes the _De Revolutionibus_ one of the most important books
in all astronomical literature, by the side of which perhaps only the
_Almagest_ and Newton’s _Principia_ (chapter IX., §§ 177 _seqq._) can
be placed, is that the apparent motions of the celestial bodies are to
a great extent not real motions, but are due to the motion of the earth
carrying the observer with it. Coppernicus tells us that he had long
been struck by the unsatisfactory nature of the current explanations of
astronomical observations, and that, while searching in philosophical
writings for some better explanation, he had found a reference of
Cicero to the opinion of Hicetas that the earth turned round on its
axis daily. He found similar views held by other Pythagoreans, while
Philolaus and Aristarchus of Samos had also held that the earth not
only rotates, but moves bodily round the sun or some other centre (cf.
chapter II., § 24). The opinion that the earth is not the sole centre
of motion, but that Venus and Mercury revolve round the sun, he found
to be an old Egyptian belief, supported also by _Martianus Capella_,
who wrote a compendium of science and philosophy in the 5th or 6th
century A.D. A more modern authority, _Nicholas of Cusa_ (1401-1464), a
mystic writer who refers to a possible motion of the earth, was ignored
or not noticed by Coppernicus. None of the writers here named, with
the possible exception of Aristarchus of Samos, to whom Coppernicus
apparently paid little attention, presented the opinions quoted as
more than vague speculations; none of them gave any substantial reasons
for, much less a proof of, their views; and Coppernicus, though he
may have been glad, after the fashion of the age, to have the support
of recognised authorities, had practically to make a fresh start and
elaborate his own evidence for his opinions.

It has sometimes been said that Coppernicus _proved_ what earlier
writers had guessed at or suggested; it would perhaps be truer to say
that he took up certain floating ideas, which were extremely vague
and had never been worked out scientifically, based on them certain
definite fundamental principles, and from these principles developed
mathematically an astronomical system which he shewed to be at least as
capable of explaining the observed celestial motions as any existing
variety of the traditional Ptolemaic system. The Coppernican system,
as it left the hands of the author, was in fact decidedly superior to
its rivals as an explanation of ordinary observations, an advantage
which it owed quite as much to the mathematical skill with which it
was developed as to its first principles; it was in many respects very
much simpler; and it avoided certain fundamental difficulties of the
older system. It was however liable to certain serious objections,
which were only overcome by fresh evidence which was subsequently
brought to light. For the predecessors of Coppernicus there was, apart
from variations of minor importance, but one scientific system which
made any serious attempt to account for known facts; for his immediate
successors there were two, the newer of which would to an impartial
mind appear on the whole the more satisfactory, and the further study
of the two systems, with a view to the discovery of fresh arguments
or fresh observations tending to support the one or the other, was
immediately suggested as an inquiry of first-rate importance.

76. The plan of the _De Revolutionibus_ bears a general resemblance to
that of the _Almagest_. In form at least the book is not primarily an
argument in favour of the motion of the earth, and it is possible to
read much of it without ever noticing the presence of this doctrine.

Coppernicus, like Ptolemy, begins with certain first principles
or postulates, but on account of their novelty takes a little more
trouble than his predecessor (cf. chapter II., § 47) to make them at
once appear probable. With these postulates as a basis he proceeds
to develop, by means of elaborate and rather tedious mathematical
reasoning, aided here and there by references to observations, detailed
schemes of the various celestial motions; and it is by the agreement
of these calculations with observations, far more than by the general
reasoning given at the beginning, that the various postulates are in
effect justified.

His first postulate, that the universe is spherical, is supported by
vague and inconclusive reasons similar to those given by Ptolemy and
others; for the spherical form of the earth he gives several of the
usual valid arguments, one of his proofs for its curvature from east to
west being the fact that eclipses visible at one place are not visible
at another. A third postulate, that the motions of the celestial bodies
are uniform circular motions or are compounded of such motions, is, as
might be expected, supported only by reasons of the most unsatisfactory
character. He argues, for example, that any want of uniformity in motion

 “must arise either from irregularity in the moving power, whether this
 be within the body or foreign to it, or from some inequality of the
 body in revolution.... Both of which things the intellect shrinks from
 with horror, it being unworthy to hold such a view about bodies which
 are constituted in the most perfect order.”

77. The discussion of the possibility that the earth may move, and may
even have more than one motion, then follows, and is more satisfactory
though by no means conclusive. Coppernicus has a firm grasp of the
principle, which Aristotle had also enunciated, sometimes known as that
of relative motion, which he states somewhat as follows:—

  “For all change in position which is seen is due to a motion
 either of the observer or of the thing looked at, or to changes in
 the position of both, provided that these are different. For when
 things are moved equally relatively to the same things, no motion is
 perceived, as between the object seen and the observer.”[49]

Coppernicus gives no proof of this principle, regarding it probably
as sufficiently obvious, when once stated, to the mathematicians and
astronomers for whom he was writing. It is, however, so fundamental
that it may be worth while to discuss it a little more fully.

[Illustration: FIG. 37.—Relative motion.]

Let, for example, the observer be at A and an object at B, then whether
the object move from B to B′, the observer remaining at rest, or the
observer move an _equal_ distance in the _opposite_ direction, from A
to A′, the object remaining at rest, the effect is to the eye exactly
the same, since in either case the distance between the observer and
object and the direction in which the object is seen, represented in
the first case by A B′ and in the second by A′ B, are the same.

Thus if in the course of a year _either_ the sun passes successively
through the positions A, B, C, D (fig. 38), the earth remaining at rest
at E, _or_ if the sun is at rest and the earth passes successively
through the positions _a_, _b_, _c_, _d_, at the corresponding times,
the sun remaining at rest at S, exactly the same effect is produced
on the eye, provided that the lines _a_ S, _b_ S, _c_ S, _d_ S are,
as in the figure, equal in length and parallel in direction to E
A, E B, E C, E D respectively. The same being true of intermediate
points, exactly the same apparent effect is produced whether the sun
describe the circle A B C D, or the earth describe at the same rate
the equal circle _a b c d_. It will be noticed further that, although
the corresponding motions in the two cases are at the same times in
_opposite_ directions (as at A and _a_), yet each circle as a whole
is described, as indicated by the arrowheads, in the _same_ direction
(contrary to that of the motion of the hands of a clock, in the figures
given). It follows in the same sort of way that an apparent motion (as
of a planet) may be explained as due partially to the motion of the
object, partially to that of the observer.

[Illustration: FIG. 38.—The relative motion of the sun and moon.]

Coppernicus gives the familiar illustration of the passenger in a boat
who sees the land apparently moving away from him, by quoting and
explaining Virgil’s line:—

  “Provehimur portu, terræque urbesque recedunt.”

[Illustration: FIG. 39.—The daily rotation of the earth.]

78. The application of the same ideas to an apparent rotation round
the observer, as in the case of the apparent daily motion of the
celestial sphere, is a little more difficult. It must be remembered
that the eye has no means of judging the direction of an object taken
by itself; it can only judge the difference between the direction of
the object and some other direction, whether that of another object or
a direction fixed in some way by the body of the observer. Thus when
after looking at a star twice at an interval of time we decide that it
has moved, this means that its direction has changed relatively to,
say, some tree or house which we had noticed nearly in its direction,
or that its direction has changed relatively to the direction in which
we are directing our eyes or holding our bodies. Such a change can
evidently be interpreted as a change of direction, either of the star
or of the line from the eye to the tree which we used as a line of
reference. To apply this to the case of the celestial sphere, let us
suppose that S represents a star on the celestial sphere, which (for
simplicity) is overhead to an observer on the earth at A, this being
determined by comparison with a line A B drawn upright on the earth.
Next, earth and celestial sphere being supposed to have a common centre
at O, let us suppose _firstly_ that the celestial sphere turns round
(in the direction of the hands of a clock) till S comes to S′, and
that the observer now sees the star on his horizon or in a direction
at right angles to the original direction A B, the angle turned
through by the celestial sphere being S O S′; and _secondly_ that, the
celestial sphere being unchanged, the earth turns round in the opposite
direction, till A B comes to A′ B′, and the star is again seen by the
observer on his horizon. Whichever of these motions has taken place,
the observer sees exactly the same apparent motion in the sky; and the
figure shews at once that the angle S O S′ through which the celestial
sphere was supposed to turn in the first case is equal to the angle A O
A′ through which the earth turns in the second case, but that the two
rotations are in opposite directions. A similar explanation evidently
applies to more complicated cases.

Hence the apparent daily rotation of the celestial sphere about an axis
through the poles would be produced equally well, either by an actual
rotation of this character, or by a rotation of the earth about an
axis also passing through the poles, and at the same rate, but in the
opposite direction, _i.e._ from west to east. This is the first motion
which Coppernicus assigns to the earth.

79. The apparent annual motion of the sun, in accordance with which
it appears to revolve round the earth in a path which is nearly a
circle, can be equally well explained by supposing the sun to be at
rest, and the earth to describe an exactly equal path round the sun,
the direction of the revolution being the same. This is virtually the
second motion which Coppernicus gives to the earth, though, on account
of a peculiarity in his geometrical method, he resolves this motion
into two others, and combines with one of these a further small motion
which is required for precession.[50]

80. Coppernicus’s conception then is that the earth revolves round the
sun in the plane of the ecliptic, while rotating daily on an axis which
continually points to the poles of the celestial sphere, and therefore
retains (save for precession) a fixed direction in space.

It should be noticed that the two motions thus assigned to the earth
are perfectly distinct; each requires its own proof, and explains a
different set of appearances. It was quite possible, with perfect
consistency, to believe in one motion without believing in the other,
as in fact a very few of the 16th-century astronomers did (chapter V.,
§ 105).

In giving his reasons for believing in the motion of the earth
Coppernicus discusses the chief objections which had been urged by
Ptolemy. To the objection that if the earth had a rapid motion of
rotation about its axis, the earth would be in danger of flying to
pieces, and the air, as well as loose objects on the surface, would be
left behind, he replies that if such a motion were dangerous to the
solid earth, it must be much more so to the celestial sphere, which,
on account of its vastly greater size, would have to move enormously
faster than the earth to complete its daily rotation; he enters also
into an obscure discussion of difference between a “natural” and an
“artificial” motion, of which the former might be expected not to
disturb anything on the earth.

Coppernicus shews that the earth is very small compared to the sphere
of the stars, because wherever the observer is on the earth the horizon
appears to divide the celestial sphere into two equal parts and the
observer appears always to be at the centre of the sphere, so that any
distance through which the observer moves on the earth is imperceptible
as compared with the distance of the stars.

81. He goes on to argue that the chief irregularity in the motion
of the planets, in virtue of which they move backwards at intervals
(chapter I., § 14, and chapter II., § 51), can readily be explained
in general by the motion of the earth and by a motion of each planet
round the sun, in its own time and at its own distance. From the fact
that Venus and Mercury were never seen very far from the sun, it could
be inferred that their paths were nearer to the sun than that of the
earth. Mercury being the nearer to the sun of the two, because never
seen so far from it in the sky as Venus. The other three planets,
being seen at times in a direction opposite to that of the sun, must
necessarily evolve round the sun in orbits larger than that of the
earth, a view confirmed by the fact that they were brightest when
opposite the sun (in which positions they would be nearest to us).
The order of their respective distances from the sun could be at once
inferred from the disturbing effects produced on their apparent motions
by the motion of the earth; Saturn being least affected must on the
whole be farthest from the earth, Jupiter next, and Mars next. The
earth thus became one of six planets revolving round the sun, the
order of distance—Mercury, Venus, Earth, Mars, Jupiter, Saturn—being
also in accordance with the rates of motion round the sun, Mercury
performing its revolution most rapidly (in about 88 days[51]), Saturn
most slowly (in about 30 years). On the Coppernican system the moon
alone still revolved round the earth, being the only celestial body the
status of which was substantially unchanged; and thus Coppernicus was
able to give the accompanying diagram of the solar system (fig. 40),
representing his view of its general arrangement (though not of the
right proportions of the different parts) and of the various motions.

[Illustration: FIG. 40.—The solar system according to Coppernicus. From
the _De Revolutionibus_.]

82. The effect of the motion of the earth round the sun on the length
of the day and other seasonal effects is discussed in some detail, and
illustrated by diagrams which are here reproduced.[52]

[Illustration: FIG. 41.—Coppernican explanation of the seasons. From
the _De Revolutionibus_.]

[Illustration: FIG. 42.—Coppernican explanation of the seasons. From
the _De Revolutionibus_.]

In fig. 41 A, B, C, D represent the centre of the earth in four
positions, occupied by it about December 23rd, March 21st, June 22nd,
and September 22nd respectively (_i.e._ at the beginnings of the four
seasons, according to astronomical reckoning); the circle F G H I in
each of its positions represents the equator of the earth, _i.e._ a
great circle on the earth the plane of which is perpendicular to the
axis of the earth and is consequently always parallel to the celestial
equator. This circle is not in the plane of the ecliptic, but tilted
up at an angle of 23-1∕2°, so that F must always be supposed _below_
and H _above_ the plane of the paper (which represents the ecliptic);
the equator cuts the ecliptic along G I. The diagram (in accordance
with the common custom in astronomical diagrams) represents the various
circles as seen from the north side of the equator and ecliptic. When
the earth is at A, the north pole (as is shewn more clearly in fig.
42, in which P, P′ denote the north pole and south pole respectively)
is turned away from the sun, E, which is on the lower or south side of
the plane of the equator, and consequently inhabitants of the northern
hemisphere see the sun for less than half the day, while those on
the southern hemisphere see the sun for more than half the day, and
those beyond the line K L (in fig. 42) see the sun during the whole
day. Three months later, when the earth’s centre is at B (fig. 41),
the sun lies in the plane of the equator, the poles of the earth are
turned neither towards nor away from the sun, but aside, and all over
the earth daylight lasts for 12 hours and night for an equal time.
Three months later still, when the earth’s centre is at C, the sun is
above the plane of the equator, and the inhabitants of the northern
hemisphere see the sun for more than half the day, those on the
southern hemisphere for less than half, while those in parts of the
earth farther north than the line M N (in fig. 42) see the sun for the
whole 24 hours. Finally, when, at the autumn equinox, the earth has
reached D (fig. 41), the sun is again in the plane of the equator, and
the day is everywhere equal to the night.

83. Coppernicus devotes the first eleven chapters of the first book
to this preliminary sketch of his system; the remainder of this book
he fills with some mathematical propositions and tables, which, as
previously mentioned (§ 74), had already been separately printed by
Rheticus. The second book contains chiefly a number of the usual
results relating to the celestial sphere and its apparent daily motion,
treated much as by earlier writers, but with greater mathematical
skill. Incidentally Coppernicus gives his measurement of the obliquity
of the ecliptic, and infers from a comparison with earlier observations
that the obliquity had decreased, which was in fact the case, though
to a much less extent than his imperfect observations indicated. The
book ends with a catalogue of stars, which is Ptolemy’s catalogue,
occasionally corrected by fresh observations, and rearranged so as to
avoid the effects of precession.[53] When, as frequently happened, the
Greek and Latin versions of the _Almagest_ gave, owing to copyists’
or printers’ errors, different results, Coppernicus appears to have
followed sometimes the Latin and sometimes the Greek version, without
in general attempting to ascertain by fresh observations which was
right.

84. The third book begins with an elaborate discussion of the
precession of the equinoxes (chapter II., § 42). From a comparison
of results obtained by Timocharis, by later Greek astronomers, and
by Albategnius, Coppernicus infers that the amount of precession has
varied, but that its average value is 50″·2 annually (almost exactly
the true value), and accepts accordingly Tabit ben Korra’s unhappy
suggestion of the trepidation (chapter III., § 58). An examination of
the data used by Coppernicus shews that the erroneous or fraudulent
observations of Ptolemy (chapter II., § 50) are chiefly responsible for
the perpetuation of this mistake.

Of much more interest than the detailed discussion of trepidation and
of geometrical schemes for representing it is the interpretation of
precession as the result of a motion of the earth’s axis. Precession
was originally recognised by Hipparchus as a motion of the celestial
equator, in which its inclination to the ecliptic was sensibly
unchanged. Now the ideas of Coppernicus make the celestial equator
dependent on the equator of the earth, and hence on its axis; it
is in fact a great circle of the celestial sphere which is always
perpendicular to the axis about which the earth rotates daily. Hence
precession, on the theory of Coppernicus, arises from a slow motion of
the axis of the earth, which moves so as always to remain inclined at
the same angle to the ecliptic, and to return to its original position
after a period of about 26,000 years (since a motion of 50″·2 annually
is equivalent to 360° or a complete circuit in that period); in other
words, the earth’s axis has a slow conical motion, the central line (or
axis) of the cone being at right angles to the plane of the ecliptic.

85. Precession being dealt with, the greater part of the remainder of
the third book is devoted to a discussion in detail of the apparent
annual motion of the sun round the earth, corresponding to the real
annual motion of the earth round the sun. The geometrical theory of
the _Almagest_ was capable of being immediately applied to the new
system, and Coppernicus, like Ptolemy, uses an eccentric. He makes the
calculations afresh, arrives at a smaller and more accurate value of
the eccentricity (about 1∕31 instead of 1∕24), fixes the position of
the apogee and perigee (chapter II., § 39), or rather of the equivalent
=aphelion= and =perihelion= (_i.e._ the points in the earth’s orbit
where it is respectively farthest from and nearest to the sun), and
thus verifies Albategnius’s discovery (chapter III., § 59) of the
motion of the line of apses. The theory of the earth’s motion is worked
out in some detail, and tables are given whereby the apparent place of
the sun at any time can be easily computed.

The fourth book deals with the theory of the moon. As has been already
noticed, the moon was the only celestial body the position of which
in the universe was substantially unchanged by Coppernicus, and it
might hence have been expected that little alteration would have
been required in the traditional theory. Actually, however, there
is scarcely any part of the subject in which Coppernicus did more to
diminish the discrepancies between theory and observation. He rejects
Ptolemy’s equant (chapter II., § 51), partly on the ground that it
produces an irregular motion unsuitable for the heavenly bodies, partly
on the more substantial ground that, as already pointed out (chapter
II., § 48), Ptolemy’s theory makes the apparent size of the moon at
times twice as great as at others. By an arrangement of epicycles
Coppernicus succeeded in representing the chief irregularities in the
moon’s motion, including evection, but without Ptolemy’s prosneusis
(chapter II., § 48) or Abul Wafa’s inequality (chapter III., § 60),
while he made the changes in the moon’s distance, and consequently in
its apparent size, not very much greater than those which actually
take place, the difference being imperceptible by the rough methods of
observation which he used.[54]

In discussing the distances and sizes of the sun and moon Coppernicus
follows Ptolemy closely (chapter II., § 49; cf. also fig. 20); he
arrives at substantially the same estimate of the distance of the moon,
but makes the sun’s distance 1,500 times the earth’s radius, thus
improving to some extent on the traditional estimate, which was based
on Ptolemy’s. He also develops in some detail the effect of parallax
on the apparent place of the moon, and the variations in the apparent
size, owing to the variations in distance: and the book ends with a
discussion of eclipses.

86. The last two books (V. and VI.) deal at length with the motion of
the planets.

In the cases of Mercury and Venus, Ptolemy’s explanation of the
motion could with little difficulty be rearranged so as to fit the
ideas of Coppernicus. We have seen (chapter II., § 51) that, minor
irregularities being ignored, the motion of either of these planets
could be represented by means of an epicycle moving on a deferent, the
centre of the epicycle being always in the direction of the sun, the
ratio of the sizes of the epicycle and deferent being fixed, but the
actual dimensions being practically arbitrary. Ptolemy preferred on the
whole to regard the epicycles of both these planets as lying between
the earth and the sun. The idea of making the sun a centre of motion
having once been accepted, it was an obvious simplification to make
the centre of the epicycle not merely lie in the direction of the sun,
but actually be the sun. In fact, if the planet in question revolved
round the sun at the proper distance and at the proper rate, the same
appearances would be produced as by Ptolemy’s epicycle and deferent,
the path of the planet round the sun replacing the epicycle, and the
apparent path of the sun round the earth (or the path of the earth
round the sun) replacing the deferent.

[Illustration: FIG. 43.—The orbits of Venus and of the earth.]

[Illustration: FIG. 44.—The synodic and sidereal periods of Venus.]

In discussing the time of revolution of a planet a distinction has to
be made, as in the case of the moon (chapter II., § 40), between the
synodic and sidereal periods of revolution. Venus, for example, is
seen as an evening star at its greatest angular distance from the sun
(as at V in fig. 43) at intervals of about 584 days. This is therefore
the time which Venus takes to return to the same position relatively
to the sun, as seen from the earth, or relatively to the earth, as
seen from the sun; this time is called the =synodic period=. But as
during this time the line E S has changed its direction, Venus is no
longer in the same position relatively to the stars, as seen either
from the sun or from the earth. If at first Venus and the earth are at
V_{1}, E_{1}; respectively, after 584 days (or about a year and seven
months) the earth will have performed rather more than a revolution and
a half round the sun and will be at E_{2}; Venus being again at the
greatest distance from the sun will therefore be at V_{2}, but will
evidently be seen in quite a different part of the sky, and will not
have performed an exact revolution round the sun. It is important to
know how long the line S V_{1} takes to return to the same position,
_i.e._ how long Venus takes to return to the same position with respect
to the stars, as seen from the sun, an interval of time known as
the =sidereal period=. This can evidently be calculated by a simple
rule-of-three sum from the data given. For Venus has in 584 days gained
a complete revolution on the earth, or has gone as far as the earth
would have gone in 584 + 365 or 949 days (fractions of days being
omitted for simplicity); hence Venus goes in 584 × 365∕949 days as far
as the earth in 365 days, _i.e._ Venus completes a revolution in 584 ×
365∕949 or 225 days. This is therefore the sidereal period of Venus.
The process used by Coppernicus was different, as he saw the advantage
of using a long period of time, so as to diminish the error due to
minor irregularities, and he therefore obtained two observations of
Venus at a considerable interval of time, in which Venus occupied very
nearly the same position both with respect to the sun and to the stars,
so that the interval of time contained very nearly an exact number of
sidereal periods as well as of synodic periods. By dividing therefore
the observed interval of time by the number of sidereal periods (which
being a whole number could readily be estimated), the sidereal period
was easily obtained. A similar process shewed that the synodic period
of Mercury was about 116 days, and the sidereal period about 88 days.

The comparative sizes of the orbits of Venus and Mercury as compared
with that of the earth could easily be ascertained from observations of
the position of either planet when most distant from the sun. Venus,
for example, appears at its greatest distance from the sun when at a
point V_{1} (fig. 44) such that V_{1} E_{1} touches the circle in which
Venus moves, and the angle E_{1} V_{1} S is then (by a known property
of a circle) a right angle. The angle S E_{1} V_{1} being observed,
the shape of the triangle S E_{1} V_{1} is known, and the ratio of
its sides can be readily calculated. Thus Coppernicus found that the
average distance of Venus from the sun was about 72 and that of Mercury
about 36, the distance of the earth from the sun being taken to be 100;
the corresponding modern figures are 72·3 and 38·7.

[Illustration: FIG. 45.—The epicycle of Jupiter.]

87. In the case of the superior planets. Mars, Jupiter, and Saturn,
it was much more difficult to recognise that their motions could be
explained by supposing them to revolve round the sun, since the centre
of the epicycle did not always lie in the direction of the sun, but
might be anywhere in the ecliptic. One peculiarity, however, in the
motion of any of the superior planets might easily have suggested their
motion round the sun, and was either completely overlooked by Ptolemy
or not recognised by him as important. It is possible that it was one
of the clues which led Coppernicus to his system. This peculiarity
is that the radius of the epicycle of the planet, _j_ J, is always
parallel to the line E S joining the earth and sun, and consequently
performs a complete revolution in a year. This connection between the
motion of the planet and that of the sun received no explanation from
Ptolemy’s theory. Now if we draw E J′ parallel to _j_ J and equal to
it in length, it is easily seen[55] that the line J′ J is equal and
parallel to E _j_, that consequently J describes a circle round J′
just as _j_ round E. Hence the motion of the planet can equally well
be represented by supposing it to move in an epicycle (represented by
the large dotted circle in the figure) of which J′ is the centre and
J′ J the radius, while the centre of the epicycle, remaining always
in the direction of the sun, describes a deferent (represented by the
small circle round E) of which the earth is the centre. By this method
of representation the motion of the superior planet is exactly like
that of an inferior planet, except that its epicycle is larger than
its deferent; the same reasoning as before shows that the motion can
be represented simply by supposing the centre J′ of the epicycle to be
actually the sun. Ptolemy’s epicycle and deferent are therefore capable
of being replaced, without affecting the position of the planet in the
sky, by a motion of the planet in a circle round the sun, while the
sun moves round the earth, or, more simply, the earth round the sun.

[Illustration: FIG. 46.—The relative sizes of the orbits of the earth
and of a superior planet.]

The synodic period of a superior planet could best be determined
by observing when the planet was in opposition, _i.e._ when it was
(nearly) opposite the sun, or, more accurately (since a planet does
not move exactly in the ecliptic), when the longitudes of the planet
and sun differed by 180° (or two right angles, chapter II., § 43).
The sidereal period could then be deduced nearly as in the case of
an inferior planet, with this difference, that the superior planet
moves more slowly than the earth, and therefore _loses_ one complete
revolution in each synodic period; or the sidereal period might be
found as before by observing when oppositions occurred nearly in
the same part of the sky.[56] Coppernicus thus obtained very fairly
accurate values for the synodic and sidereal periods, _viz._ 780 days
and 687 days respectively for Mars, 399 days and about 12 years for
Jupiter, 378 days and 30 years for Saturn (cf. fig. 40).

The calculation of the distance of a superior planet from the sun is a
good deal more complicated than that of Venus or Mercury. If we ignore
various details, the process followed by Coppernicus is to compute the
position of the planet as seen from the sun, and then to notice when
this position differs most from its position as seen from the earth,
_i.e._ when the earth and sun are farthest apart as seen from the
planet. This is clearly when (fig. 46) the line joining the planet (P)
to the earth (E) touches the circle described by the earth, so that the
angle S P E is then as great as possible. The angle P E S is a right
angle, and the angle S P E is the difference between the observed place
of the planet and its computed place as seen from the sun; these two
angles being thus known, the shape of the triangle S P E is known,
and therefore also the ratio of its sides. In this way Coppernicus
found the average distances of Mars, Jupiter, and Saturn from the sun
to be respectively about 1-1∕2, 5, and 9 times that of the earth; the
corresponding modern figures are 1·5, 5·2, 9·5.

88. The explanation of the stationary points of the planets (chapter
I., § 14) is much simplified by the ideas of Coppernicus. If we take
first an inferior planet, say Mercury (fig. 47), then when it lies
between the earth and sun, as at M (or as on Sept. 5 in fig. 7),
both the earth and Mercury are moving in the same direction, but a
comparison of the sizes of the paths of Mercury and the earth, and of
their respective times of performing complete circuits, shews that
Mercury is moving faster than the earth. Consequently to the observer
at E, Mercury appears to be moving from left to right (in the figure),
or from east to west; but this is contrary to the general direction
of motion of the planets, _i.e._ Mercury appears to be retrograding.
On the other hand, when Mercury appears at the greatest distance from
the sun, as at M_{1} and M_{2}, its own motion is directly towards or
away from the earth, and is therefore imperceptible; but the earth is
moving towards the observer’s right, and therefore Mercury appears to
be moving towards the left, or from west to east. Hence between M_{1}
and M its motion has changed from direct to retrograde, and therefore
at some intermediate point, say _m_{1}_, (about Aug, 23 in fig. 7),
Mercury appears for the moment to be stationary, and similarly it
appears to be stationary again when at some point _m_{2}_ between M and
M_{2} (about Sept. 13 in fig. 7).

[Illustration: FIG. 47.—The stationary points of Mercury.]

In the case of a superior planet, say Jupiter, the argument is nearly
the same. When in opposition at J (as on Mar. 26 in fig. 6), Jupiter
moves more slowly than the earth, and in the same direction, and
therefore appears to be moving in the opposite direction to the earth,
_i.e._ as seen from E (fig. 48), from left to right, or from east to
west, that is in the retrograde direction. But when Jupiter is in
either of the positions J_{1} or J (in which the earth appears to the
observer on Jupiter to be at its greatest distance from the sun), the
motion of the earth itself being directly to or from Jupiter produces
no effect on the apparent motion of Jupiter (since any displacement
directly to or from the observer makes no difference in the object’s
place on the celestial sphere); but Jupiter itself is actually moving
towards the left, and therefore the motion of Jupiter appears to be
also from right to left, or from west to east. Hence, as before,
between J_{1} and J and between J and J_{2} there must be points
_j_{1}_, _j_{2}_ (Jan. 24 and May 27, in fig. 6) at which Jupiter
appears for the moment to be stationary.

[Illustration: FIG. 48.—The stationary points of Jupiter.]

The actual discussion of the stationary points given by Coppernicus is
a good deal more elaborate and more technical than the outline given
here, as he not only shews that the stationary points must exist, but
shews how to calculate their exact positions.

89. So far the theory of the planets has only been sketched very
roughly, in order to bring into prominence the essential differences
between the Coppernican and the Ptolemaic explanations of their
motions, and no account has been taken of the minor irregularities for
which Ptolemy devised his system of equants, eccentrics, etc., nor of
the motion in latitude, _i.e._ to and from the ecliptic. Coppernicus,
as already mentioned, rejected the equant, as being productive of
an irregularity “unworthy” of the celestial bodies, and constructed
for each planet a fairly complicated system of epicycles. For the
motion in latitude discussed in Book VI. he supposed the orbit of each
planet round the sun to be inclined to the ecliptic at a small angle,
different for each planet, but found it necessary, in order that his
theory should agree with observation, to introduce the wholly imaginary
complication of a regular increase and decrease in the inclinations of
the orbits of the planets to the ecliptic.

The actual details of the epicycles employed are of no great interest
now, but it may be worth while to notice that for the motions of the
moon, earth, and five other planets Coppernicus required altogether
34 circles, _viz._ four for the moon, three for the earth, seven for
Mercury (the motion of which is peculiarly irregular), and five for
each of the other planets; this number being a good deal less than that
required in most versions of Ptolemy’s system: Fracastor (chapter III.,
§ 69), for example, writing in 1538, required 79 spheres, of which six
were required for the fixed stars.

90. The planetary theory of Coppernicus necessarily suffered from one
of the essential defects of the system of epicycles. It is, in fact,
always possible to choose a system of epicycles in such a way as to
make _either_ the direction of any body _or_ its distance vary in any
required manner, but not to satisfy both requirements at once. In the
case of the motion of the moon round the earth, or of the earth round
the sun, cases in which variations in distance could not readily be
observed, epicycles might therefore be expected to give a satisfactory
result, at any rate until methods of observation were sufficiently
improved to measure with some accuracy the apparent sizes of the sun
and moon, and so check the variations in their distances. But any
variation in the distance of the earth from the sun would affect not
merely the distance, but also the direction in which a planet would be
seen; in the figure, for example, when the planet is at P and the sun
at S, the apparent position of the planet, as seen from the earth, will
be different according as the earth is at E or E′. Hence the epicycles
and eccentrics of Coppernicus, which had to be adjusted in such a
way that they necessarily involved incorrect values of the distances
between the sun and earth, gave rise to corresponding errors in the
observed places of the planets. The observations which Coppernicus
used were hardly extensive or accurate enough to show this discrepancy
clearly; but a crucial test was thus virtually suggested by means
of which, when further observations of the planets had been made, a
decision could be taken between an epicyclic representation of the
motion of the planets and some other geometrical scheme.

[Illustration: FIG. 49.—The alteration in a planet’s apparent position
due to an alteration in the earth’s distance from the sun.]

91. The merits of Coppernicus are so great, and the part which he
played in the overthrow of the Ptolemaic system is so conspicuous,
that we are sometimes liable to forget that, so far from rejecting the
epicycles and eccentrics of the Greeks, he used no other geometrical
devices, and was even a more orthodox “epicyclist” than Ptolemy
himself, as he rejected the equants of the latter.[57] Milton’s famous
description (_Par. Lost_, VIII. 82-5) of

                          “The Sphere
    With Centric and Eccentric scribbled o’er,
    Cycle and Epicycle, Orb in Orb,”

applies therefore just as well to the astronomy of Coppernicus as to
that of his predecessors; and it was Kepler (chapter VII.), writing
more than half a century later, not Coppernicus, to whom the rejection
of the epicycle and eccentric is due.

92. One point which was of importance in later controversies deserves
special mention here. The basis of the Coppernican system was that a
motion of the earth carrying the observer with it produced an apparent
motion of other bodies. The apparent motions of the sun and planets
were thus shewn to be in great part explicable as the result of the
motion of the earth round the sun. Similar reasoning ought apparently
to lead to the conclusion that the fixed stars would also appear to
have an annual motion. There would, in fact, be a displacement of
the apparent position of a star due to the alteration of the earth’s
position in its orbit, closely resembling the alteration in the
apparent position of the moon due to the alteration of the observer’s
position on the earth which had long been studied under the name of
parallax (chapter II., § 43). As such a displacement had never been
observed, Coppernicus explained the apparent contradiction by supposing
the fixed stars so far off that any motion due to this cause was too
small to be noticed. If, for example, the earth moves in six months
from E to E′, the change in direction of a star at S′ is the angle E′
S′ E, which is less than that of a nearer star at S; and by supposing
the star S′ sufficiently remote, the angle E′ S′ E can be made as small
as may be required. For instance, if the distance of the star were 300
times the distance E E′, _i.e._ 600 times as far from the earth as the
sun is, the angle E S′ E′ would be less than 12′, a quantity which the
instruments of the time were barely capable of detecting.[58] But more
accurate observations of the fixed stars might be expected to throw
further light on this problem.

[Illustration: FIG. 50.—Stellar parallax.]



CHAPTER V.

THE RECEPTION OF THE COPPERNICAN THEORY AND THE PROGRESS OF OBSERVATION.

    “Preposterous wits that cannot row at ease
    On the smooth channel of our common seas;
    And such are those, in my conceit at least,
    Those clerks that think—think how absurd a jest!
    That neither heavens nor stars do turn at all
    Nor dance about this great round Earthly Ball,
    But the Earth itself, this massy globe of ours,
    Turns round about once every twice twelve hours!”
                                   DU BARTAS (Sylvester’s translation).


93. The publication of the _De Revolutionibus_ appears to have been
received much more quietly than might have been expected from the
startling nature of its contents. The book, in fact, was so written
as to be unintelligible except to mathematicians of considerable
knowledge and ability, and could not have been read at all generally.
Moreover the preface, inserted by Osiander but generally supposed to
be by the author himself, must have done a good deal to disarm the
hostile criticism due to prejudice and custom, by representing the
fundamental principles of Coppernicus as mere geometrical abstractions,
convenient for calculating the celestial motions. Although, as we
have seen (chapter IV., § 73), the contradiction between the opinions
of Coppernicus and the common interpretation of various passages in
the Bible was promptly noticed by Luther, Melanchthon, and others, no
objection was raised either by the Pope to whom the book was dedicated,
or by his immediate successors.

The enthusiastic advocacy of the Coppernican views by Rheticus has
already been referred to. The only other astronomer of note who at
once accepted the new views was his friend and colleague _Erasmus
Reinhold_ (born at Saalfeld in 1511), who occupied the chief chair
of mathematics and astronomy at Wittenberg from 1536 to 1553, and it
thus happened, curiously enough, that the doctrines so emphatically
condemned by two of the great Protestant leaders were championed
principally in what was generally regarded as the very centre of
Protestant thought.

94. Rheticus, after the publication of the _Narratio Prima_ and of an
Ephemeris or Almanack based on Coppernican principles (1550), occupied
himself principally with the calculation of a very extensive set of
mathematical tables, which he only succeeded in finishing just before
his death in 1576.

Reinhold rendered to astronomy the extremely important service of
calculating, on the basis of the _De Revolutionibus_, tables of the
motions of the celestial bodies, which were published in 1551 at the
expense of Duke Albert of Prussia and hence called _Tabulæ Prutenicæ_,
or _Prussian Tables_. Reinhold revised most of the calculations made
by Coppernicus, whose arithmetical work was occasionally at fault; but
the chief object of the tables was the development in great detail of
the work in the _De Revolutionibus_, in such a form that the places of
the chief celestial bodies at any required time could be ascertained
with ease. The author claimed for his tables that from them the places
of all the heavenly bodies could be computed for the past 3,000 years,
and would agree with all observations recorded during that period. The
tables were indeed found to be on the whole decidedly superior to their
predecessors the _Alfonsine Tables_ (chapter III., § 66), and gradually
came more and more into favour, until superseded three-quarters of a
century later by the _Rudolphine Tables_ of Kepler (chapter VII., §
148). This superiority of the new tables was only indirectly connected
with the difference in the principles on which the two sets of tables
were based, and was largely due to the facts that Reinhold was a much
better computer than the assistants of Alfonso, and that Coppernicus,
if not a better mathematician than Ptolemy, at any rate had better
mathematical tools at command. Nevertheless the tables naturally, had
great weight in inducing the astronomical world gradually to recognise
the merits of the Coppernican system, at any rate as a basis for
calculating the places of the celestial bodies.

Reinhold was unfortunately cut off by the plague in 1553, and with
him disappeared a commentary on the _De Revolutionibus_ which he had
prepared but not published.

95. Very soon afterwards we find the first signs that the Coppernican
system had spread into England. In 1556 _John Field_ published an
almanack for the following year avowedly based on Coppernicus and
Reinhold, and a passage in the _Whetstone of Witte_ (1557) by _Robert
Recorde_ (1510-1558), our first writer on algebra, shews that the
author regarded the doctrines of Coppernicus with favour, even if
he did not believe in them entirely. A few years later _Thomas
Digges_ (?-1595), in his _Alae sive Scalae Mathematicae_ (1573), an
astronomical treatise of no great importance, gave warm praise to
Coppernicus and his ideas.

96. For nearly half a century after the death of Reinhold no important
contributions were made to the Coppernican controversy. Reinhold’s
tables were doubtless slowly doing their work in familiarising men’s
minds with the new ideas, but certain definite additions to knowledge
had to be made before the evidence for them could be regarded as really
conclusive.

The serious mechanical difficulties connected with the assumption
of a rapid motion of the earth which is quite imperceptible to its
inhabitants could only be met by further progress in mechanics, and
specially in knowledge of the laws according to which the motion of
bodies is produced, kept up, changed, or destroyed; in this direction
no considerable progress was made before the time of Galilei, whose
work falls chiefly into the early 17th century (cf. chapter VI., §§
116, 130, 133).

The objection to the Coppernican scheme that the stars shewed no such
apparent annual motions as the motion of the earth should produce
(chapter IV., § 92) would also be either answered or strengthened
according as improved methods of observation did or did not reveal the
required motion.

Moreover the _Prussian Tables_, although more accurate than
the _Alfonsine_, hardly claimed, and certainly did not possess,
minute accuracy. Coppernicus had once told Rheticus that he would
be extravagantly pleased if he could make his theory agree with
observation to within 10′; but as a matter of fact discrepancies
of a much more serious character were noticed from time to time.
The comparatively small number of observations available and their
roughness made it extremely difficult, either to find the most
satisfactory numerical data necessary for the detailed development of
any theory, or to test the theory properly by comparison of calculated
with observed places of the celestial bodies. Accordingly it became
evident to more than one astronomer that one of the most pressing
needs of the science was that observations should be taken on as large
a scale as possible and with the utmost attainable accuracy. To meet
this need two schools of observational astronomy, of very unequal
excellence, developed during the latter half of the 16th century, and
provided a mass of material for the use of the astronomers of the
next generation. Fortunately too the same period was marked by rapid
progress in algebra and allied branches of mathematics. Of the three
great inventions which have so enormously diminished the labour of
numerical calculations, one, the so-called Arabic notation (chapter
III., § 64), was already familiar, the other two (decimal fractions
and logarithms) were suggested in the 16th century and were in working
order early in the 17th century.

97. The first important set of observations taken after the death of
Regiomontanus and Walther (chapter III., § 68) were due to the energy
of the Landgrave _William IV._ of Hesse (1532-1592). He was remarkable
as a boy for his love of study, and is reported to have had his
interest in astronomy created or stimulated when he was little more
than 20 by a copy of Apian’s beautiful _Astronomicum Caesareum_, the
cardboard models in which he caused to be imitated and developed in
metal-work. He went on with the subject seriously, and in 1561 had an
observatory built at Cassel, which was remarkable as being the first
which had a revolving roof, a device now almost universal. In this he
made extensive observations (chiefly of fixed stars) during the next
six years. The death of his father then compelled him to devote most
of his energy to the duties of government, and his astronomical ardour
abated. A few years later, however (1575), as the result of a short
visit from the talented and enthusiastic young Danish astronomer Tycho
Brahe (§ 99), he renewed his astronomical work, and secured shortly
afterwards the services of two extremely able assistants, _Christian
Rothmann_ (in 1577) and _Joost Bürgi_ (in 1579). Rothmann, of whose
life extremely little is known, appears to have been a mathematician
and theoretical astronomer of considerable ability, and was the author
of several improvements in methods of dealing with various astronomical
problems. He was at first a Coppernican, but shewed his independence
by calling attention to the needless complication introduced by
Coppernicus in resolving the motion of the earth into _three_ motions
when two sufficed (chapter IV., § 79). His faith in the system was,
however, subsequently shaken by the errors which observation revealed
in the _Prussian Tables_. Bürgi (1552-1632) was originally engaged by
the Landgrave as a clockmaker, but his remarkable mechanical talents
were soon turned to astronomical account, and it then appeared that he
also possessed unusual ability as a mathematician.[59]

98. The chief work of the Cassel Observatory was the formation of a
star catalogue. The positions of stars were compared with that of
the sun, Venus or Jupiter being used as connecting links, and their
positions relatively to the equator and the first point of Aries
(♈) deduced; allowance was regularly made for the errors due to the
refraction of light by the atmosphere, as well as for the parallax of
the sun, but the most notable new departure was the use of a clock
to record the time of observations and to measure the motion of the
celestial sphere. The construction of clocks of sufficient accuracy for
the purpose was rendered possible by the mechanical genius of Bürgi,
and in particular by his discovery that a clock could be regulated by
a pendulum, a discovery which he appears to have taken no steps to
publish, and which had in consequence to be made again independently
before it received general recognition.[60] By 1586 121 stars had been
carefully observed, but a more extensive catalogue which was to have
contained more than a thousand stars was never finished, owing to the
unexpected disappearance of Rothmann in 1590[61] and the death of the
Landgrave two years later.

99. The work of the Cassel Observatory was, however, overshadowed by
that carried out nearly at the same time by _Tycho_ (_Tyge_) _Brahe_.
He was born in 1546 at Knudstrup in the Danish province of Scania
(now the southern extremity of Sweden), being the eldest child of a
nobleman who was afterwards governor of Helsingborg Castle. He was
adopted as an infant by an uncle, and brought up at his country estate.
When only 13 he went to the University of Copenhagen, where he began
to study rhetoric and philosophy, with a view to a political career.
He was, however, very much interested by a small eclipse of the sun
which he saw in 1560, and this stimulus, added to some taste for the
astrological art of casting horoscopes, led him to devote the greater
part of the remaining two years spent at Copenhagen to mathematics
and astronomy. In 1562 he went on to the University of Leipzig,
accompanied, according to the custom of the time, by a tutor, who
appears to have made persevering but unsuccessful attempts to induce
his pupil to devote himself to law. Tycho, however, was now as always
a difficult person to divert from his purpose, and went on steadily
with his astronomy. In 1563 he made his first recorded observation, of
a close approach of Jupiter and Saturn, the time of which he noticed
to be predicted a whole month wrong by the _Alfonsine Tables_ (chapter
III., § 66), while the _Prussian Tables_ (§ 94) were several days
in error. While at Leipzig he bought also a few rough instruments,
and anticipated one of the great improvements afterwards carried out
systematically, by trying to estimate and to allow for the errors of
his instruments.

In 1565 Tycho returned to Copenhagen, probably on account of the war
with Sweden which had just broken out, and stayed about a year, during
the course of which he lost his uncle. He then set out again (1566)
on his travels, and visited Wittenberg, Rostock, Basle, Ingolstadt,
Augsburg, and other centres of learning, thus making acquaintance with
several of the most notable astronomers of Germany. At Augsburg he
met the brothers Hainzel, rich citizens with a taste for science, for
one of whom he designed and had constructed an enormous =quadrant=
(quarter-circle) with a radius of about 19 feet, the rim of which was
graduated to single minutes; and he began also here the construction of
his great celestial globe, five feet in diameter, on which he marked
one by one the positions of the stars as he afterwards observed them.

In 1570 Tycho returned to his father at Helsingborg, and soon after
the death of the latter (1571) went for a long visit to Steen Bille,
an uncle with scientific tastes. During this visit he seems to have
devoted most of his time to chemistry (or perhaps rather to alchemy),
and his astronomical studies fell into abeyance for a time.

100. His interest in astronomy was fortunately revived by the
sudden appearance, in November 1572, of a brilliant new star in the
constellation Cassiopeia. Of this Tycho took a number of extremely
careful observations; he noted the gradual changes in its brilliancy
from its first appearance, when it rivalled Venus at her brightest,
down to its final disappearance 16 months later. He repeatedly measured
its angular distance from the chief stars in Cassiopeia, and applied
a variety of methods to ascertain whether it had any perceptible
parallax (chapter II., §§ 43, 49). No parallax could be definitely
detected, and he deduced accordingly that the star must certainly be
farther off than the moon; as moreover it had no share in the planetary
motions, he inferred that it must belong to the region of the fixed
stars. To us of to-day this result may appear fairly commonplace, but
most astronomers of the time held so firmly to Aristotle’s doctrine
that the heavens generally, and the region of the fixed stars in
particular, were incorruptible and unchangeable, that new stars
were, like comets, almost universally ascribed to the higher regions
of our own atmosphere. Tycho wrote an account of the new star, which
he was ultimately induced by his friends to publish (1573), together
with some portions of a calendar for that year which he had prepared.
His reluctance to publish appears to have been due in great part to
a belief that it was unworthy of the dignity of a Danish nobleman to
write books! The book in question (_De Nova ... Stella_) compares very
favourably with the numerous other writings which the star called
forth, though it shews that Tycho held the common beliefs that comets
were in our atmosphere, and that the planets were carried round by
solid crystalline spheres, two delusions which his subsequent work did
much to destroy. He also dealt at some length with the astrological
importance of the star, and the great events which it foreshadowed,
utterances on which Kepler subsequently made the very sensible
criticism that “if that star did nothing else, at least it announced
and produced a great astronomer.”

In 1574 Tycho was requested to give some astronomical lectures at
the University of Copenhagen, the first of which, dealing largely
with astrology, was printed in 1610, after his death. When these were
finished, he set off again on his travels (1575). After a short visit
to Cassel (§ 97), during which he laid the foundation of a lifelong
friendship with the Landgrave, he went on to Frankfort to buy books,
thence to Basle (where he had serious thoughts of settling) and on to
Venice, then back to Augsburg and to Regensburg, where he obtained a
copy of the _Commentariolus_ of Coppernicus (chapter IV., § 73), and
finally came home by way of Saalfeld and Wittenberg.

101. The next year (1576) was the beginning of a new epoch in Tycho’s
career. The King of Denmark, Frederick II., who was a zealous patron
of science and literature, determined to provide Tycho with endowments
sufficient to enable him to carry out his astronomical work in the most
effective way. He accordingly gave him for occupation the little island
of Hveen in the Sound (now belonging to Sweden), promised money for
building a house and observatory, and supplemented the income derived
from the rents of the island by an annual payment of about £100. Tycho
paid his first visit to the island in May, soon set to work building,
and had already begun to make regular observations in his new house
before the end of the year.

[Illustration: FIG. 51.—Uraniborg. From a collection of letters
published by Tycho.]

The buildings were as remarkable for their magnificence as for their
scientific utility. Tycho never forgot that he was a Danish nobleman as
well as an astronomer, and built in a manner suitable to his rank.[62]
His chief building (fig. 51), called Uraniborg (the Castle of the
Heavens), was in the middle of a large square enclosure, laid out as a
garden, the corners of which pointed North, East, South, and West, and
contained several observatories, a library and laboratory, in addition
to living rooms. Subsequently, when the number of pupils and assistants
who came to him had increased, he erected (1584) a second building,
Stjerneborg (Star Castle), which was remarkable for having underground
observatories. The convenience of being able to carry out all necessary
work on his own premises induced him moreover to establish workshops,
where nearly all his instruments were made, and afterwards also a
printing press and paper mill. Both at Uraniborg and Stjerneborg
not only the rooms, but even the instruments which were gradually
constructed, were elaborately painted or otherwise ornamented.

102. The expenses of the establishment must have been enormous,
particularly as Tycho lived in magnificent style and probably paid
little attention to economy. His income was derived from various
sources, and fluctuated from time to time, as the King did not merely
make him a fixed annual payment, but added also temporary grants of
lands or money. Amongst other benefactions he received in 1579 one of
the canonries of the cathedral of Roskilde, the endowments of which
had been practically secularised at the Reformation. Unfortunately
most of his property was held on tenures which involved corresponding
obligations, and as he combined the irritability of a genius with
the haughtiness of a mediaeval nobleman, continual quarrels were the
result. Very soon after his arrival at Hveen his tenants complained
of work which he illegally forced from them; chapel services which his
canonry required him to keep up were neglected, and he entirely refused
to make certain recognised payments to the widow of the previous canon.
Further difficulties arose out of a lighthouse, the maintenance of
which was a duty attached to one of his estates, but was regularly
neglected. Nothing shews the King’s good feeling towards Tycho more
than the trouble which he took to settle these quarrels, often ending
by paying the sum of money under dispute. Tycho was moreover extremely
jealous of his scientific reputation, and on more than one occasion
broke out into violent abuse of some assistant or visitor whom he
accused of stealing his ideas and publishing them elsewhere.

In addition to the time thus spent in quarrelling, a good deal must
have been occupied in entertaining the numerous visitors whom his
fame attracted, and who included, in addition to astronomers, persons
of rank such as several of the Danish royal family and James VI. of
Scotland (afterwards James I. of England).

Notwithstanding these distractions, astronomical work made steady
progress, and during the 21 years that Tycho spent at Hveen he
accumulated, with the help of pupils and assistants, a magnificent
series of observations, far transcending in accuracy and extent
anything that had been accomplished by his predecessors. A good deal of
attention was also given to alchemy, and some to medicine. He seems to
have been much impressed with the idea of the unity of Nature, and to
have been continually looking out for analogies or actual connection
between the different subjects which he studied.

103. In 1577 appeared a brilliant comet, which Tycho observed with
his customary care; and, although he had not at the time his full
complement of instruments, his observations were exact enough to
satisfy him that the comet was at least three times as far off as the
moon, and thus to refute the popular belief, which he had himself
held a few years before (§ 100), that comets were generated in our
atmosphere. His observations led him also to the belief that the comet
was revolving round the sun, at a distance from it greater than that
of Venus, a conclusion which interfered seriously with the common
doctrine of the solid crystalline spheres. He had further opportunities
of observing comets in 1580, 1582, 1585, 1590, and 1596, and one of his
pupils also took observations of a comet seen in 1593. None of these
comets attracted as much general attention as that of 1577, but Tycho’s
observations, as was natural, gradually improved in accuracy.

104. The valuable results obtained by means of the new star of 1572,
and by the comets, suggested the propriety of undertaking a complete
treatise on astronomy embodying these and other discoveries. According
to the original plan, there were to be three preliminary volumes
devoted respectively to the new star, to the comet of 1577, and to
the later comets, while the main treatise was to consist of several
more volumes dealing with the theories of the sun, moon, and planets.
Of this magnificent plan comparatively little was ever executed. The
first volume, called the _Astronomiae Instauratae Progymnasmata_, or
Introduction to the New Astronomy, was hardly begun till 1588, and,
although mostly printed by 1592, was never quite finished during
Tycho’s lifetime, and was actually published by Kepler in 1602. One
question, in fact, led to another in such a way that Tycho felt himself
unable to give a satisfactory account of the star of 1572 without
dealing with a number of preliminary topics, such as the positions of
the fixed stars, precession, and the annual motion of the sun, each
of which necessitated an elaborate investigation. The second volume,
dealing with the comet of 1577, called _De Mundi aetherei recentioribus
Phaenomenis Liber secundus_ (Second book about recent appearances in
the Celestial World), was finished long before the first, and copies
were sent to friends and correspondents in 1588, though it was not
regularly published and on sale till 1603. The third volume was never
written, though some material was collected for it, and the main
treatise does not appear to have been touched.

[Illustration: FIG. 52.—Tycho’s system of the world. From his book on
the comet of 1577.]

105. The book on the comet of 1577 is of special interest, as
containing an account of Tycho’s system of the world, which was a
compromise between those of Ptolemy and of Coppernicus. Tycho was too
good an astronomer not to realise many of the simplifications which
the Coppernican system introduced, but was unable to answer two of the
serious objections; he regarded any motion of “the sluggish and heavy
earth” as contrary to “physical principles,” and he objected to the
great distance of the stars which the Coppernican system required,
because a vast empty space would be left between them and the planets,
a space which he regarded as wasteful.[63] Biblical difficulties[64]
also had some weight with him. He accordingly devised (1583) a new
system according to which the five planets revolved round the sun (C,
in fig. 52), while the sun revolved annually round the earth (A), and
the whole celestial sphere performed also a daily revolution round
the earth. The system was never worked out in detail, and, like many
compromises, met with little support; Tycho nevertheless was extremely
proud of it, and one of the most violent and prolonged quarrels of
his life (lasting a dozen years) was with _Reymers Bär_ or _Ursus_
(?-1600), who had communicated to the Landgrave in 1586 and published
two years later a system of the world very like Tycho’s. Reymers had
been at Hveen for a short time in 1584, and Tycho had no hesitation in
accusing him of having stolen the idea from some manuscript seen there.
Reymers naturally retaliated with a counter-charge of theft against
Tycho. There is, however, no good reason why the idea should not have
occurred independently to each astronomer; and Reymers made in some
respects a great improvement on Tycho’s scheme by accepting the daily
rotation of the earth, and so doing away with the daily rotation of the
celestial sphere, which was certainly one of the weakest parts of the
Ptolemaic scheme.

[Illustration: TYCHO BRAHE.]

106. The same year (1588) which saw the publication of Tycho’s book on
the comet was also marked by the death of his patron, Frederick II.
The new King Christian was a boy of 11, and for some years the country
was managed by four leading statesmen. The new government seems to
have been at first quite friendly to Tycho; a large sum was paid to
him for expenses incurred at Hveen, and additional endowments were
promised, but as time went on Tycho’s usual quarrels with his tenants
and others began to produce their effect. In 1594 he lost one of
his chief supporters at court, the Chancellor Kaas, and his successor,
as well as two or three other important officials at court, were not
very friendly, although the stories commonly told of violent personal
animosities appear to have little foundation. As early as 1591 Tycho
had hinted to a correspondent that he might not remain permanently in
Denmark, and in 1594 he began a correspondence with representatives
of the Emperor Rudolph II., who was a patron of science. But his
scientific activity during these years was as great as ever; and in
1596 he completed the printing of an extremely interesting volume of
scientific correspondence between the Landgrave, Rothmann, and himself.
The accession of the young King to power in 1596 was at once followed
by the withdrawal of one of Tycho’s estates, and in the following year
the annual payment which had been made since 1576 was stopped. It is
difficult to blame the King for these economies; he was evidently
not as much interested in astronomy as his father, and consequently
regarded the heavy expenditure at Hveen as an extravagance, and it is
also probable that he was seriously annoyed at Tycho’s maltreatment of
his tenants, and at other pieces of unruly conduct on his part. Tycho,
however, regarded the forfeiture of his annual pension as the last
straw, and left Hveen early in 1597, taking his more portable property
with him. After a few months spent in Copenhagen, he took the decisive
step of leaving Denmark for Germany, in return for which action the
King deprived him of his canonry. Tycho thereupon wrote a remonstrance
in which he pointed out the impossibility of carrying on his work
without proper endowments, and offered to return if his services were
properly appreciated. The King, however, was by this time seriously
annoyed, and his reply was an enumeration of the various causes of
complaint against Tycho which had arisen of late years. Although Tycho
made some more attempts through various friends to regain royal favour,
the breach remained final.

107. Tycho spent the winter 1597-8 with a friend near Hamburg, and,
while there, issued, under the title of _Astronomiae Instauratae
Mechanica_, a description of his instruments, together with a short
autobiography and an interesting account of his chief discoveries.
About the same time he circulated manuscript copies of a catalogue of
1,000 fixed stars, of which only 777 had been properly observed, the
rest having been added hurriedly to make up the traditional number. The
catalogue and the _Mechanica_ were both intended largely as evidence
of his astronomical eminence, and were sent to various influential
persons. Negotiations went on both with the Emperor and with the Prince
of Orange, and after another year spent in various parts of Germany,
Tycho definitely accepted an invitation of the Emperor and arrived at
Prague in June 1599.

108. It was soon agreed that he should inhabit the castle of Benatek,
some twenty miles from Prague, where he accordingly settled with
his family and smaller instruments towards the end of 1599. He at
once started observing, sent one of his sons to Hveen for his larger
instruments, and began looking about for assistants. He secured one of
the most able of his old assistants, and by good fortune was also able
to attract a far greater man, _John Kepler_, to whose skilful use of
the materials collected by Tycho the latter owes no inconsiderable part
of his great reputation. Kepler, whose life and work will be dealt with
at length in chapter VII., had recently published his first important
work, the _Mysterium Cosmographicum_ (§ 136), which had attracted the
attention of Tycho among others, and was beginning to find his position
at Gratz in Styria uncomfortable on account of impending religious
disputes. After some hesitation he joined Tycho at Benatek early in
1600. He was soon set to work at the study of Mars for the planetary
tables which Tycho was then preparing, and thus acquired special
familiarity with the observations of this planet which Tycho had
accumulated. The relations of the two astronomers were not altogether
happy, Kepler being then as always anxious about money matters, and the
disturbed state of the country rendering it difficult for Tycho to get
payment from the Emperor. Consequently Kepler very soon left Benatek
and returned to Prague, where he definitely settled after a short visit
to Gratz; Tycho also moved there towards the end of 1600, and they then
worked together harmoniously for the short remainder of Tycho’s life.
Though he was by no means an old man, there were some indications that
his health was failing, and towards the end of 1601 he was suddenly
seized with an illness which terminated fatally after a few days
(November 24th). It is characteristic of his devotion to the great work
of his life that in the delirium which preceded his death he cried out
again and again his hope that his life might not prove to have been
fruitless (_Ne frustra vixisse videar_).

109. Partly owing to difficulties between Kepler and one of Tycho’s
family, partly owing to growing political disturbances, scarcely any
use was made of Tycho’s instruments after his death, and most of them
perished during the Civil Wars in Bohemia. Kepler obtained possession
of his observations; but they have never been published except in an
imperfect form.

110. Anything like a satisfactory account of Tycho’s services to
astronomy would necessarily deal largely with technical details of
methods of observing, which would be out of place here. It may,
however, be worth while to attempt to give some general account of his
characteristics as an observer before referring to special discoveries.

Tycho realised more fully than any of his predecessors the importance
of obtaining observations which should not only be as accurate as
possible, but should be taken so often as to preserve an almost
continuous record of the positions and motions of the celestial bodies
dealt with; whereas the prevailing custom (as illustrated for example
by Coppernicus) was only to take observations now and then, either
when an astronomical event of special interest such as an eclipse or a
conjunction was occurring, or to supply some particular datum required
for a point of theory. While Coppernicus, as has been already noticed
(chapter IV., § 73), only used altogether a few dozen observations in
his book, Tycho—to take one instance—observed the sun _daily_ for many
years, and must therefore have taken some thousands of observations
of this one body, in addition to the many thousands which he took of
other celestial bodies. It is true that the Arabs had some idea of
observing continuously (cf. chapter III., § 57), but they had too
little speculative power or originality to be able to make much use
of their observations, few of which passed into the hands of European
astronomers. Regiomontanus (chapter III., § 68), if he had lived, might
probably have to a considerable extent anticipated Tycho, but his
short life was too fully occupied with the study and interpretation of
Greek astronomy for him to accomplish very much in other departments
of the subject. The Landgrave and his staff, who were in constant
communication with Tycho, were working in the same direction, though on
the whole less effectively. Unlike the Arabs, Tycho was, however, fully
impressed with the idea that observations were only a means to an end,
and that mere observations without a hypothesis or theory to connect
and interpret them were of little use.

The actual accuracy obtained by Tycho in his observations naturally
varied considerably according to the nature of the observation, the
care taken, and the period of his career at which it was made. The
places which he assigned to nine stars which were fundamental in
his star catalogue differ from their positions as deduced from the
best modern observations by angles which are in most cases less than
1′, and in only one case as great as 2′ (this error being chiefly
due to refraction (chapter II., § 46), Tycho’s knowledge of which
was necessarily imperfect). Other star places were presumably less
accurate, but it will not be far from the truth if we assume that in
most cases the errors in Tycho’s observations did not exceed 1′ or 2′.
Kepler in a famous passage speaks of an error of 8′ in a planetary
observation by Tycho as impossible. This great increase in accuracy
can only be assigned in part to the size and careful construction of
the instruments used, the characteristics on which the Arabs and other
observers had laid such stress. Tycho certainly used good instruments,
but added very much to their efficiency, partly by minor mechanical
devices, such as the use of specially constructed “sights” and of a
particular method of graduation,[65] and partly by using instruments
capable only of restricted motions, and therefore of much greater
steadiness than instruments which were able to point to any part of
the sky. Another extremely important idea was that of systematically
allowing as far as possible for the inevitable mechanical imperfections
of even the best constructed instruments, as well as for other
permanent causes of error. It had been long known, for example, that
the refraction of light through the atmosphere had the effect of
slightly raising the apparent places of stars in the sky. Tycho took a
series of observations to ascertain the amount of this displacement for
different parts of the sky, hence constructed a table of refractions
(a very imperfect one, it is true), and in future observations
regularly allowed for the effect of refraction. Again, it was known
that observations of the sun and planets were liable to be disturbed
by the effect of parallax (chapter II., §§ 43, 49), though the amount
of this correction was uncertain. In cases where special accuracy was
required, Tycho accordingly observed the body in question at least
twice, choosing positions in which parallax was known to produce nearly
opposite effects, and thus by combining the observations obtained a
result nearly free from this particular source of error. He was also
one of the first to realise fully the importance of repeating the same
observation many times under different conditions, in order that the
various accidental sources of error in the separate observations should
as far as possible neutralise one another.

111. Almost every astronomical quantity of importance was re-determined
and generally corrected by him. The annual motion of the sun’s apogee
relative to ♈, for example, which Coppernicus had estimated at 24″,
Tycho fixed at 45″, the modern value being 61″; the length of
the year he determined with an error of less than a second; and he
constructed tables of the motion of the sun which gave its place to
within 1′, previous tables being occasionally 15′ or 20′ wrong. By an
unfortunate omission he made no inquiry into the distance of the sun,
but accepted the extremely inaccurate value which had been handed down,
without substantial alteration, from astronomer to astronomer since the
time of Hipparchus (chapter II., § 41).

In the theory of the moon Tycho made several important discoveries.
He found that the irregularities in its movement were not fully
represented by the equation of the centre and the evection (chapter
II., §§ 39, 48), but that there was a further irregularity which
vanished at opposition and conjunction as well as at quadratures,
but in intermediate positions of the moon might be as great as 40′.
This irregularity, known as the =variation=, was, as has been already
mentioned (chapter III., § 60), very possibly discovered by Abul Wafa,
though it had been entirely lost subsequently. At a later stage in
his career, at latest during his visit to Wittenberg in 1598-9, Tycho
found that it was necessary to introduce a further small inequality
known as the =annual equation=, which depended on the position of the
earth in its path round the sun; this, however, he never completely
investigated. He also ascertained that the inclination of the moon’s
orbit to the ecliptic was not, as had been thought, fixed, but
oscillated regularly, and that the motion of the moon’s nodes (chapter
II., § 40) was also variable.

112. Reference has already been made to the star catalogue. Its
construction led to a study of precession, the amount of which was
determined with considerable accuracy; the same investigation led Tycho
to reject the supposed irregularity in precession which, under the name
of trepidation (chapter III., § 58), had confused astronomy for several
centuries, but from this time forward rapidly lost its popularity.

The planets were always a favourite subject of study with Tycho, but
although he made a magnificent series of observations, of immense value
to his successors, he died before he could construct any satisfactory
theory of the planetary motions. He easily discovered, however, that
their motions deviated considerably from those assigned by any of the
planetary tables, and got as far as detecting some regularity in these
deviations.



CHAPTER VI.

GALILEI.

 “Dans la Science nous sommes tous disciples de Galilée.”—TROUESSART.

 “Bacon pointed out at a distance the road to true philosophy: Galileo
 both pointed it out to others, and made himself considerable advances
 in it.”—DAVID HUME.


113. To the generation which succeeded Tycho belonged two of the best
known of all astronomers, Galilei and Kepler. Although they were nearly
contemporaries, Galilei having been born seven years earlier than
Kepler, and surviving him by twelve years, their methods of work and
their contributions to astronomy were so different in character, and
their influence on one another so slight, that it is convenient to make
some departure from strict chronological order, and to devote this
chapter exclusively to Galilei, leaving Kepler to the next.

_Galileo Galilei_ was born in 1564, at Pisa, at that time in the Grand
Duchy of Tuscany, on the day of Michel Angelo’s death and in the year
of Shakespeare’s birth. His father, Vincenzo, was an impoverished
member of a good Florentine family, and was distinguished by his skill
in music and mathematics. Galileo’s talents shewed themselves early,
and although it was originally intended that he should earn his living
by trade, Vincenzo was wise enough to see that his son’s ability and
tastes rendered him much more fit for a professional career, and
accordingly he sent him in 1581 to study medicine at the University
of Pisa. Here his unusual gifts soon made him conspicuous, and he
became noted in particular for his unwillingness to accept without
question the dogmatic statements of his teachers, which were based not
on direct evidence, but on the authority of the great writers of the
past. This valuable characteristic, which marked him throughout his
life, coupled with his skill in argument, earned for him the dislike of
some of his professors, and from his fellow-students the nickname of
The Wrangler.

114. In 1582 his keen observation led to his first scientific
discovery. Happening one day in the Cathedral of Pisa to be looking at
the swinging of a lamp which was hanging from the roof, he noticed that
as the motion gradually died away and the extent of each oscillation
became less, the time occupied by each oscillation remained sensibly
the same, a result which he verified more precisely by comparison with
the beating of his pulse. Further thought and trial shewed him that
this property was not peculiar to cathedral lamps, but that any weight
hung by a string (or any other form of pendulum) swung to and fro
in a time which depended only on the length of the string and other
characteristics of the pendulum itself, and not to any appreciable
extent on the way in which it was set in motion or on the extent of
each oscillation. He devised accordingly an instrument the oscillations
of which could be used while they lasted as a measure of time, and
which was in practice found very useful by doctors for measuring the
rate of a patient’s pulse.

115. Before very long it became evident that Galilei had no special
taste for medicine, a study selected for him chiefly as leading to a
reasonably lucrative professional career, and that his real bent was
for mathematics and its applications to experimental science. He had
received little or no formal teaching in mathematics before his second
year at the University, in the course of which he happened to overhear
a lesson on Euclid’s geometry, given at the Grand Duke’s court, and
was so fascinated that he continued to attend the course, at first
surreptitiously, afterwards openly; his interest in the subject was
thereby so much stimulated, and his aptitude for it was so marked,
that he obtained his father’s consent to abandon medicine in favour of
mathematics.

In 1585, however, poverty compelled him to quit the University without
completing the regular course and obtaining a degree, and the next four
years were spent chiefly at home, where he continued to read and to
think on scientific subjects. In the year 1586 he wrote his first known
scientific essay,[66] which was circulated in manuscript, and only
printed during the present century.

116. In 1589 he was appointed for three years to a professorship
of mathematics (including astronomy) at Pisa. A miserable stipend,
equivalent to about five shillings a week, was attached to the post,
but this he was to some extent able to supplement by taking private
pupils.

In his new position Galilei had scope for his remarkable power
of exposition, but far from being content with giving lectures
on traditional lines he also carried out a series of scientific
investigations, important both in themselves and on account of the
novelty in the method of investigation employed.

It will be convenient to discuss more fully at the end of this chapter
Galilei’s contributions to mechanics and to scientific method, and
merely to refer here briefly to his first experiments on falling
bodies, which were made at this time. Some were performed by dropping
various bodies from the top of the leaning tower of Pisa, and others
by rolling balls down grooves arranged at different inclinations.
It is difficult to us nowadays, when scientific experiments are so
common, to realise the novelty and importance at the end of the 16th
century of such simple experiments. The mediaeval tradition of carrying
out scientific investigation largely by the interpretation of texts
in Aristotle, Galen, or other great writers of the past, and by the
deduction of results from general principles which were to be found in
these writers without any fresh appeal to observation, still prevailed
almost undisturbed at Pisa, as elsewhere. It was in particular commonly
asserted, on the authority of Aristotle, that, the cause of the fall of
a heavy body being its weight, a heavier body must fall faster than a
lighter one and in proportion to its greater weight. It may perhaps be
doubted whether any one before Galilei’s time had clear enough ideas
on the subject to be able to give a definite answer to such a question
as how much farther a ten-pound weight would fall in a second than a
one-pound weight; but if so he would probably have said that it would
fall ten times as far, or else that it would require ten times as long
to fall the same distance. To actually try the experiment, to vary
its conditions, so as to remove as many accidental causes of error as
possible, to increase in some way the time of the fall so as to enable
it to be measured with more accuracy, these ideas, put into practice by
Galilei, were entirely foreign to the prevailing habits of scientific
thought, and were indeed regarded by most of his colleagues as
undesirable if not dangerous innovations. A few simple experiments were
enough to prove the complete falsity of the current beliefs in this
matter, and to establish that in general bodies of different weights
fell nearly the same distance in the same time, the difference being
not more than could reasonably be ascribed to the resistance offered by
the air.

These and other results were embodied in a tract, which, like most of
Galilei’s earlier writings, was only circulated in manuscript, the
substance of it being first printed in the great treatise on mechanics
which he published towards the end of his life (§ 133).

These innovations, coupled with the slight respect that he was in the
habit of paying to those who differed from him, evidently made Galilei
far from popular with his colleagues at Pisa, and either on this
account, or on account of domestic troubles consequent on the death of
his father (1591), he resigned his professorship shortly before the
expiration of his term of office, and returned to his mother’s home at
Florence.

117. After a few months spent at Florence he was appointed, by the
influence of a Venetian friend, to a professorship of mathematics at
Padua, which was then in the territory of the Venetian republic (1592).
The appointment was in the first instance for a period of six years,
and the salary much larger than at Pisa. During the first few years
of Galilei’s career at Padua his activity seems to have been very
great and very varied; in addition to giving his regular lectures,
to audiences which rapidly increased, he wrote tracts, for the most
part not printed at the time, on astronomy, on mechanics, and on
fortification, and invented a variety of scientific instruments.

No record exists of the exact time at which he first adopted the
astronomical views of Coppernicus, but he himself stated that in 1597
he had adopted them some years before, and had collected arguments in
their support.

In the following year his professorship was renewed for six years with
an increased stipend, a renewal which was subsequently made for six
years more, and finally for life, the stipend being increased on each
occasion.

Galilei’s first contribution to astronomical discovery was made in
1604, when a star appeared suddenly in the constellation Serpentarius,
and was shewn by him to be at any rate more distant than the planets, a
result confirming Tycho’s conclusions (chapter V., § 100) that changes
take place in the celestial regions even beyond the planets, and are
by no means confined—as was commonly believed—to the earth and its
immediate surroundings.

118. By this time Galilei had become famous throughout Italy, not only
as a brilliant lecturer, but also as a learned and original man of
science. The discoveries which first gave him a European reputation
were, however, the series of telescopic observations made in 1609 and
the following years.

Roger Bacon (chapter III., § 67) had claimed to have devised a
combination of lenses enabling distant objects to be seen as if they
were near; a similar invention was probably made by our countryman
_Leonard Digges_ (who died about 1571), and was described also by the
Italian _Porta_ in 1558. If such an instrument was actually made by
any one of the three, which is not certain, the discovery at any rate
attracted no attention and was again lost. The effective discovery
of the telescope was made in Holland in 1608 by _Hans Lippersheim_
(?-1619), a spectacle-maker of Middleburg, and almost simultaneously by
two other Dutchmen, but whether independently or not it is impossible
to say. Early in the following year the report of the invention
reached Galilei, who, though without any detailed information as to
the structure of the instrument, succeeded after a few trials in
arranging two lenses—one convex and one concave—in a tube in such a
way as to enlarge the apparent size of an object looked at; his first
instrument made objects appear three times nearer, consequently three
times greater (in breadth and height), and he was soon able to make
telescopes which in the same way magnified thirty-fold.

That the new instrument might be applied to celestial as well as
to terrestrial objects was a fairly obvious idea, which was acted
on almost at once by the English mathematician _Thomas Harriot_
(1560-1621), by _Simon Marius_ (1570-1624) in Germany, and by Galilei.
That the credit of first using the telescope for astronomical
purposes is almost invariably attributed to Galilei, though his first
observations were in all probability slightly later in date than those
of Harriot and Marius, is to a great extent justified by the persistent
way in which he examined object after object, whenever there seemed any
reasonable prospect of results following, by the energy and acuteness
with which he followed up each clue, by the independence of mind with
which he interpreted his observations, and above all by the insight
with which he realised their astronomical importance.

119. His first series of telescopic discoveries were published early
in 1610 in a little book called _Sidereus Nuncius_, or _The Sidereal
Messenger_. His first observations at once threw a flood of light
on the nature of our nearest celestial neighbour, the moon. It was
commonly believed that the moon, like the other celestial bodies, was
perfectly smooth and spherical, and the cause of the familiar dark
markings on the surface was quite unknown.[67]

[Illustration: FIG. 53.—One of Galilei’s drawings of the moon. From the
_Sidereus Nuncius_.]

Galilei discovered at once a number of smaller markings, both bright
and dark (fig. 53), and recognised many of the latter as shadows of
lunar mountains cast by the sun; and further identified bright spots
seen near the boundary of the illuminated and dark portions of the moon
as mountain-tops just catching the light of the rising or setting sun,
while the surrounding lunar area was still in darkness. Moreover, with
characteristic ingenuity and love of precision, he calculated from
observations of this nature the height of some of the more conspicuous
lunar mountains, the largest being estimated by him to be about
four miles high, a result agreeing closely with modern estimates of
the greatest height on the moon. The large dark spots he explained
(erroneously) as possibly caused by water, though he evidently had
less confidence in the correctness of the explanation than some of his
immediate scientific successors, by whom the name of =seas= was given
to these spots (chapter VIII., § 153). He noticed also the absence of
clouds. Apart however from details, the really significant results of
his observations were that the moon was in many important respects
similar to the earth, that the traditional belief in its perfectly
spherical form had to be abandoned, and that so far the received
doctrine of the sharp distinction to be drawn between things celestial
and things terrestrial was shewn to be without justification; the
importance of this in connection with the Coppernican view that the
earth, instead of being unique, was one of six planets revolving round
the sun, needs no comment.

One of Galilei’s numerous scientific opponents[68] attempted to explain
away the apparent contradiction between the old theory and the new
observations by the ingenious suggestion that the apparent valleys
in the moon were in reality filled with some _invisible_ crystalline
material, so that the moon was in fact perfectly spherical. To this
Galilei replied that the idea was so excellent that he wished to extend
its application, and accordingly maintained that the moon had on it
mountains of this same invisible substance, at least ten times as high
as any which he had observed.

120. The telescope revealed also the existence of an immense number
of stars too faint to be seen by the unaided eye; Galilei saw, for
example, 36 stars in the Pleiades, which to an ordinary eye consist
of six only. Portions of the Milky Way and various nebulous patches
of light were also discovered to consist of multitudes of faint stars
clustered together; in the cluster Præsepe (in the Crab), for example,
he counted 40 stars.

121. By far the most striking discovery announced in the _Sidereal
Messenger_ was that of the bodies now known as the moons or satellites
of Jupiter. On January 7th, 1610, Galilei turned his telescope on to
Jupiter, and noticed three faint stars which caught his attention on
account of their closeness to the planet and their arrangement nearly
in a straight line with it. He looked again next night, and noticed
that they had changed their positions relatively to Jupiter, but that
the change did not seem to be such as could result from Jupiter’s own
motion, if the new bodies were fixed stars. Two nights later he was
able to confirm this conclusion, and to infer that the new bodies
were not fixed stars, but moving bodies which accompanied Jupiter in
his movements. A fourth body was noticed on January 13th, and the
motions of all four were soon recognised by Galilei as being motions of
revolution round Jupiter as a centre. With characteristic thoroughness
he watched the motions of the new bodies night after night, and by the
date of the publication of his book had already estimated with very
fair accuracy their periods of revolution round Jupiter, which ranged
between about 42 hours and 17 days; and he continued to watch their
motions for years.

[Illustration: FIG. 54.—Jupiter and its satellites as seen on Jan. 7,
1610. From the _Sidereus Nuncius_.]

The new bodies were at first called by their discoverer Medicean
planets, in honour of his patron Cosmo de Medici, the Grand Duke of
Tuscany; but it was evident that bodies revolving round a planet, as
the planets themselves revolved round the sun, formed a new class of
bodies distinct from the known planets, and the name of =satellite=,
suggested by Kepler as applicable to the new bodies as well as to the
moon, has been generally accepted.

The discovery of Jupiter’s satellites shewed the falsity of the old
doctrine that the earth was the only centre of motion; it tended,
moreover, seriously to discredit the infallibility of Aristotle and
Ptolemy, who had clearly no knowledge of the existence of such bodies;
and again those who had difficulty in believing that Venus and Mercury
could revolve round an apparently moving body, the sun, could not
but have their doubts shaken when shewn the new satellites evidently
performing a motion of just this character; and—most important
consequence of all—the very real mechanical difficulty involved in the
Coppernican conception of the moon revolving round the moving earth and
not dropping behind was at any rate shewn not to be insuperable, as
Jupiter’s satellites succeeded in performing a precisely similar feat.

The same reasons which rendered Galilei’s telescopic discoveries of
scientific importance made them also objectionable to the supporters
of the old views, and they were accordingly attacked in a number of
pamphlets, some of which are still extant and possess a certain amount
of interest. One _Martin Horky_, for example, a young German who had
studied under Kepler, published a pamphlet in which, _after_ proving to
his own satisfaction that the satellites of Jupiter did not exist, he
discussed at some length what they were, what they were like, and why
they existed. Another writer gravely argued that because the human body
had seven openings in it—the eyes, ears, nostrils, and mouth—therefore
by analogy there must be seven planets (the sun and moon being
included) and no more. However, confirmation by other observers was
soon obtained and the pendulum even began to swing in the opposite
direction, a number of new satellites of Jupiter being announced by
various observers. None of these, however, turned out to be genuine,
and Galilei’s four remained the only known satellites of Jupiter till a
few years ago (chapter XIII., § 295).

122. The reputation acquired by Galilei by the publication of the
_Messenger_ enabled him to bring to a satisfactory issue negotiations
which he had for some time been carrying on with the Tuscan court.
Though he had been well treated by the Venetians, he had begun to feel
the burden of regular teaching somewhat irksome, and was anxious to
devote more time to research and to writing. A republic could hardly
be expected to provide him with such a sinecure as he wanted, and he
accordingly accepted in the summer of 1610 an appointment as professor
at Pisa, and also as “First Philosopher and Mathematician” to the
Grand Duke of Tuscany, with a handsome salary and no definite duties
attached to either office.

123. Shortly before leaving Padua he turned his telescope on to Saturn,
and observed that the planet appeared to consist of three parts, as
shewn in the first drawing of fig. 67 (chapter VIII., § 154). On
subsequent occasions, however, he failed to see more than the central
body, and the appearances of Saturn continued to present perplexing
variations, till the mystery was solved by Huygens in 1655 (chapter
VIII., § 154).

The first discovery made at Florence (October 1610) was that Venus,
which to the naked eye appears to vary very much in brilliancy but not
in shape, was in reality at times crescent-shaped like the new moon
and passed through phases similar to some of those of the moon. This
shewed that Venus was, like the moon, a dark body in itself, deriving
its light from the sun; so that its similarity to the earth was thereby
made more evident.

[Illustration: FIG. 55.—Sun-spots. From Galilei’s _Macchie Solari_.]

124. The discovery of dark spots on the sun completed this series of
telescopic discoveries. According to his own statement Galilei first
saw them towards the end of 1610,[69] but apparently paid no particular
attention to them at the time; and, although he shewed them as a matter
of curiosity to various friends, he made no formal announcement of the
discovery till May 1612, by which time the same discovery had been
made independently by Harriot (§ 118) in England, by _John Fabricius_
(1587-? 1615) in Holland, and by the Jesuit _Christopher Scheiner_
(1575-1650) in Germany, and had been published by Fabricius (June
1611). As a matter of fact dark spots had been seen with the naked
eye long before, but had been generally supposed to be caused by the
passage of Mercury in front of the sun. The presence on the sun of such
blemishes as black spots, the “mutability” involved in their changes in
form and position, and their formation and subsequent disappearance,
were all distasteful to the supporters of the old views, according
to which celestial bodies were perfect and unchangeable. The fact,
noticed by all the early observers, that the spots appeared to move
across the face of the sun from the eastern to the western side (_i.e._
roughly from left to right, as seen at midday by an observer in our
latitudes), gave at first sight countenance to the view, championed by
Scheiner among others, that the spots might really be small planets
revolving round the sun, and appearing as dark objects whenever they
passed between the sun and the observer. In three letters to his friend
Welser, a merchant prince of Augsburg, written in 1612 and published
in the following year,[70] Galilei, while giving a full account of his
observations, gave a crushing refutation of this view; proved that
the spots must be on or close to the surface of the sun, and that the
motions observed were exactly such as would result if the spots were
attached to the sun, and it revolved on an axis in a period of about a
month; and further, while disclaiming any wish to speak confidently,
called attention to several of their points of resemblance to clouds.

One of his arguments against Scheiner’s views is so simple and at the
same time so convincing, that it may be worth while to reproduce it as
an illustration of Galilei’s method, though the controversy itself is
quite dead.

Galilei noticed, namely, that while a spot took about fourteen days
to cross from one side of the sun to the other, and this time was the
same whether the spot passed through the centre of the sun’s disc, or
along a shorter path at some distance from it, its rate of motion was
by no means uniform, but that the spot’s motion always appeared much
slower when near the edge of the sun than when near the centre. This he
recognised as an effect of foreshortening, which would result if, and
only if, the spot were near the sun.

If, for example, in the figure, the circle represent a section of
the sun by a plane through the observer at O, and A, B, C, D, E be
points taken at equal distances along the surface of the sun, so as
to represent the positions of an object on the sun at equal intervals
of time, on the assumption that the sun revolves uniformly, then the
apparent motion from A to B, as seen by the observer at O, is measured
by the angle A O B, and is obviously much less than that from D to E,
measured by the angle D O E, and consequently an object attached to
the sun must appear to move more slowly from A to B, _i.e._ near the
sun’s edge, than from D to E, near the centre. On the other hand, if
the spot be a body revolving round the sun at some distance from it,
_e.g._ along the dotted circle _c d e_, then if _c_, _d_, _e_ be taken
at equal distances from one another, the apparent motion from _c_ to
_d_, measured again by the angle _c_ O _d_, is only very slightly less
than that from _d_ to _e_, measured by the angle _d_ O _e_. Moreover,
it required only a simple calculation, performed by Galilei in several
cases, to express these results in a numerical shape, and so to infer
from the actual observations that the spots could not be more than
a very moderate distance from the sun. The only escape from this
conclusion was by the assumption that the spots, if they were bodies
revolving round the sun, moved irregularly, in such a way as always to
be moving fastest when they happened to be between the centre of the
sun and the earth, whatever the earth’s position might be at the time,
a procedure for which, on the one hand, no sort of reason could be
given, and which, on the other, was entirely out of harmony with the
uniformity to which mediæval astronomy clung so firmly.

[Illustration: FIG. 56.—Galilei’s proof that sun-spots are not planets.]

The rotation of the sun about an axis, thus established, might
evidently have been used as an argument in support of the view that
the earth also had such a motion, but, as far as I am aware, neither
Galilei nor any contemporary noticed the analogy. Among other facts
relating to the spots observed by Galilei were the greater darkness
of the central parts, some of his drawings (see fig. 55) shewing, like
most modern drawings, a fairly well-marked line of division between
the central part (or =umbra=) and the less dark fringe (or =penumbra=)
surrounding it; he noticed also that spots frequently appeared in
groups, that the members of a group changed their positions relatively
to one another, that individual spots changed their size and shape
considerably during their lifetime, and that spots were usually
most plentiful in two regions on each side of the sun’s equator,
corresponding roughly to the tropics on our own globe, and were never
seen far beyond these limits.

Similar observations were made by other telescopists, and to Scheiner
belongs the credit of fixing, with considerably more accuracy than
Galilei, the position of the sun’s axis and equator and the time of its
rotation.

125. The controversy with Scheiner as to the nature of spots
unfortunately developed into a personal quarrel as to their respective
claims to the discovery of spots, a controversy which made Scheiner his
bitter enemy, and probably contributed not a little to the hostility
with which Galilei was henceforward regarded by the Jesuits. Galilei’s
uncompromising championship of the new scientific ideas, the slight
respect which he shewed for established and traditional authority, and
the biting sarcasms with which he was in the habit of greeting his
opponents, had won for him a large number of enemies in scientific
and philosophic circles, particularly among the large party who spoke
in the name of Aristotle, although, as Galilei was never tired of
reminding them, their methods of thought and their conclusions would in
all probability have been rejected by the great Greek philosopher if he
had been alive.

It was probably in part owing to his consciousness of a growing
hostility to his views, both in scientific and in ecclesiastical
circles, that Galilei paid a short visit to Rome in 1611, when he
met with a most honourable reception and was treated with great
friendliness by several cardinals and other persons in high places.

Unfortunately he soon began to be drawn into a controversy as to the
relative validity in scientific matters of observation and reasoning
on the one hand, and of the authority of the Church and the Bible on
the other, a controversy which began to take shape about this time and
which, though its battle-field has shifted from science to science, has
lasted almost without interruption till modern times.

In 1611 was published a tract maintaining Jupiter’s satellites to
be unscriptural. In 1612 Galilei consulted Cardinal Conti as to the
astronomical teaching of the Bible, and obtained from him the opinion
that the Bible appeared to discountenance both the Aristotelian
doctrine of the immutability of the heavens and the Coppernican
doctrine of the motion of the earth. A tract of Galilei’s on floating
bodies, published in 1612, roused fresh opposition, but on the other
hand Cardinal Barberini (who afterwards, as Urban VIII., took a leading
part in his persecution) specially thanked him for a presentation
copy of the book on sun-spots, in which Galilei, for the first time,
clearly proclaimed in public his adherence to the Coppernican system.
In the same year (1613) his friend and follower, Father Castelli, was
appointed professor of mathematics at Pisa, with special instructions
not to lecture on the motion of the earth. Within a few months Castelli
was drawn into a discussion on the relations of the Bible to astronomy,
at the house of the Grand Duchess, and quoted Galilei in support of his
views; this caused Galilei to express his opinions at some length in a
letter to Castelli, which was circulated in manuscript at the court.
To this a Dominican preacher, Caccini, replied a few months afterwards
by a violent sermon on the text, “Ye Galileans, why stand ye gazing
up into heaven?”[71] and in 1615 Galilei was secretly denounced to
the Inquisition on the strength of the letter to Castelli and other
evidence. In the same year he expanded the letter to Castelli into a
more elaborate treatise, in the form of a _Letter to the Grand Duchess
Christine_, which was circulated in manuscript, but not printed till
1636. The discussion of the bearing of particular passages of the
Bible (_e.g._ the account of the miracle of Joshua) on the Ptolemaic
and Coppernican systems has now lost most of its interest; it may,
however, be worth noticing that on the more general question Galilei
quotes with approval the saying of Cardinal Baronius, “That the
intention of the Holy Ghost is to teach us not how the heavens go, but
how to go to heaven,”[72] and the following passage gives a good idea
of the general tenor of his argument:—

 “Methinks, that in the Discussion of Natural Problemes we ought not
 to begin at the authority of places of Scripture; but at Sensible
 Experiments and Necessary Demonstrations. For ... Nature being
 inexorable and immutable, and never passing the bounds of the Laws
 assigned her, as one that nothing careth, whether her abstruse reasons
 and methods of operating be or be not exposed to the capacity of men;
 I conceive that that concerning Natural Effects, which either sensible
 experience sets before our eyes, or Necessary Demonstrations do prove
 unto us, ought not, upon any account, to be called into question, much
 less condemned upon the testimony of Texts of Scripture, which may
 under their words, couch senses seemingly contrary thereto.”[73]

126. Meanwhile his enemies had become so active that Galilei thought
it well to go to Rome at the end of 1615 to defend his cause. Early
in the next year a body of theologians known as the Qualifiers of the
Holy Office (Inquisition), who had been instructed to examine certain
Coppernican doctrines, reported:—

 “That the doctrine that the sun was the centre of the world and
 immoveable was false and absurd, formally heretical and contrary
 to Scripture, whereas the doctrine that the earth was not the
 centre of the world but moved, and has further a daily motion, was
 philosophically false and absurd and theologically at least erroneous.”

In consequence of this report it was decided to censure Galilei, and
the Pope accordingly instructed Cardinal Bellarmine “to summon Galilei
and admonish him to abandon the said opinion,” which the Cardinal
did.[74] Immediately afterwards a decree was issued condemning the
opinions in question and placing on the well-known _Index of Prohibited
Books_ three books containing Coppernican views, of which the _De
Revolutionibus_ of Coppernicus and another were only suspended “until
they should be corrected,” while the third was altogether prohibited.
The necessary corrections to the _De Revolutionibus_ were officially
published in 1620, and consisted only of a few alterations which
tended to make the essential principles; of the book appear as mere
mathematical hypotheses, convenient for calculation. Galilei seems to
have been on the whole well satisfied with the issue of the inquiry, as
far as he was personally concerned, and after obtaining from Cardinal
Bellarmine a certificate that he had neither abjured his opinions nor
done penance for them, stayed on in Rome for some months to shew that
he was in good repute there.

127. During the next few years Galilei, who was now more than fifty,
suffered a good deal from ill-health and was comparatively inactive.
He carried on, however, a correspondence with the Spanish court on
a method of ascertaining the longitude at sea by means of Jupiter’s
satellites. The essential problem in finding the longitude is to obtain
the time as given by the sun at the required place and also that at
some place the longitude of which is known. If, for example, midday at
Rome occurs an hour earlier than in London, the sun takes an hour to
travel from the meridian of Rome to that of London, and the longitude
of Rome is 15° east of that of London. At sea it is easy to ascertain
the local time, _e.g._ by observing when the sun is highest in the sky,
but the great difficulty, felt in Galilei’s time and long afterwards
(chapter X., §§ 197, 226), was that of ascertaining the time at some
standard place. Clocks were then, and long afterwards, not to be relied
upon to keep time accurately during a long ocean voyage, and some
astronomical means of determining the time was accordingly wanted.
Galilei’s idea was that if the movements of Jupiter’s satellites, and
in particular the eclipses which constantly occurred when a satellite
passed into Jupiter’s shadow, could be predicted, then a table could be
prepared giving the times, according to some standard place, say Rome,
at which the eclipses would occur, and a sailor by observing the local
time of an eclipse and comparing it with the time given in the table
could ascertain by how much his longitude differed from that of Rome.
It is, however, doubtful whether the movements of Jupiter’s satellites
could at that time be predicted accurately enough to make the method
practically useful, and in any case the negotiations came to nothing.

In 1618 three comets appeared, and Galilei was soon drawn into a
controversy on the subject with a Jesuit of the name of Grassi. The
controversy was marked by the personal bitterness which was customary,
and soon developed so as to include larger questions of philosophy and
astronomy. Galilei’s final contribution to it was published in 1623
under the title _Il Saggiatore_ (The Assayer), which dealt incidentally
with the Coppernican theory, though only in the indirect way which the
edict of 1616 rendered necessary. In a characteristic passage, for
example, Galilei says:—

 “Since the motion attributed to the earth, which I, as a pious and
 Catholic person, consider most false, and not to exist, accommodates
 itself so well to explain so many and such different phenomena, I
 shall not feel sure ... that, false as it is, it may not just as
 deludingly correspond with the phenomena of comets”;

and again, in speaking of the rival systems of Coppernicus and Tycho,
he says:—

 “Then as to the Copernican hypothesis, if by the good fortune of
 us Catholics we had not been freed from error and our blindness
 illuminated by the Highest Wisdom, I do not believe that such grace
 and good fortune could have been obtained by means of the reasons and
 observations given by Tycho.”

Although in scientific importance the _Saggiatore_ ranks far below
many others of Galilei’s writings, it had a great reputation as a
piece of brilliant controversial writing, and notwithstanding its
thinly veiled Coppernicanism, the new Pope, Urban VIII., to whom it
was dedicated, was so much pleased with it that he had it read aloud
to him at meals. The book must, however, have strengthened the hands
of Galilei’s enemies, and it was probably with a view to counteracting
their influence that he went to Rome next year, to pay his respects
to Urban and congratulate him on his recent elevation. The visit was
in almost every way a success; Urban granted to him several friendly
interviews, promised a pension for his son, gave him several presents,
and finally dismissed him with a letter of special recommendation to
the new Grand Duke of Tuscany, who had shewn some signs of being less
friendly to Galilei than his father. On the other hand, however, the
Pope refused to listen to Galilei’s request that the decree of 1616
should be withdrawn.

128. Galilei now set seriously to work on the great astronomical
treatise, the _Dialogue on the Two Chief Systems of the World, the
Ptolemaic and Coppernican_, which he had had in mind as long ago as
1610, and in which he proposed to embody most of his astronomical work
and to collect all the available evidence bearing on the Coppernican
controversy. The form of a dialogue was chosen, partly for literary
reasons, and still more because it enabled him to present the
Coppernican case as strongly as he wished through the mouths of some
of the speakers, without necessarily identifying his own opinions
with theirs. The manuscript was almost completed in 1629, and in the
following year Galilei went to Rome to obtain the necessary licence
for printing it. The censor had some alterations made and then gave
the desired permission for printing at Rome, on condition that the
book was submitted to him again before being finally printed off. Soon
after Galilei’s return to Florence the plague broke out, and quarantine
difficulties rendered it almost necessary that the book should be
printed at Florence instead of at Rome. This required a fresh licence,
and the difficulty experienced in obtaining it shewed that the Roman
censor was getting more and more doubtful about the book. Ultimately,
however, the introduction and conclusion having been sent to Rome for
approval and probably to some extent rewritten there, and the whole
work having been approved by the Florentine censor, the book was
printed and the first copies were ready early in 1632, bearing both the
Roman and the Florentine _imprimatur_.

129. The Dialogue extends over four successive days, and is carried on
by three speakers, of whom Salviati is a Coppernican and Simplicio an
Aristotelian philosopher, while Sagredo is avowedly neutral, but on
almost every occasion either agrees with Salviati at once or is easily
convinced by him, and frequently joins in casting ridicule upon the
arguments of the unfortunate Simplicio. Though many of the arguments
have now lost their immediate interest, and the book is unduly long,
it is still very readable, and the specimens of scholastic reasoning
put into the mouth of Simplicio and the refutation of them by the other
speakers strike the modern reader as excellent fooling.

Many of the arguments used had been published by Galilei in earlier
books, but gain impressiveness and cogency by being collected and
systematically arranged. The Aristotelian dogma of the immutability
of the celestial bodies is once more belaboured, and shewn to be not
only inconsistent with observations of the moon, the sun, comets, and
new stars, but to be in reality incapable of being stated in a form
free from obscurity and self-contradiction. The evidence in favour of
the earth’s motion derived from the existence of Jupiter’s satellites
and from the undoubted phases of Venus, from the suspected phases of
Mercury and from the variations in the apparent size of Mars, are once
more insisted on. The greater simplicity of the Coppernican explanation
of the daily motion of the celestial sphere and of the motion of the
planets is forcibly urged and illustrated in detail. It is pointed
out that on the Coppernican hypothesis all motions of revolution or
rotation take place in the same direction (from west to east), whereas
the Ptolemaic hypothesis requires some to be in one direction, some
in another. Moreover the apparent daily motion of the stars, which
appears simple enough if the stars are regarded as rigidly attached
to a material sphere, is shewn in a quite different aspect if, as
even Simplicio admits, no such sphere exists, and each star moves in
some sense independently. A star near the pole must then be supposed
to move far more slowly than one near the equator, since it describes
a much smaller circle in the same time; and further—an argument very
characteristic of Galilei’s ingenuity in drawing conclusions from
known facts—owing to the precession of the equinoxes (chapter II., §
42, and IV., § 84) and the consequent change of the position of the
pole among the stars, some of those stars which in Ptolemy’s time
were describing very small circles, and therefore moving slowly, must
now be describing large ones at a greater speed, and _vice versa_. An
extremely complicated adjustment of motions becomes therefore necessary
to account for observations which Coppernicus explained adequately by
the rotation of the earth and a simple displacement of its axis of
rotation.

Salviati deals also with the standing difficulty that the annual
motion of the earth ought to cause a corresponding apparent motion
of the stars, and that if the stars be assumed so far off that this
motion is imperceptible, then some of the stars themselves must be at
least as large as the earth’s orbit round the sun. Salviati points out
that the apparent or angular magnitudes of the fixed stars, avowedly
difficult to determine, are in reality almost entirely illusory, being
due in great part to an optical effect known as =irradiation=, in
virtue of which a bright object always tends to appear enlarged;[75]
and that there is in consequence no reason to suppose the stars nearly
as large as they might otherwise be thought to be. It is suggested
also that the most promising way of detecting the annual motion of
stars resulting from the motion of the earth would be by observing
the relative displacement of two stars close together in the sky (and
therefore nearly in the same direction), of which one might be presumed
from its greater brightness to be nearer than the other. It is, for
example, evident that if, in the figure, E, E′ represent two positions
of the earth in its path round the sun, and A, B two stars at different
distances, but nearly in the same direction, then to the observer at
E the star A appears to the left of B, whereas six months afterwards,
when the observer is at E′, A appears to the right of B. Such a motion
of one star with respect to another close to it would be much more
easily observed than an alteration of the same amount in the distance
of the star from some standard point such as the pole. Salviati points
out that accurate observations of this kind had not been made, and that
the telescope might be of assistance for the purpose. This method,
known as the =double-star= or =differential method of parallax=, was in
fact the first to lead—two centuries later—to a successful detection of
the motion in question (chapter XIII., § 278).

[Illustration: FIG. 57.—The differential method of parallax.]

130. Entirely new ground is broken in the _Dialogue_ when Galilei’s
discoveries of the laws of motion of bodies are applied to the problem
of the earth’s motion. His great discovery, which threw an entirely new
light on the mechanics of the solar system, was substantially the law
afterwards given by Newton as the first of his three laws of motion,
in the form: _Every body continues in its state of rest or of uniform
motion in a straight line, except in so far as it is compelled by force
applied to it to change that state._ Putting aside for the present
any discussion of _force_, a conception first made really definite by
Newton, and only imperfectly grasped by Galilei, we may interpret this
law as meaning that a body has no more inherent tendency to diminish
its motion or to stop than it has to increase its motion or to start,
and that any alteration in either the speed or the direction of a
body’s motion is to be explained by the action on it of some other
body, or at any rate by some other assignable cause. Thus a stone
thrown along a road comes to rest on account of the friction between
it and the ground, a ball thrown up into the air ascends more and more
slowly and then falls to the ground on account of that attraction
of the earth on it which we call its weight. As it is impossible to
entirely isolate a body from all others, we cannot experimentally
realise the state of things in which a body goes on moving indefinitely
in the same direction and at the same rate; it may, however, be shewn
that the more we remove a body from the influence of others, the less
alteration is there in its motion. The law is therefore, like most
scientific laws, an abstraction referring to a state of things to
which we may approximate in nature. Galilei introduces the idea in the
Dialogue by means of a ball on a smooth inclined plane. If the ball
is projected upwards, its motion is gradually retarded; if downwards,
it is continually accelerated. This is true if the plane is fairly
smooth—like a well-planed plank—and the inclination of the plane not
very small. If we imagine the experiment performed on an ideal plane,
which is supposed _perfectly_ smooth, we should expect the same results
to follow, however small the inclination of the plane. Consequently,
if the plane were quite level, so that there is no distinction between
up and down, we should expect the motion to be neither retarded nor
accelerated, but to continue without alteration. Other more familiar
examples are also given of the tendency of a body, when once in
motion, to continue in motion, as in the case of a rider whose horse
suddenly stops, or of bodies in the cabin of a moving ship which have
no tendency to lose the motion imparted to them by the ship, so that,
_e.g._, a body falls down to all appearances exactly as if the rest
of the cabin were at rest, and therefore, in reality, while falling
retains the forward motion which it shares with the ship and its
contents. Salviati states also that—contrary to general belief—a stone
dropped from the masthead of a ship in motion falls at the foot of the
mast, not behind it, but there is no reference to the experiment having
been actually performed.

This mechanical principle being once established, it becomes easy to
deal with several common objections to the supposed motion of the
earth. The case of a stone dropped from the top of a tower, which if
the earth be in reality moving rapidly from west to east might be
expected to fall to the west in its descent, is easily shewn to be
exactly parallel to the case of a stone dropped from the masthead of
a ship in motion. The motion towards the east, which the stone when
resting on the tower shares with the tower and the earth, is not
destroyed in its descent, and it is therefore entirely in accordance
with the Coppernican theory that the stone should fall as it does at
the foot of the tower.[76] Similarly, the fact that the clouds, the
atmosphere in general, birds flying in it, and loose objects on the
surface of the earth, shew no tendency to be left behind as the earth
moves rapidly eastward, but are apparently unaffected by the motion of
the earth, is shewn to be exactly parallel to the fact that the flies
in a ship’s cabin and the loose objects there are in no way affected
by the uniform onward motion of the ship (though the irregular motions
of pitching and rolling do affect them). The stock objection that
a cannon-ball shot westward should, on the Coppernican hypothesis,
carry farther than one shot eastward under like conditions, is met
in the same way; but it is further pointed out that, owing to the
imperfection of gunnery practice, the experiment could not really be
tried accurately enough to yield any decisive result.

The most unsatisfactory part of the _Dialogue_ is the fourth day’s
discussion, on the tides, of which Galilei suggests with great
confidence an explanation based merely on the motion of the earth,
while rejecting with scorn the suggestion of Kepler and others—correct
as far as it went—that they were caused by some influence emanating
from the moon. It is hardly to be wondered at that the rudimentary
mechanical and mathematical knowledge at Galilei’s command should not
have enabled him to deal correctly with a problem of which the vastly
more powerful resources of modern science can only give an imperfect
solution (cf. chapter XI., § 248, and chapter XIII., § 292).

131. The book as a whole was in effect, though not in form, a
powerful—indeed unanswerable—plea for Coppernicanism. Galilei tried to
safeguard his position, partly by the use of dialogue, and partly by
the very remarkable introduction, which was not only read and approved
by the licensing authorities, but was in all probability in part the
composition of the Roman censor and of the Pope. It reads to us like
a piece of elaborate and thinly veiled irony, and it throws a curious
light on the intelligence or on the seriousness of the Pope and the
censor, that they should have thus approved it:—

  “Judicious reader, there was published some years since in _Rome_ a
 salutiferous Edict, that, for the obviating of the dangerous Scandals
 of the present Age, imposed a reasonable Silence upon the Pythagorean
 Opinion of the Mobility of the Earth. There want not such as
 unadvisedly affirm, that the Decree was not the production of a sober
 Scrutiny, but of an informed passion; and one may hear some mutter
 that Consultors altogether ignorant of Astronomical observations ought
 not to clipp the wings of speculative wits with rash prohibitions. My
 zeale cannot keep silence when I hear these inconsiderate complaints.
 I thought fit, as being thoroughly acquainted with that prudent
 Determination, to appear openly upon the Theatre of the World as a
 Witness of the naked Truth. I was at that time in _Rome_, and had
 not only the audiences, but applauds of the most Eminent Prelates of
 that Court; nor was that Decree published without Previous Notice
 given me thereof. Therefore it is my resolution in the present case
 to give Foreign Nations to see, that this point is as well understood
 in _Italy_, and particularly in _Rome_, as Transalpine Diligence can
 imagine it to be: and collecting together all the proper speculations
 that concerne the _Copernican Systeme_ to let them know, that the
 notice of all preceded the Censure of the _Roman Court_; and that
 there proceed from this Climate not only Doctrines for the health of
 the Soul, but also ingenious Discoveries for the recreating of the
 Mind.... I hope that by these considerations the world will know, that
 if other Nations have Navigated more than we, we have not studied less
 than they; and that our returning to assert the Earth’s stability, and
 to take the contrary only for a Mathematical _Capriccio_, proceeds
 not from inadvertency of what others have thought thereof, but (had
 one no other inducements), from these reasons that Piety, Religion,
 the Knowledge of the Divine Omnipotency, and a consciousness of the
 incapacity of man’s understanding dictate unto us.”[77]

132. Naturally Galilei’s many enemies were not long in penetrating
these thin disguises, and the immense success of the book only
intensified the opposition which it excited; the Pope appears to have
been persuaded that Simplicio—the butt of the whole dialogue—was
intended for himself, a supposed insult which bitterly wounded his
vanity; and it was soon evident that the publication of the book
could not be allowed to pass without notice. In June 1632 a special
commission was appointed to inquire into the matter—an unusual
procedure, probably meant as a mark of consideration for Galilei—and
two months later the further issue of copies of the book was
prohibited, and in September a papal mandate was issued requiring
Galilei to appear personally before the Inquisition. He was evidently
frightened by the summons, and tried to avoid compliance through
the good offices of the Tuscan court and by pleading his age and
infirmities, but after considerable delay, at the end of which the
Pope issued instructions to bring him if necessary by force and in
chains, he had to submit, and set off for Rome early in 1633. Here
he was treated with unusual consideration, for whereas in general
even the most eminent offenders under trial by the Inquisition were
confined in its prisons, he was allowed to live with his friend
Niccolini, the Tuscan ambassador, throughout the trial, with the
exception of a period of about three weeks, which he spent within the
buildings of the Inquisition, in comfortable rooms belonging to one
of the officials, with permission to correspond with his friends,
to take exercise in the garden, and other privileges. At his first
hearing before the Inquisition, his reply to the charge of having
violated the decree of 1616 (§ 126) was that he had not understood
that the decree or the admonition given to him forbade the teaching
of the Coppernican theory as a mere “hypothesis,” and that his book
had not upheld the doctrine in any other way. Between his first and
second hearing the Commission, which had been examining his book,
reported that it did distinctly defend and maintain the obnoxious
doctrines, and Galilei, having been meanwhile privately advised by
the Commissary-General of the Inquisition to adopt a more submissive
attitude, admitted at the next hearing that on reading his book again
he recognised that parts of it gave the arguments for Coppernicanism
more strongly than he had at first thought. The pitiable state to which
he had been reduced was shewn by the offer which he now made to write
a continuation to the Dialogue which should as far as possible refute
his own Coppernican arguments. At the final hearing on June 21st he
was examined under threat of torture,[78] and on the next day he was
brought up for sentence. He was convicted “of believing and holding the
doctrines—false and contrary to the Holy and Divine Scriptures—that the
sun is the centre of the world, and that it does not move from east
to west, and that the earth does move and is not the centre of the
world; also that an opinion can be held and supported as probable after
it has been declared and decreed contrary to the Holy Scriptures.”
In punishment, he was required to “abjure, curse, and detest the
aforesaid errors,” the abjuration being at once read by him on his
knees; and was further condemned to the “formal prison of the Holy
Office” during the pleasure of his judges, and required to repeat the
seven penitential psalms once a week for three years. On the following
day the Pope changed the sentence of imprisonment into confinement at
a country-house near Rome belonging to the Grand Duke, and Galilei
moved there on June 24th.[79] On petitioning to be allowed to return
to Florence, he was at first allowed to go as far as Siena, and at the
end of the year was permitted to retire to his country-house at Arcetri
near Florence, on condition of not leaving it for the future without
permission, while his intercourse with scientific and other friends was
jealously watched.

[Illustration: GALILEI.]

The story of the trial reflects little credit either on Galilei or
on his persecutors. For the latter, it may be urged that they acted
with unusual leniency considering the customs of the time; and it
is probable that many of those who were concerned in the trial were
anxious to do as little injury to Galilei as possible, but were
practically forced by the party personally hostile to him to take some
notice of the obvious violation of the decree of 1616. It is easy to
condemn Galilei for cowardice, but it must be borne in mind, on the
one hand, that he was at the time nearly seventy, and much shaken
in health, and, on the other, that the Roman Inquisition, if not as
cruel as the Spanish, was a very real power in the early 17th century;
during Galilei’s lifetime (1600) Giordano Bruno had been burnt alive
at Rome for writings which, in addition to containing religious and
political heresies, supported the Coppernican astronomy and opposed
the traditional Aristotelian philosophy. Moreover, it would be unfair
to regard his submission as due merely to considerations of personal
safety, for—apart from the question whether his beloved science would
have gained anything by his death or permanent imprisonment—there
can be no doubt that Galilei was a perfectly sincere member of his
Church, and although he did his best to convince individual officers
of the Church of the correctness of his views, and to minimise the
condemnation of them passed in 1616, yet he was probably prepared,
when he found that the condemnation was seriously meant by the Pope,
the Holy Office, and others, to believe that in some senses at least
his views must be wrong, although, as a matter of observation and pure
reason, he was unable to see how or why. In fact, like many other
excellent people, he kept watertight compartments in his mind, respect
for the Church being in one and scientific investigation in another.

Copies of the sentence on Galilei and of his abjuration were at once
circulated in Italy and in Roman Catholic circles elsewhere, and a
decree of the Congregation of the Index was also issued adding the
_Dialogue_ to the three Coppernican books condemned in 1616, and to
Kepler’s _Epitome_ of the Coppernican Astronomy (chapter VII., §
145), which had been put on the _Index_ shortly afterwards. It may
be of interest to note that these five books still remained in the
edition of the _Index of Prohibited Books_ which was issued in 1819
(with appendices dated as late as 1821), but disappeared from the next
edition, that of 1835.

133. The rest of Galilei’s life may be described very briefly. With
the exception of a few months, during which he was allowed to be at
Florence for the sake of medical treatment, he remained continuously
at Arcetri, evidently pretty closely watched by the agents of the
Holy Office, much restricted in his intercourse with his friends, and
prevented from carrying on his studies in the directions which he liked
best. He was moreover very infirm, and he was afflicted by domestic
troubles, especially by the death in 1634 of his favourite child, a nun
in a neighbouring convent. But his spirit was not broken, and he went
on with several important pieces of work, which he had begun earlier
in his career. He carried a little further the study of his beloved
Medicean Planets and of the method of finding longitude based on
their movements (§ 127), and negotiated on the subject with the Dutch
government. He made also a further discovery relating to the moon, of
sufficient importance to deserve a few words of explanation.

[Illustration: FIG. 58.—The daily libration of the moon.]

It had long been well known that as the moon describes her monthly path
round the earth we see the same markings substantially in the same
positions on the disc, so that substantially the same face of the moon
is turned towards the earth. It occurred to Galilei to inquire whether
this was accurately the case, or whether, on the contrary, some change
in the moon’s disc could be observed. He saw that if, as seemed likely,
the line joining the centres of the earth and moon always passed
through the same point on the moon’s surface, nevertheless certain
alterations in an observer’s position on the earth would enable him to
see different portions of the moon’s surface from time to time. The
simplest of these alterations is due to the daily motion of the earth.
Let us suppose for simplicity that the observer is on the earth’s
equator, and that the moon is at the time in the celestial equator. Let
the larger circle in fig. 58 represent the earth’s equator, and the
smaller circle the section of the moon by the plane of the equator.
Then in about 12 hours the earth’s rotation carries the observer from
A, where he sees the moon rising, to B, where he sees it setting. When
he is at C, on the line joining the centres of the earth and moon, the
point O appears to be in the centre of the moon’s disc, and the portion
_c_ O _c′_ is visible, _c_ R _c′_ invisible. But when the observer is
at A, the point P, on the right of O, appears in the centre, and the
portion _a_ P _a′_ is visible, so that _c′ a′_ is now visible and _a
c_ invisible. In the same way, when the observer is at B, he can see
the portion _c b_, while _b′ c′_ is invisible and Q appears to be in
the centre of the disc. Thus in the course of the day the portion _a_ O
_b′_ (dotted in the figure) is constantly visible and _b_ R _a′_ (also
dotted) constantly invisible, while _a c b_ and _a′ c′ b′_ alternately
come into view and disappear. In other words, when the moon is rising
we see a little more of the side which is the then uppermost, and
when she is setting we see a little more of the other side which is
uppermost in this position. A similar explanation applies when the
observer is not on the earth’s equator, but the geometry is slightly
more complicated. In the same way, as the moon passes from south to
north of the equator and back as she revolves round the earth, we see
alternately more and less of the northern and southern half of the
moon. This set of changes—the simplest of several somewhat similar ones
which are now known as =librations= of the moon—being thus thought of
as likely to occur, Galilei set to work to test their existence by
observing certain markings of the moon usually visible near the edge,
and at once detected alterations in their distance from the edge, which
were in general accordance with his theoretical anticipations. A more
precise inquiry was however interrupted by failing sight, culminating
(at the end of 1636) in total blindness.

But the most important work of these years was the completion of the
great book, in which he summed up and completed his discoveries in
mechanics, _Mathematical Discourses and Demonstrations concerning
Two New Sciences, relating to Mechanics and to Local Motion_. It was
written in the form of a dialogue between the same three speakers who
figured in the Dialogue on the Systems, but is distinctly inferior in
literary merit to the earlier work. We have here no concern with a
large part of the book, which deals with the conditions under which
bodies are kept, at rest by forces applied to them (statics), and
certain problems relating to the resistance of bodies to fracture and
to bending, though in both of these subjects Galilei broke new ground.
More important astronomically—and probably intrinsically also—is what
he calls the science of local motion,[80] which deals with the motion
of bodies. He builds up on the basis of his early experiments (§
116) a theory of falling bodies, in which occurs for the first time
the important idea of =uniformly accelerated motion=, or =uniform
acceleration=, _i.e._ motion in which the moving body receives in every
equal interval of time an equal increase of velocity. He shews that
the motion of a falling body is—except in so far as it is disturbed
by the air—of this nature, and that, as already stated, the motion
is the same for all bodies, although his numerical estimate is not
at all accurate.[81] From this fundamental law he works out a number
of mathematical deductions, connecting the space fallen through, the
velocity, and the time elapsed, both for the case of a body falling
freely and for one falling down an inclined plane. He gives also a
correct elementary theory of projectiles, in the course of which he
enunciates more completely than before the law of inertia already
referred to (§ 130), although Galilei’s form is still much less general
than Newton’s:—

_Conceive a body projected or thrown along a horizontal plane, all
impediments being removed. Now it is clear by what we have said before
at length that its motion will be uniform and perpetual along the said
plane, if the plane extend indefinitely._

In connection with projectiles, Galilei also appears to realise that
a body may be conceived as having motions in two different directions
simultaneously, and that each may be treated as independent of the
other, so that, for example, if a bullet is shot horizontally out of
a gun, its downward motion, due to its weight, is unaffected by its
horizontal motion, and consequently it reaches the ground at the same
time as a bullet simply allowed to drop; but Galilei gives no general
statement of this principle, which was afterwards embodied by Newton in
his Second Law of Motion.

The treatise on the _Two New Sciences_ was finished in 1636, and, since
no book of Galilei’s could be printed in Italy, it was published after
some little delay at Leyden in 1638. In the same year his eyesight,
which he had to some extent recovered after his first attack of
blindness, failed completely, and four years later (January 8th, 1642)
the end came.

134. Galilei’s chief scientific discoveries have already been noticed.
The telescopic discoveries, on which much of his popular reputation
rests, have probably attracted more than their fair share of attention;
many of them were made almost simultaneously by others, and the rest,
being almost inevitable results of the invention of the telescope,
could not have been delayed long. But the skilful use which Galilei
made of them as arguments for the Coppernican system, the no less
important support which his dynamical discoveries gave to the same
cause, the lucidity and dialectic brilliance with which he marshalled
the arguments in favour of his views and demolished those of his
opponents, together with the sensational incidents of his persecution,
formed conjointly a contribution to the Coppernican controversy which
was in effect decisive. Astronomical textbooks still continued to give
side by side accounts of the Ptolemaic and of the Coppernican systems,
and the authors, at any rate if they were good Roman Catholics, usually
expressed, in some more or less perfunctory way, their adherence
to the former, but there was no real life left in the traditional
astronomy; new advances in astronomical theory were all on Coppernican
lines, and in the extensive scientific correspondence of Newton and
his contemporaries the truth of the Coppernican system scarcely ever
appears as a subject for discussion.

Galilei’s dynamical discoveries, which are only in part of astronomical
importance, are in many respects his most remarkable contribution to
science. For whereas in astronomy he was building on foundations laid
by previous generations, in dynamics it was no question of improving
or developing an existing science, but of creating a new one. From his
predecessors he inherited nothing but erroneous traditions and obscure
ideas; and when these had been discarded, he had to arrive at clear
fundamental notions, to devise experiments and make observations, to
interpret his experimental results, and to follow out the mathematical
consequences of the simple laws first arrived at. The positive results
obtained may not appear numerous, if viewed from the standpoint of our
modern knowledge, but they sufficed to constitute a secure basis for
the superstructure which later investigators added.

It is customary to associate with our countryman Francis Bacon
(1561-1627) the reform in methods of scientific discovery which
took place during the seventeenth century, and to which much of the
rapid progress in the natural sciences made since that time must be
attributed. The value of Bacon’s theory of scientific discovery is
very differently estimated by different critics, but there can be no
question of the singular ill-success which attended his attempts to
apply it in particular cases, and it may fairly be questioned whether
the scientific methods constantly referred to incidentally by Galilei,
and brilliantly exemplified by his practice, do not really contain a
large part of what is valuable in the Baconian philosophy of science,
while at the same time avoiding some of its errors. Reference has
already been made on several occasions to Galilei’s protests against
the current method of dealing with scientific questions by the
interpretation of passages in Aristotle, Ptolemy, or other writers;
and to his constant insistence on the necessity of appealing directly
to actual observation of facts. But while thus agreeing with Bacon in
these essential points, he differed from him in the recognition of
the importance, both of deducing new results from established ones by
mathematical or other processes of exact reasoning, and of using such
deductions, when compared with fresh experimental results, as a means
of verifying hypotheses provisionally adopted. This method of proof,
which lies at the base of nearly all important scientific discovery,
can hardly be described better than by Galilei’s own statement of it,
as applied to a particular case:—

 “Let us therefore take this at present as a _Postulatum_, the truth
 whereof we shall afterwards find established, when we shall see other
 conclusions built upon this _Hypothesis_, to answer and most exactly
 to agree with Experience.”[82]



CHAPTER VII.

KEPLER.

 “His celebrated laws were the outcome of a lifetime of speculation,
 for the most part vain and groundless.... But Kepler’s name was
 destined to be immortal, on account of the patience with which he
 submitted his hypotheses to comparison with observation, the candour
 with which he acknowledged failure after failure, and the perseverance
 and ingenuity with which he renewed his attack upon the riddles of
 nature.”
                                                                JEVONS.


135. John Kepler, or Keppler,[83] was born in 1571, seven years
after Galilei, at Weil in Würtemberg; his parents were in reduced
circumstances, though his father had some claims to noble descent.
Though Weil itself was predominantly Roman Catholic, the Keplers were
Protestants, a fact which frequently stood in Kepler’s way at various
stages of his career. But the father could have been by no means
zealous in his faith, for he enlisted in the army of the notorious Duke
of Alva when it was engaged in trying to suppress the revolt of the
Netherlands against Spanish persecution.

John Kepler’s childhood was marked by more than the usual number of
illnesses, and his bodily weaknesses, combined with a promise of great
intellectual ability, seemed to point to the Church as a suitable
career for him. After attending various elementary schools with great
irregularity—due partly to ill-health, partly to the requirements of
manual work at home—he was sent in 1584 at the public expense to the
monastic school at Adelberg, and two years later to the more advanced
school or college of the same kind at Maulbronn, which was connected
with the University of Tübingen, then one of the great centres of
Protestant theology.

In 1588 he obtained the B.A. degree, and in the following year entered
the philosophical faculty at Tübingen.

There he came under the influence of Maestlin, the professor of
mathematics, by whom he was in private taught the principles of the
Coppernican system, though the professorial lectures were still on the
traditional lines.

In 1591 Kepler graduated as M.A., being second out of fourteen
candidates, and then devoted himself chiefly to the study of theology.

136. In 1594, however, the Protestant Estates of Styria applied to
Tübingen for a lecturer on mathematics (including astronomy) for the
high school of Gratz, and the appointment was offered to Kepler. Having
no special knowledge of the subject and as yet no taste for it, he
naturally hesitated about accepting the offer, but finally decided
to do so, expressly stipulating, however, that he should not thereby
forfeit his claims to ecclesiastical preferment in Würtemberg. The
demand for higher mathematics at Gratz seems to have been slight;
during his first year Kepler’s mathematical lectures were attended
by very few students, and in the following year by none, so that to
prevent his salary from being wasted he was set to teach the elements
of various other subjects. It was moreover one of his duties to
prepare an annual almanack or calendar, which was expected to contain
not merely the usual elementary astronomical information such as we
are accustomed to in the calendars of to-day, but also astrological
information of a more interesting character, such as predictions of
the weather and of remarkable events, guidance as to unlucky and lucky
times, and the like. Kepler’s first calendar, for the year 1595,
contained some happy weather-prophecies, and he acquired accordingly
a considerable popular reputation as a prophet and astrologer, which
remained throughout his life.

Meanwhile his official duties evidently left him a good deal of
leisure, which he spent with characteristic energy in acquiring as
thorough a knowledge as possible of astronomy, and in speculating on
the subject.

According to his own statement, “there were three things in particular,
_viz._ the number, the size, and the motion of the heavenly bodies,
as to which he searched zealously for reasons why they were as they
were and not otherwise”; and the results of a long course of wild
speculation on the subject led him at last to a result with which he
was immensely pleased—a numerical relation connecting the distances
of the several planets from the sun with certain geometrical bodies
known as the regular solids (of which the cube is the best known), a
relation which is not very accurate numerically, and is of absolutely
no significance or importance.[84] This discovery, together with a
detailed account of the steps which led to it, as well as of a number
of other steps which led nowhere, was published in 1596 in a book a
portion of the title of which may be translated as _The Forerunner of
Dissertations on the Universe, containing the Mystery of the Universe_,
commonly referred to as the _Mysterium Cosmographicum_. The contents
were probably much more attractive and seemed more valuable to Kepler’s
contemporaries than to us, but even to those who were least inclined
to attach weight to its conclusions, the book shewed evidence of
considerable astronomical knowledge and very great ingenuity; and both
Tycho Brahe and Galilei, to whom copies were sent, recognised in the
author a rising astronomer likely to do good work.

137. In 1597 Kepler married. In the following year the religious
troubles, which had for some years been steadily growing, were
increased by the action of the Archduke Ferdinand of Austria
(afterwards the Emperor Ferdinand II.), who on his return from a
pilgrimage to Loretto started a vigorous persecution of Protestants
in his dominions, one step in which was an order that all Protestant
ministers and teachers in Styria should quit the country at once
(1598). Kepler accordingly fled to Hungary, but returned after a
few weeks by special permission of the Archduke, given apparently
on the advice of the Jesuit party, who had hopes of converting the
astronomer. Kepler’s hearers had, however, mostly been scattered by
the persecution, it became difficult to ensure regular payment of
his stipend, and the rising tide of Catholicism made his position
increasingly insecure. Tycho’s overtures were accordingly welcome,
and in 1600 he paid a visit to him, as already described (chapter V.,
§ 108), at Benatek and Prague. He returned to Gratz in the autumn,
still uncertain whether to accept Tycho’s offer or not, but being then
definitely dismissed from his position at Gratz on account of his
Protestant opinions, he returned finally to Prague at the end of the
year.

138. Soon after Tycho’s death Kepler was appointed his successor as
mathematician to the Emperor Rudolph (1602), but at only half his
predecessor’s salary, and even this was paid with great irregularity,
so that complaints as to arrears and constant pecuniary difficulties
played an important part in his future life, as they had done during
the later years at Gratz. Tycho’s instruments never passed into his
possession, but as he had little taste or skill for observing, the loss
was probably not great; fortunately, after some difficulties with the
heirs, he secured control of the greater part of Tycho’s incomparable
series of observations, the working up of which into an improved theory
of the solar system was the main occupation of the next 25 years of
his life. Before, however, he had achieved any substantial result in
this direction, he published several minor works—for example, two
pamphlets on a new star which appeared in 1604, and a treatise on
the applications of optics to astronomy (published in 1604 with a
title beginning _Ad Vitellionem Paralipomena quibus Astronomiae Pars
Optica Traditur_ ...), the most interesting and important part of
which was a considerable improvement in the theory of astronomical
refraction (chapter II., § 46, and chapter V., § 110). A later optical
treatise (the _Dioptrice_ of 1611) contained a suggestion for the
construction of a telescope by the use of two convex lenses, which
is the form now most commonly adopted, and is a notable improvement
on Galilei’s instrument (chapter VI., § 118), one of the lenses of
which is concave; but Kepler does not seem himself to have had enough
mechanical skill to actually construct a telescope on this plan, or
to have had access to workmen capable of doing so for him; and it is
probable that Galilei’s enemy Scheiner (chapter VI., §§ 124, 125) was
the first person to use (about 1613) an instrument of this kind.

[Illustration: KEPLER.]

139. It has already been mentioned (chapter V., § 108) that when Tycho
was dividing the work of his observatory among his assistants he
assigned to Kepler the study of the planet Mars, probably as presenting
more difficulties than the subjects assigned to the others. It had been
known since the time of Coppernicus that the planets, including the
earth, revolved round the sun in paths that were at any rate not very
different from circles, and that the deviations from uniform circular
motion could be represented roughly by systems of eccentrics and
epicycles. The deviations from uniform circular motion were, however,
notably different in amount in different planets, being, for example,
very small in the case of Venus, relatively large in the case of Mars,
and larger still in that of Mercury. The _Prussian Tables_ calculated
by Reinhold on a Coppernican basis (chapter V., § 94) were soon found
to represent the actual motions very imperfectly, errors of 4° and 5°
having been noted by Tycho and Kepler, so that the principles on which
the tables were calculated were evidently at fault.

The solution of the problem was clearly more likely to be found by the
study of a planet in which the deviations from circular motion were as
great as possible. In the case of Mercury satisfactory observations
were scarce, whereas in the case of Mars there was an abundant series
recorded by Tycho, and hence it was true insight on Tycho’s part to
assign to his ablest assistant this particular planet, and on Kepler’s
to continue the research with unwearied patience. The particular system
of epicycles used by Coppernicus (chapter IV., § 87) having proved
defective, Kepler set to work to devise other geometrical schemes,
the results of which could be compared with observation. The places
of Mars as seen on the sky being a combined result of the motions
of Mars and of the earth in their respective orbits round the sun,
the irregularities of the two orbits were apparently inextricably
mixed up, and a great simplification was accordingly effected when
Kepler succeeded, by an ingenious combination of observations taken
at suitable times, in disentangling the irregularities due to the
earth from those due to the motion of Mars itself, and thus rendering
it possible to concentrate his attention on the latter. His fertile
imagination suggested hypothesis after hypothesis, combination after
combination of eccentric, epicycle, and equant; he calculated the
results of each and compared them rigorously with observation; and
at one stage he arrived at a geometrical scheme which was capable of
representing the observations with errors not exceeding 8′.[85] A man
of less intellectual honesty, or less convinced of the necessity of
subordinating theory to fact when the two conflict, might have rested
content with this degree of accuracy, or might have supposed Tycho’s
refractory observations to be in error. Kepler, however, thought
otherwise:—

 “Since the divine goodness has given to us in Tycho Brahe a most
 careful observer, from whose observations the error of 8′ is shewn
 in this calculation, ... it is right that we should with gratitude
 recognise and make use of this gift of God.... For if I could have
 treated 8′ of longitude as negligible I should have already corrected
 sufficiently the hypothesis ... discovered in chapter XVI. But as
 they could not be neglected, these 8′ alone have led the way towards
 the complete reformation of astronomy, and have been made the
 subject-matter of a great part of this work.”[86]

140. He accordingly started afresh, and after trying a variety of
other combinations of circles decided that the path of Mars must
be an oval of some kind. At first he was inclined to believe in an
egg-shaped oval, larger at one end than at the other, but soon had to
abandon this idea. Finally he tried the simplest known oval curve,
the =ellipse=,[87] and found to his delight that it satisfied the
conditions of the problem, if the sun were taken to be at a focus of
the ellipse described by Mars.

It was further necessary to formulate the law of variation of the rate
of motion of the planet in different parts of its orbit. Here again
Kepler tried a number of hypotheses, in the course of which he fairly
lost his way in the intricacies of the mathematical questions involved,
but fortunately arrived, after a dubious process of compensation of
errors, at a simple law which agreed with observation. He found that
the planet moved fast when near the sun and slowly when distant from
it, in such a way that the area described or swept out in any time
by the line joining the sun to Mars was always proportional to the
time. Thus in fig. 60[88] the motion of Mars is most rapid at the
point A nearest to the focus S where the sun is, least rapid at A′,
and the shaded and unshaded portions of the figure represent equal
areas each corresponding to the motion of the planet during a month.
Kepler’s triumph at arriving at this result is expressed by the figure
of victory in the corner of the diagram (fig. 61) which was used in
establishing the last stage of his proof.

[Illustration: FIG. 60.—Kepler’s second law.]

141. Thus were established for the case of Mars the two important
results generally known as Kepler’s first two laws:—

1. _The planet describes an ellipse, the sun being in one focus._

2. _The straight line joining the planet to the sun sweeps out equal
areas in any two equal intervals of time._

The full history of this investigation, with the results already stated
and a number of developments and results of minor importance, together
with innumerable digressions and quaint comments on the progress of the
inquiry, was published in 1609 in a book of considerable length, the
_Commentaries on the Motions of Mars_.[89]

[Illustration: FIG. 61.—Diagram used by Kepler to establish his laws of
planetary motion. From the _Commentaries on Mars_.]

142. Although the two laws of planetary motion just given were only
fully established for the case of Mars, Kepler stated that the earth’s
path also must be an oval of some kind, and was evidently already
convinced—aided by his firm belief in the harmony of Nature—that
all the planets moved in accordance with the same laws. This view
is indicated in the dedication of the book to the Emperor Rudolph,
which gives a fanciful account of the work as a struggle against the
rebellious War-God Mars, as the result of which he is finally brought
captive to the feet of the Emperor and undertakes to live for the
future as a loyal subject. As, however, he has many relations in the
ethereal spaces—his father Jupiter, his grandfather Saturn, his dear
sister Venus, his faithful brother Mercury—and he yearns for them
and they for him _on account of the similarity of their habits_, he
entreats the Emperor to send out an expedition as soon as possible
to capture them also, and with that object to provide Kepler with the
“sinews of war” in order that he may equip a suitable army.

Although the money thus delicately asked for was only supplied very
irregularly, Kepler kept steadily in view the expedition for which it
was to be used, or, in plainer words, he worked steadily at the problem
of extending his elliptic theory to the other planets, and constructing
the tables of the planetary motions, based on Tycho’s observations, at
which he had so long been engaged.

143. In 1611 his patron Rudolph was forced to abdicate the imperial
crown in favour of his brother Matthias, who had little interest in
astronomy, or even in astrology; and as Kepler’s position was thus
rendered more insecure than ever, he opened negotiations with the
Estates of Upper Austria, as the result of which he was promised a
small salary, on condition of undertaking the somewhat varied duties
of teaching mathematics at the high school of Linz, the capital, of
constructing a new map of the province, and of completing his planetary
tables. For the present, however, he decided to stay with Rudolph.

In the same year Kepler lost his wife, who had long been in weak bodily
and mental health.

In the following year (1612) Rudolph died, and Kepler then moved
to Linz and took up his new duties there, though still holding the
appointment of mathematician to the Emperor and occasionally even
receiving some portion of the salary of the office. In 1613 he married
again, after a careful consideration, recorded in an extraordinary
but very characteristic letter to one of his friends, of the relative
merits of eleven ladies whom he regarded as possible; and the provision
of a proper supply of wine for his new household led to the publication
of a pamphlet, of some mathematical interest, dealing with the proper
way of measuring the contents of a cask with curved sides.[90]

144. In the years 1618-1621, although in some ways the most disturbed
years of his life, he published three books of importance—an _Epitome
of the Copernican Astronomy_, the _Harmony of the World_,[91] and a
treatise on _Comets_.

The second and most important of these, published in 1619, though the
leading idea in it was discovered early in 1618, was regarded by Kepler
as a development of his early _Mysterium Cosmographicum_ (§ 136). His
speculative and mystic temperament led him constantly to search for
relations between the various numerical quantities occurring in the
solar system; by a happy inspiration he thought of trying to get a
relation connecting the sizes of the orbits of the various planets
with their times of revolution round the sun, and after a number of
unsuccessful attempts discovered a simple and important relation,
commonly known as Kepler’s third law:—

_The squares of the times of revolution of any two planets (including
the earth) about the sun are proportional to the cubes of their mean
distances from the sun._

If, for example, we express the times of revolution of the various
planets in terms of any one, which may be conveniently taken to be that
of the earth, namely a year, and in the same way express the distances
in terms of the distance of the earth from the sun as a unit, then the
times of revolution of the several planets taken in the order Mercury,
Venus, Earth, Mars, Jupiter, Saturn are approximately ·24, ·615, 1,
1·88, 11·86, 29·457, and their distances from the sun are respectively
·387, ·723, 1, 1·524, 5·203, 9·539; if now we take the squares of
the first series of numbers (the square of a number being the number
multiplied by itself) and the cubes of the second series (the cube of
a number being the number multiplied by itself twice, or the square
multiplied again by the number), we get the two series of numbers given
approximately by the table:—

  ┌———————————┬——————————┬————————┬————————┬——————-┬——————————┬————————┐
  │           │ Mercury. │ Venus. │ Earth. │ Mars. │ Jupiter. │ Saturn.│
  ├———————————┼——————————┼————————┼————————┼——————-┼——————————┼————————┤
  │Square of }│          │        │        │       │          │        │
  │ periodic }│   ·058   │  ·378  │   1    │ 3·54  │  140·7   │  867·7 │
  │ time     }│          │        │        │       │          │        │
  ├———————————┼——————————┼————————┼————————┼——————-┼——————————┼————————┤
  │Cube of   }│          │        │        │       │          │        │
  │ mean     }│   ·058   │  ·378  │   1    │ 3·54  │  140·8   │  867·9 │
  │ distance }│          │        │        │       │          │        │
  └———————————┴——————————┴————————┴————————┴——————-┴——————————┴————————┘

Here it will be seen that the two series of numbers, in the upper and
lower row respectively, agree completely for as many decimal places
as are given, except in the cases of the two outer planets, where the
lower numbers are slightly in excess of the upper. For this discrepancy
Newton afterwards assigned a reason (chapter IX., § 186), but with the
somewhat imperfect knowledge of the times of revolution and distances
which Kepler possessed the discrepancy was barely capable of detection,
and he was therefore justified—from his standpoint—in speaking of the
law as “precise.”[92]

[Illustration: FIG. 62.—The “music of the spheres,” according to
Kepler. From the _Harmony of the World_.]

It should be noticed further that Kepler’s law requires no knowledge of
the actual distances of the several planets from the sun, but only of
their relative distances, _i.e._ the number of times farther off from
the sun or nearer to the sun any planet is than any other. In other
words, it is necessary to have or to be able to construct a map of the
solar system correct in its _proportions_, but it is quite unnecessary
for this purpose to know the _scale_ of the map.

Although the _Harmony of the World_ is a large book, there is scarcely
anything of value in it except what has already been given. A good
deal of space is occupied with repetitions of the earlier speculations
contained in the _Mysterium Cosmographicum_, and most of the rest is
filled with worthless analogies between the proportions of the solar
system and the relations between various musical scales.

He is bold enough to write down in black and white the “music of the
spheres” (in the form shewn in fig. 62), while the nonsense which he
was capable of writing may be further illustrated by the remark which
occurs in the same part of the book: “The Earth sings the notes M I,
F A, M I, so that you may guess from them that in this abode of ours
MIsery (_miseria_) and FAmine (_fames_) prevail.”

145. The _Epitome of the Copernican Astronomy_, which appeared in
parts in 1618, 1620, and 1621, although there are no very striking
discoveries in it, is one of the most attractive of Kepler’s books,
being singularly free from the extravagances which usually render his
writings so tedious. It contains within moderately short compass, in
the form of question and answer, an account of astronomy as known at
the time, expounded from the Coppernican standpoint, and embodies
both Kepler’s own and Galilei’s latest discoveries. Such a textbook
supplied a decided want, and that this was recognised by enemies as
well as by friends was shewn by its prompt appearance in the Roman
_Index of Prohibited Books_ (cf. chapter VI., §§ 126, 132). The
_Epitome_ contains the first clear statement that the two fundamental
laws of planetary motion established for the case of Mars (§ 141)
were true also for the other planets (no satisfactory proof being,
however, given), and that they applied also to the motion of the moon
round the earth, though in this case there were further irregularities
which complicated matters. The theory of the moon is worked out in
considerable detail, both evection (chapter II., § 48) and variation
(chapter III., § 60; chapter V., § 111) being fully dealt with,
though the “annual equation” which Tycho had just begun to recognise
at the end of his life (chapter V., § 111) is not discussed. Another
interesting development of his own discoveries is the recognition that
his third law of planetary motion applied also to the movements of
the four satellites round Jupiter, as recorded by Galilei and Simon
Marius (chapter VI., § 118). Kepler also introduced in the _Epitome_ a
considerable improvement in the customary estimate of the distance of
the earth from the sun, from which those of the other planets could at
once be deduced.

If, as had been generally believed since the time of Hipparchus and
Ptolemy, the distance of the sun were 1,200 times the radius of the
earth, then the parallax (chapter II., §§ 43, 49) of the sun would
at times be as much as 3′, and that of Mars, which in some positions
is much nearer to the earth, proportionally larger. But Kepler had
been unable to detect any parallax of Mars, and therefore inferred
that the distances of Mars and of the sun must be greater than had
been supposed. Having no exact data to go on, he produced out of his
imagination and his ideas of the harmony of the solar system a distance
about three times as great as the traditional one. He argued that,
as the earth was the abode of measuring creatures, it was reasonable
to expect that the measurements of the solar system would bear some
simple relation to the dimensions of the earth. Accordingly he assumed
that the volume of the sun was as many times greater than the volume
of the earth as the distance of the sun was greater than the radius of
the earth, and from this quaint assumption deduced the value of the
distance already stated, which, though an improvement on the old value,
was still only about one-seventh of the true distance.

The _Epitome_ contains also a good account of eclipses both of the
sun and moon, with the causes, means of predicting them, etc. The
faint light (usually reddish) with which the face of the eclipsed moon
often shines is correctly explained as being sunlight which has passed
through the atmosphere of the earth, and has there been bent from a
straight course so as to reach the moon, which the light of the sun in
general is, owing to the interposition of the earth, unable to reach.
Kepler mentions also a ring of light seen round the eclipsed sun in
1567, when the eclipse was probably total, not annular (chapter II.,
§ 43), and ascribes it to some sort of luminous atmosphere round the
sun, referring to a description in Plutarch of the same appearance.
This seems to have been an early observation, and a rational though of
course very imperfect explanation, of that remarkable solar envelope
known as the =corona= which has attracted so much attention in the
last half-century (chapter XIII., § 301).

146. The treatise on _Comets_ (1619) contained an account of a comet
seen in 1607, afterwards famous as Halley’s comet (chapter X., §
200), and of three comets seen in 1618. Following Tycho, Kepler held
firmly the view that comets were celestial not terrestrial bodies, and
accounted for their appearance and disappearance by supposing that
they moved in straight lines, and therefore after having once passed
near the earth receded indefinitely into space; he does not appear to
have made any serious attempt to test this theory by comparison with
observation, being evidently of opinion that the path of a body which
would never reappear was not a suitable object for serious study.
He agreed with the observation made by Fracastor and Apian (chapter
III., § 69) that comets’ tails point away from the sun, and explained
this by the supposition that the tail is formed by rays of the sun
which penetrate the body of the comet and carry away with them some
portion of its substance, a theory which, allowance being made for the
change in our view’s as to the nature of light, is a curiously correct
anticipation of modern theories of comets’ tails (chapter XIII., § 304).

In a book intended to have a popular sale it was necessary to make
the most of the “meaning” of the appearance of a comet, and of its
influence on human affairs, and as Kepler was writing when the Thirty
Years’ War had just begun, while religious persecutions and wars
had been going on in Europe almost without interruption during his
lifetime, it was not difficult to find sensational events which had
happened soon after or shortly before the appearance of the comets
referred to. Kepler himself was evidently not inclined to attach much
importance to such coincidences; he thought that possibly actual
contact with a comet’s tail might produce pestilence, but beyond that
was not prepared to do more than endorse the pious if somewhat neutral
opinion that one of the uses of a comet is to remind us that we are
mortal. His belief that comets are very numerous is expressed in the
curious form: “There are as many arguments to prove the annual motion
of the earth round the sun as there are comets in the heavens.”

147. Meanwhile Kepler’s position at Linz had become more and more
uncomfortable, owing to the rising tide of the religious and political
disturbances which finally led to the outbreak of the Thirty Years’
War in 1618; but notwithstanding this he had refused in 1617 an offer
of a chair of mathematics at Bologna, partly through attachment to
his native country and partly through a well-founded distrust of the
Papal party in Italy. Three years afterwards he rejected also the
overtures made by the English ambassador, with a view to securing him
as an ornament to the court of James I., one of his chief grounds for
refusal in this case being a doubt whether he would not suffer from
being cooped up within the limits of an island. In 1619 the Emperor
Matthias died, and was succeeded by Ferdinand II., who as Archduke had
started the persecution of the Protestants at Gratz (§ 137) and who
had few scientific interests. Kepler was, however, after some delay,
confirmed in his appointment as Imperial Mathematician. In 1620 Linz
was occupied by the Imperialist troops, and by 1626 the oppression of
the Protestants by the Roman Catholics had gone so far that Kepler made
up his mind to leave, and, after sending his family to Regensburg, went
himself to Ulm.

148. At Ulm Kepler published his last great work. For more than a
quarter of a century he had been steadily working out in detail, on the
basis of Tycho’s observations and of his own theories, the motions of
the heavenly bodies, expressing the results in such convenient tabular
form that the determination of the place of any body at any required
time, as well as the investigation of other astronomical events such
as eclipses, became merely a matter of calculation according to fixed
rules; this great undertaking, in some sense the summing up of his own
and of Tycho’s work, was finally published in 1627 as the _Rudolphine
Tables_ (the name being given in honour of his former patron), and
remained for something like a century the standard astronomical tables.

It had long been Kepler’s intention, after finishing the tables,
to write a complete treatise on astronomy, to be called the _New
Almagest_; but this scheme was never fairly started, much less carried
out.

149. After a number of unsuccessful attempts to secure the arrears of
his salary, he was told to apply to Wallenstein, the famous Imperialist
general, then established in Silesia in a semi-independent position,
who was keenly interested in astrology and usually took about with
him one or more representatives of the art. Kepler accordingly joined
Wallenstein in 1628, and did astrology for him, in addition to
writing some minor astronomical and astrological treatises. In 1630
he travelled to Regensburg, where the Diet was then sitting, to press
in person his claims for various arrears of salary; but, worn out by
anxiety and by the fatigues of the journey, he was seized by a fever a
few days after his arrival, and died on November 15th (N.S.), 1630, in
his 59th year.

The inventory of his property, made after his death, shews that he was
in possession of a substantial amount, so that the effect of extreme
poverty which his letters convey must have been to a considerable
extent due to his over-anxious and excitable temperament.

150. In addition to the great discoveries already mentioned Kepler
made a good many minor contributions to astronomy, such as new
methods of finding the longitude, and various improvements in methods
of calculation required for astronomical problems. He also made
speculations of some interest as to possible causes underlying the
known celestial motions. Whereas the Ptolemaic system required a number
of motions round mere geometrical points, centres of epicycles or
eccentrics, equants, etc., unoccupied by any real body, and many such
motions were still required by Coppernicus, Kepler’s scheme of the
solar system placed a real body, the sun, at the most important point
connected with the path of each planet, and dealt similarly with the
moon’s motion round the earth and with that of the four satellites
round Jupiter. Motions of revolution came in fact to be associated
not with some central _point_ but with some central _body_, and it
became therefore an inquiry of interest to ascertain if there were
any connection between the motion and the central body. The property
possessed by a magnet of attracting a piece of iron at some little
distance from it suggested a possible analogy to Kepler, who had read
with care and was evidently impressed by the treatise _On the Magnet_
(De Magnete) published in 1600 by our countryman _William Gilbert_ of
Colchester (1540-1603). He suggested that the planets might thus be
regarded as connected with the sun, and therefore as sharing to some
extent the sun’s own motion of revolution. In other words, a certain
“carrying virtue” spread out from the sun, with or like the rays of
light and heat, and tried to carry the planets round with the sun.

 “There is therefore a conflict between the carrying power of the sun
 and the impotence or material sluggishness (_inertia_) of the planet;
 each enjoys some measure of victory, for the former moves the planet
 from its position and the latter frees the planet’s body to some
 extent from the bonds in which it is thus held, ... but only to be
 captured again by another portion of this rotatory virtue.”[93]

The annexed diagram is given by Kepler in illustration of this rather
confused and vague theory.

[Illustration: FIG. 63.—Kepler’s idea of gravity. From the _Epitome_.]

He believed also in a more general “gravity,” which he defined[94] as
“a mutual bodily affection between allied bodies tending towards their
union or junction,” and regarded the tides as due to an action of this
sort between the moon and the water of the earth. But the speculative
ideas thus thrown out, which it is possible to regard as anticipations
of Newton’s discovery of universal gravitation, were not in any way
developed logically, and Kepler’s mechanical ideas were too imperfect
for him to have made real progress in this direction.

151. There are few astronomers about whose merits such different
opinions have been held as about Kepler. There is, it is true, a
general agreement as to the great importance of his three laws of
planetary motion, and as to the substantial value of the _Rudolphine
Tables_ and of various minor discoveries. These results, however, fill
but a small part of Kepler’s voluminous writings, which are encumbered
with masses of wild speculation, of mystic and occult fancies, of
astrology, weather prophecies, and the like, which are not only
worthless from the standpoint of modern astronomy, but which—unlike
many erroneous or imperfect speculations—in no way pointed towards the
direction in which the science was next to make progress, and must
have appeared almost as unsound to sober-minded contemporaries like
Galilei as to us. Hence as one reads chapter after chapter without
a lucid still less a correct idea, it is impossible to refrain from
regrets that the intelligence of Kepler should have been so wasted,
and it is difficult not to suspect at times that some of the valuable
results which lie imbedded in this great mass of tedious speculation
were arrived at by a mere accident. On the other hand, it must not
be forgotten that such accidents have a habit of happening only to
great men, and that if Kepler loved to give reins to his imagination
he was equally impressed with the necessity of scrupulously comparing
speculative results with observed facts, and of surrendering without
demur the most beloved of his fancies if it was unable to stand this
test. If Kepler had burnt three-quarters of what he printed, we should
in all probability have formed a higher opinion of his intellectual
grasp and sobriety of judgment, but we should have lost to a great
extent the impression of extraordinary enthusiasm and industry, and of
almost unequalled intellectual honesty, which we now get from a study
of his works.



CHAPTER VIII.

FROM GALILEI TO NEWTON.

    “And now the lofty telescope, the scale
    By which they venture heaven itself t’assail.
    Was raised, and planted full against the moon.”
                                                         _Hudibras_


152. Between the publication of Galilei’s _Two New Sciences_ (1638)
and that of Newton’s _Principia_ (1687) a period of not quite half a
century elapsed; during this interval no astronomical discovery of
first-rate importance was published, but steady progress was made on
lines already laid down.

On the one hand, while the impetus given to exact observation by Tycho
Brahe had not yet spent itself, the invention of the telescope and
its gradual improvement opened out an almost indefinite field for
possible discovery of new celestial objects of interest. On the other
hand, the remarkable character of the three laws in which Kepler had
summed up the leading characteristics of the planetary motions could
hardly fail to suggest to any intelligent astronomer the question _why_
these particular laws should hold, or, in other words, to stimulate
the inquiry into the possibility of shewing them to be necessary
consequences of some simpler and more fundamental law or laws, while
Galilei’s researches into the laws of motion suggested the possibility
of establishing some connection between the causes underlying these
celestial motions and those of ordinary terrestrial objects.

153. It has been already mentioned how closely Galilei was followed
by other astronomers (if not in some cases actually anticipated) in
most of his telescopic discoveries. To his rival Christopher Scheiner
(chapter VI., §§ 124, 125) belongs the credit of the discovery of
bright cloud-like objects on the sun, chiefly visible near its edge,
and from their brilliancy named =faculae= (little torches). Scheiner
made also a very extensive series of observations of the motions and
appearances of spots.

The study of the surface of the moon was carried on with great care
by _John Hevel_ of Danzig (1611-1687), who published in 1647 his
_Selenographia_, or description of the moon, magnificently illustrated
by plates engraved as well as drawn by himself. The chief features of
the moon—mountains, craters, and the dark spaces then believed to be
seas—were systematically described and named, for the most part after
corresponding features of our own earth. Hevel’s names for the chief
mountain ranges, _e.g._ the _Apennines_ and the _Alps_, and for the
seas, _e.g._ _Mare Serenitatis_ or Pacific Ocean, have lasted till
to-day; but similar names given by him to single mountains and craters
have disappeared, and they are now called after various distinguished
men of science and philosophers, _e.g._ _Plato_ and _Coppernicus_,
in accordance with a system introduced by _John Baptist Riccioli_
(1598-1671) in his bulky treatise on astronomy called the _New
Almagest_ (1651).

Hevel, who was an indefatigable worker, published two large books
on comets, _Prodromus Cometicus_ (1654) and _Cometographia_ (1668),
containing the first systematic account of all recorded comets. He
constructed also a catalogue of about 1,500 stars, observed on the
whole with accuracy rather greater than Tycho’s, though still without
the use of the telescope; he published in addition an improved set of
tables of the sun, and a variety of other calculations and observations.

154. The planets were also watched with interest by a number of
observers, who detected at different times bright or dark markings on
Jupiter, Mars, and Venus. The two appendages of Saturn which Galilei
had discovered in 1610 and had been unable to see two years later
(chapter VI., § 123) were seen and described by a number of astronomers
under a perplexing variety of appearances, and the mystery was only
unravelled, nearly half a century after Galilei’s first observation,
by the greatest astronomer of this period, _Christiaan Huygens_
(1629-1695), a native of the Hague. Huygens possessed remarkable
ability, both practical and theoretical, in several different
directions, and his contributions to astronomy were only a small
part of his services to science. Having acquired the art of grinding
lenses with unusual accuracy, he was able to construct telescopes of
much greater power than his predecessors. By the help of one of these
instruments he discovered in 1655 a satellite of Saturn (_Titan_).
With one of those remnants of mediaeval mysticism from which even
the soberest minds of the century freed themselves with the greatest
difficulty, he asserted that, as the total number of planets and
satellites now reached the perfect number 12, no more remained to be
discovered—a prophecy which has been abundantly falsified since (§ 160;
chapter XII., §§ 253, 255; chapter XIII., §§ 289, 294, 295).

Using a still finer telescope, and aided by his acuteness in
interpreting his observations, Huygens made the much more interesting
discovery that the puzzling appearances seen round Saturn were due to
a thin ring (fig. 64) inclined at a considerable angle (estimated by
him at 31°) to the plane of the ecliptic, and therefore also to the
plane in which Saturn’s path round the sun lies. This result was first
announced—according to the curious custom of the time—by an anagram,
in the same pamphlet in which the discovery of the satellite was
published, _De Saturni Luna Observatio Nova_ (1656); and three years
afterwards (1659) the larger _Systema Saturnium_ appeared, in which the
interpretation of the anagram was given, and the varying appearances
seen both by himself and by earlier observers were explained with
admirable lucidity and thoroughness. The ring being extremely thin is
invisible either when its edge is presented to the observer or when it
is presented to the sun, because in the latter position the rest of the
ring catches no light. Twice in the course of Saturn’s revolution round
the sun (at B and D in fig. 66), _i.e._ at intervals of about 15 years,
the plane of the ring passes for a short time through or very close
both to the earth and to the sun, and at these two periods the ring is
consequently invisible (fig. 65). Near these positions (as at Q, R, S,
T) the ring appears much foreshortened, and presents the appearance
of two arms projecting from the body of Saturn; farther off still
the ring appears wider and the opening becomes visible; and about
seven years before and after the periods of invisibility (at A and C)
the ring is seen at its widest. Huygens gives for comparison with his
own results a number of drawings by earlier observers (reproduced in
fig. 67), from which it may be seen how near some of them were to the
discovery of the ring.

[Illustration: FIG. 64.—Saturn’s ring, as drawn by Huygens. From the
_Systema Saturnium_.]

[Illustration: FIG. 65.—Saturn, with the ring seen edge-wise. From the
_Systema Saturnium_.]

[Illustration: FIG. 66.—The phases of Saturn’s ring. From the _Systema
Saturnium_.]

155. To our countryman _William Gascoigne_ (1612?-1644) is due the
first recognition that the telescope could be utilised, not merely
for observing generally the appearances of celestial bodies, but also
as an instrument of precision, which would give the directions of
stars, etc., with greater accuracy than is possible with the naked
eye, and would magnify small angles in such a way as to facilitate the
measurement of angular distances between neighbouring stars, of the
diameters of the planets, and of similar quantities. He was unhappily
killed when quite a young man at the battle of Marston Moor (1644),
but his letters, published many years afterwards shew that by 1640
he was familiar with the use of telescopic “sights,” for determining
with accuracy the position of a star, and that he had constructed a
so-called =micrometer=[95] with which he was able to measure angles
of a few seconds. Nothing was known of his discoveries at the time,
and it was left for Huygens to invent independently a micrometer of an
inferior kind (1658), and for _Adrien Auzout_ (?-1691) to introduce
as an improvement (about 1666) an instrument almost identical with
Gascoigne’s.

The systematic use of telescopic sights for the regular work of an
observatory was first introduced about 1667 by Auzout’s friend and
colleague _Jean Picard_ (1620-1682).

[Illustration: FIG. 67.—Early drawings of Saturn. From the _Systema
Saturnium_.]

156. With Gascoigne should be mentioned his friend _Jeremiah Horrocks_
(1617?-1641), who was an enthusiastic admirer of Kepler and had made
a considerable improvement in the theory of the moon, by taking the
elliptic orbit as a basis and then allowing for various irregularities.
He was the first observer of a transit of Venus, _i.e._ a passage of
Venus over the disc of the sun, an event which took place in 1639,
contrary to the prediction of Kepler in the _Rudolphine Tables_, but
in accordance with the rival tables of _Philips von Lansberg_
(1561-1632) which Horrocks had verified for the purpose. It was not,
however, till long afterwards that Halley pointed out the importance
of the transit of Venus as a means of ascertaining the distance of the
sun from the earth (chapter X., § 202). It is also worth noticing that
Horrocks suggested the possibility of the irregularities of the moon’s
motion being due to the disturbing action of the sun, and that he also
had some idea of certain irregularities in the motion of Jupiter and
Saturn, now known to be due to their mutual attraction (chapter X., §
204; chapter XI., § 243).

157. Another of Huygens’s discoveries revolutionised the art of exact
astronomical observation. This was the invention of the pendulum-clock
(made 1656, patented in 1657). It has been already mentioned how the
same discovery was made by Bürgi, but virtually lost (see chapter V., §
98); and how Galilei again introduced the pendulum as a time-measurer
(chapter VI., § 114). Galilei’s pendulum, however, could only be used
for measuring very short times, as there was no mechanism to keep it in
motion, and the motion soon died away. Huygens attached a pendulum to
a clock driven by weights, so that the clock kept the pendulum going
and the pendulum regulated the clock.[96] Henceforward it was possible
to take reasonably accurate time-observations, and, by noticing the
interval between the passage of two stars across the meridian, to
deduce, from the known rate of motion of the celestial sphere, their
angular distance east and west of one another, thus helping to fix the
position of one with respect to the other. It was again Picard (§ 155)
who first recognised the astronomical importance of this discovery, and
introduced regular time-observations at the new Observatory of Paris.

158. Huygens was not content with this practical use of the pendulum,
but worked out in his treatise called _Oscillatorium Horologium_ or
_The Pendulum Clock_ (1673) a number of important results in the theory
of the pendulum, and in the allied problems connected with the motion
of a body in a circle or other curve. The greater part of these
investigations lie outside the field of astronomy, but his formula
connecting the time of oscillation of a pendulum with its length and
the intensity of gravity[97] (or, in other words, the rate of falling
of a heavy body) afforded a practical means of measuring gravity, of
far greater accuracy than any direct experiments on falling bodies;
and his study of circular motion, leading to the result that a body
moving in a circle must be acted on by some force towards the _centre_,
the magnitude of which depended in a definite way on the speed of the
body and the size of the circle,[98] is of fundamental importance in
accounting for the planetary motions by gravitation.

159. During the 17th century also the first measurements of the earth
were made which were a definite advance on those of the Greeks and
Arabs (chapter II., §§ 36, 45, and chapter III., § 57). _Willebrord
Snell_ (1591-1626), best known by his discovery of the law of
refraction of light, made a series of measurements in Holland in 1617,
from which the length of a degree of a meridian appeared to be about
67 miles, an estimate subsequently altered to about 69 miles by one
of his pupils, who corrected some errors in the calculations, the
result being then within a few hundred feet of the value now accepted.
Next, _Richard Norwood_ (1590?-1675) measured the distance from London
to York, and hence obtained (1636) the length of the degree with an
error of less than half a mile. Lastly, Picard in 1671 executed some
measurements near Paris leading to a result only a few yards wrong. The
length of a degree being known, the circumference and radius of the
earth can at once be deduced.

160. Auzout and Picard were two members of a group of observational
astronomers working at Paris, of whom the best known, though probably
not really the greatest, was _Giovanni Domenico Cassini_ (1625-1712).
Born in the north of Italy, he acquired a great reputation, partly
by some rather fantastic schemes for the construction of gigantic
instruments, partly by the discovery of the rotation of Jupiter
(1665), of Mars (1666), and possibly of Venus (1667), and also by his
tables of the motions of Jupiter’s moons (1668). The last caused Picard
to procure for him an invitation from Louis XIV. (1669) to come to
Paris and to exercise a general superintendence over the Observatory,
which was then being built and was substantially completed in 1671.
Cassini was an industrious observer and a voluminous writer, with a
remarkable talent for impressing the scientific public as well as the
Court. He possessed a strong sense of the importance both of himself
and of his work, but it is more than doubtful if he had as clear ideas
as Picard of the really important pieces of work which ought to be
done at a public observatory, and of the way to set about them. But,
notwithstanding these defects, he rendered valuable services to various
departments of astronomy. He discovered four new satellites of Saturn:
_Japetus_ in 1671, _Rhea_ in the following year, _Dione_ and _Thetis_
in 1684; and also noticed in 1675 a dark marking in Saturn’s ring,
which has subsequently been more distinctly recognised as a division
of the ring into two, an inner and an outer, and is known as Cassini’s
division (see fig. 95 facing p. 384). He also improved to some extent
the theory of the sun, calculated a fresh table of atmospheric
refraction which was an improvement on Kepler’s (chapter VII., § 138),
and issued in 1693 a fresh set of tables of Jupiter’s moons, which were
much more accurate than those which he had published in 1668, and much
the best existing.

161. It was probably at the suggestion of Picard or Cassini that one
of their fellow astronomers, _John Richer_ (?-1696), otherwise almost
unknown, undertook (1671-3) a scientific expedition to Cayenne (in
latitude 5° N.). Two important results were obtained. It was found that
a pendulum of given length beat more slowly at Cayenne than at Paris,
thus shewing that the intensity of gravity was less near the equator
than in higher latitudes. This fact suggested that the earth was not a
perfect sphere, and was afterwards used in connection with theoretical
investigations of the problem of the earth’s shape (cf. chapter IX., §
187). Again, Richer’s observations of the position of Mars in the sky,
combined with observations taken at the same time Cassini, Picard,
and others in France, led to a reasonably accurate estimate of the
distance of Mars and hence of that of the sun. Mars was at the time in
opposition (chapter II., § 43), so that it was nearer to the earth than
at other times (as shewn in fig. 68), and therefore favourably situated
for such observations. The principle of the method is extremely simple
and substantially identical with that long used in the case of the moon
(chapter II., § 49). One observer is, say, at Paris (P, in fig. 69),
and observes the direction in which Mars appears, _i.e._ the direction
of the line P M; the other at Cayenne (C) observes similarly the
direction of the line C M. The line C P, joining Paris and Cayenne, is
known geographically; the shape of the triangle C P M and the length of
one of its sides being thus known, the lengths of the other sides are
readily calculated.

[Illustration: FIG. 68.—Mars in opposition.]

[Illustration: FIG. 69.—The parallax of a planet.]

The result of an investigation of this sort is often most conveniently
expressed by means of a certain angle, from which the distance in
terms of the radius of the earth, and hence in miles, can readily be
deduced when desired.

The parallax of a heavenly body such as the moon, the sun, or a planet,
being in the first instance defined generally (chapter II., § 43) as
the angle (O M P) between the lines joining the heavenly body to the
observer and to the centre of the earth, varies in general with the
position of the observer. It is evidently greatest when the observer
is in such a position, as at Q, that the line M Q touches the earth;
in this position M is on the observer’s horizon. Moreover the angle O
Q M being a right angle, the shape of the triangle and the ratio of
its sides are completely known when the angle O M Q is known. Since
this angle is the parallax of M, when on the observer’s horizon, it
is called the =horizontal parallax= of M, but the word horizontal is
frequently omitted. It is easily seen by a figure that the more distant
a body is the smaller is its horizontal parallax; and with the small
parallaxes with which we are concerned in astronomy, the distance and
the horizontal parallax can be treated as inversely proportional to
one another; so that if, for example, one body is twice as distant as
another, its parallax is half as great, and so on.

It may be convenient to point out here that the word “parallax” is
used in a different though analogous sense when a fixed star is in
question. The apparent displacement of a fixed star due to the earth’s
motion (chapter IV., § 92), which was not actually detected till long
afterwards (chapter XIII., § 278), is called =annual= or =stellar
parallax= (the adjective being frequently omitted); and the name is
applied in particular to the greatest angle between the direction of
the star as seen from the sun and as seen from the earth in the course
of the year. If in fig. 69 we regard M as representing a star, O the
sun, and the circle as being the earth’s path round the sun, then the
angle O M Q is the annual parallax of M.

In this particular case Cassini deduced from Richer’s observations,
by some rather doubtful processes, that the sun’s parallax was about
9″·5, corresponding to a distance from the earth of about 87,000,000
miles, or about 360 times the distance of the moon, the most probable
value, according to modern estimates (chapter XIII., § 284), being
a little less than 93,000,000. Though not really an accurate result,
this was an enormous improvement on anything that had gone before, as
Ptolemy’s estimate of the sun’s distance, corresponding to a parallax
of 3′, had survived up to the earlier part of the 17th century,
and although it was generally believed to be seriously wrong, most
corrections of it had been purely conjectural (chapter VII., §§ 145).

162. Another famous discovery associated with the early days of the
Paris Observatory was that of the velocity of light. In 1671 Picard
paid a visit to Denmark to examine what was left of Tycho Brahe’s
observatory at Hveen, and brought back a young Danish astronomer,
_Olaus Roemer_ (1644-1710), to help him at Paris. Roemer, in studying
the motion of Jupiter’s moons, observed (1675) that the intervals
between successive eclipses of a moon (the eclipse being caused by the
passage of the moon into Jupiter’s shadow) were regularly less when
Jupiter and the earth were approaching one another than when they were
receding. This he saw to be readily explained by the supposition that
light travels through space at a definite though very great speed. Thus
if Jupiter is approaching the earth, the time which the light from one
of his moons takes to reach the earth is gradually decreasing, and
consequently the interval between successive eclipses as seen by us
is apparently diminished. From the difference of the intervals thus
observed and the known rates of motion of Jupiter and of the earth, it
was thus possible to form a rough estimate of the rate at which light
travels. Roemer also made a number of instrumental improvements of
importance, but they are of too technical a character to be discussed
here.

163. One great name belonging to the period dealt with in this chapter
remains to be mentioned, that of _René Descartes_[99] (1596-1650).
Although he ranks as a great philosopher, and also made some extremely
important advances in pure mathematics, his astronomical writings
were of little value and in many respects positively harmful. In his
_Principles of Philosophy_ (1644) he gave, among some wholly erroneous
propositions, a fuller and more general statement of the first law
of motion discovered by Galilei (chapter VI., §§ 130, 133), but did
not support it by any evidence of value. The same book contained an
exposition of his famous theory of vortices, which was an attempt to
explain the motions of the bodies of the solar system by means of a
certain combination of vortices or eddies. The theory was unsupported
by any experimental evidence, and it was not formulated accurately
enough to be capable of being readily tested by comparison with actual
observation; and, unlike many erroneous theories (such as the Greek
epicycles), it in no way led up to or suggested the truer theories
which followed it. But “Cartesianism,” both in philosophy and in
natural science, became extremely popular, especially in France, and
its vogue contributed notably to the overthrow of the authority of
Aristotle, already shaken by thinkers like Galilei and Bacon, and thus
rendered men’s minds a little more ready to receive new ideas: in this
indirect way, as well as by his mathematical discoveries, Descartes
probably contributed something to astronomical progress.



CHAPTER IX.

UNIVERSAL GRAVITATION.

    “Nature and Nature’s laws lay hid in night;
    God said ‘Let Newton be!’ and all was light.”
                                                               POPE.


164. Newton’s life may be conveniently divided into three portions.
First came 22 years (1643-1665) of boyhood and undergraduate life;
then followed his great productive period, of almost exactly the same
length, culminating in the publication of the _Principia_ in 1687;
while the rest of his life (1687-1727), which lasted nearly as long as
the other two periods put together, was largely occupied with official
work and studies of a non-scientific character, and was marked by no
discoveries ranking with those made in his middle period, though some
of his earlier work received important developments and several new
results of decided interest were obtained.

165. Isaac Newton was born at Woolsthorpe, near Grantham, in
Lincolnshire, on January 4th, 1643;[100] this was very nearly a year
after the death of Galilei, and a few months after the beginning of
our Civil Wars. His taste for study does not appear to have developed
very early in life, but ultimately became so marked that, after some
unsuccessful attempts to turn him into a farmer, he was entered at
Trinity College, Cambridge, in 1661.

Although probably at first rather more backward than most
undergraduates, he made extremely rapid progress in mathematics and
allied subjects, and evidently gave his teachers some trouble by
the rapidity with which he absorbed what little they knew. He met
with Euclid’s _Elements of Geometry_ for the first time while an
undergraduate, but is reported to have soon abandoned it as being “a
trifling book,” in favour of more advanced reading. In January 1665
graduated in the ordinary course as Bachelor of Arts.

166. The external events of Newton’s life during the next 22 years may
be very briefly dismissed. He was elected a Fellow in 1667, became
M.A. in due course in the following year, and was appointed Lucasian
Professor of Mathematics, in succession to his friend Isaac Barrow, in
1669. Three years later he was elected a Fellow of the recently founded
Royal Society. With the exception of some visits to his Lincolnshire
home, he appears to have spent almost the whole period in quiet study
at Cambridge, and the history of his life is almost exclusively the
history of his successive discoveries.

167. His scientific work falls into three main groups, astronomy
(including dynamics), optics, and pure mathematics. He also spent a
good deal of time on experimental work in chemistry, as well as on
heat and other branches of physics, and in the latter half of his life
devoted much attention to questions of chronology and theology; in none
of these subjects, however, did he produce results of much importance.

168. In forming an estimate of Newton’s genius it is of course
important to bear in mind the range of subjects with which he dealt;
from our present point of view, however, his mathematics only presents
itself as a tool to be used in astronomical work; and only those of
his optical discoveries which are of astronomical importance need be
mentioned here. In 1668 he constructed a =reflecting telescope=, that
is, a telescope in which the rays of light from the object viewed
are concentrated by means of a curved mirror instead of by a lens,
as in the =refracting telescopes= of Galilei and Kepler. Telescopes
on this principle, differing however in some important particulars
from Newton’s, had already been described in 1663 by _James Gregory_
(1638-1675), with whose ideas Newton was acquainted, but it does
not appear that Gregory had actually made an instrument. Owing to
mechanical difficulties in construction, half a century elapsed before
reflecting telescopes were made which could compete with the best
refractors of the time, and no important astronomical discoveries were
made with them before the time of William Herschel (chapter XII.), more
than a century after the original invention.

Newton’s discovery of the effect of a prism in resolving a beam of
white light into different colours is in a sense the basis of the
method of spectrum analysis (chapter XIII., § 299), to which so many
astronomical discoveries of the last 40 years are due.

169. The ideas by which Newton is best known in each of his three great
subjects—gravitation, his theory of colours, and fluxions—seem to have
occurred to him and to have been partly thought out within less than
two years after he took his degree, that is before he was 24. His own
account—written many years afterwards—gives a vivid picture of his
extraordinary mental activity at this time:—

  “In the beginning of the year 1665 I found the method of
 approximating Series and the Rule for reducing any dignity of any
 Binomial into such a series. The same year in May I found the method
 of tangents of Gregory and Slusius, and in November had the direct
 method of Fluxions, and the next year in January had the Theory of
 Colours, and in May following I had entrance into the inverse method
 of Fluxions. And the same year I began to think of gravity extending
 to the orb of the Moon, and having found out how to estimate the force
 with which [a] globe revolving within a sphere presses the surface of
 the sphere, from Kepler’s Rule of the periodical times of the Planets
 being in a sesquialterate proportion of their distances from the
 centers of their orbs I deduced that the forces which keep the Planets
 in their orbs must [be] reciprocally as the squares of their distances
 from the centers about which they revolve: and thereby compared the
 force requisite to keep the Moon in her orb with the force of gravity
 at the surface of the earth, and found them answer pretty nearly. All
 this was in the two plague years of 1665 and 1666, for in those days I
 was in the prime of my age for invention, and minded Mathematicks and
 Philosophy more than at any time since.”[101]

170. He spent a considerable part of this time (1665-1666) at
Woolsthorpe, on account of the prevalence of the plague.

The well-known story, that he was set meditating on gravity by the
fall of an apple in the orchard, is based on good authority, and is
perfectly credible in the sense that the apple may have reminded him at
that particular time of certain problems connected with gravity. That
the apple seriously suggested to him the existence of the problems or
any key to their solution is wildly improbable.

Several astronomers had already speculated on the “cause” of the known
motions of the planets and satellites; that is they had attempted to
exhibit these motions as consequences of some more fundamental and
more general laws. Kepler, as we have seen (chapter VII., § 150), had
pointed out that the motions in question should not be considered as
due to the influence of mere geometrical points, such as the centres
of the old epicycles, but to that of other bodies; and in particular
made some attempt to explain the motion of the planets as due to a
special kind of influence emanating from the sun. He went, however,
entirely wrong by looking for a force to keep up the motion of the
planets and as it were push them along. Galilei’s discovery that the
motion of a body goes on indefinitely unless there is some cause at
work to alter or stop it, at once put a new aspect on this as on other
mechanical problems; but he himself did not develop his idea in this
particular direction. _Giovanni Alfonso Borelli_ (1608-1679), in a book
on Jupiter’s satellites published in 1666, and therefore about the
time of Newton’s first work on the subject, pointed out that a body
revolving in a circle (or similar curve) had a tendency to recede from
the centre, and that in the case of the planets this might be supposed
to be counteracted by some kind of attraction towards the sun. We have
then here the idea— in a very indistinct form certainly—that the
motion of a planet is to be explained, not by a force acting in the
direction in which it is moving, but by a force directed towards the
sun, that is about at right angles to the direction of the planet’s
motion. Huygens carried this idea much further—though without special
reference to astronomy—and obtained (chapter VIII., § 158) a numerical
measure for the tendency of a body moving in a circle to recede from
the centre, a tendency which had in some way to be counteracted if the
body was not to fly away. Huygens published his work in 1673, some
years after Newton had obtained his corresponding result, but before
he had published anything; and there can be no doubt that the two men
worked quite independently.

[Illustration: FIG. 70.—Motion in a circle.]

171. Viewed as a purely general question, apart from its astronomical
applications, the problem may be said to be to examine under what
conditions a body can revolve with uniform speed in a circle.

Let A represent the position at a certain instant of a body which is
revolving with uniform speed in a circle of centre O. Then at this
instant the body is moving in the direction of the tangent A _a_ to the
circle. Consequently by Galilei’s First Law (chapter VI., §§ 130, 133),
if left to itself and uninfluenced by any other body, it would continue
to move with the same speed and in the same direction, _i.e._ along the
line A _a_, and consequently would be found after some time at such a
point as _a_. But actually it is found to be at B on the circle. Hence
some influence must have been at work to bring it to B instead of to
_a_. But B is nearer to the centre of the circle than _a_ is; hence
some influence must be at work tending constantly to draw the body
towards O, or counteracting the tendency which it has, in virtue of the
First Law of Motion, to get farther and farther away from O. To express
either of these tendencies numerically we want a more complex idea
than that of velocity or rate of motion, namely =acceleration= or rate
of change of velocity, an idea which Galilei added to science in his
discussion of the law of falling bodies (chapter VI., §§ 116, 133). A
falling body, for example, is moving after one second with the velocity
of about 32 feet per second, after two seconds with the velocity of
64, after three seconds with the velocity of 96, and so on; thus in
every second it gains a downward velocity of 32 feet per second; and
this may be expressed otherwise by saying that the body has a downward
acceleration of 32 feet per second per second. A further investigation
of the motion in a circle shews that the motion is completely explained
if the moving body has, in addition to its original velocity, an
acceleration of a certain magnitude _directed towards the centre of
the circle_. It can be shewn further that the acceleration may be
numerically expressed by taking the square of the velocity of the
moving body (expressed, say, in feet per second), and dividing this by
the radius of the circle in feet. If, for example, the body is moving
in a circle having a radius of four feet, at the rate of ten feet a
second, then the acceleration towards the centre is (10 × 10)∕4 = 25
feet per second per second.

These results, with others of a similar character, were first published
by Huygens—not of course precisely in this form—in his book on the
_Pendulum Clock_ (chapter VIII., § 158); and discovered independently
by Newton in 1666.

If then a body is seen to move in a circle, its motion becomes
intelligible if some other body can be discovered which produces this
acceleration. In a common case, such as when a stone is tied to a
string and whirled round, this acceleration is produced by the string
which pulls the stone; in a spinning-top the acceleration of the outer
parts is produced by the forces binding them on to the inner part, and
so on.

172. In the most important cases of this kind which occur in astronomy,
a planet is known to revolve round the sun in a path which does not
differ much from a circle. If we assume for the present that the path
is actually a circle, the planet must have an acceleration towards the
centre, and it is possible to attribute this to the influence of the
central body, the sun. In this way arises the idea of attributing to
the sun the power of influencing in some way a planet which revolves
round it, so as to give it an acceleration towards the sun; and the
question at once arises of how this “influence” differs at different
distances. To answer this question Newton made use of Kepler’s Third
Law (chapter VII., § 144). We have seen that, according to this
law, the squares of the times of revolution of any two planets are
proportional to the cubes of their distances from the sun; but the
velocity of the planet may be found by dividing the length of the
path it travels in its revolution round the sun by the time of the
revolution, and this length is again proportional to the distance of
the planet from the sun. Hence the velocities of the two planets are
proportional to their distances from the sun, divided by the times
of revolution, and consequently the squares of the velocities are
proportional to the squares of the distances from the sun divided
by the squares of the times of revolution. Hence, by Kepler’s law,
the squares of the velocities are proportional to the squares of the
distances divided by the cubes of the distances, that is the squares
of the velocities are _inversely_ proportional to the distances, the
more distant planet having the less velocity and _vice versa_. Now by
the formula of Huygens the acceleration is measured by the square of
the velocity divided by the radius of the circle (which in this case
is the distance of the planet from the sun). The accelerations of the
two planets towards the sun are therefore inversely proportional to the
distances each multiplied by itself, that is are inversely proportional
to the squares of the distances. Newton’s first result therefore is:
that the motions of the planets—regarded as moving in circles, and in
strict accordance with Kepler’s Third Law—can be explained as due to
the action of the sun, if the sun is supposed capable of producing on a
planet an acceleration towards the sun itself which is proportional to
the inverse square of its distance from the sun; _i.e._ at twice the
distance it is 1∕4 as great, at three times the distance 1∕9 as great,
at ten times the distance 1∕100 as great, and so on.

The argument may perhaps be made clearer by a numerical example. In
round numbers Jupiter’s distance from the sun is five times as great as
that of the earth, and Jupiter takes 12 years to perform a revolution
round the sun, whereas the earth takes one. Hence Jupiter goes in
12 years five times as far as the earth goes in one, and Jupiter’s
velocity is therefore about 5∕12 that of the earth’s, or the two
velocities are in the ratio of 5 to 12; the squares of the velocities
are therefore as 5 × 5 to 12 × 12, or as 25 to 144. The accelerations
of Jupiter and of the earth towards the sun are therefore as 25 ÷ 5
to 144, or as 5 to 144; hence Jupiter’s acceleration towards the sun
is about 1∕28 earth, and if we had taken more accurate figures this
fraction would have come out more nearly 1∕25. Hence at five times the
distance the acceleration is 25 times less.

This =law of the inverse square=, as it may be called, is also the law
according to which the light emitted from the sun or any other bright
body varies, and would on this account also be not unlikely to suggest
itself in connection with any kind of influence emitted from the sun.

173. The next step in Newton’s investigation was to see whether the
motion of the moon round the earth could be explained in some similar
way. By the same argument as before, the moon could be shewn to have
an acceleration towards the earth. Now a stone if let drop falls
downwards, that is in the direction of the centre of the earth, and, as
Galilei had shewn (chapter VI., § 133), this motion is one of uniform
acceleration; if, in accordance with the opinion generally held at
that time, the motion is regarded as being due to the earth, we may
say that the earth has the power of giving an acceleration towards
its own centre to bodies near its surface. Newton noticed that this
power extended at any rate to the tops of mountains, and it occurred
to him that it might possibly extend as far as the moon and so give
rise to the required acceleration. Although, however, the acceleration
of falling bodies, as far as was known at the time, was the same
for terrestrial bodies wherever situated, it was probable that at
such a distance as that of the moon the acceleration caused by the
earth would be much less. Newton assumed as a working hypothesis that
the acceleration diminished according to the same law which he had
previously arrived at in the case of the sun’s action on the planets,
that is that the acceleration produced by the earth on any body is
inversely proportional to the square of the distance of the body from
the centre of the earth.

It may be noticed that a difficulty arises here which did not present
itself in the corresponding case of the planets. The distances of the
planets from the sun being large compared with the size of the sun, it
makes little difference whether the planetary distances are measured
from the centre of the sun or from any other point in it. The same is
true of the moon and earth; but when we are comparing the action of the
earth on the moon with that on a stone situated on or near the ground,
it is clearly of the utmost importance to decide whether the distance
of the stone is to be measured from the nearest point of the earth, a
few feet off, from the centre of the earth, 4000 miles off, or from
some other point. Provisionally at any rate Newton decided on measuring
from the centre of the earth.

It remained to verify his conjecture in the case of the moon by a
numerical calculation; this could easily be done if certain things were
known, _viz._ the acceleration of a falling body on the earth, the
distance of the surface of the earth from its centre, the distance of
the moon, and the time taken by the moon to perform a revolution round
the earth. The first of these was possibly known with fair accuracy;
the last was well known; and it was also known that the moon’s distance
was about 60 times the radius of the earth. How accurately Newton at
this time knew the size of the earth is uncertain. Taking moderately
accurate figures, the calculation is easily performed. In a month of
about 27 days the moon moves about 60 times as far as the distance
round the earth; that is she moves about 60 × 24,000 miles in 27 days,
which is equivalent to about 3,300 feet per second. The acceleration
of the moon is therefore measured by the square of this, divided by
the distance of the moon (which is 60 times the radius of the earth,
or 20,000,000 feet); that is, it is (3,300 × 3,300)∕(60 × 20,000,000),
which reduces to about 1∕110. Consequently, if the law of the inverse
square holds, the acceleration of a falling body at the surface of the
earth, which is 60 times nearer to the centre than the moon is, should
be (60 × 60)∕110, or between 32 and 33; but the actual acceleration
of falling bodies is rather more than 32. The argument is therefore
satisfactory, and Newton’s hypothesis is so far verified.

The analogy thus indicated between the motion of the moon round the
earth and the motion of a falling stone may be illustrated by a
comparison, due to Newton, of the moon to a bullet shot horizontally
out of a gun from a high place on the earth. Let the bullet start from
B in fig. 71, then moving at first horizontally it will describe a
curved path and reach the ground at a point such as C, at some distance
from the point A, vertically underneath its starting-point. If it were
shot out with a greater velocity, its path at first would be flatter
and it would reach the ground at a point C′ beyond C; if the velocity
were greater still, it would reach the ground at C″ or at C‴; and
it requires only a slight effort of the imagination to conceive that,
with a still greater velocity to begin with, it would miss the earth
altogether and describe a circuit round it, such as B D E. This is
exactly what the moon does, the only difference being that the moon is
at a much greater distance than we have supposed the bullet to be, and
that her motion has not been produced by anything analogous to the gun;
but the motion being once there it is immaterial how it was produced
or whether it was ever produced in the past. We may in fact say of the
moon “that she is a falling body, only she is going so fast and is so
far off that she falls quite round to the other side of the earth,
instead of hitting it; and so goes on for ever.”[102]

[Illustration: FIG. 71.—The moon as a projectile.]

In the memorandum already quoted (§ 169) Newton speaks of the
hypothesis as fitting the facts “pretty nearly”; but in a letter of
earlier date (June 20th, 1686) he refers to the calculation as not
having been made accurately enough. It is probable that he used a
seriously inaccurate value of the size of the earth, having overlooked
the measurements of Snell and Norwood (chapter VIII., § 159); it is
known that even at a later stage he was unable to deal satisfactorily
with the difficulty above mentioned, as to whether the earth might
for the purposes of the problem be identified with its centre; and he
was of course aware that the moon’s path differed considerably from a
circle. The view, said to have been derived from Newton’s conversation
many years afterwards, that he was so dissatisfied with his results
as to regard his hypothesis as substantially defective, is possible,
but by no means certain; whatever the cause may have been, he laid the
subject aside for some years without publishing anything on it, and
devoted himself chiefly to optics and mathematics.

174. Meanwhile the problem of the planetary motions was one of the
numerous subjects of discussion among the remarkable group of men who
were the leading spirits of the Royal Society, founded in 1662. _Robert
Hooke_ (1635-1703), who claimed credit for most of the scientific
discoveries of the time, suggested with some distinctness, not later
than 1674, that the motions of the planets might be accounted for
by attraction between them and the sun, and referred also to the
possibility of the earth’s attraction on bodies varying according
to the law of the inverse square. _Christopher Wren_ (1632-1723),
better known as an architect than as a man of science, discussed some
questions of this sort with Newton in 1677, and appears also to have
thought of a law of attraction of this kind. A letter of Hooke’s to
Newton, written at the end of 1679, dealing amongst other things
with the curve which a falling body would describe, the rotation of
the earth being taken into account, stimulated Newton, who professed
that at this time his “affection to philosophy” was “worn out,” to go
on with his study of the celestial motions. Picard’s more accurate
measurement of the earth (chapter VIII., § 159) was now well known,
and Newton repeated his former calculation of the moon’s motion, using
Picard’s improved measurement, and found the result more satisfactory
than before.

175. At the same time (1679) Newton made a further discovery of the
utmost importance by overcoming some of the difficulties connected with
motion in a path other than a circle.

He shewed that if a body moved round a central body, in such a way
that the line joining the two bodies sweeps out equal areas in equal
times, as in Kepler’s Second Law of planetary motion (chapter VII.,
§ 141), then the moving body is acted on by an attraction directed
exactly towards the central body; and further that if the path is an
ellipse, with the central body in one focus, as in Kepler’s First
Law of planetary motion, then this attraction must vary in different
parts of the path as the inverse square of the distance between the
two bodies. Kepler’s laws of planetary motion were in fact shewn to
lead necessarily to the conclusions that the sun exerts on a planet an
attraction inversely proportional to the square of the distance of the
planet from the sun, and that such an attraction affords a sufficient
explanation of the motion of the planet.

Once more, however, Newton published nothing and “threw his
calculations by, being upon other studies.”

176. Nearly five years later the matter was again brought to his
notice, on this occasion by _Edmund Halley_ (chapter X., §§ 199-205),
whose friendship played henceforward an important part in Newton’s
life, and whose unselfish devotion to the great astronomer forms a
pleasant contrast to the quarrels and jealousies prevalent at that
time between so many scientific men. Halley, not knowing of Newton’s
work in 1666, rediscovered, early in 1684, the law of the inverse
square, as a consequence of Kepler’s Third Law, and shortly afterwards
discussed with Wren and Hooke what was the curve in which a body would
move if acted on by an attraction varying according to this law; but
none of them could answer the question.[103] Later in the year Halley
visited Newton at Cambridge and learnt from him the answer. Newton had,
characteristically enough, lost his previous calculation, but was able
to work it out again and sent it to Halley a few months afterwards.
This time fortunately his attention was not diverted to other topics;
he worked out at once a number of other problems of motion, and devoted
his usual autumn course of University lectures to the subject. Perhaps
the most interesting of the new results was that Kepler’s Third Law,
from which the law of the inverse square had been deduced in 1666,
only on the supposition that the planets moved in circles, was equally
consistent with Newton’s law when the paths of the planets were taken
to be ellipses.

177. At the end of the year 1684 Halley went to Cambridge again and
urged Newton to publish his results. In accordance with this request
Newton wrote out, and sent to the Royal Society, a tract called
_Propositiones de Motu_, the 11 propositions of which contained the
results already mentioned and some others relating to the motion
of bodies under attraction to a centre. Although the propositions
were given in an abstract form, it was pointed out that certain
of them applied to the case of the planets. Further pressure from
Halley persuaded Newton to give his results a more permanent form by
embodying them in a larger book. As might have been expected, the
subject grew under his hands, and the great treatise which resulted
contained an immense quantity of material not contained in the _De
Motu_. By the middle of 1686 the rough draft was finished, and some
of it was ready for press. Halley not only undertook to pay the
expenses, but superintended the printing and helped Newton to collect
the astronomical data which were necessary. After some delay in the
press, the book finally appeared early in July 1687, under the title
_Philosophiae Naturalis Principia Mathematica_.

178. The _Principia_, as it is commonly called, consists of three books
in addition to introductory matter: the first book deals generally
with problems of the motion of bodies, solved for the most part in
an abstract form without special reference to astronomy; the second
book deals with the motion of bodies through media which resist
their motion, such as ordinary fluids, and is of comparatively small
astronomical importance, except that in it some glaring inconsistencies
in the Cartesian theory of vortices are pointed out; the third book
applies to the circumstances of the actual solar system the results
already obtained, and is in fact an explanation of the motions of the
celestial bodies on Newton’s mechanical principles.

179. The introductory portion, consisting of “Definitions” and “Axioms,
or Laws of Motion,” forms a very notable contribution to dynamics,
being in fact the first coherent statement of the fundamental laws
according to which the motions of bodies are produced or changed.
Newton himself does not appear to have regarded this part of his book
as of very great importance, and the chief results embodied in it,
being overshadowed as it were by the more striking discoveries in other
parts of the book, attracted comparatively little attention. Much of it
must be passed over here, but certain results of special astronomical
importance require to be mentioned.

Galilei, as we have seen (chapter VI., §§ 130, 133), was the first to
enunciate the law that a body when once in motion continues to move in
the same direction and at the same speed unless some cause is at work
to make it change its motion. This law is given by Newton in the form
already quoted in § 130, as the first of three fundamental laws, and is
now commonly known as the First Law of Motion.

Galilei also discovered that a falling body moves with continually
changing velocity, but with a uniform acceleration (chapter VI., §
133), and that this acceleration is the same for all bodies (chapter
VI., § 116). The tendency of a body to fall having been generally
recognised as due to the earth, Galilei’s discovery involved the
recognition that one effect of one body on another may be an
acceleration produced in its motion. Newton extended this idea by
shewing that the earth produced an acceleration in the motion of the
moon, and the sun in the motion of the planets, and was led to the
general idea of acceleration in a body’s motion, which might be due
in a variety of ways to the action of other bodies, and which could
conveniently be taken as a measure of the effect produced by one body
on another.

180. To these ideas Newton added the very important and difficult
conception of =mass=.

If we are comparing two different bodies of the same material but
of different sizes, we are accustomed to think of the larger one as
heavier than the other. In the same way we readily think of a ball
of lead as being heavier than a ball of wood of the same size. The
most prominent idea connected with “heaviness” and “lightness” is
that of the muscular effort required to support or to lift the body
in question; a greater effort, for example, is required to hold the
leaden ball than the wooden one. Again, the leaden ball if supported by
an elastic string stretches it farther than does the wooden ball; or
again, if they are placed in the scales of a balance, the lead sinks
and the wood rises. All these effects we attribute to the “weight” of
the two bodies, and the weight we are mostly accustomed to attribute
in some way to the action of the earth on the bodies. The ordinary
process of weighing a body in a balance shews, further, that we are
accustomed to think of weight as a measurable quantity. On the other
hand, we know from Galilei’s result, which Newton tested very carefully
by a series of pendulum experiments, that the leaden and the wooden
ball, if allowed to drop, fall with the same acceleration. If therefore
we measure the effect which the earth produces on the two balls by
their acceleration, then the earth affects them equally; but if we
measure it by the power which they have of stretching strings, or by
the power which one has of supporting the other in a balance, then the
effect which the earth produces on the leaden ball is greater than that
produced on the wooden ball. Taken in this way, the action of the earth
on either ball may be spoken of as weight, and the weight of a body can
be measured by comparing it in a balance with standard bodies.

The difference between two such bodies as the leaden and wooden ball
may, however, be recognised in quite a different way. We can easily
see, for example, that a greater effort is needed to set the one in
motion than the other; or that if each is tied to the end of a string
of given kind and whirled round at a given rate, the one string is
more tightly stretched than the other. In these cases the attraction
of the earth is of no importance, and we recognise a distinction
between the two bodies which is independent of the attraction of the
earth. This distinction Newton regarded as due to a difference in the
quantity of matter or material in the two bodies, and to this quantity
he gave the name of mass. It may fairly be doubted whether anything is
gained by this particular definition of mass, but the really important
step was the distinct recognition of mass as a property of bodies,
of fundamental importance in dynamical questions, and capable of
measurement.

Newton, developing Galilei’s idea, gave as one measurement of the
action exerted by one body on another the product of the mass by the
acceleration produced—a quantity for which he used different names,
now replaced by force. The _weight_ of a body was thus identified with
the force exerted on it by the earth. Since the earth produces the
same acceleration in all bodies at the same place, it follows that the
masses of bodies at the same place are proportional to their weights;
thus if two bodies are compared at the same place, and the weight of
one (as shewn, for example, by a pair of scales) is found to be ten
times that of the other, then its mass is also ten times as great.
But such experiments as those of Richer at Cayenne (chapter VIII., §
161) shewed that the acceleration of falling bodies was less at the
equator than in higher latitudes; so that if a body is carried from
London or Paris to Cayenne, its weight is altered but its mass remains
the same as before. Newton’s conception of the earth’s gravitation as
extending as far as the moon gave further importance to the distinction
between mass and weight; for if a body were removed from the earth to
the moon, then its mass would be unchanged, but the acceleration due
to the earth’s attraction would be 60 × 60 times less, and its weight
diminished in the same proportion.

Rules are also given for the effect produced on a body’s motion by the
simultaneous action of two or more forces.[104]

A further principle of great importance, of which only very indistinct
traces are to be found before Newton’s time, was given by him as the
Third Law of Motion in the form: “To every action there is always an
equal and contrary reaction; or, the mutual actions of any two bodies
are always equal and oppositely directed.” Here action and reaction are
to be interpreted primarily in the sense of force. If a stone rests on
the hand, the force with which the stone presses the hand downwards is
equal to that with which the hand presses the stone upwards; if the
earth attracts a stone downwards with a certain force, then the stone
attracts the earth upwards with the same force, and so on. It is to
be carefully noted that if, as in the last example, two bodies are
acting on one another, the _accelerations_ produced are not the same,
but since force is measured by the product of mass and acceleration,
the body with the larger mass receives the lesser acceleration. In the
case of a stone and the earth, the mass of the latter being enormously
greater,[105] its acceleration is enormously less than that of the
stone, and is therefore (in accordance with our experience) quite
insensible.

181. When Newton began to write the _Principia_ he had probably
satisfied himself (§ 173) that the attracting power of the earth
extended as far as the moon, and that the acceleration thereby produced
in any body—whether the moon, or whether a body close to the earth—is
inversely proportional to the square of the distance from the centre
of the earth. With the ideas of force and mass this result may be
stated in the form: _the earth attracts any body with a force inversely
proportional to the square of the distance on the earth’s centre, and
also proportional to the mass of the body_.

In the same way Newton had established that the motions of the planets
could be explained by an attraction towards the sun producing an
acceleration inversely proportional to the square of the distance from
the sun’s centre, not only in the _same_ planet in different parts of
its path, but also in _different_ planets. Again, it follows from this
that the sun attracts any planet with a force inversely proportional
to the square of the distance of the planet from the sun’s centre, and
also proportional to the mass of the planet.

But by the Third Law of Motion a body experiencing an attraction
towards the earth must in turn exert an equal attraction on the earth;
similarly a body experiencing an attraction towards the sun must exert
an equal attraction on the sun. If, for example, the mass of Venus is
seven times that of Mars, then the force with which the sun attracts
Venus is seven times as great as that with which it would attract
Mars if placed at the same distance; and therefore also the force
with which Venus attracts the sun is seven times as great as that
with which Mars would attract the sun if at an equal distance from
it. Hence, in all the cases of attraction hitherto considered and in
which the comparison is possible, the force is proportional not only
to the mass of the attracted body, but also to that of the attracting
body, as well as being inversely proportional to the square of the
distance. Gravitation thus appears no longer as a property peculiar to
the central body of a revolving system, but as belonging to a planet in
just the same way as to the sun, and to the moon or to a stone in just
the same way as to the earth.

Moreover, the fact that separate bodies on the surface of the earth
are attracted by the earth, and therefore in turn attract it, suggests
that this power of attracting other bodies which the celestial bodies
are shewn to possess does not belong to each celestial body as a whole,
but to the separate particles making it up, so that, for example, the
force with which Jupiter and the sun mutually attract one another is
the result of compounding the forces with which the separate particles
making up Jupiter attract the separate particles making up the sun.
Thus is suggested finally the law of gravitation in its most general
form: _every particle of matter attracts every other particle with a
force proportional to the mass of each, and inversely proportional to
the square of the distance between them_.[106]

182. In all the astronomical cases already referred to the attractions
between the various celestial bodies have been treated as if they were
accurately directed towards their centres, and the distance between
the bodies has been taken to be the distance between their centres.
Newton’s doubts on this point, in the case of the earth’s attraction
of bodies, have been already referred to (§ 173); but early in 1685
he succeeded in justifying this assumption. By a singularly beautiful
and simple course of reasoning he shewed (_Principia_, Book I.,
propositions 70, 71) that, if a body is spherical in form and equally
dense throughout, it attracts any particle external to it exactly as
if its whole mass were concentrated at its centre. He shewed, further,
that the same is true for a sphere of variable density, provided it can
be regarded as made up of a series of spherical shells, having a common
centre, each of uniform density throughout, different shells being,
however, of different densities. For example, the result is true for
a hollow indiarubber ball as well as for a solid one, but is not true
for a sphere made up of a hemisphere of wood and a hemisphere of iron
fastened together.

183. The law of gravitation being thus provisionally established,
the great task which lay before Newton, and to which he devotes the
greater part of the first and third books of the _Principia_, was
that of deducing from it and the “laws of motion” the motions of the
various members of the solar system, and of shewing, if possible, that
the motions so calculated agreed with those observed. If this were
successfully done, it would afford a verification of the most delicate
and rigorous character of Newton’s principles.

The conception of the solar system as a mechanism, each member of
which influences the motion of every other member in accordance with
one universal law of attraction, although extremely simple in itself,
is easily seen to give rise to very serious difficulties when it is
proposed actually to calculate the various motions. If in dealing
with the motion of a planet such as Mars it were possible to regard
Mars as acted on only by the attraction of the sun, and to ignore the
effects of the other planets, then the problem would be completely
solved by the propositions which Newton established in 1679 (§ 175);
and by their means the position of Mars at any time could be calculated
with any required degree of accuracy. But in the case which actually
exists the motion of Mars is affected by the forces with which all the
other planets (as well as the satellites) attract it, and these forces
in turn depend on the position of Mars (as well as upon that of the
other planets) and hence upon the motion of Mars. A problem of this
kind in all its generality is quite beyond the powers of any existing
mathematical methods. Fortunately, however, the mass of even the
largest of the planets is so very much less than that of the sun, that
the motion of any one planet is only slightly affected by the others;
and it may be regarded as moving very nearly as it would move if the
other planets did not exist, the effect of these being afterwards
allowed for as producing disturbances or =perturbations= in its path.
Although even in this simplified form the problem of the motion of
the planets is one of extreme difficulty (cf. chapter XI., § 228), and
Newton was unable to solve it with anything like completeness, yet he
was able to point out certain general effects which must result from
the mutual action of the planets, the most interesting being the slow
forward motion of the apses of the earth’s orbit, which had long ago
been noticed by observing astronomers (chapter III., § 59). Newton also
pointed out that Jupiter, on account of its great mass, must produce a
considerable perturbation in the motion of its neighbour Saturn, and
thus gave some explanation of an irregularity first noted by Horrocks
(chapter VIII., § 156).

184. The motion of the moon presents special difficulties, but
Newton, who was evidently much interested in the problems of lunar
theory, succeeded in overcoming them much more completely than the
corresponding ones connected with the planets.

The moon’s motion round the earth is primarily due to the attraction
of the earth; the perturbations due to the other planets are
insignificant; but the sun, which though at a very great distance has
an enormously greater mass than the earth, produces a very sensible
disturbing effect on the moon’s motion. Certain irregularities, as we
have seen (chapter II., §§ 40, 48; chapter V., § 111), had already been
discovered by observation. Newton was able to shew that the disturbing
action of the sun would necessarily produce perturbations of the same
general character as those thus recognised, and in the case of the
motion of the moon’s nodes and of her apogee he was able to get a very
fairly accurate numerical result;[107] and he also discovered a number
of other irregularities, for the most part very small, which had not
hitherto been noticed. He indicated also the existence of certain
irregularities in the motions of Jupiter’s and Saturn’s moons analogous
to those which occur in the case of our moon.

185. One group of results of an entirely novel character resulted from
Newton’s theory of gravitation. It became for the first time possible
to estimate the _masses_ of some of the celestial bodies, by comparing
the attractions exerted by them on other bodies with that exerted by
the earth.

The case of Jupiter may be given as an illustration. The time of
revolution of Jupiter’s outermost satellite is known to be about 16
days 16 hours, and its distance from Jupiter was estimated by Newton
(not very correctly) at about four times the distance of the moon from
the earth. A calculation exactly like that of § 172 or § 173 shews
that the acceleration of the satellite due to Jupiter’s attraction
is about ten times as great as the acceleration of the moon towards
the earth, and that therefore, the distance being four times as
great, Jupiter attracts a body with a force 10 × 4 × 4 times as great
as that with which the earth attracts a body at the same distance;
consequently Jupiter’s mass is 160 times that of the earth. This
process of reasoning applies also to Saturn, and in a very similar
way a comparison of the motion of a planet, Venus for example, round
the sun with the motion of the moon round the earth gives a relation
between the masses of the sun and earth. In this way Newton found the
mass of the sun to be 1067, 3021, and 169282 times greater than that
of Jupiter, Saturn, and the earth, respectively. The corresponding
figures now accepted are not far from 1047, 3530, 324439. The large
error in the last number is due to the use of an erroneous value of the
distance of the sun—then not at all accurately known—upon which depend
the other distances in the solar system, except those connected with
the earth-moon system. As it was necessary for the employment of this
method to be able to observe the motion of some other body attracted
by the planet in question, it could not be applied to the other three
planets (Mars, Venus, and Mercury), of which no satellites were known.

186. From the equality of action and reaction it follows that, since
the sun attracts the planets, they also attract the sun, and the sun
consequently is in motion, though—owing to the comparative smallness of
the planets—only to a very small extent. It follows that Kepler’s Third
Law is not strictly accurate, deviations from it becoming sensible in
the case of the large planets Jupiter and Saturn (cf. chapter VII., §
144). It was, however, proved by Newton that in any system of bodies,
such as the solar system, moving about in any way under the influence
of their mutual attractions, there is a particular point, called the
=centre of gravity=, which can always be treated as at rest; the sun
moves relatively to this point, but so little that the distance between
the centre of the sun and the centre of gravity can never be much more
than the diameter of the sun.

It is perhaps rather curious that this result was not seized upon by
some of the supporters of the Church in the condemnation of Galilei,
now rather more than half a century old; for if it was far from
supporting the view that the earth is at the centre of the world, it at
any rate negatived that part of the doctrine of Coppernicus and Galilei
which asserted the sun to be _at rest_ in the centre of the world.
Probably no one who was capable of understanding Newton’s book was a
serious supporter of any anti-Coppernican system, though some still
professed themselves obedient to the papal decrees on the subject.[108]

187. The variation of the time of oscillation of a pendulum in
different parts of the earth, discovered by Richer in 1672 (chapter
VIII., § 161), indicated that the earth was probably not a sphere.
Newton pointed out that this departure from the spherical form was
a consequence of the mutual gravitation of the particles making up
the earth and of the earth’s rotation. He supposed a canal of water
to pass from the pole to the centre of the earth, and then from the
centre to a point on the equator (B O _a_ A in fig. 72), and then
found the condition that these two columns of water O B, O A, each
being attracted towards the centre of the earth, should balance. This
method involved certain assumptions as to the inside of the earth, of
which little can be said to be known even now, and consequently, though
Newton’s general result, that the earth is flattened at the poles and
bulges out at the equator, was right, the actual numerical expression
which he found was not very accurate. If, in the figure, the dotted
line is a circle the radius of which is equal to the distance of the
pole B from the centre of the earth O, then the actual surface of the
earth extends at the equator beyond this circle as far as A, where,
according to Newton, _a_ A is about 1∕230 of O B or O A, and according
to modern estimates, based on actual measurement of the earth as well
as upon theory (chapter X., § 221), it is about 1∕293 of O A. Both
Newton’s fraction and the modern one are so small that the resulting
flattening cannot be made sensible in a figure; in fig. 72 the length
_a_ A is made, for the sake of distinctness, nearly 30 times as great
as it should be.

[Illustration: FIG. 72.—The spheroidal form of the earth.]

Newton discovered also in a similar way the flattening of Jupiter,
which, owing to its more rapid rotation, is considerably more flattened
than the earth; this was also detected telescopically by Domenico
Cassini four years after the publication of the _Principia_.

188. The discovery of the form of the earth led to an explanation of
the precession of the equinoxes, a phenomenon which had been discovered
1,800 years before (chapter II., § 42), but had remained a complete
mystery ever since.

If the earth is a perfect sphere, then its attraction on any other body
is exactly the same as if its mass were all concentrated at its centre
(§ 182), and so also the attraction on it of any other body such as
the sun or moon is equivalent to a single force passing through the
centre O of the earth; but this is no longer true if the earth is not
spherical. In fact the action of the sun or moon on the spherical part
of the earth, inside the dotted circle in fig. 72, is equivalent to
a force through O, and has no tendency to turn the earth in any way
about its centre; but the attraction on the remaining portion is of a
different character, and Newton shewed that from it resulted a motion
of the axis of the earth of the same general character as precession.
The amount of the precession as calculated by Newton did as a matter of
fact agree pretty closely with the observed amount, but this was due to
the accidental compensation of two errors, arising from his imperfect
knowledge of the form and construction of the earth, as well as from
erroneous estimates of the distance of the sun and of the mass of the
moon, neither of which quantities Newton was able to measure with any
accuracy.[109] It was further pointed out that the motion in question
was necessarily not quite uniform, but that, owing to the unequal
effects of the sun in different positions, the earth’s axis would
oscillate to and fro every six months, though to a very minute extent.

189. Newton also gave a general explanation of the tides as due to
the disturbing action of the moon and sun, the former being the more
important. If the earth be regarded as made of a solid spherical
nucleus, covered by the ocean, then the moon attracts different
parts unequally, and in particular the attraction, measured by the
acceleration produced, on the water nearest to the moon is greater
than that on the solid earth, and that on the water farthest from
the moon is less. Consequently the water moves on the surface of the
earth, the general character of the motion being the same as if the
portion of the ocean on the side towards the moon were attracted and
that on the opposite side repelled. Owing to the rotation of the earth
and the moon’s motion, the moon returns to nearly the same position
with respect to any place on the earth in a period which exceeds a
day by (on the average) about 50 minutes, and consequently Newton’s
argument shewed that low tides (or high tides) due to the moon would
follow one another at any given place at intervals equal to about half
this period; or, in other words, that two tides would in general occur
daily, but that on each day any particular phase of the tides would
occur on the average about 50 minutes later than on the preceding day,
a result agreeing with observation. Similar but smaller tides were
shewn by the same argument to arise from the action of the sun, and the
actual tide to be due to the combination of the two. It was shewn that
at new and full moon the lunar and solar tides would be added together,
whereas at the half moon they would tend to counteract one another, so
that the observed fact of greater tides every fortnight received an
explanation. A number of other peculiarities of the tides were also
shewn to result from the same principles.

Newton ingeniously used observations of the height of the tide when
the sun and moon acted together and when they acted in opposite ways
to compare the tide-raising powers of the sun and moon, and hence
to estimate the mass of the moon in terms of that of the sun, and
consequently in terms of that of the earth (§ 185). The resulting mass
of the moon was about twice what it ought to be according to modern
knowledge, but as before Newton’s time no one knew of any method of
measuring the moon’s mass even in the roughest way, and this result
had to be disentangled from the innumerable complications connected
with both the theory and with observation of the tides, it cannot
but be regarded as a remarkable achievement. Newton’s theory of the
tides was based on certain hypotheses which had to be made in order to
render the problem at all manageable, but which were certainly not
true, and consequently, as he was well aware, important modifications
would necessarily have to be made, in order to bring his results into
agreement with actual facts. The mere presence of land not covered
by water is, for example, sufficient by itself to produce important
alterations in tidal effects at different places. Thus Newton’s theory
was by no means equal to such a task as that of predicting the times
of high tide at any required place, or the height of any required
tide, though it gave a satisfactory explanation of many of the general
characteristics of tides.

190. As we have seen (chapter V., § 103; chapter VII., § 146), comets
until quite recently had been commonly regarded as terrestrial objects
produced in the higher regions of our atmosphere, and even the
more enlightened astronomers who, like Tycho, Kepler, and Galilei,
recognised them as belonging to the celestial bodies, were unable to
give an explanation of their motions and of their apparently quite
irregular appearances and disappearances. Newton was led to consider
whether a comet’s motion could not be explained, like that of a planet,
by gravitation towards the sun. If so then, as he had proved near the
beginning of the _Principia_, its path must be either an ellipse or
one of two other allied curves, the =parabola= and =hyperbola=. If a
comet moved in an ellipse which only differed slightly from a circle,
then it would never recede to any very great distance from the centre
of the solar system, and would therefore be regularly visible, a result
which was contrary to observation. If, however, the ellipse was very
elongated, as shewn in fig. 73, then the period of revolution might
easily be very great, and, during the greater part of it, the comet
would be so far from the sun and consequently also from the earth as
to be invisible. If so the comet would be seen for a short time and
become invisible, only to reappear after a very long time, when it
would naturally be regarded as a new comet. If again the path of the
comet were a parabola (which may be regarded as an ellipse indefinitely
elongated), the comet would not return at all, but would merely be seen
once when in that part of its path which is near the sun. But if a
comet moved in a parabola, with the sun in a focus, then its positions
when not very far from the sun would be almost the same as if it moved
in an elongated ellipse (see fig. 73), and consequently it would hardly
be possible to distinguish the two cases. Newton accordingly worked
out the case of motion in a parabola, which is mathematically the
simpler, and found that, in the case of a comet which had attracted
much attention in the winter 1680-1, a parabolic path could be found,
the calculated places of the comet in which agreed closely with those
observed. In the later editions of the _Principia_ the motions of a
number of other comets were investigated with a similar result. It was
thus established that in many cases a comet’s path is either a parabola
or an elongated ellipse, and that a similar result was to be expected
in other cases. This reduction to rule of the apparently arbitrary
motions of comets, and their inclusion with the planets in the same
class of bodies moving round the sun under the action of gravitation,
may fairly be regarded as one of the most striking of the innumerable
discoveries contained in the _Principia_.

[Illustration: FIG. 73.—An elongated ellipse and a parabola.]

In the same section Newton discussed also at some length the nature of
comets and in particular the structure of their tails, arriving at the
conclusion, which is in general agreement with modern theories (chapter
XIII., § 304), that the tail is formed by a stream of finely divided
matter of the nature of smoke, rising up from the body of the comet,
and so illuminated by the light of the sun when tolerably near it as to
become visible.

191. The _Principia_ was published, as we have seen, in 1687. Only a
small edition seems to have been printed, and this was exhausted in
three or four years. Newton’s earlier discoveries, and the presentation
to the Royal Society of the tract _De Motu_ (§ 177), had prepared the
scientific world to look for important new results in the _Principia_,
and the book appears to have been read by the leading Continental
mathematicians and astronomers, and to have been very warmly received
in England. The Cartesian philosophy had, however, too firm a hold to
be easily shaken; and Newton’s fundamental principle, involving as it
did the idea of an action between two bodies separated by an interval
of empty space, seemed impossible of acceptance to thinkers who had
not yet fully grasped the notion of judging a scientific theory by the
extent to which its consequences agree with observed facts. Hence even
so able a man as Huygens (chapter VIII., §§ 154, 157, 158), regarded
the idea of gravitation as “absurd,” and expressed his surprise that
Newton should have taken the trouble to make such a number of laborious
calculations with no foundation but this principle, a remark which
shewed Huygens to have had no conception that the agreement of the
results of these calculations with actual facts was proof of the
soundness of the principle. Personal reasons also contributed to the
Continental neglect of Newton’s work, as the famous quarrel between
Newton and Leibniz as to their respective claims to the invention of
what Newton called fluxions and Leibniz the differential method (out of
which the differential and integral calculus have developed) grew in
intensity and fresh combatants were drawn into it on both sides. Half
a century in fact elapsed before Newton’s views made any substantial
progress on the Continent (cf. chapter XI., § 229). In our country the
case was different; not only was the _Principia_ read with admiration
by the few who were capable of understanding it, but scholars like
Bentley, philosophers like Locke, and courtiers like Halifax all made
attempts to grasp Newton’s general ideas, even though the details
of his mathematics were out of their range. It was moreover soon
discovered that his scientific ideas could be used with advantage as
theological arguments.

192. One unfortunate result of the great success of the _Principia_
was that Newton was changed from a quiet Cambridge professor, with
abundant leisure and a slender income, into a public character, with a
continually increasing portion of his time devoted to public business
of one sort or another.

Just before the publication of the _Principia_ he had been appointed
one of the representatives of his University to defend its rights
against the encroachments of James II., and two years later he sat
as member for the University in the Convention Parliament, though he
retired after its dissolution.

Notwithstanding these and many other distractions, he continued to
work at the theory of gravitation, paying particular attention to
the lunar theory, a difficult subject with his treatment of which he
was never quite satisfied.[110] He was fortunately able to obtain
from time to time first-rate observations of the moon (as well as of
other bodies) from the Astronomer Royal Flamsteed (chapter X., §§
197-8), though Newton’s continual requests and Flamsteed’s occasional
refusals led to strained relations at intervals. It is possible that
about this time Newton contemplated writing a new treatise, with more
detailed treatment of various points discussed in the _Principia_;
and in 1691 there was already some talk of a new edition of the
_Principia_, possibly to be edited by some younger mathematician. In
any case nothing serious in this direction was done for some years,
perhaps owing to a serious illness, apparently some nervous disorder,
which attacked Newton in 1692 and lasted about two years. During this
illness, as he himself said, “he had not his usual consistency of
mind,” and it is by no means certain that he ever recovered his full
mental activity and power.

[Illustration: NEWTON.]

Soon after recovering from this illness he made some preparations
for a new edition of the _Principia_, besides going on with the lunar
theory, but the work was again interrupted in 1695, when he received
the valuable appointment of Warden to the Mint, from which he was
promoted to the Mastership four years later. He had, in consequence, to
move to London (1696), and much of his time was henceforward occupied
by official duties. In 1701 he resigned his professorship at Cambridge,
and in the same year was for the second time elected the Parliamentary
representative of the University. In 1703 he was chosen President of
the Royal Society, an office which he held till his death, and in 1705
he was knighted on the occasion of a royal visit to Cambridge.

During this time he published (1704) his treatise on _Optics_, the
bulk of which was probably written long before, and in 1709 he finally
abandoned the idea of editing the _Principia_ himself, and arranged
for the work to be done by _Roger Cotes_ (1682-1716), the brilliant
young mathematician whose untimely death a few years later called from
Newton the famous eulogy, “If Mr. Cotes had lived we might have known
something.” The alterations to be made were discussed in a long and
active correspondence between the editor and author, the most important
changes being improvements and additions to the lunar theory, and to
the discussions of precession and of comets, though there were also a
very large number of minor changes; and the new edition appeared in
1713. A third edition, edited by Pemberton, was published in 1726,
but this time Newton, who was over 80, took much less part, and the
alterations were of no great importance. This was Newton’s last piece
of scientific work, and his death occurred in the following year (March
3rd, 1727).

193. It is impossible to give an adequate idea of the immense magnitude
of Newton’s scientific discoveries except by a free use of the
mathematical technicalities in which the bulk of them were expressed.
The criticism passed on him by his personal enemy Leibniz that, “Taking
mathematics from the beginning of the world to the time when Newton
lived, what he had done was much the better half,” and the remark of
his great successor Lagrange (chapter XI., § 237), “Newton was the
greatest genius that ever existed, and the most fortunate, for we
cannot find more than once a system of the world to establish,” shew
the immense respect for his work felt by those who were most competent
to judge it.

With these magnificent eulogies it is pleasant to compare Newton’s
own grateful recognition of his predecessors, “If I have seen further
than other men, it is because I have stood upon the shoulders of the
giants,” and his modest estimate of his own performances:—

 “I do not know what I may appear to the world; but to myself I seem
 to have been only like a boy playing on the seashore, and diverting
 myself in now and then finding a smoother pebble or a prettier shell
 than ordinary, whilst the great ocean of truth lay all undiscovered
 before me.”

194. It is sometimes said, in explanation of the difference between
Newton’s achievements and those of earlier astronomers, that whereas
they discovered _how_ the celestial bodies moved, he shewed _why_ the
motions were as they were, or, in other words, that they _described_
motions while he _explained_ them or ascertained their cause. It
is, however, doubtful whether this distinction between How and Why,
though undoubtedly to some extent convenient, has any real validity.
Ptolemy, for example, represented the motion of a planet by a certain
combination of epicycles; his scheme was equivalent to a particular
method of describing the motion; but if any one had asked him why the
planet would be in a particular position at a particular time, he
might legitimately have answered that it was so because the planet was
connected with this particular system of epicycles, and its place could
be deduced from them by a rigorous process of calculation. But if any
one had gone further and asked why the planet’s epicycles were as they
were, Ptolemy could have given no answer. Moreover, as the system of
epicycles differed in some important respects from planet to planet,
Ptolemy’s system left unanswered a number of questions which obviously
presented themselves. Then Coppernicus gave a partial answer to some
of these questions. To the question why certain of the planetary
motions, corresponding to certain epicycles, existed, he would have
replied that it was because of certain motions of the earth, from
which these (apparent) planetary motions could be deduced as necessary
consequences. But the same information could also have been given as a
mere descriptive statement that the earth moves in certain ways and the
planets move in certain other ways. But again, if Coppernicus had been
asked why the earth rotated on its axis, or why the planets revolved
round the sun, he could have given no answer; still less could he have
said why the planets had certain irregularities in their motions,
represented by his epicycles.

Kepler again described the same motions very much more simply and
shortly by means of his three laws of planetary motion; but if any
one had asked why a planet’s motion varied in certain ways, he might
have replied that it was because all planets moved in ellipses so as
to sweep out equal areas in equal times. _Why_ this was so Kepler was
unable to say, though he spent much time in speculating on the subject.
This question was, however, answered by Newton, who shewed that the
planetary motions were necessary consequences of his law of gravitation
and his laws of motion. Moreover from these same laws, which were
extremely simple in statement and few in number, followed as necessary
consequences the motion of the moon and many other astronomical
phenomena, and also certain familiar terrestrial phenomena, such as
the behaviour of falling bodies; so that a large number of groups of
observed facts, which had hitherto been disconnected from one another,
were here brought into connection as necessary consequences of certain
fundamental laws. But again Newton’s view of the solar system might
equally well be put as a mere descriptive statement that the planets,
etc., move with accelerations of certain magnitudes towards one
another. As, however, the actual position or rate of motion of a planet
at any time can only be deduced by an extremely elaborate calculation
from Newton’s laws, they are not at all obviously equivalent to the
observed celestial motions, and we do not therefore at all easily think
of them as being merely a description.

Again Newton’s laws at once suggest the question why bodies attract one
another in this particular way; and this question, which Newton fully
recognised as legitimate, he was unable to answer. Or again we might
ask why the planets are of certain sizes, at certain distances from
the sun, etc., and to these questions again Newton could give no answer.

But whereas the questions left unanswered by Ptolemy, Coppernicus,
and Kepler were in whole or in part answered by their successors,
that is, their unexplained facts or laws were shewn to be necessary
consequences of other simpler and more general laws, it happens that up
to the present day no one has been able to answer, in any satisfactory
way, these questions which Newton left unanswered. In this particular
direction, therefore, Newton’s laws mark the boundary of our present
knowledge. But if any one were to succeed this year or next in shewing
gravitation to be a consequence of some still more general law, this
new law would still bring with it a new Why.

If, however, Newton’s laws cannot be regarded as an ultimate
explanation of the phenomena of the solar system, except in the
historic sense that they have not yet been shewn to depend on other
more fundamental laws, their success in “explaining,” with fair
accuracy, such an immense mass of observed results in all parts of the
solar system, and their universal character, gave a powerful impetus
to the idea of accounting for observed facts in other departments
of science, such as chemistry and physics, in some similar way as
the consequence of forces acting between bodies, and hence to the
conception of the material universe as made up of a certain number
of bodies, each acting on one another with definite forces in such a
way that all the changes which can be observed to go on are necessary
consequences of these forces, and are capable of prediction by any one
who has sufficient knowledge of the forces and sufficient mathematical
skill to develop their consequences.

Whether this conception of the material universe is adequate or not,
it has undoubtedly exercised a very important influence on scientific
discovery as well as on philosophical thought, and although it was
never formulated by Newton, and parts of it would probably have been
repudiated by him, there are indications that some such ideas were in
his head, and those who held the conception most firmly undoubtedly
derived their ideas directly or indirectly from him.

195. Newton’s scientific method did not differ essentially from that
followed by Galilei (chapter VI., § 134), which has been variously
described as =complete induction= or as the =inverse deductive method=,
the difference in name corresponding to a difference in the stress laid
upon different parts of the same general process. Facts are obtained
by observation or experiment; a hypothesis or provisional theory is
devised to account for them; from this theory are obtained, if possible
by a rigorous process of deductive reasoning, certain consequences
capable of being compared with actual facts, and the comparison is then
made. In some cases the first process may appear as the more important,
but in Newton’s work the really convincing part of the proof of his
results lay in the verification involved in the two last processes.
This has perhaps been somewhat obscured by his famous remark,
_Hypotheses non fingo_ (I do not invent hypotheses), dissociated from
its context. The words occur in the conclusion of the _Principia_,
after he has been speaking of universal gravitation:—

 “I have not yet been able to deduce (_deducere_) from phenomena
 the reason of these properties of gravitation, and I do not invent
 hypotheses. For any thing which cannot be deduced from phenomena
 should be called a hypothesis.”

Newton probably had in his mind such speculations as the Cartesian
vortices, which could not be deduced directly from observations, and
the consequences of which either could not be worked out and compared
with actual facts or were inconsistent with them. Newton in fact
rejected hypotheses which were unverifiable, but he constantly made
hypotheses, suggested by observed facts, and verified by the agreement
of their consequences with fresh observed facts. The extension of
gravity to the moon (§ 173) is a good example: he was acquainted with
certain facts as to the motion of falling bodies and the motion of the
moon; it occurred to him that the earth’s attraction might extend as
far as the moon, and certain other facts connected with Kepler’s Third
Law suggested the law of the inverse square. If this were right, the
moon’s acceleration towards the earth ought to have a certain value,
which could be obtained by calculation. The calculation was made and
found to agree roughly with the actual motion of the moon.

Moreover it may be fairly urged, in illustration of the great
importance of the process of verification, that Newton’s fundamental
laws were not rigorously established by him, but that the deficiencies
in his proofs have been to a great extent filled up by the elaborate
process of verification that has gone on since. For the motions of
the solar system, as deduced by Newton from gravitation and the laws
of motion, only agreed roughly with observation; many outstanding
discrepancies were left; and though there was a strong presumption that
these were due to the necessary imperfections of Newton’s processes
of calculation, an immense expenditure of labour and ingenuity on the
part of a series of mathematicians has been required to remove these
discrepancies one by one, and as a matter of fact there remain even
to-day a few small ones which are unexplained (chapter XIII., § 290).



CHAPTER X.

OBSERVATIONAL ASTRONOMY IN THE 18TH CENTURY.

 “Through Newton theory had made a great advance and was ahead of
 observation; the latter now made efforts to come once more level with
 theory.”—BESSEL.


196. Newton virtually created a new department of astronomy,
=gravitational astronomy=, as it is often called, and bequeathed to his
successors the problem of deducing more fully than he had succeeded in
doing the motions of the celestial bodies from their mutual gravitation.

To the solution of this problem Newton’s own countrymen contributed
next to nothing throughout the 18th century, and his true successors
were a group of Continental mathematicians whose work began soon after
his death, though not till nearly half a century after the publication
of the _Principia_.

This failure of the British mathematicians to develop Newton’s
discoveries may be explained as due in part to the absence or scarcity
of men of real ability, but in part also to the peculiarity of the
mathematical form in which Newton presented his discoveries. The
_Principia_ is written almost entirely in the language of geometry,
modified in a special way to meet the requirements of the case; nearly
all subsequent progress in gravitational astronomy has been made by
mathematical methods known as =analysis=. Although the distinction
between the two methods cannot be fully appreciated except by those
who have used them both, it may perhaps convey some impression of the
differences between them to say that in the geometrical treatment
of an astronomical problem each step of the reasoning is expressed
in such a way as to be capable of being interpreted in terms of the
original problem, whereas in the analytical treatment the problem is
first expressed by means of algebraical symbols; these symbols are
manipulated according to certain purely formal rules, no regard being
paid to the interpretation of the intermediate steps, and the final
algebraical result, if it can be obtained, yields on interpretation
the solution of the original problem. The geometrical solution of
a problem, if it can be obtained, is frequently shorter, clearer,
and more elegant; but, on the other hand, each special problem has
to be considered separately, whereas the analytical solution can be
conducted to a great extent according to fixed rules applicable in a
larger number of cases. In Newton’s time modern analysis was only just
coming into being, some of the most important parts of it being in
fact the creation of Leibniz and himself, and although he sometimes
used analysis to solve an astronomical problem, it was his practice to
translate the result into geometrical language before publication; in
doing so he was probably influenced to a large extent by a personal
preference for the elegance of geometrical proofs, partly also by an
unwillingness to increase the numerous difficulties contained in the
_Principia_, by using mathematical methods which were comparatively
unfamiliar. But though in the hands of a master like Newton geometrical
methods were capable of producing astonishing results, the lesser men
who followed him were scarcely ever capable of using his methods to
obtain results beyond those which he himself had reached. Excessive
reverence for Newton and all his ways, combined with the estrangement
which long subsisted between British and foreign mathematicians, as the
result of the fluxional controversy (chapter IX., § 191), prevented
the former from using the analytical methods which were being rapidly
perfected by Leibniz’s pupils and other Continental mathematicians. Our
mathematicians remained, therefore, almost isolated during the whole
of the 18th century, and with the exception of some admirable work by
_Colin Maclaurin_ (1698-1746), which carried Newton’s theory of the
figure of the earth a stage further, nothing of importance was done in
our country for nearly a century after Newton’s death to develop the
theory of gravitation beyond the point at which it was left in the
_Principia_.

In other departments of astronomy, however, important progress was made
both during and after Newton’s lifetime, and by a curious inversion,
while Newton’s ideas were developed chiefly by French mathematicians,
the Observatory of Paris, at which Picard and others had done such
admirable work (chapter VIII., §§ 160-2), produced little of real
importance for nearly a century afterwards, and a large part of
the best observing work of the 18th century was done by Newton’s
countrymen. It will be convenient to separate these two departments of
astronomical work, and to deal in the next chapter with the development
of the theory of gravitation.

197. The first of the great English observers was Newton’s contemporary
_John Flamsteed_, who was born near Derby in 1646 and died at Greenwich
in 1720.[111] Unfortunately the character of his work was such that,
marked as it was by no brilliant discoveries, it is difficult to
present it in an attractive form or to give any adequate idea of its
real extent and importance. He was one of those laborious and careful
investigators, the results of whose work are invaluable as material for
subsequent research, but are not striking in themselves.

He made some astronomical observations while quite a boy, and wrote
several papers, of a technical character, on astronomical subjects,
which attracted some attention. In 1675 appointed a member of a
Committee to report on a method for finding the longitude at sea which
had been offered to the Government by a certain Frenchman of the name
of _St. Pierre_. The Committee, acting largely on Flamsteed’s advice,
reported unfavourably on the method in question, and memorialised
Charles II. in favour of founding a national observatory, in order that
better knowledge of the celestial bodies might lead to a satisfactory
method of finding the longitude, a problem which the rapid increase
of English shipping rendered of great practical importance. The King
having agreed, Flamsteed was in the same year appointed to the new
office of Astronomer Royal, with a salary of £100 a year, and the
warrant for building an Observatory at Greenwich was signed on June
12th, 1675. About a year was occupied in building it, and Flamsteed
took up his residence there and began work in July 1676, five years
after Cassini entered upon his duties at the Observatory of Paris
(chapter VIII., § 160). The Greenwich Observatory was, however, on
a very different scale from the magnificent sister institution. The
King had, it is true, provided Flamsteed with a building and a very
small salary, but furnished him neither with instruments nor with an
assistant. A few instruments he possessed already, a few more were
given to him by rich friends, and he gradually made at his own expense
some further instrumental additions of importance. Some years after
his appointment the Government provided him with “a silly, surly
labourer” to help him with some of the rough work, but he was compelled
to provide more skilled assistance out of his own pocket, and this
necessity in turn compelled him to devote some part of his valuable
time to taking pupils.

198. Flamsteed’s great work was the construction of a more accurate
and more extensive star catalogue than any that existed; he also
made a number of observations of the moon, of the sun, and to a less
extent of other bodies. Like Tycho, the author of the last great star
catalogue (chapter V., § 107), he found problems continually presenting
themselves in the course of his work which had to be solved before his
main object could be accomplished, and we accordingly owe to him the
invention of several improvements in practical astronomy, the best
known being his method of finding the position of the first point of
Aries (chapter II., § 42), one of the fundamental points with reference
to which all positions on the celestial sphere are defined. He was the
first astronomer to use a clock systematically for the determination of
one of the two fundamental quantities (the right ascension) necessary
to fix the position of a star, a method which was first suggested
and to some extent used by Picard (chapter VIII., § 157), and, as
soon as he could get the necessary instruments, he regularly used the
telescopic sights of Gascoigne and Auzout (chapter VIII., § 155),
instead of making naked-eye observations. Thus while Hevel (chapter
VIII., § 153) was the last and most accurate observer of the old
school, employing methods not differing essentially from those which
had been in use for centuries, Flamsteed belongs to the new school,
and his methods differ rather in detail than in principle from those
now in vogue for similar work at Greenwich, Paris, or Washington. This
adoption of new methods, together with the most scrupulous care in
details, rendered Flamsteed’s observations considerably more accurate
than any made in his time or earlier, the first definite advance
afterwards being made by Bradley (§ 218).

Flamsteed compared favourably with many observers by not merely taking
and recording observations, but by performing also the tedious process
known as reduction (§ 218), whereby the results of the observation are
put into a form suitable for use by other astronomers; this process
is usually performed in modern observatories by assistants, but in
Flamsteed’s case had to be done almost exclusively by the astronomer
himself. From this and other causes he was extremely slow in publishing
observations; we have already alluded (chapter IX., § 192) to the
difficulty which Newton had in extracting lunar observations from him,
and after a time a feeling that the object for which the Observatory
had been founded was not being fulfilled became pretty general among
astronomers. Flamsteed always suffered from bad health as well as
from the pecuniary and other difficulties which have been referred
to; moreover he was much more anxious that his observations should
be kept back till they were as accurate as possible, than that they
should be published in a less perfect form and used for the researches
which he once called “Mr. Newton’s crotchets”; consequently he took
remonstrances about the delay in the publication of his observations
in bad part. Some painful quarrels occurred between Flamsteed on the
one hand and Newton and Halley on the other. The last straw was the
unauthorised publication in 1712, under the editorship of Halley, of a
volume of Flamsteed’s observations, a proceeding to which Flamsteed not
unnaturally replied by calling Halley a “malicious thief.” Three years
later he succeeded in getting hold of all the unsold copies and in
destroying them, but fortunately he was also stimulated to prepare for
publication an authentic edition. The _Historia Coelestis Britannica_,
as he called the book, contained an immense series of observations made
both before and during his career at Greenwich, but the most important
and permanently valuable part was a catalogue of the places of nearly
3,000 stars.[112]

Flamsteed himself only lived just long enough to finish the second of
the three volumes; the third was edited by his assistants _Abraham
Sharp_ (1651-1742) and _Joseph Crosthwait_; and the whole was published
in 1725. Four years later still appeared his valuable Star-Atlas, which
long remained in common use.

The catalogue was not only three times as extensive as Tycho’s, which
it virtually succeeded, but was also very much more accurate. It
has been estimated[113] that, whereas Tycho’s determinations of the
positions of the stars were on the average about 1′ in error, the
corresponding errors in Flamsteed’s case were about 10″. This quantity
is the apparent diameter of a shilling seen from a distance of about
500 yards; so that if two marks were made at opposite points on the
edge of the coin, and it were placed at a distance of 500 yards, the
two marks might be taken to represent the true direction of an average
star and its direction as given in Flamsteed’s catalogue. In some cases
of course the error might be much greater and in others considerably
less.

Flamsteed contributed to astronomy no ideas of first-rate importance;
he had not the ingenuity of Picard and of Roemer in devising
instrumental improvements, and he took little interest in the
theoretical work of Newton;[114] but by unflagging industry and
scrupulous care he succeeded in bequeathing to his successors an
immense treasure of observations, executed with all the accuracy that
his instrumental means permitted.

199. Flamsteed was succeeded as Astronomer Royal by Edmund Halley, whom
we have already met with (chapter IX., § 176) as Newton’s friend and
helper.

Born in 1656, ten years after Flamsteed, he studied astronomy in his
schooldays, and published a paper on the orbits of the planets as
early as 1676. In the same year he set off for St. Helena (in latitude
16° S.) in order to make observations of stars which were too near
the south pole to be visible in Europe. The climate turned out to be
disappointing, and he was only able after his return to publish (1678)
a catalogue of the places of 341 southern stars, which constituted,
however, an important addition to precise knowledge of the stars. The
catalogue was also remarkable as being the first based on telescopic
observation, though the observations do not seem to have been taken
with all the accuracy which his instruments rendered attainable. During
his stay at St. Helena he also took a number of pendulum observations
which confirmed the results obtained a few years before by Richer at
Cayenne (chapter VIII., § 161), and also observed a transit of Mercury
across the sun, which occurred in November 1677.

After his return to England he took an active part in current
scientific questions, particularly in those connected with astronomy,
and made several small contributions to the subject. In 1684, as
we have seen, he first came effectively into contact with Newton,
and spent a good part of the next few years in helping him with the
_Principia_.

200. Of his numerous contributions to astronomy, which touched almost
every branch of the subject, his work on comets is the best known and
probably the most important. He observed the comets of 1680 and 1682;
he worked out the paths both of these and of a number of other recorded
comets in accordance with Newton’s principles, and contributed a good
deal of the material contained in the sections of the _Principia_
dealing with comets, particularly in the later editions. In 1705 he
published a _Synopsis of Cometary Astronomy_ in which no less than 24
cometary orbits were calculated. Struck by the resemblance between
the paths described by the comets of 1531, 1607, and 1682, and by
the approximate equality in the intervals between their respective
appearances and that of a fourth comet seen in 1456, he was shrewd
enough to conjecture that the three later comets, if not all four, were
really different appearances of the same comet, which revolved round
the sun in an elongated ellipse in a period of about 75 or 76 years.
He explained the difference between the 76 years which separate the
appearances of the comet in 1531 and 1607, and the slightly shorter
period which elapsed between 1607 and 1682, as probably due to the
perturbations caused by planets near which the comet had passed;
and finally predicted the probable reappearance of the same comet
(which now deservedly bears his name) about 76 years after its last
appearance, _i.e._ about 1758, though he was again aware that planetary
perturbation might alter the time of its appearance; and the actual
appearance of the comet about the predicted time (chapter XI., § 231)
marked an important era in the progress of our knowledge of these
extremely troublesome and erratic bodies.

201. In 1693 Halley read before the Royal Society a paper in which
he called attention to the difficulty of reconciling certain ancient
eclipses with the known motion of the moon, and referred to the
possibility of some slight increase in the moon’s average rate of
motion round the earth.

This irregularity, now known as the =secular acceleration of the moon’s
mean motion=, was subsequently more definitely established as a fact
of observation; and the difficulties met with in explaining it as a
result of gravitation have rendered it one of the most interesting of
the moon’s numerous irregularities (cf. chapter XI., § 240, and chapter
XIII., § 287).

202. Halley also rendered good service to astronomy by calling
attention to the importance of the expected transits of Venus across
the sun in 1761 and 1769 as a means of ascertaining the distance of
the sun. The method had been suggested rather vaguely by Kepler,
and more definitely by James Gregory in his _Optics_ published in
1663. The idea was first suggested to Halley by his observation of
the transit of Mercury in 1677. In three papers published by the
Royal Society he spoke warmly of the advantages of the method, and
discussed in some detail the places and means most suitable for
observing the transit of 1761. He pointed out that the desired result
could be deduced from a comparison of the durations of the transit of
Venus, as seen from different stations on the earth, _i.e._ of the
intervals between the first appearance of Venus on the sun’s disc and
the final disappearance, as seen at two or more different stations.
He estimated, moreover, that this interval of time, which would be
several hours in length, could be measured with an error of only about
two seconds, and that in consequence the method might be relied upon
to give the distance of the sun to within about 1∕500 part of its
true value. As the current estimates of the sun’s distance differed
among one another by 20 or 30 per cent., the new method, expounded
with Halley’s customary lucidity and enthusiasm, not unnaturally
stimulated astronomers to take great trouble to carry out Halley’s
recommendations. The results, as we shall see (§ 227), were, however,
by no means equal to Halley’s expectations.

203. In 1718 Halley called attention to the fact that three well-known
stars, Sirius, Procyon, and Arcturus, had changed their angular
distances from the ecliptic since Greek times, and that Sirius had
even changed its position perceptibly since the time of Tycho Brahe.
Moreover comparison of the places of other stars shewed that the
changes could not satisfactorily be attributed to any motion of the
ecliptic, and although he was well aware that the possible errors of
observation were such as to introduce a considerable uncertainty into
the amounts involved, he felt sure that such errors could not wholly
account for the discrepancies noticed, but that the stars in question
must have really shifted their positions in relation to the rest; and
he naturally inferred that it would be possible to detect similar
=proper motions= (as they are now called) in other so-called “fixed”
stars.

204. He also devoted a good deal of time to the standing astronomical
problem of improving the tables of the moon and planets, particularly
the former. He made observations of the moon as early as 1683, and by
means of them effected some improvement in the tables. In 1676 he had
already noted defects in the existing tables of Jupiter and Saturn, and
ultimately satisfied himself of the existence of certain irregularities
in the motion of these two planets, suspected long ago by Horrocks
(chapter VIII., § 156); these irregularities he attributed correctly
to the perturbations of the two planets by one another, though he was
not mathematician enough to work out the theory; from observation,
however, he was able to estimate the irregularities in question with
fair accuracy and to improve the planetary tables by making allowance
for them. But neither the lunar nor the planetary tables were ever
completed in a form which Halley thought satisfactory. By 1719 they
were printed, but kept back from publication, in hopes that subsequent
improvements might be effected. After his appointment as Astronomer
Royal in succession to Flamsteed (1720) he devoted special attention
to getting fresh observations for this purpose, but he found the
Observatory almost bare of instruments, those used by Flamsteed having
been his private property, and having been removed as such by his heirs
or creditors. Although Halley procured some instruments, and made with
them a number of observations, chiefly of the moon, the age (63) at
which he entered upon his office prevented him from initiating much, or
from carrying out his duties with great energy, and the observations
taken were in consequence only of secondary importance, while the
tables for the improvement of which they were specially designed were
only finally published in 1752, ten years after the death of their
author. Although they thus appeared many years after the time at which
they were virtually prepared and owed little to the progress of science
during the interval, they at once became and for some time remained the
standard tables for both the lunar and planetary motions (cf. § 226,
and chapter XI., § 247).

205. Halley’s remarkable versatility in scientific work is further
illustrated by the labour which he expended in editing the writings
of the great Greek geometer Apollonius (chapter II., § 38) and the
star catalogue of Ptolemy (chapter II., § 50). He was also one of the
first of modern astronomers to pay careful attention to the effects
to be observed during a total eclipse of the sun, and in the vivid
description which he wrote of the eclipse of 1715, besides referring
to the mysterious corona, which Kepler and others had noticed before
(chapter VII., § 145), he called attention also to “a very narrow
streak of a dusky but strong Red Light,” which was evidently a portion
of that remarkable envelope of the sun which has been so extensively
studied in modern times (chapter XIII., § 301) under the name of the
chromosphere.

It is worth while to notice, as an illustration of Halley’s unselfish
enthusiasm for science and of his power of looking to the future, that
two of his most important pieces of work, by which certainly he is
now best known, necessarily appeared during his lifetime as of little
value, and only bore their fruit after his death (1742), for his comet
only returned in 1759, when he had been dead 17 years, and the first of
the pair of transits of Venus, from which he had shewn how to deduce
the distance of the sun, took place two years later still (§ 227).

206. The third Astronomer Royal, _James Bradley_, is popularly known as
the author of two memorable discoveries, _viz._ the aberration of light
and the nutation of the earth’s axis. Remarkable as these are both in
themselves and on account of the ingenious and subtle reasoning and
minutely accurate observations by means of which they were made, they
were in fact incidents in a long and active astronomical career, which
resulted in the execution of a vast mass of work of great value.

The external events of Bradley’s life may be dealt with very briefly.
Born in 1693, he proceeded in due course to Oxford (B.A. 1714, M.A.
1717), but acquired his first knowledge of astronomy and his marked
taste for the subject from his uncle _James Pound_, for many years
rector of Wansted in Essex, who was one of the best observers of
the time. Bradley lived with his uncle for some years after leaving
Oxford, and carried out a number of observations in concert with him.
The first recorded observation of Bradley’s is dated 1715, and by
1718 he was sufficiently well thought of in the scientific world to
receive the honour of election as a Fellow of the Royal Society. But,
as his biographer[115] remarks, “it could not be foreseen that his
astronomical labours would lead to any establishment in life, and it
became necessary for him to embrace a profession.” He accordingly took
orders, and was fortunate enough to be presented almost at once to two
livings, the duties attached to which do not seem to have interfered
appreciably with the prosecution of his astronomical studies at Wansted.

In 1721 he was appointed Savilian Professor of Astronomy at Oxford,
and resigned his livings. The work of the professorship appears to
have been very light, and for more than ten years he continued to
reside chiefly at Wansted, even after his uncle’s death in 1724.
In 1732 he took a house in Oxford and set up there most of his
instruments, leaving, however, at Wansted the most important of all,
the “zenith-sector,” with which his two famous discoveries were made.
Ten years afterwards Halley’s death rendered the post of Astronomer
Royal vacant, and Bradley received the appointment.

The work of the Observatory had been a good deal neglected by Halley
during the last few years of his life, and Bradley’s first care was to
effect necessary repairs in the instruments. Although the equipment
of the Observatory with instruments worthy of its position and of the
state of science at the time was a work of years, Bradley had some
of the most important instruments in good working order within a few
months of his appointment, and observations were henceforward made
systematically. Although the 20 remaining years of his life (1742-1762)
were chiefly spent at Greenwich in the discharge of the duties of
his office and in researches connected with them, he retained his
professorship at Oxford, and continued to make observations at Wansted
at least up till 1747.

[Illustration: BRADLEY.]

207. The discovery of aberration resulted from an attempt to detect the
parallactic displacement of stars which should result from the annual
motion of the earth. Ever since the Coppernican controversy had called
attention to the importance of the problem (cf. chapter IV., § 92
and chapter VI., § 129), it had naturally exerted a fascination on
the minds of observing astronomers, many of whom had tried to detect
the motion in question, and some of whom (including the “universal
claimant” Hooke) professed to have succeeded. Actually, however, all
previous attempts had been failures, and Bradley was no more successful
than his predecessors in this particular undertaking, but was able to
deduce from his observations two results of great interest and of an
entirely unexpected character.

The problem which Bradley set himself was to examine whether any
star could be seen to have in the course of the year a slight motion
relative to others or relative to fixed points on the celestial sphere
such as the pole. It was known that such a motion, if it existed, must
be very small, and it was therefore evident that extreme delicacy in
instrumental adjustments and the greatest care in observation would
have to be employed. Bradley worked at first in conjunction with his
friend _Samuel Molyneux_ (1689-1728), who had erected a telescope at
Kew. In accordance with the method adopted in a similar investigation
by Hooke, whose results it was desired to test, the telescope was fixed
in a nearly vertical position, so chosen that a particular star in
the Dragon (γ Draconis) would be visible through it when it crossed
the meridian, and the telescope was mounted with great care so as to
maintain an invariable position throughout the year. If then the star
in question were to undergo any motion which altered its distance from
the pole, there would be a corresponding alteration in the position in
which it would be seen in the field of view of the telescope. The first
observations were taken on December 14th, 1725 (N.S.), and by December
28th Bradley believed that he had already noticed a slight displacement
of the star towards the south. This motion was clearly verified on
January 1st, and was then observed to continue; in the following March
the star reached its extreme southern position, and then began to move
northwards again. In September it once more altered its direction of
motion, and by the end of the year had completed the cycle of its
changes and returned to its original position, the greatest change in
position amounting to nearly 40′.

The star was thus observed to go through _some_ annual motion. It
was, however, at once evident to Bradley that this motion was not the
parallactic motion of which he was in search, for the position of
the star was such that parallax would have made it appear farthest
south in December and farthest north in June, or in each case three
months earlier than was the case in the actual observations. Another
explanation which suggested itself was that the earth’s axis might
have a to-and-fro oscillatory motion or nutation which would alter
the position of the celestial pole and hence produce a corresponding
alteration in the position of the star. Such a motion of the celestial
pole would evidently produce opposite effects on two stars situated on
opposite sides of it, as any motion which brought the pole nearer to
one star of such a pair would necessarily move it away from the other.
Within a fortnight of the decisive observation made on January 1st a
star[116] had already been selected for the application of this test,
with the result which can best be given in Bradley’s own words:—

 “A nutation of the earth’s axis was one of the first things that
 offered itself upon this occasion, but it was soon found to be
 insufficient; for though it might have accounted for the change of
 declination in γ Draconis, yet it would not at the same time agree
 with the phaenomena in other stars; particularly in a small one
 almost opposite in right ascension to γ Draconis, at about the same
 distance from the north pole of the equator: for though this star
 seemed to move the same way as a nutation of the earth’s axis would
 have made it, yet, it changing its declination but about half as
 much as γ Draconis in the same time, (as appeared upon comparing the
 observations of both made upon the same days, at different seasons of
 the year,) this plainly proved that the apparent motion of the stars
 was not occasioned by a real nutation, since, if that had been the
 cause, the alteration in both stars would have been near equal.”

One or two other explanations were tested and found insufficient, and
as the result of a series of observations extending over about two
years, the phenomenon in question, although amply established, still
remained quite unexplained.

By this time Bradley had mounted an instrument of his own at Wansted,
so arranged that it was possible to observe through it the motions of
stars other than γ Draconis.

Several stars were watched carefully throughout a year, and the
observations thus obtained gave Bradley a fairly complete knowledge of
the geometrical laws according to which the motions varied both from
star to star and in the course of the year.

208. The true explanation of =aberration=, as the phenomenon in
question was afterwards called, appears to have occurred to him about
September, 1728, and was published to the Royal Society, after some
further verification, early in the following year. According to a
well-known story,[117] he noticed, while sailing on the Thames, that a
vane on the masthead appeared to change its direction every time that
the boat altered its course, and was informed by the sailors that this
change was not due to any alteration in the wind’s direction, but to
that of the boat’s course. In fact the apparent direction of the wind,
as shewn by the vane, was not the true direction of the wind, but
resulted from a combination of the motions of the wind and of the boat,
being more precisely that of the motion of the wind _relative_ to the
boat. Replacing in imagination the wind by light coming from a star,
and the boat shifting its course by the earth moving round the sun
and continually changing its direction of motion, Bradley arrived at
an explanation which, when worked out in detail, was found to account
most satisfactorily for the apparent changes in the direction of a
star which he had been studying. His own account of the matter is as
follows:—

 “At last I conjectured that all the phaenomena hitherto mentioned
 proceeded from the progressive motion of light and the earth’s annual
 motion in its orbit. For I perceived that, if light was propagated in
 time, the apparent place of a fixed object would not be the same when
 the eye is at rest, as when it is moving in any other direction than
 that of the line passing through the eye and object; and that when the
 eye is moving

in different directions, the apparent place of the object would be
different.

“I considered this matter in the following manner. I imagined C A to
be a ray of light, falling perpendicularly upon the line B D; then if
the eye is at rest at A, the object must appear in the direction A C,
whether light be propagated in time or in an instant. But if the eye
is moving from B towards A, and light is propagated in time, with a
velocity that is to the velocity of the eye, as C A to B A; then light
moving from C to A, whilst the eye moves from B to A, that particle of
it by which the object will be discerned when the eye in its motion
comes to A, is at C when the eye is at B. Joining the points B, C, I
supposed the line C B to be a tube (inclined to the line B D in the
angle D B C) of such a diameter as to admit of but one particle of
light; then it was easy to conceive that the particle of light at C (by
which the object must be seen when the eye, as it moves along, arrives
at A) would pass through the tube B C, if it is inclined to B D in the
angle D B C, and accompanies the eye in its motion from B to A; and
that it could not come to the eye, placed behind such a tube, if it had
any other inclination to the line B D....

[Illustration: FIG. 74.—The aberration of light. From Bradley’s paper
in the _Phil. Trans._]

“Although therefore the true or real place of an object is
perpendicular to the line in which the eye is moving, yet the visible
place will not be so, since that, no doubt, must be in the direction
of the tube; but the difference between the true and apparent place
will be (caeteris paribus) greater or less, according to the different
proportion between the velocity of light and that of the eye. So that
if we could suppose that light was propagated in an instant, then there
would be no difference between the real and visible place of an object,
although the eye were in motion; for in that case, A C being infinite
with respect to A B, the angle A C B (the difference between the true
and visible place) vanishes. But if light be propagated in time, (which
I presume will readily be allowed by most of the philosophers of this
age,) then it is evident from the foregoing considerations, that there
will be always a difference between the real and visible place of an
object, unless the eye is moving either directly towards or from the
object.”

Bradley’s explanation shews that the apparent position of a star is
determined by the motion of the star’s light _relative_ to the earth,
so that the star appears slightly nearer to the point on the celestial
sphere towards which the earth is moving than would otherwise be
the case. A familiar illustration of a precisely analogous effect
may perhaps be of service. Any one walking on a rainy but windless
day protects himself most effectually by holding his umbrella, not
immediately over his head, but a little in front, exactly as he would
do if he were at rest and there were a slight wind blowing in his face.
In fact, if he were to ignore his own motion and pay attention only to
the direction in which he found it advisable to point his umbrella,
he would believe that there was a slight head-wind blowing the rain
towards him.

[Illustration:  FIG. 75.—The aberration of light.]

209. The passage quoted from Bradley’s paper deals only with the
simple case in which the star is at right angles to the direction of
the earth’s motion. He shews elsewhere that if the star is in any
other direction the effect is of the same kind but less in amount. In
Bradley’s figure (fig. 74) the amount of the star’s displacement from
its true position is represented by the angle B C A, which depends on
the proportion between the lines A C and A B; but if (as in fig. 75)
the earth is moving (without change of speed) in the direction A B′
instead of A B, so that the direction of the star is oblique to it, it
is evident from the figure that the star’s displacement, represented by
the angle A C B′, is less than before; and the amount varies according
to a simple mathematical law[118] with the angle between the two
directions. It follows therefore that the displacement in question is
different for different stars, as Bradley’s observations had already
shewn, and is, moreover, different for the same star in the course of
the year, so that a star appears to describe a curve which is very
nearly an ellipse (fig. 76), the centre (S) corresponding to the
position which the star would occupy if aberration did not exist. It
is not difficult to see that, wherever a star is situated, the earth’s
motion is twice a year, at intervals of six months, at right angles to
the direction of the star, and that at these times the star receives
the greatest possible displacement from its mean position, and is
consequently at the ends of the greatest axis of the ellipse which it
describes, as at A and A′, whereas at intermediate times it undergoes
its least displacement, as at B and B′. The greatest displacement S
A, or half of A A′, which is the same for all stars, is known as the
=constant of aberration=, and was fixed by Bradley at between 20″
and 20-1∕2″, the value at present accepted being 20″·47. The least
displacement, on the other hand, S B, or half of B B′, was shewn to
depend in a simple way upon the star’s distance from the ecliptic,
being greatest for stars farthest from the ecliptic.

[Illustration: FIG. 76.—The aberrational ellipse.]

210. The constant of aberration, which is represented by the angle A
C B in fig. 74, depends only on the ratio between A C and A B, which
are in turn proportional to the velocities of light and of the earth.
Observations of aberration give then the ratio of these two velocities.
From Bradley’s value of the constant of aberration it follows by an
easy calculation that the velocity of light is about 10,000 times
that of the earth; Bradley also put this result into the form that
light travels from the sun to the earth in 8 minutes 13 seconds. From
observations of the eclipses of Jupiter’s moons, Roemer and others had
estimated the same interval at from 8 to 11 minutes (chapter VIII., §
162); and Bradley was thus able to get a satisfactory confirmation of
the truth of his discovery. Aberration being once established, the same
calculation could be used to give the most accurate measure of the
velocity of light in terms of the dimensions of the earth’s orbit, the
determination of aberration being susceptible of considerably greater
accuracy than the corresponding measurements required for Roemer’s
method.

211. One difficulty in the theory of aberration deserves mention.
Bradley’s own explanation, quoted above, refers to light as a material
substance shot out from the star or other luminous body. This was in
accordance with the corpuscular theory of light, which was supported
by the great weight of Newton’s authority and was commonly accepted
in the 18th century. Modern physicists, however, have entirely
abandoned the corpuscular theory, and regard light as a particular
form of wave-motion transmitted through ether. From this point of
view Bradley’s explanation and the physical illustrations given are
far less convincing; the question becomes in fact one of considerable
difficulty, and the most careful and elaborate of modern investigations
cannot be said to be altogether satisfactory. The curious inference
may be drawn that, if the more correct modern notions of the nature
of light had prevailed in Bradley’s time, it must have been very much
more difficult, if not impracticable, for him to have thought of his
explanation of the stellar motions which he was studying; and thus an
erroneous theory led to a most important discovery.

212. Bradley had of course not forgotten the original object of his
investigation. He satisfied himself, however, that the agreement
between the observed positions of γ Draconis and those which resulted
from aberration was so close that any displacement of a star due
to parallax which might exist must certainly be less than 2″, and
probably not more than 1∕2″, so that the large parallax amounting to
nearly 30″, which Hooke claimed to have detected, must certainly be
rejected as erroneous.

From the point of view of the Coppernican controversy, however,
Bradley’s discovery was almost as good as the discovery of a
parallax; since if the earth were at rest no explanation of the least
plausibility could be given of aberration.

213. The close agreement thus obtained between theory and observation
would have satisfied an astronomer less accurate and careful than
Bradley. But in his paper on aberration (1729) we find him writing:—

 “I have likewise met with some small varieties in the declination of
 other stars in different years which do not seem to proceed from the
 same cause.... But whether these small alterations proceed from a
 regular cause, or are occasioned by any change in the materials, etc.,
 of my instrument, I am not yet able fully to determine.”

The slender clue thus obtained was carefully followed up and led to
a second striking discovery, which affords one of the most beautiful
illustrations of the important results which can be deduced from the
study of “residual phenomena.” Aberration causes a star to go through
a cyclical series of changes in the course of a year; if therefore
at the end of a year a star is found not to have returned to its
original place, some other explanation of the motion has to be sought.
Precession was one known cause of such an alteration; but Bradley
found, at the end of his first year’s set of observations at Wansted,
that the alterations in the positions of various stars differed by a
minute amount (not exceeding 2″) from those which would have resulted
from the usual estimate of precession; and that, although an alteration
in the value of precession would account for the observed motions of
some of these stars, it would have increased the discrepancy in the
case of others. A nutation or nodding of the earth’s axis had, as we
have seen (§ 207), already presented itself to him as a possibility;
and although it had been shewn to be incapable of accounting for the
main phenomenon—due to aberration—it might prove to be a satisfactory
explanation of the much smaller residual motions. It soon occurred to
Bradley that such a nutation might be due to the action of the moon, as
both observation and the Newtonian explanation of precession indicated:—

 “I suspected that the moon’s action upon the equatorial parts of
 the earth might produce these effects: for if the precession of
 the equinox be, according to Sir Isaac Newton’s principles, caused
 by the actions of the sun and moon upon those parts, the plane of
 the moon’s orbit being at one time above ten degrees more inclined
 to the plane of the equator than at another, it was reasonable to
 conclude, that the part of the whole annual precession, which arises
 from her action, would in different years be varied in its quantity;
 whereas the plane of the ecliptic, wherein the sun appears, keeping
 always nearly the same inclination to the equator, that part of the
 precession which is owing to the sun’s action may be the same every
 year; and from hence it would follow, that although the mean annual
 precession, proceeding from the joint actions of the sun and moon,
 were 50″, yet the apparent annual precession might sometimes exceed
 and sometimes fall short of that mean quantity, according to the
 various situations of the nodes of the moon’s orbit.”

Newton in his discussion of precession (chapter IX., § 188;
_Principia_, Book III., proposition 21) had pointed out the existence
of a small irregularity with a period of six months. But it is evident,
on looking at this discussion of the effect of the solar and lunar
attractions on the protuberant parts of the earth, that the various
alterations in the positions of the sun and moon relative to the earth
might be expected to produce irregularities, and that the uniform
precessional motion known from observation and deduced from gravitation
by Newton was, as it were, only a smoothing out of a motion of a much
more complicated character. Except for the allusion referred to, Newton
made no attempt to discuss these irregularities, and none of them had
as yet been detected by observation.

Of the numerous irregularities of this class which are now known,
and which may be referred to generally as =nutation=, that indicated
by Bradley in the passage just quoted is by far the most important.
As soon as the idea of an irregularity depending on the position of
the moon’s nodes occurred to him, he saw that it would be desirable
to watch the motions of several stars during the whole period (about
19 years) occupied by the moon’s nodes in performing the circuit of
the ecliptic and returning to the same position. This inquiry was
successfully carried out between 1727 and 1747 with the telescope
mounted at Wansted. When the moon’s nodes had performed half their
revolution, _i.e._ after about nine years, the correspondence between
the displacements of the stars and the changes in the moon’s orbit was
so close that Bradley was satisfied with the general correctness of his
theory, and in 1737 he communicated the result privately to Maupertuis
(§ 221), with whom he had had some scientific correspondence.
Maupertuis appears to have told others, but Bradley himself waited
patiently for the completion of the period which he regarded as
necessary for the satisfactory verification of his theory, and only
published his results definitely at the beginning of 1748.

[Illustration: FIG. 77.—Precession and nutation.]

214. Bradley’s observations established the existence of certain
alterations in the positions of various stars, which could be accounted
for by supposing that, on the one hand, the distance of the pole from
the ecliptic fluctuated, and that, on the other, the precessional
motion of the pole was not uniform, but varied slightly in speed.
_John Machin_ (?-1751), one of the best English mathematicians of the
time, pointed out that these effects would be produced if the pole
were supposed to describe on the celestial sphere a minute circle in
a period of rather less than 19 years—being that of the revolution
of the nodes of the moon’s orbit—round the position which it would
occupy if there were no nutation, but a uniform precession. Bradley
found that this hypothesis fitted his observations, but that it would
be better to replace the circle by a slightly flattened ellipse, the
greatest and least axes of which he estimated at about 18″ and 16″
respectively.[119] This ellipse would be about as large as a shilling
placed in a slightly oblique position at a distance of 300 yards from
the eye. The motion of the pole was thus shewn to be a double one;
as the result of precession and nutation combined it describes round
the pole of the ecliptic “a gently undulated ring,” as represented in
the figure, in which, however, the undulations due to nutation are
enormously exaggerated.

215. Although Bradley was aware that nutation must be produced by the
action of the moon, he left the theoretical investigation of its cause
to more skilled mathematicians than himself.

In the following year (1749) the French mathematician D’Alembert
(chapter XI., § 232) published a treatise[120] in which not only
precession, but also a motion of nutation agreeing closely with that
observed by Bradley, were shewn by a rigorous process of analysis to be
due to the attraction of the moon on the protuberant parts of the earth
round the equator (cf. chapter IX., § 187), while Newton’s explanation
of precession was confirmed by the same piece of work. Euler (chapter
XI., § 236) published soon afterwards another investigation of the
same subject; and it has been studied afresh by many mathematical
astronomers since that time, with the result that Bradley’s nutation
is found to be only the most important of a long series of minute
irregularities in the motion of the earth’s axis.

216. Although aberration and nutation have been discussed first,
as being the most important of Bradley’s discoveries, other
investigations were carried out by him before or during the same time.

The earliest important piece of work which he accomplished was in
connection with Jupiter’s satellites. His uncle had devoted a good deal
of attention to this subject, and had drawn up some tables dealing
with the motion of the first satellite, which were based on those of
Domenico Cassini, but contained a good many improvements. Bradley seems
for some years to have made a practice of frequently observing the
eclipses of Jupiter’s satellites, and of noting discrepancies between
the observations and the tables; and he was thus able to detect several
hitherto unnoticed peculiarities in the motions, and thereby to form
improved tables. The most interesting discovery was that of a period
of 437 days, after which the motions of the three inner satellites
recurred with the same irregularities. Bradley, like Pound, made use of
Roemer’s suggestion (chapter VIII., § 162) that light occupied a finite
time in travelling from Jupiter to the earth, a theory which Cassini
and his school long rejected. Bradley’s tables of Jupiter’s satellites
were embodied in Halley’s planetary and lunar tables, printed in 1719,
but not published till more than 30 years afterwards (§ 204). Before
that date the Swedish astronomer _Pehr Vilhelm Wargentin_ (1717-1783)
had independently discovered the period of 437 days, which he utilised
for the construction of an extremely accurate set of tables for the
satellites published in 1746.

In this case as in that of nutation Bradley knew that his mathematical
powers were unequal to giving an explanation on gravitational
principles of the inequalities which observation had revealed to him,
though he was well aware of the importance of such an undertaking, and
definitely expressed the hope “that some geometer,[121] in imitation
of the great Newton, would apply himself to the investigation of these
irregularities, from the certain and demonstrative principles of
gravity.”

On the other hand, he made in 1726 an interesting practical application
of his superior knowledge of Jupiter’s satellites by determining,
in accordance with Galilei’s method (chapter VI., § 127), but with
remarkable accuracy, the longitudes of Lisbon and of New York.

217. Among Bradley’s minor pieces of work may be mentioned his
observations of several comets and his calculation of their respective
orbits according to Newton’s method; the construction of improved
tables of refraction, which remained in use for nearly a century;
a share in pendulum experiments carried out in England and Jamaica
with the object of verifying the variation of gravity in different
latitudes; a careful testing of Mayer’s lunar tables (§ 226), together
with improvements of them; and lastly, some work in connection with the
reform of the calendar made in 1752 (cf, chapter II., § 22).

218. It remains to give some account of the magnificent series of
observations carried out during Bradley’s administration of the
Greenwich Observatory.

These observations fall into two chief divisions of unequal merit,
those after 1749 having been made with some more accurate instruments
which a grant from the government enabled him at that time to procure.

The main work of the Observatory under Bradley consisted in taking
observations of fixed stars, and to a lesser extent of other bodies,
as they passed the meridian, the instruments used (the “mural
quadrant” and the “transit instrument”) being capable of motion only
in the meridian, and being therefore steadier and susceptible of
greater accuracy than those with more freedom of movement. The most
important observations taken during the years 1750-1762, amounting to
about 60,000, were published long after Bradley’s death in two large
volumes which appeared in 1798 and 1805. A selection of them had
been used earlier as the basis of a small star catalogue, published
in the _Nautical Almanac_ for 1773; but it was not till 1818 that
the publication of Bessel’s _Fundamenta Astronomiae_ (chapter XIII.,
§ 277), a catalogue of more than 3000 stars based on Bradley’s
observations, rendered these observations thoroughly available for
astronomical work. One reason for this apparently excessive delay is to
be found in Bradley’s way of working. Allusion has already been made
to a variety of causes which prevent the apparent place of a star, as
seen in the telescope and noted at the time, from being a satisfactory
permanent record of its position. There are various instrumental
errors, and errors due to refraction; again, if a star’s places at
two different times are to be compared, precession must be taken into
account; and Bradley himself unravelled in aberration and nutation
two fresh sources of error. In order therefore to put into a form
satisfactory for permanent reference a number of star observations,
it is necessary to make corrections which have the effect of allowing
for these various sources of error. This process of =reduction=, as
it is technically called, involves a certain amount of rather tedious
calculation, and though in modern observatories the process has been
so far systematised that it can be carried out almost according to
fixed rules by comparatively unskilled assistants, in Bradley’s time
it required more judgment, and it is doubtful if his assistants could
have performed the work satisfactorily, even if their time had not been
fully occupied with other duties. Bradley himself probably found the
necessary calculations tedious, and preferred devoting his energies to
work of a higher order. It is true that Delambre, the famous French
historian of astronomy, assures his readers that he had never found
the reduction of an observation tedious if performed the same day,
but a glance at any of his books is enough to shew his extraordinary
fondness for long calculations of a fairly elementary character, and
assuredly Bradley is not the only astronomer whose tastes have in this
respect differed fundamentally from Delambre’s. Moreover reducing
an observation is generally found to be a duty that, like answering
letters, grows harder to perform the longer it is neglected; and it
is not only less interesting but also much more difficult for an
astronomer to deal satisfactorily with some one else’s observations
than with his own. It is not therefore surprising that after Bradley’s
death a long interval should have elapsed before an astronomer appeared
with both the skill and the patience necessary for the complete
reduction of Bradley’s 60,000 observations.

A variety of circumstances combined to make Bradley’s observations
decidedly superior to those of his predecessors. He evidently
possessed in a marked degree the personal characteristics—of eye and
judgment—which make a first-rate observer; his instruments were mounted
in the best known way for securing accuracy, and were constructed by
the most skilful makers; he made a point of studying very carefully the
defects of his instruments, and of allowing for them; his discoveries
of aberration and nutation enabled him to avoid sources of error,
amounting to a considerable number of seconds, which his predecessors
could only have escaped imperfectly by taking the average of a number
of observations; and his improved tables of refraction still further
added to the correctness of his results.

Bessel estimates that the errors in Bradley’s observations of
the declination of stars were usually less than 4″, while the
corresponding errors in right ascension, a quantity which depends
ultimately on a time-observation, were less than 15″, or one second of
time. His observations thus shewed a considerable advance in accuracy
compared with those of Flamsteed (§ 198), which represented the best
that had hitherto been done.

219. The next Astronomer Royal was _Nathaniel Bliss_ (1700-1764), who
died after two years. He was in turn succeeded by _Nevil Maskelyne_
(1732-1811), who carried on for nearly half a century the tradition of
accurate observation which Bradley had established at Greenwich, and
made some improvements in methods.

To him is also due the first serious attempt to measure the density and
hence the mass of the earth. By comparing the attraction exerted by the
earth with that of the sun and other bodies, Newton, as we have seen
(chapter IX., § 185), had been able to connect the masses of several
of the celestial bodies with that of the earth. To connect the mass of
the whole earth with that of a given terrestrial body, and so express
it in pounds or tons, was a problem of quite a different kind. It is of
course possible to examine portions of the earth’s surface and compare
their density with that of, say, water; then to make some conjecture,
based on rough observations in mines, etc., as to the rate at which
density increases as we go from the surface towards the centre of the
earth, and hence to infer the average density of the earth. Thus the
mass of the whole earth is compared with that of a globe of water of
the same size, and, the size being known, is expressible in pounds or
tons.

By a process of this sort Newton had in fact, with extraordinary
insight, estimated that the density of the earth was between five and
six times as great as that of water.[122]

It was, however, clearly desirable to solve the problem in a less
conjectural manner, by a direct comparison of the gravitational
attraction exerted by the earth with that exerted by a known mass—a
method that would at the same time afford a valuable test of Newton’s
theory of the gravitating properties of portions of the earth, as
distinguished from the whole earth. In their Peruvian expedition (§
221), Bouguer and La Condamine had noticed certain small deflections
of the plumb-line, which indicated an attraction by Chimborazo, near
which they were working; but the observations were too uncertain to
be depended on. Maskelyne selected for his purpose Schehallien in
Perthshire, a narrow ridge running east and west. The direction of
the plumb-line was observed (1774) on each side of the ridge, and a
change in direction amounting to about 12″ was found to be caused by
the attraction of the mountain. As the direction of the plumb-line
depends on the attraction of the earth as a whole and on that of the
mountain, this deflection at once led to a comparison of the two
attractions. Hence an intricate calculation performed by _Charles
Hutton_ (1737-1823) led to a comparison of the average densities of the
earth and mountain, and hence to the final conclusion (published in
1778) that the earth’s density was about 4-1∕2 times that of water. As
Hutton’s estimate of the density of the mountain was avowedly almost
conjectural, this result was of course correspondingly uncertain.

A few years later _John Michell_ (1724-1793) suggested, and the famous
chemist and electrician _Henry Cavendish_ (1731-1810) carried out
(1798), an experiment in which the mountain was replaced by a pair of
heavy balls, and their attraction on another body was compared with
that of the earth, the result being that the density of the earth was
found to be about 5-1∕2 times that of water.

The =Cavendish experiment=, as it is often called, has since been
repeated by various other experimenters in modified forms, and one or
two other methods, too technical to be described here, have also been
devised. All the best modern experiments give for the density numbers
converging closely on 5-1∕2, thus verifying in a most striking way both
Newton’s conjecture and Cavendish’s original experiment.

With this value of the density the mass of the earth is a
little more than 13 billion billion pounds, or more precisely
13,136,000,000,000,000,000,000,000 lbs.

220. While Greenwich was furnishing the astronomical world with a
most valuable series of observations, the Paris Observatory had not
fulfilled its early promise. It was in fact suffering, like English
mathematics, from the evil effects of undue adherence to the methods
and opinions of a distinguished man. Domenico Cassini happened to hold
several erroneous opinions in important astronomical matters; he was
too good a Catholic to be a genuine Coppernican, he had no belief in
gravitation, he was firmly persuaded that the earth was flattened at
the equator instead of at the poles, and he rejected Roemer’s discovery
of the velocity of light. After his death in 1712 the directorship of
the Observatory passed in turn to three of his descendants, the last
of whom resigned office in 1793; and several members of the Maraldi
family, into which his sister had married, worked in co-operation with
their cousins. Unfortunately a good deal of their energy was expended,
first in defending, and afterwards in gradually withdrawing from, the
errors of their distinguished head. _Jacques Cassini_ for example, the
second of the family (1677-1756), although a Coppernican, was still a
timid one, and rejected Kepler’s law of areas; his son again, commonly
known as _Cassini de Thury_ (1714-1784), still defended the ancestral
errors as to the form of the earth; while the fourth member of the
family, _Count Cassini_ (1748-1845), was the first of the family to
accept the Newtonian idea of gravitation.

Some planetary and other observations of value were made by the
Cassini-Maraldi school, but little of this work was of first-rate
importance.

221. A series of important measurements of the earth, in which the
Cassinis had a considerable share, were made during the 18th century,
almost entirely by Frenchmen, and resulted in tolerably exact knowledge
of the earth’s size and shape.

The variation of the length of the seconds pendulum observed by Richer
in his Cayenne expedition (chapter VIII., § 161) had been the first
indication of a deviation of the earth from a spherical form. Newton
inferred, both from these pendulum experiments and from an independent
theoretical investigation (chapter IX., § 187), that the earth was
spheroidal, being flattened towards the poles; and this view was
strengthened by the satisfactory explanation of precession to which it
led (chapter IX., § 188).

On the other hand, a comparison of various measurements of arcs of the
meridian in different latitudes gave some support to the view that
the earth was elongated towards the poles and flattened towards the
equator, a view championed with great ardour by the Cassini school.
It was clearly important that the question should be settled by more
extensive and careful earth-measurements.

The essential part of an ordinary measurement of the earth consists
in ascertaining the distance in miles between two places on the same
meridian, the latitudes of which differ by a known amount. From
these two data the length of an arc of a meridian corresponding to a
difference of latitude of 1° at once follows. The =latitude= of a place
is the angle which the vertical at the place makes with the equator,
or, expressed in a slightly different form, is the angular distance of
the zenith from the celestial equator. The =vertical= at any place may
be defined as a direction perpendicular to the surface of still water
at the place in question, and may be regarded as perpendicular to the
true surface of the earth, accidental irregularities in its form such
as hills and valleys being ignored.[123]

The difference of latitude between two places, north and south of one
another, is consequently the angle between the verticals there. Fig.
78 shews the verticals, marked by the arrowheads, at places on the
same meridian in latitudes differing by 10°; so that two consecutive
verticals are inclined in every case at an angle of 10°.

[Illustration: FIG. 78.—The varying curvature of the earth.]

If, as in fig. 78, the shape of the earth is drawn in accordance with
Newton’s views, the figure shews at once that the arcs A A_{1}, A_{1}
A_{2}, etc., each of which corresponds to 10° of latitude, steadily
_increase_ as we pass from a point A on the equator to the pole B. If
the opposite hypothesis be adopted, which will be illustrated by the
same figure if we now regard A as the pole and B as a point on the
equator, then the successive arcs _decrease_ as we pass from equator
to pole. A comparison of the measurements made by Eratosthenes in
Egypt (chapter II., § 36) with some made in Europe (chapter VIII., §
159) seemed to indicate that a degree of the meridian near the equator
was longer than one in higher latitudes; and a similar conclusion was
indicated by a comparison of different portions of an extensive French
arc, about 9° in length, extending from Dunkirk to the Pyrenees, which
was measured under the superintendence of the Cassinis in continuation
of Picard’s arc, the result being published by J. Cassini in 1720.
In neither case, however, were the data sufficiently accurate to
justify the conclusion; and the first decisive evidence was obtained
by measurement of arcs in places differing far more widely in latitude
than any that had hitherto been available. The French Academy organised
an expedition to Peru, under the management of three Academicians,
_Pierre Bouguer_ (1698-1758), _Charles Marie de La Condamine_
(1701-1774), and _Louis Godin_ (1704-1760), with whom two Spanish naval
officers also co-operated.

The expedition started in 1735, and, owing to various difficulties, the
work was spread out over nearly ten years. The most important result
was the measurement, with very fair accuracy, of an arc of about 3° in
length, close to the equator; but a number of pendulum experiments of
value were also performed, and a good many miscellaneous additions to
knowledge were made.

But while the Peruvian party were still at their work a similar
expedition to Lapland, under the Academician _Pierre Louis Moreau de
Maupertuis_ (1698-1759), had much more rapidly (1736-7), if somewhat
carelessly, effected the measurement of an arc of nearly 1° close to
the arctic circle.

From these measurements it resulted that the lengths of a degree
of a meridian about latitude 2° S. (Peru), about latitude 47° N.
(France) and about latitude 66° N. (Lapland) were respectively 362,800
feet, 364,900 feet, and 367,100 feet.[124] There was therefore clear
evidence, from a comparison of any two of these arcs, of an increase
of the length of a degree of a meridian as the latitude increases; and
the general correctness of Newton’s views as against Cassini’s was thus
definitely established.

The extent to which the earth deviates from a sphere is usually
expressed by a fraction known as the =ellipticity=, which is the
difference between the lines C A, C B of fig. 78 divided by the greater
of them. From comparison of the three arcs just mentioned several very
different values of the ellipticity were deduced, the discrepancies
being partly due to different theoretical methods of interpreting the
results and partly to errors in the arcs.

A measurement, made by _Jöns Svanberg_ (1771-1851) in 1801-3, of an
arc near that of Maupertuis has in fact shewn that his estimate of the
length of a degree was about 1,000 feet too large.

A large number of other arcs have been measured in different parts
of the earth at various times during the 18th and 19th centuries.
The details of the measurements need not be given, but to prevent
recurrence to the subject it is convenient to give here the results,
obtained by a comparison of these different measurements, that the
ellipticity is very nearly 1∕292, and the greatest radius of the earth
(C A in fig. 78) a little less than 21,000,000 feet or 4,000 miles. It
follows from these figures that the length of a degree in the latitude
of London contains, to use Sir John Herschel’s ingenious mnemonic,
almost exactly as many thousand feet as the year contains days.

222. Reference has already been made to the supremacy of Greenwich
during the 18th century in the domain of exact observation. France,
however, produced during this period one great observing astronomer
who actually accomplished much, and under more favourable external
conditions might almost have rivalled Bradley.

_Nicholas Louis de Lacaille_ was born in 1713. After he had devoted
a good deal of time to theological studies with a view to an
ecclesiastical career, his interests were diverted to astronomy and
mathematics. He was introduced to Jacques Cassini, and appointed one of
the assistants at the Paris Observatory.

In 1738 and the two following years he took an active part in the
measurement of the French arc, then in process of verification.
While engaged in this work he was appointed (1739) to a poorly paid
professorship at the Mazarin College, at which a small observatory was
erected. Here it was his regular practice to spend the whole night, if
fine, in observation, while “to fill up usefully the hours of leisure
which bad weather gives to observers only too often” he undertook a
variety of extensive calculations and wrote innumerable scientific
memoirs. It is therefore not surprising that he died comparatively
early (1762) and that his death was generally attributed to overwork.

223. The monotony of Lacaille’s outward life was broken by the
scientific expedition to the Cape of Good Hope (1750-1754) organised by
the Academy of Sciences and placed under his direction.

The most striking piece of work undertaken during this expedition was
a systematic survey of the southern skies, in the course of which more
than 10,000 stars were observed.

These observations, together with a carefully executed catalogue
of nearly 2,000 of the stars[125] and a star-map, were published
posthumously in 1763 under the title _Coelum Australe Stelliferum_, and
entirely superseded Halley’s much smaller and less accurate catalogue
(§ 199). Lacaille found it necessary to make 14 new constellations
(some of which have since been generally abandoned), and to restore
to their original places the stars which the loyal Halley had made
into King Charles’s Oak. Incidentally Lacaille observed and described
42 nebulae, nebulous stars, and star-clusters, objects the systematic
study of which was one of Herschel’s great achievements (chapter XII.,
§§ 259-261).

He made a large number of pendulum experiments, at Mauritius as well
as at the Cape, with the usual object of determining in a new part
of the world the acceleration due to gravity, and measured an arc of
the meridian extending over rather more than a degree. He made also
careful observations of the positions of Mars and Venus, in order that
from comparison of them with simultaneous observations in northern
latitudes he might get the parallax of the sun (chapter VIII., §
161). These observations of Mars compared with some made in Europe by
Bradley and others, and a similar treatment of Venus, both pointed to
a solar parallax slightly in excess of 10″, a result less accurate
than Cassini’s (chapter VIII., § 161), though obtained by more reliable
processes.

A large number of observations of the moon, of which those made by
him at the Cape formed an important part, led, after an elaborate
discussion in which the spheroidal form of the earth was taken into
account, to an improved value of the moon’s distance, first published
in 1761.

Lacaille also used his observations of fixed stars to improve our
knowledge of refraction, and obtained a number of observations of the
sun in that part of its orbit which it traverses in our winter months
(the summer of the southern hemisphere), and in which it is therefore
too near the horizon to be observed satisfactorily in Europe.

The results of this—one of the most fruitful scientific expeditions
ever undertaken—were published in separate memoirs or embodied in
various books published after his return to Paris.

224. In 1757, under the title _Astronomiae Fundamenta_, appeared a
catalogue of 400 of the brightest stars, observed and reduced with
the most scrupulous care, so that, notwithstanding the poverty of
Lacaille’s instrumental outfit, the catalogue was far superior to any
of its predecessors, and was only surpassed by Bradley’s observations
as they were gradually published. It is characteristic of Lacaille’s
unselfish nature that he did not have the _Fundamenta_ sold in the
ordinary way, but distributed copies gratuitously to those interested
in the subject, and earned the money necessary to pay the expenses of
publication by calculating some astronomical almanacks.

Another catalogue, of rather more than 500 stars situated in the
zodiac, was published posthumously.

In the following year (1758) he published an excellent set of Solar
Tables, based on an immense series of observations and calculations.
These were remarkable as the first in which planetary perturbations
were taken into account.

Among Lacaille’s minor contributions to astronomy may be mentioned:
improved methods of calculating cometary orbits and the actual
calculation of the orbits of a large number of recorded comets, the
calculation of all eclipses visible in Europe since the year 1, a
warning that the transit of Venus would be capable of far less accurate
observation than Halley had expected (§ 202), observations of the
actual transit of 1761 (§ 227), and a number of improvements in
methods of calculation and of utilising observations.

In estimating the immense mass of work which Lacaille accomplished
during an astronomical career of about 22 years, it has also to be
borne in mind that he had only moderately good instruments at his
observatory, and _no assistant_, and that a considerable part of his
time had to be spent in earning the means of living and of working.

225. During the period under consideration Germany also produced one
astronomer, primarily an observer, of great merit, _Tobias Mayer_
(1723-1762). He was appointed professor of mathematics and political
economy at Göttingen in 1751, apparently on the understanding that he
need not lecture on the latter subject, of which indeed he seems to
have professed no knowledge; three years later he was put in charge
of the observatory, which had been erected 20 years before. He had at
least one fine instrument,[126] and following the example of Tycho,
Flamsteed, and Bradley, he made a careful study of its defects, and
carried further than any of his predecessors the theory of correcting
observations for instrumental errors.[127]

He improved Lacaille’s tables of the sun, and made a catalogue of
998 zodiacal stars, published posthumously in 1775; by a comparison
of star places recorded by Roemer (1706) with his own and Lacaille’s
observations he obtained evidence of a considerable number of proper
motions (§ 203); and he made a number of other less interesting
additions to astronomical knowledge.

226. But Mayer’s most important work was on the moon. At the beginning
of his career he made a careful study of the position of the craters
and other markings, and was thereby able to get a complete geometrical
explanation of the various librations of the moon (chapter VI., § 133),
and to fix with accuracy the position of the axis about which the moon
rotates. A map of the moon based on his observations was published with
other posthumous works in 1775.

[Illustration: FIG. 79.—Tobias Mayer’s map of the moon.]

Much more important, however, were his lunar theory and the tables
based on it. The intrinsic mathematical interest of the problem of the
motion of the moon, and its practical importance for the determination
of longitude, had caused a great deal of attention to be given to the
subject by the astronomers of the 18th century. A further stimulus
was also furnished by the prizes offered by the British Government in
1713 for a method of finding the longitude at sea, _viz._ £20,000 for
a method reliable to within half a degree, and smaller amounts for
methods of less accuracy.

All the great mathematicians of the period made attempts at deducing
the moon’s motions from gravitational principles. Mayer worked out a
theory in accordance with methods used by Euler (chapter XI., § 233),
but made a much more liberal and also more skilful use of observations
to determine various numerical quantities, which pure theory gave
either not at all or with considerable uncertainty. He accordingly
succeeded in calculating tables of the moon (published with those of
the sun in 1753) which were a notable improvement on those of any
earlier writer. After making further improvements, he sent them in
1755 to England. Bradley, to whom the Admiralty submitted them for
criticism, reported favourably of their accuracy; and a few years
later, after making some alterations in the tables on the basis of his
own observations, he recommended to the Admiralty a longitude method
based on their use which he estimated to be in general capable of
giving the longitude within about half a degree.

Before anything definite was done, Mayer died at the early age of
39, leaving behind him a new set of tables, which were also sent to
England. Ultimately £3,000 was paid to his widow in 1765; and both
his _Theory of the Moon_[128] and his improved Solar and Lunar Tables
were published in 1770 at the expense of the Board of Longitude. A
later edition, improved by Bradley’s former assistant _Charles Mason_
(1730-1787), appeared in 1787.

A prize was also given to Euler for his theoretical work; while
£3,000 and subsequently £10,000 more were awarded to _John Harrison_
for improvements in the chronometer, which rendered practicable an
entirely different method of finding the longitude (chapter VI., § 127).

227. The astronomers of the 18th century had two opportunities of
utilising a transit of Venus for the determination of the distance of
the sun, as recommended by Halley (§ 202).

A passage or transit of Venus across the sun’s disc is a phenomenon
of the same nature as an eclipse of the sun by the moon, with the
important difference that the apparent magnitude of the planet is too
small to cause any serious diminution in the sun’s light, and it merely
appears as a small black dot on the bright surface of the sun.

If the path of Venus lay in the ecliptic, then at every inferior
conjunction, occurring once in 584 days, she would necessarily pass
between the sun and earth and would appear to transit. As, however,
the paths of Venus and the earth are inclined to one another, at
inferior conjunction Venus is usually far enough “above” or “below”
the ecliptic for no transit to occur. With the present position of
the two paths—which planetary perturbations are only very gradually
changing—transits of Venus occur in pairs eight years apart, while
between the latter of one pair and the earlier of the next pair elapse
alternately intervals of 105-1∕2 and of 121-1∕2 years. Thus transits
have taken place in December 1631 and 1639, June 1761 and 1769,
December 1874 and 1882, and will occur again in 2004 and 2012, 2117 and
2125, and so on.

The method of getting the distance of the sun from a transit of Venus
may be said not to differ essentially from that based on observations
of Mars (chapter VIII., § 161).

The observer’s object in both cases is to obtain the difference in
direction of the planet as seen from different places on the earth.
Venus, however, when at all near the earth, is usually too near the
sun in the sky to be capable of minutely exact observation, but when
a transit occurs the sun’s disc serves as it were as a dial-plate on
which the position of the planet can be noted. Moreover the measurement
of minute angles, an art not yet carried to very great perfection
in the 18th century, can be avoided by time-observations, as the
difference in the times at which Venus enters (or leaves) the sun’s
disc as seen at different stations, or the difference in the durations
of the transit, can be without difficulty translated into difference of
direction, and the distances of Venus and the sun can be deduced.[129]

Immense trouble was taken by Governments, Academies, and private
persons in arranging for the observation of the transits of 1761 and
1769. For the former observing parties were sent as far as to Tobolsk,
St. Helena, the Cape of Good Hope, and India, while observations were
also made by astronomers at Greenwich, Paris, Vienna, Upsala, and
elsewhere in Europe. The next transit was observed on an even larger
scale, the stations selected ranging from Siberia to California, from
the Varanger Fjord to Otaheiti (where no less famous a person than
Captain Cook was placed), and from Hudson’s Bay to Madras.

The expeditions organised on this occasion by the American
Philosophical Society may be regarded as the first of the contributions
made by America to the science which has since owed so much to her;
while the Empress Catherine bore witness to the newly acquired
civilisation of her country by arranging a number of observing stations
on Russian soil.

The results were far more in accordance with Lacaille’s anticipations
than with Halley’s. A variety of causes prevented the moments of
contact between the discs of Venus and the sun from being observed
with the precision that had been hoped. By selecting different sets
of observations, and by making different allowances for the various
probable sources of error, a number of discordant results were obtained
by various calculators. The values of the parallax (chapter VIII., §
161) of the sun deduced from the earlier of the two transits ranged
between about 8″ and 10″; while those obtained in 1769, though much
more consistent, still varied between about 8″ and 9″, corresponding
to a variation of about 10,000,000 miles in the distance of the sun.

The whole set of observations were subsequently very elaborately
discussed in 1822-4 and again in 1835 by _Johann Franz Encke_
(1791-1865), who deduced a parallax of 8″·571, corresponding to a
distance of 95,370,000 miles, a number which long remained classical.
The uncertainty of the data is, however, shewn by the fact that other
equally competent astronomers have deduced from the observations of
1769 parallaxes of 8″·8 and 8″·9.

No account has yet been given of William Herschel, perhaps the most
famous of all observers, whose career falls mainly into the last
quarter of the 18th century and the earlier part of the 19th century.
As, however, his work was essentially different from that of almost all
the astronomers of the 18th century, and gave a powerful impulse to a
department of astronomy hitherto almost ignored, it is convenient to
postpone to a later chapter (XII.) the discussion of his work.



CHAPTER XI.

GRAVITATIONAL ASTRONOMY IN THE 18TH CENTURY.

 “Astronomy, considered in the most general way, is a great problem
 of mechanics, the arbitrary data of which are the elements of the
 celestial movements; its solution depends both on the accuracy of
 observations and on the perfection of analysis.”
                            LAPLACE, Preface to the _Mécanique Céleste_.


228. The solar system, as it was known at the beginning of the 18th
century, contained 18 recognised members: the sun, six planets, ten
satellites (one belonging to the earth, four to Jupiter, and five to
Saturn), and Saturn’s ring.

Comets were known to have come on many occasions into the region of
space occupied by the solar system, and there were reasons to believe
that one of them at least (chapter X., § 200) was a regular visitor;
they were, however, scarcely regarded as belonging to the solar system,
and their action (if any) on its members was ignored, a neglect which
subsequent investigation has completely justified. Many thousands of
fixed stars had also been observed, and their places on the celestial
sphere determined; they were known to be at very great though unknown
distances from the solar system, and their influence on it was regarded
as insensible.

The motions of the 18 members of the solar system were tolerably
well known; their actual distances from one another had been roughly
estimated, while the _proportions_ between most of the distances were
known with considerable accuracy. Apart from the entirely anomalous
ring of Saturn, which may for the present be left out of consideration,
most of the bodies of the system were known from observation to be
nearly spherical in form, and the rest were generally supposed to be so
also.

Newton had shewn, with a considerable degree of probability, that these
bodies attracted one another according to the law of gravitation; and
there was no reason to suppose that they exerted any other important
influence on one another’s motions.[130]

The problem which presented itself, and which may conveniently be
called Newton’s problem, was therefore:—

_Given these 18 bodies, and their positions and motions at any time,
to deduce from their mutual gravitation by a process of mathematical
calculation their positions and motions at any other time; and to shew
that these agree with those actually observed._

Such a calculation would necessarily involve, among other quantities,
the masses of the several bodies; it was evidently legitimate to assume
these at will in such a way as to make the results of calculation agree
with those of observation. If this were done successfully the masses
would thereby be determined. In the same way the commonly accepted
estimates of the dimensions of the solar system and of the shapes of
its members might be modified in any way not actually inconsistent with
direct observation.

The general problem thus formulated can fortunately be reduced to
somewhat simpler ones.

Newton had shewn (chapter IX., § 182) that an ordinary sphere
attracted other bodies and was attracted by them, as if its mass were
concentrated at its centre; and that the effects of deviation from
a spherical form became very small at a considerable distance from
the body. Hence, except in special cases, the bodies of the solar
system could be treated as spheres, which could again be regarded as
concentrated at their respective centres. It will be convenient for the
sake of brevity to assume for the future that all “bodies” referred to
are of this sort, unless the contrary is stated or implied. The effects
of deviations from spherical form could then be treated separately
when required, as in the cases of precession and of other motions of a
planet or satellite about its centre, and of the corresponding action
of a non-spherical planet on its satellites; to this group of problems
belongs also that of the tides and other cases of the motion of parts
of a body of any form relative to the rest.

Again, the solar system happens to be so constituted that each body’s
motion can be treated as determined primarily by one other body only.
A planet, for example, moves nearly as if no other body but the sun
existed, and the moon’s motion relative to the earth is roughly the
same as if the other bodies of the solar system were non-existent.

The problem of the motion of two mutually gravitating spheres was
completely solved by Newton, and was shewn to lead to Kepler’s first
two laws. Hence each body of the solar system could be regarded as
moving nearly in an ellipse round some one body, but as slightly
disturbed by the action of others. Moreover, by a general mathematical
principle applicable in problems of motion, the effect of a number of
small disturbing causes acting conjointly is nearly the same as that
which results from adding together their separate effects. Hence each
body could, without great error, be regarded as disturbed by one body
at a time; the several disturbing effects could then be added together,
and a fresh calculation could be made to further diminish the error.
The kernel of Newton’s problem is thus seen to be a special case of the
so-called =problem of three bodies=, viz.:—

_Given at any time the positions and motions of three mutually
gravitating bodies, to determine their positions and motions at any
other time._

Even this apparently simple problem in its general form entirely
transcends the powers, not only of the mathematical methods of the
early 18th century, but also of those that have been devised since.
Certain special cases have been solved, so that it has been shewn to be
possible to suppose three bodies initially moving in such a way that
their future motion can be completely determined. But these cases do
not occur in nature.

In the case of the solar system the problem is simplified, not only by
the consideration already mentioned that one of the three bodies can
always be regarded as exercising only a small influence on the relative
motion of the other two, but also by the facts that the orbits of
the planets and satellites do not differ much from circles, and that
the planes of their orbits are in no case inclined at large angles
to any one of them, such as the ecliptic; in other words, that the
eccentricities and inclinations are small quantities.

Thus simplified, the problem has been found to admit of solutions of
considerable accuracy by methods of approximation.[131]

In the case of the system formed by the sun, earth, and moon, the
characteristic feature is the great distance of the sun, which is
the disturbing body, from the other two bodies; in the case of the
sun and two planets, the enormous mass of the sun as compared with
the disturbing planet is the important factor. Hence the methods of
treatment suitable for the two cases differ, and two substantially
distinct branches of the subject, =lunar theory= and =planetary
theory=, have developed. The problems presented by the motions of the
satellites of Jupiter and Saturn, though allied to those of the lunar
theory, differ in some important respects, and are usually treated
separately.

229. As we have seen, Newton made a number of important steps towards
the solution of his problem, but little was done by his successors
in his own country. On the Continent also progress was at first very
slow. The _Principia_ was read and admired by most of the leading
mathematicians of the time, but its principles were not accepted,
and Cartesianism remained the prevailing philosophy. A forward step
is marked by the publication by the Paris Academy of Sciences in
1720 of a memoir written by the _Chevalier de Louville_ (1671-1732)
on the basis of Newton’s principles; ten years later the Academy
awarded a prize to an essay on the planetary motions written by _John
Bernouilli_ (1667-1748) on Cartesian principles, a Newtonian essay
being put second. In 1732 Maupertuis (chapter X., § 221) published
a treatise on the figure of the earth on Newtonian lines, and the
appearance six years later of Voltaire’s extremely readable _Éléments
de la Philosophie de Newton_ had a great effect in popularising the
new ideas. The last official recognition of Cartesianism in France
seems to have been in 1740, when the prize offered by the Academy for
an essay on the tides was shared between a Cartesian and three eminent
Newtonians (§ 230).

The rapid development of gravitational astronomy that ensued between
this time and the beginning of the 19th century was almost entirely
the work of five great Continental mathematicians, Euler, Clairaut,
D’Alembert, Lagrange, and Laplace, of whom the eldest was born in 1707
and the youngest died in 1827, within a month of the centenary of
Newton’s death. Euler was a Swiss, Lagrange was of Italian birth but
French by extraction and to a great extent by adoption, and the other
three were entirely French. France therefore during nearly the whole of
the 18th century reigned supreme in gravitational astronomy, and has
not lost her supremacy even to-day, though during the present century
America, England, Germany, Italy, and other countries have all made
substantial contributions to the subject.

It is convenient to consider first the work of the three first-named
astronomers, and to treat later Lagrange and Laplace, who carried
gravitational astronomy to a decidedly higher stage of development than
their predecessors.

230. _Leonhard Euler_ was born at Basle in 1707, 14 years later
than Bradley and six years earlier than Lacaille. He was the
son of a Protestant minister who had studied mathematics under
_James Bernouilli_ (1654-1705), the first of a famous family of
mathematicians. Leonhard Euler himself was a favourite pupil of John
Bernouilli (the younger brother of James), and was an intimate friend
of his two sons, one of whom, _Daniel_ (1700-1782), was not only a
distinguished mathematician like his father and uncle, but was also
the first important Newtonian outside Great Britain. Like so many
other astronomers, Euler began by studying theology, but was induced
both by his natural tastes and by the influence of the Bernouillis to
turn his attention to mathematics. Through the influence of Daniel
Bernouilli, who had recently been appointed to a professorship at
St. Petersburg, Euler received and accepted an invitation to join the
newly created Academy of Sciences there (1727). This first appointment
carried with it a stipend, and the duties were the general promotion
of science; subsequently Euler undertook more definite professorial
work, but most of his energy during the whole of his career was devoted
to writing mathematical papers, the majority of which were published
by the St. Petersburg Academy. Though he took no part in politics,
Russian autocracy appears to have been oppressive to him, reared as
he had been among Swiss and Protestant surroundings; and in 1741 he
accepted an invitation from Frederick the Great, a despot of a less
pronounced type, to come to Berlin, and assist in reorganising the
Academy of Sciences there. On being reproached one day by the Queen for
his taciturn and melancholy demeanour, he justified his silence on the
ground that he had just come from a country where speech was liable
to lead to hanging;[132] but notwithstanding this frank criticism he
remained on good terms with the Russian court, and continued to draw
his stipend as a member of the St. Petersburg Academy and to contribute
to its Transactions. Moreover, after 25 years spent at Berlin, he
accepted a pressing invitation from the Empress Catherine II. and
returned to Russia (1766).

He had lost the use of one eye in 1735, a disaster which called from
him the remark that he would henceforward have less to distract him
from his mathematics; the second eye went soon after his return
to Russia, and with the exception of a short time during which an
operation restored the partial use of one eye he remained blind till
the end of his life. But this disability made little difference to his
astounding scientific activity; and it was only after nearly 17 years
of blindness that as a result of a fit of apoplexy “he ceased to live
and to calculate” (1783).

Euler was probably the most versatile as well as the most prolific of
mathematicians of all time. There is scarcely any branch of modern
analysis to which he was not a large contributor, and his extraordinary
powers of devising and applying methods of calculation were employed
by him with great success in each of the existing branches of applied
mathematics; problems of abstract dynamics, of optics, of the motion
of fluids, and of astronomy were all in turn subjected to his analysis
and solved. The extent of his writings is shewn by the fact that, in
addition to several books, he wrote about 800 papers on mathematical
and physical subjects; it is estimated that a complete edition of his
works would occupy 25 quarto volumes of about 600 pages each.

Euler’s first contribution to astronomy was an essay on the tides which
obtained a share of the Academy prize for 1740 already referred to,
Daniel Bernouilli and Maclaurin (chapter X., § 196) being the other two
Newtonians. The problem of the tides was, however, by no means solved
by any of the three writers.

He gave two distinct solutions of the problem of three bodies in a form
suitable for the lunar theory, and made a number of extremely important
and suggestive though incomplete contributions to planetary theory. In
both subjects his work was so closely connected with that of Clairaut
and D’Alembert that it is more convenient to discuss it in connection
with theirs.

231. _Alexis Claude Clairaut_, born at Paris in 1713, belongs to the
class of precocious geniuses. He read the Infinitesimal Calculus and
Conic Sections at the age of ten, presented a scientific memoir to the
Academy of Sciences before he was 13, and published a book containing
some important contributions to geometry when he was 18, thereby
winning his admission to the Academy.

Shortly afterwards he took part in Maupertuis’ expedition to Lapland
(chapter X., § 221), and after publishing several papers of minor
importance produced in 1743 his classical work on the figure of the
earth. In this he discussed in a far more complete form than either
Newton or Maclaurin the form which a rotating body like the earth
assumes under the influence of the mutual gravitation of its parts,
certain hypotheses of a very general nature being made as to the
variations of density in the interior; and deduced formulae for the
changes in different latitudes of the acceleration due to gravity,
which are in satisfactory agreement with the results of pendulum
experiments.

Although the subject has since been more elaborately and more
generally treated by later writers, and a good many additions have been
made, few if any results of fundamental importance have been added to
those contained in Clairaut’s book.

He next turned his attention to the problem of three bodies, obtained
a solution suitable for the moon, and made some progress in planetary
theory.

[Illustration: FIG. 80.—The path of Halley’s comet.]

Halley’s comet (chapter X., § 200) was “due” about 1758; as the time
approached Clairaut took up the task of computing the perturbations
which it would probably have experienced since its last appearance,
owing to the influence of the two great planets, Jupiter and Saturn,
close to both of which it would have passed. An extremely laborious
calculation shewed that the comet would have been retarded about 100
days by Saturn and about 518 days by Jupiter, and he accordingly
announced to the Academy towards the end of 1758 that the comet might
be expected to pass its perihelion (the point of its orbit nearest
the sun, P in fig. 80) about April 13th of the following year, though
owing to various defects in his calculation there might be an error
of a month either way. The comet was anxiously watched for by the
astronomical world, and was actually discovered by an amateur, _George
Palitzsch_ (1723-1788) of Saxony, on Christmas Day, 1758; it passed its
perihelion just a month and a day before the time assigned by Clairaut.

Halley’s brilliant conjecture was thus justified; a new member was
added to the solar system, and hopes were raised—to be afterwards
amply fulfilled—that in other cases also the motions of comets might
be reduced to rule, and calculated according to the same principles as
those of less erratic bodies. The superstitions attached to comets were
of course at the same time still further shaken.

Clairaut appears to have had great personal charm and to have been a
conspicuous figure in Paris society. Unfortunately his strength was not
equal to the combined claims of social and scientific labours, and he
died in 1765 at an age when much might still have been hoped from his
extraordinary abilities.[133]

232. _Jean-le-Rond D’Alembert_ was found in 1717 as an infant on the
steps of the church of St. Jean-le-Rond in Paris, but was afterwards
recognised, and to some extent provided for, by his father, though
his home was with his foster parents. After receiving a fair school
education, he studied law and medicine, but then turned his attention
to mathematics. He first attracted notice in mathematical circles by
a paper written in 1738, and was admitted to the Academy of Sciences
two years afterwards. His earliest important work was the _Traité de
Dynamique_ (1743), which contained, among other contributions to the
subject, the first statement of a dynamical principle which bears his
name, and which, though in one sense only a corollary from Newton’s
Third Law of Motion, has proved to be of immense service in nearly all
general dynamical problems, astronomical or otherwise. During the next
few years he made a number of contributions to mathematical physics,
as well as to the problem of three bodies; and published in 1749 his
work on precession and nutation, already referred to (chapter X., §
215). From this time onwards he began to give an increasing part of his
energies to work outside mathematics. For some years he collaborated
with Diderot in producing the famous French Encyclopaedia, which began
to appear in 1751, and exercised so great an influence on contemporary
political and philosophic thought. D’Alembert wrote the introduction,
which was read to the _Académie Française_[134] in 1754 on the occasion
of his admission to that distinguished body, as well as a variety of
scientific and other articles. In the later part of his life, which
ended in 1783, he wrote little on mathematics, but published a number
of books on philosophical, literary, and political subjects;[135] as
secretary of the Academy he also wrote obituary notices (_éloges_)
of some 70 of its members. He was thus, in Carlyle’s words, “of
great faculty, especially of great clearness and method; famous in
Mathematics; no less so, to the wonder of some, in the intellectual
provinces of Literature.”

D’Alembert and Clairaut were great rivals, and almost every work of the
latter was severely criticised by the former, while Clairaut retaliated
though with much less zeal and vehemence. The great popular reputation
acquired by Clairaut through his work on Halley’s comet appears to have
particularly excited D’Alembert’s jealousy. The rivalry, though not a
pleasant spectacle, was, however, useful in leading to the detection
and subsequent improvement of various weak points in the work of each.
In other respects D’Alembert’s personal characteristics appear to have
been extremely pleasant. He was always a poor man, but nevertheless
declined magnificent offers made to him by both Catherine II. of
Russia and Frederick the Great of Prussia, and preferred to keep his
independence, though he retained the friendship of both sovereigns and
accepted a small pension from the latter. He lived extremely simply,
and notwithstanding his poverty was very generous to his foster-mother,
to various young students, and to many others with whom he came into
contact.

233. Euler, Clairaut, and D’Alembert all succeeded in obtaining
independently and nearly simultaneously solutions of the problem of
three bodies in a form suitable for lunar theory. Euler published in
1746 some rather imperfect Tables of the Moon, which shewed that he
must have already obtained his solution. Both Clairaut and D’Alembert
presented to the Academy in 1747 memoirs containing their respective
solutions, with applications to the moon as well as to some planetary
problems. In each of these memoirs occurred the same difficulty which
Newton had met with: the calculated motion of the moon’s apogee was
only about half the observed result. Clairaut at first met this
difficulty by assuming an alteration in the law of gravitation, and
got a result which seemed to him satisfactory by assuming gravitation
to vary partly as the inverse square and partly as the inverse cube of
the distance.[136] Euler also had doubts as to the correctness of the
inverse square. Two years later, however (1749), on going through his
original calculation again, Clairaut discovered that certain terms,
which had appeared unimportant at the beginning of the calculation and
had therefore been omitted, became important later on. When these were
taken into account, the motion of the apogee as deduced from theory
agreed very nearly with that observed. This was the first of several
cases in which a serious discrepancy between theory and observation
has at first discredited the law of gravitation, but has subsequently
been explained away, and has thereby given a new verification of its
accuracy. When Clairaut had announced his discovery, Euler arrived by a
fresh calculation at substantially the same result, while D’Alembert by
carrying the approximation further obtained one that was slightly more
accurate. A fresh calculation of the motion of the moon by Clairaut won
the prize on the subject offered by the St. Petersburg Academy, and
was published in 1752, with the title _Théorie de la Lune_. Two years
later he published a set of lunar tables, and just before his death
(1765) he brought out a revised edition of the _Théorie de la Lune_ in
which he embodied a new set of tables.

D’Alembert followed his paper of 1747 by a complete lunar theory
(with a moderately good set of tables), which, though substantially
finished in 1751, was only published in 1754 as the first volume of his
_Recherches sur différens points importans du système du Monde_. In
1756 he published an improved set of tables, and a few months afterward
a third volume of _Recherches_ with some fresh developments of the
theory. The second volume of his _Opuscules Mathématiques_ (1762)
contained another memoir on the subject with a third set of tables,
which were a slight improvement on the earlier ones.

Euler’s first lunar theory (_Theoria Motuum Lunae_) was published in
1753, though it had been sent to the St. Petersburg Academy a year or
two earlier. In an appendix[137] he points out with characteristic
frankness the defects from which his treatment seems to him to suffer,
and suggests a new method of dealing with the subject. It was on this
theory that Tobias Mayer based his tables, referred to in the preceding
chapter (§ 226). Many years later Euler devised an entirely new way
of attacking the subject, and after some preliminary papers dealing
generally with the method and with special parts of the problem, he
worked out the lunar theory in great detail, with the help of one of
his sons and two other assistants, and published the whole, together
with tables, in 1772. He attempted, but without success, to deal in
this theory with the secular acceleration of the mean motion which
Halley had detected (chapter X., § 201).

In any mathematical treatment of an astronomical problem some data
have to be borrowed from observation, and of the three astronomers
Clairaut seems to have been the most skilful in utilising observations,
many of which he obtained from Lacaille. Hence his tables represented
the actual motions of the moon far more accurately than those of
D’Alembert, and were even superior in some points to those based
on Euler’s very much more elaborate second theory; Clairaut’s last
tables were seldom in error more than 1-1∕2′, and would hence serve
to determine the longitude to within about 3∕4°. Clairaut’s tables
were, however, never much used, since Tobias Mayer’s as improved by
Bradley were found in practice to be a good deal more accurate; but
Mayer borrowed so extensively from observation that his formulae cannot
be regarded as true deductions from gravitation in the same sense in
which Clairaut’s were. Mathematically Euler’s second theory is the most
interesting and was of the greatest importance as a basis for later
developments. The most modern lunar theory[138] is in some sense a
return to Euler’s methods.

234. Newton’s lunar theory may be said to have given a _qualitative_
account of the lunar inequalities known by observation at the time when
the _Principia_ was published, and to have indicated others which had
not yet been observed. But his attempts to explain these irregularities
_quantitatively_ were only partially successful.

Euler, Clairaut, and D’Alembert threw the lunar theory into an
entirely new form by using analytical methods instead of geometrical;
one advantage of this was that by the expenditure of the necessary
labour calculations could in general be carried further when required
and lead to a higher degree of accuracy. The result of their more
elaborate development was that—with one exception—the inequalities
known from observation were explained with a considerable degree of
accuracy quantitatively as well as qualitatively; and thus tables, such
as those of Clairaut, based on theory, represented the lunar motions
very closely. The one exception was the secular acceleration: we have
just seen that Euler failed to explain it; D’Alembert was equally
unsuccessful, and Clairaut does not appear to have considered the
question.

235. The chief inequalities in planetary motion which observation had
revealed up to Newton’s time were the forward motion of the apses of
the earth’s orbit and a very slow diminution in the obliquity of the
ecliptic. To these may be added the alterations in the rates of motion
of Jupiter and Saturn discovered by Halley (chapter X., § 204).

Newton had shewn generally that the perturbing effect of another planet
would cause displacements in the apses of any planetary orbit, and
an alteration in the relative positions of the planes in which the
disturbing and disturbed planet moved; but he had made no detailed
calculations. Some effects of this general nature, in addition to those
already known, were, however, indicated with more or less distinctness
as the result of observation in various planetary tables published
between the date of the _Principia_ and the middle of the 18th century.

The irregularities in the motion of the earth, shewing themselves as
irregularities in the apparent motion of the sun, and those of Jupiter
and Saturn, were the most interesting and important of the planetary
inequalities, and prizes for essays on one or another subject were
offered several times by the Paris Academy.

The perturbations of the moon necessarily involved—by the principle of
action and reaction—corresponding though smaller perturbations of the
earth; these were discussed on various occasions by Clairaut and Euler,
and still more fully by D’Alembert.

In Clairaut’s paper of 1747 (§ 233) he made some attempt to apply his
solution of the problem of three bodies to the case of the sun, earth,
and Saturn, which on account of Saturn’s great distance from the sun
(nearly ten times that of the earth) is the planetary case most like
that of the earth, moon, and sun (cf. § 228).

Ten years later he discussed in some detail the perturbations of the
earth due to Venus and to the moon. This paper was remarkable as
containing the first attempt to estimate masses of celestial bodies
by observation of perturbations due to them. Clairaut applied this
method to the moon and to Venus, by calculating perturbations in the
earth’s motion due to their action (which necessarily depended on their
masses), and then comparing the results with Lacaille’s observations
of the sun. The mass of the moon was thus found to be about 1∕67
and that of Venus 2∕3 that of the earth; the first result was a
considerable improvement on Newton’s estimate from tides (chapter IX.,
§ 189), and the second, which was entirely new, previous estimates
having been merely conjectural, is in tolerable agreement with modern
measurements.[139] It is worth noticing as a good illustration of the
reciprocal influence of observation and mathematical theory that, while
Clairaut used Lacaille’s observations for his theory, Lacaille in turn
used Clairaut’s calculations of the perturbations of the earth to
improve his tables of the sun published in 1758.

Clairaut’s method of solving the problem of three bodies was also
applied by _Joseph Jérôme Le François Lalande_ (1732-1807), who is
chiefly known as an admirable populariser of astronomy but was also an
indefatigable calculator and observer, to the perturbations of Mars by
Jupiter, of Venus by the earth, and of the earth by Mars, but with only
moderate success.

D’Alembert made some progress with the general treatment of planetary
perturbations in the second volume of his _Recherches_, and applied his
methods to Jupiter and Saturn.

236. Euler carried the general theory a good deal further in a series
of papers beginning in 1747. He made several attempts to explain
the irregularities of Jupiter and Saturn, but never succeeded in
representing the observations satisfactorily. He shewed, however, that
the perturbations due to the other planets would cause the earth’s
apse line to advance about 13″ annually, and the obliquity of the
ecliptic to diminish by about 48″ annually, both results being in fair
accordance both with observations and with more elaborate calculations
made subsequently. He indicated also the existence of various other
planetary irregularities, which for the most part had not previously
been observed.

In an essay to which the Academy awarded a prize in 1756, but which was
first published in 1771, he developed with some completeness a method
of dealing with perturbations which he had indicated in his lunar
theory of 1753. As this method, known as that of the =variation of the
elements= or =parameters=, played a very important part in subsequent
researches, it may be worth while to attempt to give a sketch of it.

If perturbations are ignored, a planet can be regarded as moving in an
ellipse with the sun in one focus. The size and shape of the ellipse
can be defined by the length of its axis and by the eccentricity; the
plane in which the ellipse is situated is determined by the position of
the line, called the line of =nodes=, in which it cuts a fixed plane,
usually taken to be the ecliptic, and by the inclination of the two
planes. When these four quantities are fixed, the ellipse may still
turn about its focus in its own plane, but if the direction of the apse
line is also fixed the ellipse is completely determined. If, further,
the position of the planet in its ellipse at any one time is known, the
motion is completely determined and its position at any other time can
be calculated. There are thus six quantities known as =elements= which
completely determine the motion of a planet not subject to perturbation.

When perturbations are taken into account, the path described by
a planet in any one revolution is no longer an ellipse, though it
differs very slightly from one; while in the case of the moon the
deviations are a good deal greater. But if the motions of a planet at
two widely different epochs are compared, though on each occasion the
path described is very nearly an ellipse, the ellipses differ in some
respects. For example, between the time of Ptolemy (A.D. 150) and that
of Euler the direction of the apse line of the earth’s orbit altered by
about 5°, and some of the other elements also varied slightly. Hence
in dealing with the motion of a planet through a long period of time
it is convenient to introduce the idea of an elliptic path which is
gradually changing its position and possibly also its size and shape.
One consequence is that the actual path described in the course of a
considerable number of revolutions is a curve no longer bearing much
resemblance to an ellipse. If, for example, the apse line turns round
uniformly while the other elements remain unchanged, the path described
is like that shewn in the figure.

[Illustration: FIG. 81.—A varying ellipse.]

Euler extended this idea so as to represent any perturbation of a
planet, whether experienced in the course of one revolution or in a
longer time, by means of changes in an elliptic orbit. For wherever a
planet may be and whatever (within certain limits[140]) be its speed
or direction of motion some ellipse can be found, having the sun in
one focus, such that the planet can be regarded as moving in it for a
short time. Hence as the planet describes a perturbed orbit it can be
regarded as moving at any instant in an ellipse, which, however, is
continually altering its position or other characteristics. Thus the
problem of discussing the planet’s motion becomes that of determining
the elements of the ellipse which represents its motion at any time.
Euler shewed further how, when the position of the perturbing planet
was known, the corresponding rates of change of the elements of the
varying ellipse could be calculated, and made some progress towards
deducing from these data the actual elements; but he found the
mathematical difficulties too great to be overcome except in some of
the simpler cases, and it was reserved for the next generation of
mathematicians, notably Lagrange, to shew the full power of the method.

237. _Joseph Louis Lagrange_ was born at Turin in 1736, when Clairaut
was just starting for Lapland and D’Alembert was still a child; he was
descended from a French family three generations of which had lived in
Italy. He shewed extraordinary mathematical talent, and when still a
mere boy was appointed professor at the Artillery School of his native
town, his pupils being older than himself. A few years afterwards
he was the chief mover in the foundation of a scientific society,
afterwards the Turin Academy of Sciences, which published in 1759 its
first volume of Transactions, containing several mathematical articles
by Lagrange, which had been written during the last few years. One of
these[141] so impressed Euler, who had made a special study of the
subject dealt with, that he at once obtained for Lagrange the honour of
admission to the Berlin Academy.

In 1764 Lagrange won the prize offered by the Paris Academy for an
essay on the libration of the moon. In this essay he not only gave
the first satisfactory, though still incomplete, discussion of the
librations (chapter vi., § 133) of the moon due to the non-spherical
forms of both the earth and moon, but also introduced an extremely
general method of treating dynamical problems,[142] which is the basis
of nearly all the higher branches of dynamics which have been developed
up to the present day.

Two years later (1766) Frederick II., at the suggestion of D’Alembert,
asked Lagrange to succeed Euler (who had just returned to St.
Petersburg) as the head of the mathematical section of the Berlin
Academy, giving as a reason that the greatest king in Europe wished to
have the greatest mathematician in Europe at his court. Lagrange
accepted this magnificently expressed invitation and spent the next 21
years at Berlin.

[Illustration: LAGRANGE.]

During this period he produced an extraordinary series of papers on
astronomy, on general dynamics, and on a variety of subjects in pure
mathematics. Several of the most important of the astronomical papers
were sent to Paris and obtained prizes offered by the Academy; most of
the other papers—about 60 in all—were published by the Berlin Academy.
During this period he wrote also his great _Mécanique Analytique_, one
of the most beautiful of all mathematical books, in which he developed
fully the general dynamical ideas contained in the earlier paper on
libration. Curiously enough he had great difficulty in finding a
publisher for his masterpiece, and it only appeared in 1788 in Paris.
A year earlier he had left Berlin in consequence of the death of
Frederick, and accepted an invitation from Louis XVI. to join the Paris
Academy. About this time he suffered from one of the fits of melancholy
with which he was periodically seized and which are generally supposed
to have been due to overwork during his career at Turin. It is said
that he never looked at the _Mécanique Analytique_ for two years
after its publication, and spent most of the time over chemistry
and other branches of natural science as well as in non-scientific
pursuits. In 1790 he was made president of the Commission appointed to
draw up a new system of weights and measures, which resulted in the
establishment of the metric system; and the scientific work connected
with this undertaking gradually restored his interest in mathematics
and astronomy. He always avoided politics, and passed through the
Revolution uninjured, unlike his friend Lavoisier the great chemist
and Bailly the historian of astronomy, both of whom were guillotined
during the Terror. He was in fact held in great honour by the various
governments which ruled France up to the time of his death; in 1793 he
was specially exempted from a decree of banishment directed against
all foreigners; subsequently he was made professor of mathematics,
first at the École Normale (1795), the École Polytechnique (1797),
the last appointment being retained till his death in 1813. During
this period of his life he published, in addition to a large number
of papers on astronomy and mathematics, three important books on pure
mathematics,[143] and at the time of his death had not quite finished
a second edition of the _Mécanique Analytique_, the second volume
appearing posthumously.

238. _Pierre Simon Laplace_, the son of a small farmer, was born at
Beaumont in Normandy in 1749, being thus 13 years younger than his
great rival Lagrange. Thanks to the help of well-to-do neighbours,
he was first a pupil and afterwards a teacher at the Military School
of his native town. When he was 18 he went to Paris with a letter of
introduction to D’Alembert, and, when no notice was taken of it, wrote
him a letter on the principles of mechanics which impressed D’Alembert
so much that he at once took interest in the young mathematician and
procured him an appointment at the Military School at Paris. From this
time onwards Laplace lived continuously at Paris, holding various
official positions. His first paper (on pure mathematics) was published
in the Transactions of the Turin Academy for the years 1766-69, and
from this time to the end of his life he produced an uninterrupted
series of papers and books on astronomy and allied departments of
mathematics.

Laplace’s work on astronomy was to a great extent incorporated in his
_Mécanique Céleste_, the five volumes of which appeared at intervals
between 1799 and 1825. In this great treatise he aimed at summing up
all that had been done in developing gravitational astronomy since the
time of Newton. The only other astronomical book which he published
was the _Exposition du Système du Monde_ (1796), one of the most
perfect and charmingly written popular treatises on astronomy ever
published, in which the great mathematician never uses either an
algebraical formula or a geometrical diagram. He published also in 1812
an elaborate treatise on the theory of probability or chance,[144] on
which nearly all later developments of the subject have been based, and
in 1819 a more popular _Essai Philosophique_ on the same subject.

[Illustration: P. S. LAPLACE.]

Laplace’s personality seems to have been less attractive than that of
Lagrange. He was vain of his reputation as a mathematician and not
always generous to rival discoverers. To Lagrange, however, he was
always friendly, and he was also kind in helping young mathematicians
of promise. While he was perfectly honest and courageous in upholding
his scientific and philosophical opinions, his politics bore an
undoubted resemblance to those of the Vicar of Bray, and were professed
by him with great success. He was appointed a member of the Commission
for Weights and Measures, and afterwards of the Bureau des Longitudes,
and was made professor at the École Normale when it was founded. When
Napoleon became First Consul, Laplace asked for and obtained the
post of Home Secretary, but—fortunately for science—was considered
quite incompetent, and had to retire after six weeks (1799)[145]; as
a compensation he was made a member of the newly created Senate. The
third volume of the _Mécanique Céleste_, published in 1802, contained
a dedication to the “Heroic Pacificator of Europe,” at whose hand he
subsequently received various other distinctions, and by whom he was
created a Count when the Empire was formed. On the restoration of the
Bourbons in 1814 he tendered his services to them, and was subsequently
made a Marquis. In 1816 he also received a very unusual honour for a
mathematician (shared, however, by D’Alembert) by being elected one of
the Forty “Immortals” of the _Académie Française_; this distinction
he seems to have owed in great part to the literary excellence of the
_Système du Monde_.

Notwithstanding these distractions he worked steadily at mathematics
and astronomy, and even after the completion of the _Mécanique Céleste_
wrote a supplement to it which was published after his death (1827).

His last words, “_Ce que nous connaissons est peu de chose, ce que nous
ignorons est immense_,” coming as they did from one who had added so
much to knowledge, shew his character in a pleasanter aspect than it
sometimes presented during his career.

239. With the exception of Lagrange’s paper on libration, nearly all
his and Laplace’s important contributions to astronomy were made when
Clairaut’s and D’Alembert’s work was nearly finished, though Euler’s
activity continued for nearly 20 years more. Lagrange, however,
survived him by 30 years and Laplace by more than 40; and together they
carried astronomical science to a far higher stage of development than
their three predecessors.

240. To the lunar theory Lagrange contributed comparatively little
except general methods, applicable to this as to other problems of
astronomy; but Laplace devoted great attention to it. Of his special
discoveries in the subject the most notable was his explanation of
the secular acceleration of the moon’s mean motion (chapter X., §
201), which had puzzled so many astronomers. Lagrange had attempted to
explain it (1774), and had failed so completely that he was inclined
to discredit the early observations on which the existence of the
phenomenon was based. Laplace, after trying ordinary methods without
success, attempted to explain it by supposing that gravitation was
an effect not transmitted instantaneously, but that, like light, it
took time to travel from the attracting body to the attracted one; but
this also failed. Finally he traced it (1787) to an indirect planetary
effect. For, as it happens, certain perturbations which the moon
experiences owing to the action of the sun depend among other things
on the eccentricity of the earth’s orbit; this is one of the elements
(§ 236) which is being altered by the action of the planets, and has
for many centuries been very slowly decreasing; the perturbation in
question is therefore being very slightly altered, and the moon’s
average rate of motion is in consequence very slowly increasing, or
the length of the month decreasing. The whole effect is excessively
minute, and only becomes perceptible in the course of a long time.
Laplace’s calculation shewed that the moon would, in the course of a
century, or in about 1,300 complete revolutions, gain about 10″ (more
exactly 10″·2) owing to this cause, so that her place in the sky would
differ by that amount from what it would be if this disturbing cause
did not exist; in two centuries the angle gained would be 40″, in
three centuries 90″, and so on. This may be otherwise expressed by
saying that the length of the month diminishes by about one-thirtieth
of a second in the course of a century. Moreover, as Laplace shewed (§
245), the eccentricity of the earth’s orbit will not go on diminishing
indefinitely, but after an immense period to be reckoned in thousands
of years will begin to increase, and the moon’s motion will again
become slower in consequence.

Laplace’s result agreed almost exactly with that indicated by
observation; and thus the last known discrepancy of importance in the
solar system between theory and observation appeared to be explained
away; and by a curious coincidence this was effected just a hundred
years after the publication of the _Principia_.

Many years afterwards, however, Laplace’s explanation was shewn to be
far less complete than it appeared at the time (chapter XIII., § 287).

The same investigation revealed to Laplace the existence of alterations
of a similar character, and due to the same cause, of other elements in
the moon’s orbit, which, though not previously noticed, were found to
be indicated by ancient eclipse observations.

241. The third volume of the _Mécanique Céleste_ contains a general
treatment of the lunar theory, based on a method entirely different
from any that had been employed before, and worked out in great
detail. “My object,” says Laplace, “in this book is to exhibit in the
one law of universal gravitation the source of all the inequalities
of the motion of the moon, and then to employ this law as a means of
discovery, to perfect the theory of this motion and to deduce from it
several important elements in the system of the moon.” Laplace himself
calculated no lunar tables, but the Viennese astronomer _John Tobias
Bürg_ (1766-1834) made considerable use of his formulae, together with
an immense number of Greenwich observations, for the construction
of lunar tables, which were sent to the Institute of France in 1801
(before the publication of Laplace’s complete lunar theory), and
published in a slightly amended form in 1806. A few years later (1812)
_John Charles Burckhardt_ (1773-1825), a German who had settled in
Paris and worked under Laplace and Lalande, produced a new set of
tables based directly on the formulae of the _Mécanique Céleste_.
These were generally accepted in lieu of Bürg’s, which had been in
their turn an improvement on Mason’s and Mayer’s.

Later work on lunar theory may conveniently be regarded as belonging to
a new period of astronomy (chapter XIII., § 286).

242. Observation had shewn the existence of inequalities in the
planetary and lunar motions which seemed to belong to two different
classes. On the one hand were inequalities, such as most of those
of the moon, which went through their cycle of changes in a single
revolution or a few revolutions of the disturbing body; and on the
other such inequalities as the secular acceleration of the moon’s
mean motion or the motion of the earth’s apses, in which a continuous
disturbance was observed always acting in the same direction, and
shewing no signs of going through a periodic cycle of changes.

The mathematical treatment of perturbations soon shewed the
desirability of adopting different methods of treatment for two classes
of inequalities, which corresponded roughly, though not exactly,
to those just mentioned, and to which the names of =periodic= and
=secular= gradually came to be attached. The distinction plays a
considerable part in Euler’s work (§ 236), but it was Lagrange who
first recognised its full importance, particularly for planetary
theory, and who made a special study of secular inequalities.

When the perturbations of one planet by another are being studied,
it becomes necessary to obtain a mathematical expression for the
disturbing force which the second planet exerts. This expression
depends in general both on the elements of the two orbits, and on the
positions of the planets at the time considered. It can, however, be
divided up into two parts, one of which depends on the positions of the
planets (as well as on the elements), while the other depends only on
the elements of the two orbits, and is independent of the positions in
their paths which the planets may happen to be occupying at the time.
Since the positions of planets in their orbits change rapidly, the
former part of the disturbing force changes rapidly, and produces in
general, at short intervals of time, effects in opposite directions,
first, for example, accelerating and then retarding the motion of the
disturbed planet; and the corresponding inequalities of motion are the
periodic inequalities, which for the most part go through a complete
cycle of changes in the course of a few revolutions of the planets,
or even more rapidly. The other part of the disturbing force remains
nearly unchanged for a considerable period, and gives rise to changes
in the elements which, though in general very small, remain for a long
time without sensible alteration, and therefore continually accumulate,
becoming considerable with the lapse of time: these are the secular
inequalities.

Speaking generally, we may say that the periodical inequalities are
temporary and the secular inequalities permanent in their effects, or
as Sir John Herschel expresses it:—

 “The secular inequalities are, in fact, nothing but what remains after
 the mutual destruction of a much larger amount (as it very often is)
 of periodical. But these are in their nature transient and temporary;
 they disappear in short periods, and leave no trace. The planet is
 temporarily withdrawn from its orbit (its slowly varying orbit), but
 forthwith returns to it, to deviate presently as much the other way,
 while the varied orbit accommodates and adjusts itself to the average
 of these excursions on either side of it.”[146]

“Temporary” and “short” are, however, relative terms. Some periodical
inequalities, notably in the case of the moon, have periods of only a
few days, and the majority which are of importance extend only over
a few years; but some are known which last for centuries or even
thousands of years, and can often be treated as secular when we only
want to consider an interval of a few years. On the other hand, most of
the known secular inequalities are not really permanent, but fluctuate
like the periodical ones, though only in the course of immense periods
of time to be reckoned usually by tens of thousands of years.

One distinction between the lunar and planetary theories is that in the
former periodic inequalities are comparatively large and, especially
for practical purposes such as computing the position of the moon a few
months hence, of great importance; whereas the periodic inequalities
of the planets are generally small and the secular inequalities are the
most interesting.

The method of treating the elements of the elliptic orbits as variable
is specially suitable for secular inequalities; but for periodic
inequalities it is generally better to treat the body as being
disturbed from an elliptic path, and to study these deviations.

 “The simplest way of regarding these various perturbations consists
 in imagining a planet moving in accordance with the laws of elliptic
 motion, on an ellipse the elements of which vary by insensible
 degrees; and to conceive at the same time that the true planet
 oscillates round this fictitious planet in a very small orbit the
 nature of which depends on its periodic perturbations.”[147]

The former method, due as we have seen in great measure to Euler, was
perfected and very generally used by Lagrange, and often bears his name.

243. It was at first naturally supposed that the slow alteration in
the rates of the motions of Jupiter and Saturn (§§ 235, 236, and
chapter X., § 204) was a secular inequality; Lagrange in 1766 made an
attempt to explain it on this basis which, though still unsuccessful,
represented the observations better than Euler’s work. Laplace in his
first paper on secular inequalities (1773) found by the use of a more
complete analysis that the secular alterations in the rates of motions
of Jupiter and Saturn appeared to vanish entirely, and attempted to
explain the motions by the hypothesis, so often used by astronomers
when in difficulties, that a comet had been the cause.

In 1773 _John Henry Lambert_ (1728-1777) discovered from a study of
observations that, whereas Halley had found Saturn to be moving more
slowly than in ancient times, it was now moving faster than in Halley’s
time—a conclusion which pointed to a fluctuating or periodic cause of
some kind.

Finally in 1784 Laplace arrived at the true explanation. Lagrange had
observed in 1776 that if the times of revolution of two planets are
exactly proportional to two whole numbers, then part of the periodic
disturbing force produces a secular change in their motions, acting
continually in the same direction; though he pointed out that such
a case did not occur in the solar system. If moreover the times of
revolution are _nearly_ proportional to two whole numbers (neither
of which is very large), then part of the periodic disturbing force
produces an irregularity that is not strictly secular, but has a very
long period; and a disturbing force so small as to be capable of
being ordinarily overlooked may, if it is of this kind, be capable of
producing a considerable effect.[148] Now Jupiter and Saturn revolve
round the sun in about 4,333 days and 10,759 days respectively; five
times the former number is 21,665, twice the latter is 21,518, which
is very little less. Consequently the exceptional case occurs; and on
working it out Laplace found an appreciable inequality with a period of
about 900 years, which explained the observations satisfactorily.

The inequalities of this class, of which several others have been
discovered, are known as =long inequalities=, and may be regarded
as connecting links between secular inequalities and periodical
inequalities of the usual kind.

244. The discovery that the observed inequality of Jupiter and Saturn
was not secular may be regarded as the first step in a remarkable
series of investigations on secular inequalities carried out by
Lagrange and Laplace, for the most part between 1773 and 1784,
leading to some of the most interesting and general results in the
whole of gravitational astronomy. The two astronomers, though living
respectively in Berlin and Paris, were in constant communication, and
scarcely any important advance was made by the one which was not at
once utilised and developed by the other.

The central problem was that of the secular alterations in the
elements of a planet’s orbit regarded as a varying ellipse. Three of
these elements, the axis of the ellipse, its eccentricity, and the
inclination of its plane to a fixed plane (usually the ecliptic), are
of much greater importance than the other three. The first two are the
elements on which the size and shape of the orbit depend, and the first
also determines (by Kepler’s Third Law) the period of revolution and
average rate of motion of the planet;[149] the third has an important
influence on the mutual relations of the two planets. The other three
elements are chiefly of importance for periodical inequalities.

It should be noted moreover that the eccentricities and inclinations
were in all cases (except those specially mentioned) considered as
small quantities; and thus all the investigations were approximate,
these quantities and the disturbing forces themselves being treated as
small.

245. The basis of the whole series of investigations was a long paper
published by Lagrange in 1766, in which he explained the method of
variation of elements, and gave formulae connecting their rates of
change with the disturbing forces.

In his paper of 1773 Laplace found that what was true of Jupiter and
Saturn had a more general application, and proved that in the case of
any planet, disturbed by any other, the axis was not only undergoing
no secular change at the present time, but could not have altered
appreciably since “the time when astronomy began to be cultivated.”

In the next year Lagrange obtained an expression for the secular change
in the inclination, _valid for all time_. When this was applied to
the case of Jupiter and Saturn, which on account of their superiority
in size and great distance from the other planets could be reasonably
treated as forming with the sun a separate system, it appeared that
the changes in the inclinations would always be of a periodic nature,
so that they could never pass beyond certain fixed limits, not
differing much from the existing values. The like result held for the
system formed by the sun, Venus, the earth, and Mars. Lagrange noticed
moreover that there were cases, which, as he said, fortunately did not
appear to exist in the system of the world, in which, on the contrary,
the inclinations might increase indefinitely. The distinction depended
on the masses of the bodies in question; and although all the planetary
masses were somewhat uncertain, and those assumed by Lagrange for
Venus and Mars almost wholly conjectural, it did not appear that any
reasonable alteration in the estimated masses would affect the general
conclusion arrived at.

Two years later (1775) Laplace, much struck by the method which
Lagrange had used, applied it to the discussion of the secular
variations of the eccentricity, and found that these were also of a
periodic nature, so that the eccentricity also could not increase or
decrease indefinitely.

In the next year Lagrange, in a remarkable paper of only 14 pages,
proved that whether the eccentricities and inclinations were treated
as small or not, and whatever the masses of the planets might be,
the changes in the length of the axis of any planetary orbit were
necessarily all periodic, so that for all time the length of the axis
could only fluctuate between certain definite limits. This result was,
however, still based on the assumption that the disturbing forces could
be treated as small.

Next came a series of five papers published between 1781 and 1784 in
which Lagrange summed up his earlier work, revised and improved his
methods, and applied them to periodical inequalities and to various
other problems.

Lastly in 1784 Laplace, in the same paper in which he explained the
long inequality of Jupiter and Saturn, established by an extremely
simple method two remarkable relations between the eccentricities and
inclinations of the planets, or any similar set of bodies.

The first relation is:—

_If the mass of each planet be multiplied by the square root of the
axis of its orbit and by the square of the eccentricity, then the sum
of these products for all the planets is invariable save for periodical
inequalities._

The second is precisely similar, save that eccentricity is replaced by
inclination.[150]

The first of these propositions establishes the existence of what
may be called a stock or fund of eccentricity shared by the planets
of the solar system. If the eccentricity of any one orbit increases,
that of some other orbit must undergo a corresponding decrease. Also
the fund can never be overdrawn. Moreover observation shews that the
eccentricities of all the planetary orbits are small; consequently the
whole fund is small, and the share owned at any time by any one planet
must be small.[151] Consequently the eccentricity of the orbit of a
planet of which the mass and distance from the sun are considerable
can never increase much, and a similar conclusion holds for the
inclinations of the various orbits.

One remarkable characteristic of the solar system is presupposed in
these two propositions; namely, that all the planets revolve round the
sun in the same direction, which to an observer supposed to be on the
north side of the orbits appears to be contrary to that in which the
hands of a clock move. If any planet moved in the opposite direction,
the corresponding parts of the eccentricity and inclination funds
would have to be subtracted instead of being added; and there would be
nothing to prevent the fund from being overdrawn.

A somewhat similar restriction is involved in Laplace’s earlier results
as to the impossibility of permanent changes in the eccentricities,
though a system might exist in which his result would still be true if
one or more of its members revolved in a different direction from the
rest, but in this case there would have to be certain restrictions on
the proportions of the orbits not required in the other case.

Stated briefly, the results established by the two astronomers were
that the changes in axis, eccentricity, and inclination of any
planetary orbit are all permanently restricted within certain definite
limits. The perturbations caused by the planets make all these
quantities undergo fluctuations of limited extent, some of which,
caused by the periodic disturbing forces, go through their changes
in comparatively short periods, while others, due to secular forces,
require vast intervals of time for their completion.

It may thus be said that the stability of the solar system was
established, as far as regards the particular astronomical causes taken
into account.

Moreover, if we take the case of the earth, as an inhabited planet,
any large alteration in the axis, that is in the average distance from
the sun, would produce a more than proportional change in the amount
of heat and light received from the sun; any great increase in the
eccentricity would increase largely that part (at present very small)
of our seasonal variations of heat and cold which are due to varying
distance from the sun; while any change in position of the ecliptic,
which was unaccompanied by a corresponding change of the equator, and
had the effect of increasing the angle between the two, would largely
increase the variations of temperature in the course of the year. The
stability shewn to exist is therefore a guarantee against certain
kinds of great climatic alterations which might seriously affect the
habitability of the earth.

It is perhaps just worth while to point out that the results
established by Lagrange and Laplace were mathematical consequences,
obtained by processes involving the neglect of certain small quantities
and therefore not perfectly rigorous, of certain definite hypotheses to
which the actual conditions of the solar system bear a tolerably close
resemblance. Apart from causes at present unforeseen, it is therefore
not unreasonable to expect that for a very considerable period of time
the motions of the actual bodies forming the solar system may be very
nearly in accordance with these results; but there is no valid reason
why certain disturbing causes, ignored or rejected by Laplace and
Lagrange on account of their insignificance, should not sooner or later
produce quite appreciable effects (cf. chapter XIII., § 293).

246. A few of Laplace’s numerical results as to the secular variations
of the elements may serve to give an idea of the magnitudes dealt with.

The line of apses of each planet moves in the same direction; the most
rapid motion, occurring in the case of Saturn, amounted to about 15″
per annum, or rather less than half a degree in a century. If this
motion were to continue uniformly, the line of apses would require no
less than 80,000 years to perform a complete circuit and return to its
original position. The motion of the line of nodes (or line in which
the plane of the planet’s orbit meets that of the ecliptic) was in
general found to be rather more rapid. The annual alteration in the
inclination of any orbit to the ecliptic in no case exceeded a fraction
of a second; while the change of eccentricity of Saturn’s orbit, which
was considerably the largest, would, if continued for four centuries,
have only amounted to 1∕1000.

247. The theory of the secular inequalities has been treated at some
length on account of the general nature of the results obtained.
For the purpose of predicting the places of the planets at moderate
distances of time the periodical inequalities are, however, of greater
importance. These were also discussed very fully both by Lagrange and
Laplace, the detailed working out in a form suitable for numerical
calculation being largely due to the latter. From the formulæ given
by Laplace and collected in the _Mécanique Céleste_ several sets of
solar and planetary tables were calculated, which were in general found
to represent closely the observed motions, and which superseded the
earlier tables based on less developed theories.[152]

248. In addition to the lunar and planetary theories nearly all the
minor problems of gravitational astronomy were rediscussed by Laplace,
in many cases with the aid of methods due to Lagrange, and their
solution was in all cases advanced.

The theory of Jupiter’s satellites, which with Jupiter form a sort of
miniature solar system but with several characteristic peculiarities,
was fully dealt with; the other satellites received a less complete
discussion. Some progress was also made with the theory of Saturn’s
ring by shewing that it could not be a uniform solid body.

Precession and nutation were treated much more completely than by
D’Alembert; and the allied problems of the irregularities in the
rotation of the moon and of Saturn’s ring were also dealt with.

The figure of the earth was considered in a much more general way than
by Clairaut, without, however, upsetting the substantial accuracy of
his conclusions; and the theory of the tides was entirely reconstructed
and greatly improved, though a considerable gap between theory and
observation still remained.

The theory of perturbations was also modified so as to be applicable
to comets, and from observation of a comet (known as Lexell’s) which
had appeared in 1770 and was found to have passed close to Jupiter in
1767 it was inferred that its orbit had been completely changed by the
attraction of Jupiter, but that, on the other hand, it was incapable
of exercising any appreciable disturbing influence on Jupiter or its
satellites.

As, on the one hand, the complete calculation of the perturbations of
the various bodies of the solar system presupposes a knowledge of their
masses, so reciprocally if the magnitudes of these disturbances can be
obtained from observation they can be used to determine or to correct
the values of the several masses. In this way the masses of Mars and
of Jupiter’s satellites, as well as of Venus (§ 235), were estimated,
and those of the moon and the other planets revised. In the case of
Mercury, however, no perturbation of any other planet by it could be
satisfactorily observed, and—except that it was known to be small—its
mass remained for a long time a matter of conjecture. It was only some
years after Laplace’s death that the effect produced by it on a comet
enabled its mass to be estimated (1842), and the mass is even now very
uncertain.

249. By the work of the great mathematical astronomers of the 18th
century, the results of which were summarised in the _Mécanique
Céleste_, it was shewn to be possible to account for the observed
motions of the bodies of the solar system with a tolerable degree of
accuracy by means of the law of gravitation.

Newton’s problem (§ 228) was therefore approximately solved, and the
agreement between theory and observation was in most cases close enough
for the practical purpose of predicting for a moderate time the places
of the various celestial bodies. The outstanding discrepancies between
theory and observation were for the most part so small as compared with
those that had already been removed as to leave an almost universal
conviction that they were capable of explanation as due to errors of
observation, to want of exactness in calculation, or to some similar
cause.

250. Outside the circle of professed astronomers and mathematicians
Laplace is best known, not as the author of the _Mécanique Céleste_,
but as the inventor of the =Nebular Hypothesis=.

This famous speculation was published (in 1796) in his popular book
the _Système du Monde_ already mentioned, and was almost certainly
independent of a somewhat similar but less detailed theory which had
been suggested by the philosopher _Immanuel Kant_ in 1755.

Laplace was struck with certain remarkable characteristics of the solar
system. The seven planets known to him when he wrote revolved round
the sun in the same direction, the fourteen satellites revolved round
their primaries still in the same direction,[153] and such motions of
rotation of sun, planets, and satellites about their axes as were known
followed the same law. There were thus some 30 or 40 motions all in the
same direction. If these motions of the several bodies were regarded
as the result of chance and were independent of one another, this
uniformity would be a coincidence of a most extraordinary character,
as unlikely as that a coin when tossed the like number of times should
invariably come down with the same face uppermost.

These motions of rotation and revolution were moreover all in planes
but slightly inclined to one another; and the eccentricities of all
the orbits were quite small, so that they were nearly circular.

Comets, on the other hand, presented none of these peculiarities; their
paths were very eccentric, they were inclined at all angles to the
ecliptic, and were described in either direction.

Moreover there were no known bodies forming a connecting link in these
respects between comets and planets or satellites.[154]

From these remarkable coincidences Laplace inferred that the various
bodies of the solar system must have had some common origin. The
hypothesis which he suggested was that they had condensed out of a body
that might be regarded either as the sun with a vast atmosphere filling
the space now occupied by the solar system, or as a fluid mass with a
more or less condensed central part or nucleus; while at an earlier
stage the central condensation might have been almost non-existent.

Observations of Herschel’s (chapter XII., §§ 259-61) had recently
revealed the existence of many hundreds of bodies known as nebulae,
presenting very nearly such appearances as might have been expected
from Laplace’s primitive body. The differences in structure which
they shewed, some being apparently almost structureless masses of
some extremely diffused substance, while others shewed decided signs
of central condensation, and others again looked like ordinary stars
with a slight atmosphere round them, were also strongly suggestive of
successive stages in some process of condensation.

Laplace’s suggestion then was that the solar system had been formed by
condensation out of a nebula; and a similar explanation would apply to
the fixed stars, with the planets (if any) which surrounded them.

He then sketched, in a somewhat imaginative way, the process whereby a
nebula, if once endowed with a rotatory motion, might, as it condensed,
throw off a series of rings, and each of these might in turn condense
into a planet with or without satellites; and gave on this hypothesis
plausible reasons for many of the peculiarities of the solar system.

So little is, however, known of the behaviour of a body like Laplace’s
nebula when condensing and rotating that it is hardly worth while to
consider the details of the scheme.

That Laplace himself, who has never been accused of underrating the
importance of his own discoveries, did not take the details of his
hypothesis nearly as seriously as many of its expounders, may be
inferred both from the fact that he only published it in a popular
book, and from his remarkable description of it as “these conjectures
on the formation of the stars and of the solar system, conjectures
which I present with all the distrust (_défiance_) which everything
which is not a result of observation or of calculation ought to
inspire.”[155]



CHAPTER XII.

HERSCHEL.

                     “Coelorum perrupit claustra.”
                                                  HERSCHEL’S _Epitaph_.


251. _Frederick William Herschel_ was born at Hanover on November 15th,
1738, two years after Lagrange and nine years before Laplace. His
father was a musician in the Hanoverian army, and the son, who shewed a
remarkable aptitude for music as well as a decided taste for knowledge
of various sorts, entered his father’s profession as a boy (1753).
On the breaking out of the Seven Years’ War he served during part of
a campaign, but his health being delicate his parents “determined to
remove him from the service—a step attended by no small difficulties,”
and he was accordingly sent to England (1757), to seek his fortune as a
musician.

After some years spent in various parts of the country, he moved (1766)
to Bath, then one of the great centres of fashion in England. At first
oboist in Linley’s orchestra, then organist of the Octagon Chapel,
he rapidly rose to a position of great popularity and distinction,
both as a musician and as a music-teacher. He played, conducted, and
composed, and his private pupils increased so rapidly that the number
of lessons which he gave was at one time 35 a week. But this activity
by no means exhausted his extraordinary energy; he had never lost his
taste for study, and, according to a contemporary biographer, “after a
fatiguing day of 14 or 16 hours spent in his vocation, he would retire
at night with the greatest avidity to _unbend the mind_, if it may be
so called, with a few propositions in Maclaurin’s Fluxions, or other
books of that sort.” His musical studies had long ago given him an
interest in mathematics, and it seems likely that the study of Robert
Smith’s _Harmonics_ led him to the _Compleat System of Optics_ of the
same author, and so to an interest in the construction and use of
telescopes. The astronomy that he read soon gave him a desire to see
for himself what the books described; first he hired a small reflecting
telescope, then thought of buying a larger instrument, but found that
the price was prohibitive. Thus he was gradually led to attempt the
construction of his own telescopes (1773). His brother Alexander, for
whom he had found musical work at Bath, and who seems to have had
considerable mechanical talent but none of William’s perseverance,
helped him in this undertaking, while his devoted sister _Caroline_
(1750-1848), who had been brought over to England by William in 1772,
not only kept house, but rendered a multitude of minor services. The
operation of grinding and polishing the mirror for a telescope was one
of the greatest delicacy, and at a certain stage required continuous
labour for several hours. On one occasion Herschel’s hand never left
the polishing tool for 16 hours, so that “by way of keeping him alive”
Caroline was “obliged to feed him by putting the victuals by bits into
his mouth,” and in less extreme cases she helped to make the operation
less tedious by reading aloud: it is with some feeling of relief that
we hear that on these occasions the books read were not on mathematics,
optics, or astronomy, but were such as _Don Quixote_, the _Arabian
Nights_, and the novels of Sterne and Fielding.

252. After an immense number of failures Herschel succeeded in
constructing a tolerable reflecting telescope—soon to be followed by
others of greater size and perfection—and with this he made his first
recorded observation, of the Orion nebula, in March 1774.

This observation, made when he was in his 36th year, may be
conveniently regarded as the beginning of his astronomical career,
though for several years more music remained his profession, and
astronomy could only be cultivated in such leisure time as he could
find or make for himself; his biographers give vivid pictures of his
extraordinary activity during this period, and of his zeal in using
odd fragments of time, such as intervals between the acts at a theatre,
for his beloved telescopes.

A letter written by him in 1783 gives a good account of the spirit in
which he was at this time carrying out his astronomical work:—

 “I determined to accept nothing on faith, but to see with my own eyes
 what others had seen before me.... I finally succeeded in completing a
 so-called Newtonian instrument, 7 feet in length. From this I advanced
 to one of 10 feet, and at last to one of 20, for I had fully made up
 my mind to carry on the improvement of my telescopes as far as it
 could possibly be done. When I had carefully and thoroughly perfected
 the great instrument in all its parts, I made systematic use of it in
 my observations of the heavens, first forming a determination never to
 pass by any, the smallest, portion of them without due investigation.”

In accordance with this last resolution he executed on four separate
occasions, beginning in 1775, each time with an instrument of greater
power than on the preceding, a review of the whole heavens, in which
everything that appeared in any way remarkable was noticed and if
necessary more carefully studied. He was thus applying to astronomy
methods comparable with those of the naturalist who aims at drawing up
a complete list of the flora or fauna of a country hitherto little known

253. In the course of the second of these reviews, made with a
telescope of the Newtonian type, 7 feet in length, he made the
discovery (March 13th, 1781) which gave him a European reputation and
enabled him to abandon music as a profession and to devote the whole of
his energies to science.

 “In examining the small stars in the neighbourhood of η _Geminorum_ I
 perceived one that appeared visibly larger than the rest; being struck
 with its uncommon appearance I compared it to η _Geminorum_ and the
 small star in the quartile between _Auriga_ and _Gemini_ and finding
 it so much larger than either of them, I suspected it to be a comet.”

If Herschel’s suspicion had been correct the discovery would have
been of far less interest than it actually was, for when the new body
was further observed and attempts were made to calculate its path,
it was found that no ordinary cometary orbit would in any way fit
its motion, and within three or four months of its discovery it was
recognised—first by _Anders Johann Lexell_ (1740-1784)—as being no
comet but a new planet, revolving round the sun in a nearly circular
path, at a distance about 19 times that of the earth and nearly double
that of Saturn.

No new planet had been discovered in historic times, and Herschel’s
achievement was therefore absolutely unique; even the discovery of
satellites inaugurated by Galilei (chapter VI., § 121) had come to a
stop nearly a century before (1684), when Cassini had detected his
second pair of satellites of Saturn (chapter VIII., § 160). Herschel
wished to exercise the discoverer’s right of christening by calling
the new planet after his royal patron _Georgium Sidus_, but though the
name was used for some time in England, Continental astronomers never
accepted it, and after an unsuccessful attempt to call the new body
_Herschel_, it was generally agreed to give a name similar to those of
the other planets, and _Uranus_ was proposed and accepted.

Although by this time Herschel had published two or three scientific
papers and was probably known to a slight extent in English scientific
circles, the complete obscurity among Continental astronomers of the
author of this memorable discovery is curiously illustrated by a
discussion in the leading astronomical journal (Bode’s _Astronomisches
Jahrbuch_) as to the way to spell his name, _Hertschel_ being perhaps
the best and _Mersthel_ the worst of several attempts.

254. This obscurity was naturally dissipated by the discovery of
Uranus. Distinguished visitors to Bath, among them the Astronomer
Royal Maskelyne (chapter X., § 219), sought his acquaintance; before
the end of the year he was elected a Fellow of the Royal Society, in
addition to receiving one of its medals, and in the following spring
he was summoned to Court to exhibit himself, his telescopes, and his
stars to George III. and to various members of the royal family. As the
outcome of this visit he received from the King an appointment as royal
astronomer, with a salary of £200 a year.

[Illustration: WILLIAM HERSCHEL.]

With this appointment his career as a musician came to an end, and in
August 1782 the brother and sister left Bath for good, and settled
first in a dilapidated house at Datchet, then, after a few months
(1785-6) spent at Clay Hall in Old Windsor, at Slough in a house now
known as Observatory House and memorable in Arago’s words as “le lieu
du monde où il a été fait le plus de découvertes.”

255. Herschel’s modest salary, though it would have sufficed for his
own and his sister’s personal wants, was of course insufficient to
meet the various expenses involved in making and mounting telescopes.
The skill which he had now acquired in the art was, however, such that
his telescopes were far superior to any others which were available,
and as his methods were his own, there was a considerable demand for
instruments made by him. Even while at Bath he had made and sold a
number, and for years after moving to the neighbourhood of Windsor he
derived a considerable income from this source, the royal family and a
number of distinguished British and foreign astronomers being among his
customers.

The necessity for employing his valuable time in this way fortunately
came to an end in 1788, when he married a lady with a considerable
fortune; Caroline lived henceforward in lodgings close to her brother,
but worked for him with unabated zeal.

By the end of 1783 Herschel had finished a telescope 20 feet in length
with a great mirror 18 inches in diameter, and with this instrument
most of his best work was done; but he was not yet satisfied that he
had reached the limit of what was possible. During the last winter
at Bath he and his brother had spent a great deal of labour in an
unsuccessful attempt to construct a 30-foot telescope; the discovery
of Uranus and its consequences prevented the renewal of the attempt
for some time, but in 1785 he began a 40-foot telescope with a mirror
four feet in diameter, the expenses of which were defrayed by a special
grant from the King. While it was being made Herschel tried a new
form of construction of reflecting telescopes, suggested by Lemaire
in 1732 but never used, by which a considerable gain of brilliancy
was effected, but at the cost of some loss of distinctness. This
=Herschelian= or =front-view= construction, as it is called, was
first tried with the 20-foot, and led to the discovery (January 11th,
1787) of two satellites of Uranus, _Oberon_ and _Titania_; it was
henceforward regularly employed. After several mishaps the 40-foot
telescope (fig. 82) was successfully constructed. On the first evening
on which it was employed (August 28th, 1789) a sixth satellite of
Saturn (_Enceladus_) was detected, and on September 17th a much fainter
seventh satellite (_Mimas_). Both satellites were found to be nearer to
the planet than any of the five hitherto discovered, Mimas being the
nearer of the two (cf. fig. 91).

Although for the detection of extremely faint objects such as these
satellites the great telescope was unequalled, for many kinds of work
and for all but the very clearest evenings a smaller instrument was as
good, and being less unwieldy was much more used. The mirror of the
great telescope deteriorated to some extent, and after 1811, Herschel’s
hand being then no longer equal to the delicate task of repolishing it,
the telescope ceased to be used though it was left standing till 1839,
when it was dismounted and closed up.

256. From the time of his establishment at Slough till he began to
lose his powers through old age the story of Herschel’s life is little
but a record of the work he did. It was his practice to employ in
observing the whole of every suitable night; his daylight hours were
devoted to interpreting his observations and to writing the papers in
which he embodied his results. His sister was nearly always present
as his assistant when he was observing, and also did a good deal of
cataloguing, indexing, and similar work for him. After leaving Bath
she also did some observing on her own account, though only when
her brother was away or for some other reason did not require her
services; she specialised on comets, and succeeded from first to last
in discovering no less than eight. To form any adequate idea of the
discomfort and even danger attending the nights spent in observing, it
is necessary to realise that the great telescopes used were erected
in the open air, that for both the Newtonian and Herschelian forms of
reflectors the observer has to be near the upper end of the telescope,
and therefore at a considerable height above the ground. In the
40-foot, for example, ladders 50 feet in length were used to reach
the platform on which the observer was stationed. Moreover from the
nature of the case satisfactory observations could not be taken in the
presence either of the moon or of artificial light. It is not therefore
surprising that Caroline Herschel’s journals contain a good many
expressions of anxiety for her brother’s welfare on these occasions,
and it is perhaps rather a matter of wonder that so few serious
accidents occurred.

[Illustration: FIG. 82.—Herschel’s forty-foot telescope.]

In addition to doing his real work Herschel had to receive a large
number of visitors who came to Slough out of curiosity or genuine
scientific interest to see the great man and his wonderful telescopes.
In 1801 he went to Paris, where he made Laplace’s acquaintance and also
saw Napoleon, whose astronomical knowledge he rated much below that of
George III., while “his general air was something like affecting to
know more than he did know.”

In the spring of 1807 he had a serious illness; and from that time
onwards his health remained delicate, and a larger proportion of his
time was in consequence given to indoor work. The last of the great
series of papers presented to the Royal Society appeared in 1818, when
he was almost 80, though three years later he communicated a list of
double stars to the newly founded Royal Astronomical Society. His last
observation was taken almost at the same time, and he died rather more
than a year afterwards (August 21st, 1822), when he was nearly 84.

He left one son, John, who became an astronomer only less distinguished
than his father (chapter XIII., §§ 306-8). Caroline Herschel after her
beloved brother’s death returned to Hanover, chiefly to be near other
members of her family; here she executed one important piece of work by
cataloguing in a convenient form her brother’s lists of nebulae, and
for the remaining 26 years of her long life her chief interest seems to
have been in the prosperous astronomical career of her nephew John.

257. The incidental references to Herschel’s work that have been made
in describing his career have shewn him chiefly as the constructor of
giant telescopes far surpassing in power any that had hitherto been
used, and as the diligent and careful observer of whatever could be
seen with them in the skies. Sun and moon, planets and fixed stars,
were all passed in review, and their peculiarities noted and described.
But this merely descriptive work was in Herschel’s eyes for the most
part means to an end, for, as he said in 1811, “a knowledge of the
construction of the heavens has always been the ultimate object of my
observations.”

Astronomy had for many centuries been concerned almost wholly with
the positions of the various heavenly bodies on the celestial sphere,
that is with their directions. Coppernicus and his successors had
found that the apparent motions on the celestial sphere of the members
of the solar system could only be satisfactorily explained by taking
into account their actual motions in space, so that the solar system
came to be effectively regarded as consisting of bodies at different
distances from the earth and separated from one another by so many
miles. But with the fixed stars the case was quite different: for, with
the unimportant exception of the proper motions of a few stars (chapter
X., § 203), all their known apparent motions were explicable as the
result of the motion of the earth; and the relative or actual distances
of the stars scarcely entered into consideration. Although the belief
in a real celestial sphere to which the stars were attached scarcely
survived the onslaughts of Tycho Brahe and Galilei, and any astronomer
of note in the latter part of the 17th or in the 18th century would,
if asked, have unhesitatingly declared the stars to be at different
distances from the earth, this was in effect a mere pious opinion which
had no appreciable effect on astronomical work.

The geometrical conception of the stars as represented by points on
a celestial sphere was in fact sufficient for ordinary astronomical
purposes, and the attention of great observing astronomers such as
Flamsteed, Bradley, and Lacaille was directed almost entirely towards
ascertaining the positions of these points with the utmost accuracy or
towards observing the motions of the solar system. Moreover the group
of problems which Newton’s work suggested naturally concentrated the
attention of eighteenth-century astronomers on the solar system, though
even from this point of view the construction of star catalogues had
considerable value as providing reference points which could be used
for fixing the positions of the members of the solar system.

Almost the only exception to this general tendency consisted in the
attempts—hitherto unsuccessful—to find the parallaxes and hence
the distances of some of the fixed stars, a problem which, though
originally suggested by the Coppernican controversy, had been
recognised as possessing great intrinsic interest.

Herschel therefore struck out an entirely new path when he began to
study the sidereal system _per se_ and the mutual relations of its
members. From this point of view the sun, with its attendant planets,
became one of an innumerable host of stars, which happened to have
received a fictitious importance from the accident that we inhabited
one member of its system.

258. A complete knowledge of the positions in space of the stars would
of course follow from the measurement of the parallax (chapter VI., §
129 and chapter X., § 207) of each. The failure of such astronomers
as Bradley to get the parallax of any one star was enough to shew
the hopelessness of this general undertaking, and, although Herschel
did make an attack on the parallax problem (§ 263), he saw that the
question of stellar distribution in space, if to be answered at all,
required some simpler if less reliable method capable of application on
a large scale.

Accordingly he devised (1784) his method of =star-gauging=. The most
superficial view of the sky shews that the stars visible to the naked
eye are very unequally distributed on the celestial sphere; the same
is true when the fainter stars visible in a telescope are taken into
account. If two portions of the sky of the same apparent or angular
magnitude are compared, it may be found that the first contains many
times as many stars as the second. If we realise that the stars are
not actually on a sphere but are scattered through space at different
distances from us, we can explain this inequality of distribution
on the _sky_ as due to either a real inequality of distribution in
_space_, or to a difference in the distance to which the sidereal
system extends in the directions in which the two sets of stars lie.
The first region on the sky may correspond to a region of space in
which the stars are really clustered together, or may represent a
direction in which the sidereal system extends to a greater distance,
so that the accumulation of layer after layer of stars lying behind one
another produces the apparent density of distribution. In the same way,
if we are standing in a wood and the wood appears less thick in one
direction than in another, it may be because the trees are really more
thinly planted there or because in that direction the edge of the wood
is nearer.

[Illustration: FIG. 83.—Section of the sidereal system. From Herschel’s
paper in the _Philosophical Transactions_.]

In the absence of any _a priori_ knowledge of the actual clustering of
the stars in space, Herschel chose the former of these two hypotheses;
that is, he treated the apparent density of the stars on any particular
part of the sky as a measure of the depth to which the sidereal systems
extended in that direction, and interpreted from this point of view the
results of a vast series of observations. He used a 20-foot telescope
so arranged that he could see with it a circular portion of the sky 15′
in diameter (one-quarter the area of the sun or full moon), turned the
telescope to different parts of the sky, and counted the stars visible
in each case. To avoid accidental irregularities he usually took the
average of several neighbouring fields, and published in 1785 the
results of gauges thus made in 683[156] regions, while he subsequently
added 400 others which he did not think it necessary to publish.
Whereas in some parts of the sky he could see on an average only one
star at a time, in others nearly 600 were visible, and he estimated
that on one occasion about 116,000 stars passed through the field of
view of his telescope in a quarter of an hour. The general result was,
as rough naked-eye observation suggests, that stars are most plentiful
in and near the Milky Way and least so in the parts of the sky most
remote from it. Now the Milky Way forms on the sky an ill-defined
band never deviating much from a great circle (sometimes called
the =galactic= circle); so that on Herschel’s hypothesis the space
occupied by the stars is shaped roughly like a disc or grindstone,
of which according to his figures the diameter is about five times
the thickness. Further, the Milky Way is during part of its length
divided into two branches, the space between the two branches being
comparatively free of stars. Corresponding to this subdivision there
has therefore to be assumed a cleft in the “grindstone.”

This “grindstone” theory of the universe had been suggested in 1750 by
_Thomas Wright_ (1711-1786) in his _Theory of the Universe_, and again
by Kant five years later; but neither had attempted, like Herschel, to
collect numerical data and to work out consistently and in detail the
consequences of the fundamental hypothesis.

That the assumption of uniform distribution of stars in space could
not be true in detail was evident to Herschel from the beginning. A
star cluster, for example, in which many thousands of faint stars
are collected together in a very small space on the sky, would have
to be interpreted as representing a long projection or spike full of
stars, extending far beyond the limits of the adjoining portions of
the sidereal system, and pointing directly away from the position
occupied by the solar system. In the same way certain regions in the
sky which are found to be bare of stars would have to be regarded as
tunnels through the stellar system. That even one or two such spikes or
tunnels should exist would be improbable enough, but as star clusters
were known in considerable numbers before Herschel began his work, and
were discovered by him in hundreds, it was impossible to explain their
existence on this hypothesis, and it became necessary to assume that
a star cluster occupied a region of space in which stars were really
closer together than elsewhere.

Moreover further study of the arrangement of the stars, particularly
of those in the Milky Way, led Herschel gradually to the belief that
his original assumption was a wider departure from the truth than he
had at first supposed; and in 1811, nearly 30 years after he had begun
star-gauging, he admitted a definite change of opinion:—

  “I must freely confess that by continuing my sweeps of the heavens
 my opinion of the arrangement of the stars ... has undergone a gradual
 change.... For instance, an equal scattering of the stars may be
 admitted in certain calculations; but when we examine the Milky Way,
 or the closely compressed clusters of stars of which my catalogues
 have recorded so many instances, this supposed equality of scattering
 must be given up.”

The method of star-gauging was intended primarily to give information
as to the limits of the sidereal system—or the visible portions of
it. Side by side with this method Herschel constantly made use of the
brightness of a star as a probable test of nearness. If two stars give
out actually the same amount of light, then that one which is nearer
to us will appear the brighter; and on the assumption that no light
is absorbed or stopped in its passage through space, the apparent
brightness of the two stars will be inversely as the _square_ of
their respective distances. Hence, if we receive nine times as much
light from one star as from another, and if it is assumed that this
difference is merely due to difference of distance, then the first star
is three times as far off as the second, and so on.

That the stars as a whole give out the same amount of light, so that
the difference in their apparent brightness is due to distance only,
is an assumption of the same general character as that of equal
distribution. There must necessarily be many exceptions, but, in
default of more exact knowledge, it affords a rough-and-ready method of
estimating with some degree of probability relative distances of stars.

To apply this method it was necessary to have some means of comparing
the amount of light received from different stars. This Herschel
effected by using telescopes of different sizes. If the same star is
observed with two reflecting telescopes of the same construction but of
different sizes, then the light transmitted by the telescope to the eye
is proportional to the area of the mirror which collects the light, and
hence to the square of the diameter of the mirror. Hence the apparent
brightness of a star as viewed through a telescope is proportional on
the one hand to the inverse square of the distance, and on the other
to the square of the diameter of the mirror of the telescope; hence
the distance of the star is, as it were, exactly counterbalanced by
the diameter of the mirror of the telescope. For example, if one star
viewed in a telescope with an eight-inch mirror and another viewed in
the great telescope with a four-foot mirror appear equally bright,
then the second star is—on the fundamental assumption—six times as far
off.

In the same way the size of the mirror necessary to make a star just
visible was used by Herschel as a measure of the distance of the
star, and it was in this sense that he constantly referred to the
“space-penetrating power” of his telescope. On this assumption he
estimated the faintest stars visible to the naked eye to be about
twelve times as remote as one of the brightest stars, such as Arcturus,
while Arcturus if removed to 900 times its present distance would just
be visible in the 20-foot telescope which he commonly used, and the
40-foot would penetrate about twice as far into space.

Towards the end of his life (1817) Herschel made an attempt to compare
statistically his two assumptions of uniform distribution in space and
of uniform actual brightness, by counting the number of stars of each
degree of apparent brightness and comparing them with the numbers that
would result from uniform distribution in space if apparent brightness
depended only on distance. The inquiry only extended as far as stars
visible to the naked eye and to the brighter of the telescopic stars,
and indicated the existence of an excess of the fainter stars of these
classes, so that either these stars are more closely packed in space
than the brighter ones, or they are in reality smaller or less luminous
than the others; but no definite conclusions as to the arrangement of
the stars were drawn.

259. Intimately connected with the structure of the sidereal system
was the question of the distribution and nature of =nebulae= (cf.
figs. 100, 102, facing pp. 397, 400) and =star clusters= (cf. fig.
104, facing p. 405). When Herschel began his work rather more than
100 such bodies were known, which had been discovered for the most
part by the French observers Lacaille (chapter X., § 223) and _Charles
Messier_ (1730-1817). Messier may be said to have been a comet-hunter
by profession; finding himself liable to mistake nebulae for comets,
he put on record (1781) the positions of 103 of the former. Herschel’s
discoveries—carried out much more systematically and with more powerful
instrumental appliances—were on a far larger scale. In 1786 he
presented to the Royal Society a catalogue of 1,000 new nebulae and
clusters, three years later a second catalogue of the same extent, and
in 1802 a third comprising 500. Each nebula was carefully observed, its
general appearance as well as its position being noted and described,
and to obtain a general idea of the distribution of nebulae on the sky
the positions were marked on a star map. The differences in brightness
and in apparent structure led to a division into eight classes; and at
quite an early stage of his work (1786) he gave a graphic account of
the extraordinary varieties in form which he had noted:—

 “I have seen double and treble nebulae, variously arranged; large
 ones with small, seeming attendants; narrow but much extended, lucid
 nebulae or bright dashes; some of the shape of a fan, resembling an
 electric brush, issuing from a lucid point; others of the cometic
 shape, with a seeming nucleus in the center; or like cloudy stars,
 surrounded with a nebulous atmosphere; a different sort again contain
 a nebulosity of the milky kind, like that wonderful inexplicable
 phenomenon about θ Orionis; while others shine with a fainter mottled
 kind of light, which denotes their being resolvable into stars.”

260. But much the most interesting problem in classification was that
of the relation between nebulae and star clusters. The Pleiades,
for example, appear to ordinary eyes as a group of six stars close
together, but many short-sighted people only see there a portion of
the sky which is a little brighter than the adjacent region; again,
the nebulous patch of light, as it appears to the ordinary eye, known
as _Praesepe_ (in the Crab), is resolved by the smallest telescope
into a cluster of faint stars. In the same way there are other objects
which in a small telescope appear cloudy or nebulous, but viewed in an
instrument of greater power are seen to be star clusters. In particular
Herschel found that many objects which to Messier were purely nebulous
appeared in his own great telescopes to be undoubted clusters, though
others still remained nebulous. Thus in his own words:—

 “Nebulae can be selected so that an insensible gradation shall
 take place from a coarse cluster like the Pleiades down to a milky
 nebulosity like that in Orion, every intermediate step being
 represented.”

These facts suggested obviously the inference that the difference
between nebulae and star clusters was merely a question of the power of
the telescope employed, and accordingly Herschel’s next sentence is:—

 “This tends to confirm the hypothesis that all are composed of stars
 more or less remote.”

The idea was not new, having at any rate been suggested, rather on
speculative than on scientific grounds, in 1755 by Kant, who had
further suggested that a single nebula or star cluster is an assemblage
of stars comparable in magnitude and structure with the whole of those
which constitute the Milky Way and the other separate stars which we
see. From this point of view the sun is one star in a cluster, and
every nebula which we see is a system of the same order. This “island
universe” theory of nebulae, as it has been called, was also at first
accepted by Herschel, so that he was able once to tell Miss Burney that
he had discovered 1,500 new universes.

Herschel, however, was one of those investigators who hold theories
lightly, and as early as 1791 further observation had convinced him
that these views were untenable, and that some nebulae at least were
essentially distinct from star clusters. The particular object which
he quotes in support of his change of view was a certain nebulous
star—that is, a body resembling an ordinary star but surrounded by a
circular halo gradually diminishing in brightness.

 “Cast your eye,” he says, “on this cloudy star, and the result will be
 no less decisive.... Your judgement, I may venture to say, will be,
 that the _nebulosity about the star is not of a starry nature_.”

If the nebulosity were due to an aggregate of stars so far off as to be
separately indistinguishable, then the central body would have to be a
star of almost incomparably greater dimensions than an ordinary star;
if, on the other hand, the central body were of dimensions comparable
with those of an ordinary star, the nebulosity must be due to something
other than a star cluster. In either case the object presented features
markedly different from those of a star cluster of the recognised kind;
and of the two alternative explanations Herschel chose the latter,
considering the nebulosity to be “a shining fluid, of a nature totally
unknown to us.” One exception to his earlier views being thus admitted,
others naturally followed by analogy, and henceforward he recognised
nebulae of the “shining fluid” class as essentially different from star
clusters, though it might be impossible in many cases to say to which
class a particular body belonged.

The evidence accumulated by Herschel as to the distribution of nebulae
also shewed that, whatever their nature, they could not be independent
of the general sidereal system, as on the “island universe” theory. In
the first place observation soon shewed him that an individual nebula
or cluster was usually surrounded by a region of the sky comparatively
free from stars; this was so commonly the case that it became his
habit while sweeping for nebulae, after such a bare region had passed
through the field of his telescope, to warn his sister to be ready
to take down observations of nebulae. Moreover, as the position of a
large number of nebulae came to be known and charted, it was seen that,
whereas clusters were common near the Milky Way, nebulae which appeared
incapable of resolution into clusters were scarce there, and shewed on
the contrary a decided tendency to be crowded together in the regions
of the sky most remote from the Milky Way—that is, round the poles of
the galactic circle (§ 258). If nebulae were external systems, there
would of course be no reason why their distribution on the sky should
shew any connection either with the scarcity of stars generally or with
the position of the Milky Way.

It is, however, rather remarkable that Herschel did not in this respect
fully appreciate the consequences of his own observations, and up to
the end of his life seems to have considered that some nebulae and
clusters were external “universes,” though many were part of our own
system.

261. As early as 1789 Herschel had thrown out the idea that the
different kinds of nebulae and clusters were objects of the same kind
at different stages of development, some “clustering power” being at
work converting a diffused nebula into a brighter and more condensed
body; so that condensation could be regarded as a sign of “age.” And he
goes on:—

 “This method of viewing the heavens seems to throw them into a new
 kind of light. They are now seen to resemble a luxuriant garden,
 which contains the greatest variety of productions, in different
 flourishing beds; and one advantage we may at least reap from it is,
 that we can, as it were, extend the range of our experience to an
 immense duration. For, to continue the simile I have borrowed from the
 vegetable kingdom, is it not almost the same thing, whether we live
 successively to witness the germination, blooming, foliage, fecundity,
 fading, withering and corruption of a plant, or whether a vast number
 of specimens, selected from every stage through which the plant passes
 in the course of its existence, be brought at once to our view?”

His change of opinion in 1791 as to the nature of nebulae led
to a corresponding modification of his views of this process of
condensation. Of the star already referred to (§ 260) he remarked
that its nebulous envelope “was more fit to produce a star by its
condensation than to depend upon the star for its existence.” In 1811
and 1814 he published a complete theory of a possible process whereby
the shining fluid constituting a diffused nebula might gradually
condense—the denser portions of it being centres of attraction—first
into a denser nebula or compressed star cluster, then into one or more
nebulous stars, lastly into a single star or group of stars. Every
supposed stage in this process was abundantly illustrated from the
records of actual nebulae and clusters which he had observed.

In the latter paper he also for the first time recognised that the
clusters in and near the Milky Way really belonged to it, and were not
independent systems that happened to lie in the same direction as seen
by us.

262. On another allied point Herschel also changed his mind towards
the end of his life. When he first used his great 20-foot telescope to
explore the Milky Way, he thought that he had succeeded in completely
resolving its faint cloudy light into component stars, and had thus
penetrated to the end of the Milky Way; but afterwards he was convinced
that this was not the case, but that there remained cloudy portions
which—whether on account of their remoteness or for other reasons—his
telescopes were unable to resolve into stars (cf. fig. 104, facing p.
405).

In both these respects therefore the structure of the Milky Way
appeared to him finally less simple than at first.

263. One of the most notable of Herschel’s discoveries was a
bye-product of an inquiry of an entirely different character. Just
as Bradley in trying to find the parallax of a star discovered
aberration and nutation (chapter X., § 207), so also the same problem
in Herschel’s hands led to the discovery of double stars. He proposed
to employ Galilei’s differential or double-star method (chapter VI.,
§ 129), in which the minute shift of a star’s position, due to the
earth’s motion round the sun, is to be detected not by measuring its
angular distance from standard points on the celestial sphere such as
the pole or the zenith, but by observing the variations in its distance
from some star close to it, which from its faintness or for some other
reason might be supposed much further off and therefore less affected
by the earth’s motion.

With this object in view Herschel set to work to find pairs of stars
close enough together to be suitable for his purpose, and, with his
usual eagerness to see and to record all that could be seen, gathered
in an extensive harvest of such objects. The limit of distance between
the two members of a pair beyond which he did not think it worth
while to go was 2′, an interval imperceptible to the naked eye except
in cases of quite abnormally acute sight. In other words, the two
stars—even if bright enough to be visible—would always appear as _one_
to the ordinary eye. A first catalogue of such pairs, each forming
what may be called a =double star=, was published early in 1782 and
contained 269, of which 227 were new discoveries; a second catalogue of
434 was presented to the Royal Society at the end of 1784; and his last
paper, sent to the Royal Astronomical Society in 1821 and published
in the first volume of its memoirs, contained a list of 145 more. In
addition to the position of each double star the angular distance
between the two members, the direction of the line joining them, and
the brightness of each were noted. In some cases also curious contrasts
in the colour of the two components were observed. There were also not
a few cases in which not merely two, but three, four, or more stars
were found close enough to one another to be reckoned as forming a
multiple star.

Herschel had begun with the idea that a double star was due to a
merely accidental coincidence in the direction of two stars which had
no connection with one another and one of which might be many times
as remote as the other. It had, however, been pointed out by Michell
(chapter X., § 219), as early as 1767, that even the few double stars
then known afforded examples of coincidences which were very improbable
as the result of mere random distribution of stars. A special case
may be taken to make the argument clearer, though Michell’s actual
reasoning was not put into a numerical form. The bright star Castor (in
the Twins) had for some time been known to consist of two stars, α and
β, rather less than 5″ apart. Altogether there are about 50 stars of
the same order of brightness as α, and 400 like β. Neither set of stars
shews any particular tendency to be distributed in any special way over
the celestial sphere. So that the question of probabilities becomes:
if there are 50 stars of one sort and 400 of another distributed at
random over the whole celestial sphere, the two distributions having no
connection with one another, what is the chance that one of the first
set of stars should be within 5″ of one of the second set? The chance
is about the same as that, if 50 grains of wheat and 400 of barley are
scattered at random in a field of 100 acres, one grain of wheat should
be found within half an inch of a grain of barley. The odds against
such a possibility are clearly very great and can be shewn to be more
than 300,000 to one. These are the odds against the existence—without
some real connection between the members—of a _single_ double star like
Castor; but when Herschel began to discover double stars by the hundred
the improbability was enormously increased. In his first paper Herschel
gave as his opinion that “it is much too soon to form any theories of
small stars revolving round large ones,” a remark shewing that the idea
had been considered; and in 1784 Michell returned to the subject, and
expressed the opinion that the odds in favour of a physical relation
between the members of Herschel’s newly discovered double stars were
“beyond arithmetic.”

264. Twenty years after the publication of his first catalogue
Herschel was of Michell’s opinion, but was now able to support it
by evidence of an entirely novel and much more direct character. A
series of observations of Castor, presented in two papers published
in the _Philosophical Transactions_ in 1803 and 1804, which were
fortunately supplemented by an observation of Bradley’s in 1759, had
shewn a progressive alteration in the direction of the line joining
its two components, of such a character as to leave no doubt that the
two stars were revolving round one another; and there were five other
cases in which a similar motion was observed. In these six cases it
was thus shewn that the double star was really formed by a connected
pair of stars near enough to influence one another’s motion. A double
star of this kind is called a =binary star= or a =physical double
star=, as distinguished from a merely optical double star, the two
members of which have no connection with one another. In three cases,
including Castor, the observations were enough to enable the period of
a complete revolution of one star round another, assumed to go on at a
uniform rate, to be at any rate roughly estimated, the results given by
Herschel being 342 years for Castor,[157] 375 and 1,200 years for the
other two. It was an obvious inference that the motion of revolution
observed in a binary star was due to the mutual gravitation of its
members, though Herschel’s data were not enough to determine with any
precision the law of the motion, and it was not till five years after
his death that the first attempt was made to shew that the orbit of
a binary star was such as would follow from, or at any rate would be
consistent with, the mutual gravitation of its members (chapter XIII.,
§ 309: cf. also fig. 101). This may be regarded as the first direct
evidence of the extension of the law of gravitation to regions outside
the solar system.

Although only a few double stars were thus definitely shewn to be
binary, there was no reason why many others should not be so also,
their motion not having been rapid enough to be clearly noticeable
during the quarter of a century or so over which Herschel’s
observations extended; and this probability entirely destroyed the
utility of double stars for the particular purpose for which Herschel
had originally sought them. For if a double star is binary, then the
two members are approximately at the same distance from the earth and
therefore equally affected by the earth’s motion, whereas for the
purpose of finding the parallax it is essential that one should be
much more remote than the other. But the discovery which he had made
appeared to him far more interesting than that which he had attempted
but failed to make; in his own picturesque language, he had, like Saul,
gone out to seek his father’s asses and had found a kingdom.

265. It had been known since Halley’s time (chapter X., § 203) that
certain stars had proper motions on the celestial sphere, relative to
the general body of stars. The conviction, that had been gradually
strengthening among astronomers, that the sun is only one of the fixed
stars, suggested the possibility that the sun, like other stars,
might have a motion in space. Thomas Wright, Lambert, and others had
speculated on the subject, and Tobias Mayer (chapter X., §§ 225-6) had
shewn how to look for such a motion.

If a single star appears to move, then by the principle of relative
motion (chapter IV., § 77) this may be explained equally well by a
motion of the star or by a motion of the observer, or by a combination
of the two; and since in this problem the internal motions of the solar
system may be ignored, this motion of the observer may be identified
with that of the sun. When the proper motions of several stars are
observed, a motion of the sun only is in general inadequate to explain
them, but they may be regarded as due either solely to the motions in
space of the stars or to combinations of these with some motion of
the sun. If now the stars be regarded as motionless and the sun be
moving towards a particular point on the celestial sphere, then by an
obvious effect of perspective the stars near that point will appear to
recede from it and one another on the celestial sphere, while those in
the opposite region will approach one another, the magnitude of these
changes depending on the rapidity of the sun’s motion and on the
nearness of the stars in question. The effect is exactly of the same
nature as that produced when, on looking along a street at night, two
lamps on opposite sides of the street at some distance from us appear
close together, but as we walk down the street towards them they appear
to become more and more separated from one another. In the figure, for
example, L and L′ as seen from B appear farther apart than when seen
from A.

[Illustration: FIG. 84.—Illustrating the effect of the sun’s motion in
space.]

If the observed proper motions of stars examined are not of this
character, they cannot be explained as due _merely_ to the motion of
the sun; but if they shew some tendency to move in this way, then the
observations can be most simply explained by regarding the sun as in
motion, and by assuming that the discrepancies between the effects
resulting from the assumed motion of the sun and the observed proper
motions are due to the motions in space of the several stars.

From the few proper motions which Mayer had at his command he was,
however, unable to derive any indication of a motion of the sun.

Herschel used the proper motions, published by Maskelyne and Lalande,
of 14 stars (13 if the double star Castor be counted as only one),
and with extraordinary insight detected in them a certain uniformity
of motion of the kind already described, such as would result from a
motion of the sun. The point on the celestial sphere towards which the
sun was assumed to be moving, the =apex= as he called it, was taken to
be the point marked by the star λ in the constellation Hercules. A
motion of the sun in this direction would, he found, produce in the 14
stars apparent motions which were in the majority of cases in general
agreement with those observed.[158] This result was published in 1783,
and a few months later _Pierre Prévost_ (1751-1839) deduced a very
similar result from Tobias Mayer’s collection of proper motions. More
than 20 years later (1805) Herschel took up the question again, using
six of the brightest stars in a collection of the proper motions of 36
published by Maskelyne in 1790, which were much more reliable than any
earlier ones, and employing more elaborate processes of calculation;
again the apex was placed in the constellation Hercules, though at a
distance of nearly 30° from the position given in 1783. Herschel’s
results were avowedly to a large extent speculative, and were received
by contemporary astronomers with a large measure of distrust; but a
number of far more elaborate modern investigations of the same subject
have confirmed the general correctness of his work, the earlier of his
two estimates appearing, however, to be the more accurate. He also made
some attempts in the same papers and in a third (published in 1806) to
estimate the speed as well as the direction of the sun’s motion; but
the work necessarily involved so many assumptions as to the probable
distances of the stars—which were quite unknown—that it is not worth
while to quote results more definite than the statement made in the
paper of 1783, that “We may in a general way estimate that the solar
motion can certainly not be less than that which the earth has in her
annual orbit.”

266. The question of the comparative brightness of stars was, as
we have seen (§ 258), of importance in connection with Herschel’s
attempts to estimate their relative distances from the earth and their
arrangement in space; it also presented itself in connection with
inquiries into the variability of the light of stars. Two remarkable
cases of variability had been for some time known. A star in the Whale
(ο _Ceti_ or _Mira_) had been found to be at times invisible to the
naked eye and at other times to be conspicuous; a Dutch astronomer,
_Phocylides Holwarda_ (1618-1651), first clearly recognised its
variable character (1639), and _Ismael Boulliau_ or _Bullialdus_
(1605-1694) in 1667 fixed its period at about eleven months, though it
was found that its fluctuations were irregular both in amount and in
period. Its variations formed the subject of the first paper published
by Herschel in the _Philosophical Transactions_ (1780). An equally
remarkable variable star is that known as _Algol_ (or β _Persei_), the
fluctuations of which were found to be performed with almost absolute
regularity. Its variability had been noted by _Geminiano Montanari_
(1632-1687) in 1669, but the regularity of its changes was first
detected in 1783 by _John Goodricke_ (1764-1786), who was soon able to
fix its period at very nearly 2 days 20 hours 49 minutes. Algol, when
faintest, gives about one-quarter as much light as when brightest,
the change from the first state to the second being effected in about
ten hours; whereas Mira varies its light several hundredfold, but
accomplishes its changes much more slowly.

At the beginning of Herschel’s career these and three or four others of
less interest were the only stars definitely recognised as variable,
though a few others were added soon afterwards. Several records also
existed of so-called “new” stars, which had suddenly been noticed in
places where no star had previously been observed, and which for the
most part rapidly became inconspicuous again (cf. chapter II., § 42;
chapter V., § 100; chapter VII., § 138); such stars might evidently be
regarded as variable stars, the times of greatest brightness occurring
quite irregularly or at long intervals. Moreover various records of
the brightness of stars by earlier astronomers left little doubt that
a good many must have varied sensibly in brightness. For example, a
small star in the Great Bear (close to the middle star of the “tail”)
was among the Arabs a noted test of keen sight, but is perfectly
visible even in our duller climate to persons with ordinary eyesight;
and Castor, which appeared the brighter of the two Twins to Bayer when
he published his Atlas (1603), was in the 18th century (as now) less
bright than Pollux.

Herschel made a good many definite measurements of the amounts of
light emitted by stars of various magnitudes, but was not able to
carry out any extensive or systematic measurements on this plan. With
a view to the future detection of such changes of brightness as have
just been mentioned, he devised and carried out on a large scale
the extremely simple method of =sequences=. If a group of stars are
observed and their order of brightness noted at two different times,
then any alteration in the order will shew that the brightness of one
or more has changed. So that if a number of stars are observed in sets
in such a way that each star is recorded as being less bright than
certain stars near it and brighter than certain other stars, materials
are thereby provided for detecting at any future time any marked
amount of variation of brightness. Herschel prepared on this plan, at
various times between 1796 and 1799, four catalogues of comparative
brightness based on naked-eye observations and comprising altogether
about 3,000 stars. In the course of the work a good many cases of
slight variability were noticed; but the most interesting discovery
of this kind was that of the variability of the well-known star α
_Herculis_, announced in 1796. The period was estimated at 60 days,
and the star thus seemed to form a connecting link between the known
variables which like Algol had periods of a very few days and those (of
which Mira was the best known) with periods of some hundreds of days.
As usual, Herschel was not content with a mere record of observations,
but attempted to explain the observed facts by the supposition that
a variable star had a rotation and that its surface was of unequal
brightness.

267. The novelty of Herschel’s work on the fixed stars, and the very
general character of the results obtained, have caused this part of his
researches to overshadow in some respects his other contributions to
astronomy.

Though it was no part of his plan to contribute to that precise
knowledge of the motions of the bodies of the solar system which
absorbed the best energies of most of the astronomers of the 18th
century—whether they were observers or mathematicians—he was a careful
and successful observer of the bodies themselves.

His discoveries of Uranus, of two of its satellites, and of two
new satellites of Saturn have been already mentioned in connection
with his life (§§ 253, 255). He believed himself to have seen also
(1798) four other satellites of Uranus, but their existence was
never satisfactorily verified; and the second pair of satellites now
known to belong to Uranus, which were discovered by Lassell in 1847
(chapter XIII., § 295), do not agree in position and motion with any
of Herschel’s four. It is therefore highly probable that they were
mere optical illusions due to defects of his mirror, though it is
not impossible that he may have caught glimpses of one or other of
Lassell’s satellites and misinterpreted the observations.

Saturn was a favourite object of study with Herschel from the very
beginning of his astronomical career, and seven papers on the subject
were published by him between 1790 and 1806. He noticed and measured
the deviation of the planet’s form from a sphere (1790); he observed
various markings on the surface of the planet itself, and seems to
have seen the inner ring, now known from its appearance as the crape
ring (chapter XIII., § 295), though he did not recognise its nature.
By observations of some markings at some distance from the equator
he discovered (1790) that Saturn rotated on an axis, and fixed the
period of rotation at about 10 h. 16 m. (a period differing only by
about 2 minutes from modern estimates), and by similar observations of
the ring (1790) concluded that it rotated in about 10-1∕2 hours, the
axis of rotation being in each case perpendicular to the plane of the
ring. The satellite Japetus, discovered by Cassini in 1671 (chapter
VIII., § 160), had long been recognised as variable in brightness, the
light emitted being several times as much at one time as at another.
Herschel found that these variations were not only perfectly regular,
but recurred at an interval equal to that of the satellite’s period
of rotation round its primary (1792), a conclusion which Cassini had
thought of but rejected as inconsistent with his observations. This
peculiarity was obviously capable of being explained by supposing that
different portions of Japetus had unequal power of reflecting light,
and that like our moon it turned on its axis once in every revolution,
in such a way as always to present the same face towards its primary,
and in consequence each face in turn to an observer on the earth. It
was natural to conjecture that such an arrangement was general among
satellites, and Herschel obtained (1797) some evidence of variability
in the satellites of Jupiter, which appeared to him to support this
hypothesis.

Herschel’s observations of other planets were less numerous and
important. He rightly rejected the supposed observations by Schroeter
(§ 271) of vast mountains on Venus, and was only able to detect some
indistinct markings from which the planet’s rotation on an axis could
be somewhat doubtfully inferred. He frequently observed the familiar
bright bands on Jupiter commonly called belts, which he was the first
to interpret (1793) as bands of cloud. On Mars he noted the periodic
diminution of the white caps on the two poles, and observed how in
these and other respects Mars was of all planets the one most like the
earth.

268. Herschel made also a number of careful observations on the sun,
and based on them a famous theory of its structure. He confirmed the
existence of various features of the solar surface which had been
noted by the earlier telescopists such as Galilei, Scheiner, and
Hevel, and added to them in some points of detail. Since Galilei’s
time a good many suggestions as to the nature of spots had been
thrown out by various observers, such as that they were clouds,
mountain-tops, volcanic products, etc., but none of these had been
supported by any serious evidence. Herschel’s observations of the
appearances of spots suggested to him that they were depressions in
the surface of the sun, a view which derived support from occasional
observations of a spot when passing over the edge of the sun as a
distinct depression or notch there. Upon this somewhat slender basis
of fact he constructed (1795) an elaborate theory of the nature of
the sun, which attracted very general notice by its ingenuity and
picturesqueness and commanded general assent in the astronomical world
for more than half a century. The interior of the sun was supposed
to be a cold dark solid body, surrounded by two cloud-layers, of
which the outer was the =photosphere= or ordinary surface of the sun,
intensely hot and luminous, and the inner served as a fire-screen to
protect the interior. The umbra (chapter VI., § 124) of a spot was the
dark interior seen through an opening in the clouds, and the penumbra
corresponded to the inner cloud-layer rendered luminous by light from
above.

 “The sun viewed in this light appears to be nothing else than a
 very eminent, large, and lucid planet, evidently the first or, in
 strictness of speaking, the only primary one of our system; ... it is
 most probably also inhabited, like the rest of the planets, by beings
 whose organs are adapted to the peculiar circumstances of that vast
 globe.”

That spots were depressions had been suggested more than twenty years
before (1774) by _Alexander Wilson_ of Glasgow (1714-1786), and
supported by evidence different from any adduced by Herschel and in
some ways more conclusive. Wilson noticed, first in the case of a large
spot seen in 1769, and afterwards in other cases, that as the sun’s
rotation carries a spot across its disc from one edge to another, its
appearance changes exactly as it would do in accordance with ordinary
laws of perspective if the spot were a saucer-shaped depression, of
which the bottom formed the umbra and the sloping sides the penumbra,
since the penumbra appears narrowest on the side nearest the centre
of the sun and widest on the side nearest the edge. Hence Wilson
inferred, like Herschel, but with less confidence, that the body of
the sun is dark. In the paper referred to Herschel shews no signs of
being acquainted with Wilson’s work, but in a second paper (1801),
which contained also a valuable series of observations of the detailed
markings on the solar surface, he refers to Wilson’s “geometrical
proof” of the depression of the umbra of a spot.

Although it is easy to see now that Herschel’s theory was a rash
generalisation from slight data, it nevertheless explained—with fair
success—most of the observations made up to that time.

Modern knowledge of heat, which was not accessible to Herschel, shews
us the fundamental impossibility of the continued existence cf a body
with a cold interior and merely a shallow ring of hot and luminous
material round it; and the theory in this form is therefore purely of
historic interest (cf. also chapter XIII., §§ 298, 303).

269. Another suggestive idea of Herschel’s was the analogy between
the sun and a variable star, the known variation in the number of
spots and possibly of other markings on the sun suggesting to him
the probability of a certain variability in the total amount of
solar light and heat emitted. The terrestrial influence of this he
tried to measure—in the absence of precise meteorological data—with
characteristic ingenuity by the price of wheat, and some evidence
was adduced to shew that at times when sun-spots had been noted to
be scarce—corresponding according to Herschel’s view to periods
of diminished solar activity—wheat had been dear and the weather
presumably colder. In reality, however, the data were insufficient to
establish any definite conclusions.

270. In addition to carrying out the astronomical researches already
sketched, and a few others of less importance, Herschel spent some
time, chiefly towards the end of his life, in working at light and
heat; but the results obtained, though of considerable value, belong
rather to physics than to astronomy, and need not be dealt with here.

271. It is natural to associate Herschel’s wonderful series of
discoveries with his possession of telescopes of unusual power and
with his formulation of a new programme of astronomical inquiry; and
these were certainly essential elements. It is, however, significant,
as shewing how important other considerations were, that though a
great number of his telescopes were supplied to other astronomers,
and though his astronomical programme when once suggested was open to
all the world to adopt, hardly any of his contemporaries executed any
considerable amount of work comparable in scope to his own.

Almost the only astronomer of the period whose work deserves mention
beside Herschel’s, though very inferior to it both in extent and in
originality, was _Johann Hieronymus Schroeter_ (1745-1816).

Holding an official position at Lilienthal, near Bremen, he devoted his
leisure during some thirty years to a scrutiny of the planets and of
the moon, and to a lesser extent of other bodies.

As has been seen in the case of Venus (§ 267), his results were not
always reliable, but notwithstanding some errors he added considerably
to our knowledge of the appearances presented by the various planets,
and in particular studied the visible features of the moon with a
minuteness and accuracy far exceeding that of any of his predecessors,
and made some attempt to deduce from his observations data as to its
physical condition. His two volumes on the moon (_Selenotopographische
Fragmente_, 1791 and 1802), and other minor writings, are a storehouse
of valuable detail, to which later workers have been largely indebted.



CHAPTER XIII.

THE NINETEENTH CENTURY.

 “The greater the sphere of our knowledge, the larger is the surface of
 its contact with the infinity of our ignorance.”


272. The last three chapters have contained some account of progress
made in three branches of astronomy which, though they overlap and
exercise an important influence on one another, are to a large extent
studied by different men and by different methods, and have different
aims. The difference is perhaps best realised by thinking of the
work of a great master in each department, Bradley, Laplace, and
Herschel. So great is the difference that Delambre in his standard
history of astronomy all but ignores the work of the great school of
mathematical astronomers who were his contemporaries and immediate
predecessors, not from any want of appreciation of their importance,
but because he regards their work as belonging rather to mathematics
than to astronomy; while Bessel (§ 277), in saying that the function of
astronomy is “to assign the places on the sky where sun, moon, planets,
comets, and stars have been, are, and will be,” excludes from its scope
nearly everything towards which Herschel’s energies were directed.

Current modern practice is, however, more liberal in its use of
language than either Delambre or Bessel, and finds it convenient to
recognise all three of the subjects or groups of subjects referred to
as integral parts of one science.

The mutual relation of gravitational astronomy and what has been for
convenience called observational astronomy has been already referred
to (chapter X., § 196). It should, however, be noticed that the
latter term has in this book hitherto been used chiefly for only one
part of the astronomical work which concerns itself primarily with
observation. Observing played at least as large a part in Herschel’s
work as in Bradley’s, but the aims of the two men were in many ways
different. Bradley was interested chiefly in ascertaining as accurately
as possible the apparent positions of the fixed stars on the celestial
sphere, and the positions and motions of the bodies of the solar
system, the former undertaking being in great part subsidiary to the
latter. Herschel, on the other hand, though certain of his researches,
_e.g._ into the parallax of the fixed stars and into the motions of the
satellites of Uranus, were precisely like some of Bradley’s, was far
more concerned with questions of the appearances, mutual relations, and
structure of the celestial bodies in themselves. This latter branch of
astronomy may conveniently be called =descriptive astronomy=, though
the name is not altogether appropriate to inquiries into the physical
structure and chemical constitution of celestial bodies which are often
put under this head, and which play an important part in the astronomy
of the present day.

273. Gravitational astronomy and exact observational astronomy have
made steady progress during the nineteenth century, but neither has
been revolutionised, and the advances made have been to a great extent
of such a nature as to be barely intelligible, still less interesting,
to those who are not experts. The account of them to be given in this
chapter must therefore necessarily be of the slightest character, and
deal either with general tendencies or with isolated results of a less
technical character than the rest.

Descriptive astronomy, on the other hand, which can be regarded
as being almost as much the creation of Herschel as gravitational
astronomy is of Newton, has not only been greatly developed on the
lines laid down by its founder, but has received—chiefly through the
invention of spectrum analysis (§ 299)—extensions into regions not
only unthought of but barely imaginable a century ago. Most of the
results of descriptive astronomy—unlike those of the older branches
of the subject—are readily intelligible and fairly interesting to
those who have but little knowledge of the subject; in particular they
are as yet to a considerable extent independent of the mathematical
ideas and language which dominate so much of astronomy and render
it unattractive or inaccessible to many. Moreover, not only can
descriptive astronomy be appreciated and studied, but its progress can
materially be assisted, by observers who have neither knowledge of
higher mathematics nor any elaborate instrumental equipment.

Accordingly, while the successors of Laplace and Bradley have been
for the most part astronomers by profession, attached to public
observatories or to universities, an immense mass of valuable
descriptive work has been done by amateurs who, like Herschel in the
earlier part of his career, have had to devote a large part of their
energies to professional work of other kinds, and who, though in some
cases provided with the best of instruments, have in many others been
furnished with only a slender instrumental outfit. For these and
other reasons one of the most notable features of nineteenth century
astronomy has been a great development, particularly in this country
and in the United States, of general interest in the subject, and the
establishment of a large number of private observatories devoted almost
entirely to the study of special branches of descriptive astronomy.
The nineteenth century has accordingly witnessed the acquisition of an
unprecedented amount of detailed astronomical knowledge. But the wealth
of material thus accumulated has outrun our powers of interpretation,
and in a number of cases our knowledge of some particular department
of descriptive astronomy consists, on the one hand of an immense
series of careful observations, and on the other of one or more highly
speculative theories, seldom capable of explaining more than a small
portion of the observed facts.

In dealing with the progress of modern descriptive astronomy the
proverbial difficulty of seeing the wood on account of the trees
is therefore unusually great. To give an account within the limits
of a single chapter of even the most important facts added to our
knowledge would be a hopeless endeavour; fortunately it would also
be superfluous, as they are to be found in many easily accessible
textbooks on astronomy, or in treatises on special parts of the
subject. All that can be attempted is to give some account of the chief
lines on which progress has been made, and to indicate some general
conclusions which seem to be established on a tolerably secure basis.

274. The progress of exact observation has of course been based very
largely on instrumental advances. Not only have great improvements
been made in the extremely delicate work of making large lenses, but
the graduated circles and other parts of the mounting of a telescope
upon which accuracy of measurement depends can also be constructed
with far greater exactitude and certainty than at the beginning of the
century. New methods of mounting telescopes and of making and recording
observations have also been introduced, all contributing to greater
accuracy. For certain special problems photography is found to present
great advantages as compared with eye-observations, though its most
important applications have so far been to descriptive astronomy.

275. The necessity for making allowance for various known sources of
errors in observation, and for diminishing as far as possible the
effect of errors due to unknown causes, had been recognised even by
Tycho Brahe (chapter V., § 110), and had played an important part in
the work of Flamsteed and Bradley (chapter X., §§ 198, 218). Some
further important steps in this direction were taken in the earlier
part of this century. The method of =least squares=, established
independently by two great mathematicians, _Adrien Marie Legendre_
(1752-1833) of Paris and _Carl Friedrich Gauss_ (1777-1855) of
Göttingen,[159] was a systematic method of combining observations,
which gave slightly different results, in such a way as to be as near
the truth as possible. Any ordinary physical measurement, _e.g._ of a
length, however carefully executed, is necessarily imperfect; if the
same measurement is made several times, even under almost identical
conditions, the results will in general differ slightly; and the
question arises of combining these so as to get the most satisfactory
result. The common practice in this simple case has long been to
take the arithmetical mean or average of the different results. But
astronomers have constantly to deal with more complicated cases in
which _two_ or more unknown quantities have to be determined from
observations of different quantities, as, for example, when the
elements of the orbit of a planet (chapter XI., § 236) have to be
found from observations of the planet’s position at different times.
The method of least squares gives a rule for dealing with such cases,
which was a generalisation of the ordinary rule of averages for the
case of a single unknown quantity; and it was elaborated in such a
way as to provide for combining observations of different value, such
as observations taken by observers of unequal skill or with different
instruments, or under more or less favourable conditions as to weather,
etc. It also gives a simple means of testing, by means of their mutual
consistency, the value of a series of observations, and comparing
their probable accuracy with that of some other series executed under
different conditions. The method of least squares and the special case
of the “average” can be deduced from a certain assumption as to the
general character of the causes which produce the error in question;
but the assumption itself cannot be justified _a priori_; on the other
hand, the satisfactory results obtained from the application of the
rule to a great variety of problems in astronomy and in physics has
shewn that in a large number of cases unknown causes of error must be
approximately of the type considered. The method is therefore very
widely used in astronomy and physics wherever it is worth while to take
trouble to secure the utmost attainable accuracy.

276. Legendre’s other contributions to science were almost entirely
to branches of mathematics scarcely affecting astronomy. Gauss, on
the other hand, was for nearly half a century head of the observatory
of Göttingen, and though his most brilliant and important work was in
pure mathematics, while he carried out some researches of first-rate
importance in magnetism and other branches of physics, he also made
some further contributions of importance to astronomy. These were for
the most part processes of calculation of various kinds required for
utilising astronomical observations, the best known being a method of
calculating the orbit of a planet from three complete observations
of its position, which was published in his _Theoria Motus_ (1809).
As we have seen (chapter XI., § 236), the complete determination of
a planet’s orbit depends on six independent elements: any complete
observation of the planet’s position in the sky, at any time, gives
two quantities, _e.g._ the right ascension and declination (chapter
II., § 33); hence three complete observations give six equations and
are theoretically adequate to determine the elements of the orbit; but
it had not hitherto been found necessary to deal with the problem in
this form. The orbits of all the planets but Uranus had been worked
out gradually by the use of a series of observations extending over
centuries; and it was feasible to use observations taken at particular
times so chosen that certain elements could be determined without
any accurate knowledge of the others; even Uranus had been under
observation for a considerable time before its path was determined
with anything like accuracy; and in the case of comets not only was
a considerable series of observations generally available, but the
problem was simplified by the fact that the orbit could be taken to
be nearly or quite a parabola instead of an ellipse (chapter IX., §
190). The discovery of the new planet Ceres on January 1st, 1801 (§
294), and its loss when it had only been observed for a few weeks,
presented virtually a new problem in the calculation of an orbit.
Gauss applied his new methods—including that of least squares—to the
observations available, and with complete success, the planet being
rediscovered at the end of the year nearly in the position indicated by
his calculations.

277. The theory of the “reduction” of observations (chapter X., § 218)
was first systematised and very much improved by _Friedrich Wilhelm
Bessel_ (1784-1846), who was for more than thirty years the director
of the new Prussian observatory at Königsberg. His first great work
was the reduction and publication of Bradley’s Greenwich observations
(chapter X., § 218). This undertaking involved an elaborate study of
such disturbing causes as precession, aberration, and refraction,
as well as of the errors of Bradley’s instruments. Allowance was
made for these on a uniform and systematic plan, and the result was
the publication in 1818, under the title _Fundamenta Astronomiae_,
of a catalogue of the places of 3,222 stars as they were in 1755.
A special problem dealt with in the course of the work was that of
refraction. Although the complete theoretical solution was then as now
unattainable, Bessel succeeded in constructing a table of refractions
which agreed very closely with observation and was presented in such a
form that the necessary correction for a star in almost any position
could be obtained with very little trouble. His general methods of
reduction—published finally in his _Tabulae Regiomontanae_ (1830)—also
had the great advantage of arranging the necessary calculations in
such a way that they could be performed with very little labour and
by an almost mechanical process, such as could easily be carried out
by a moderately skilled assistant. In addition to editing Bradley’s
observations, Bessel undertook a fresh series of observations of his
own, executed between the years 1821 and 1833, upon which were based
two new catalogues, containing about 62,000 stars, which appeared after
his death.

[Illustration: FIG. 85.—_61 Cygni_ and the two neighbouring stars used
by Bessel.]

[Illustration: FIG. 86.—The parallax of _61 Cygni_.]

278. The most memorable of Bessel’s special pieces of work was the
first definite detection of the parallax of a fixed star. He abandoned
the test of brightness as an indication of nearness, and selected a
star (_61 Cygni_) which was barely visible to the naked eye but was
remarkable for its large proper motion (about 5″ per annum); evidently
if a star is moving at an assigned rate (in miles per hour) through
space, the nearer to the observer it is the more rapid does its motion
appear to be, so that apparent rapidity of motion, like brightness,
is a probable but by no means infallible indication of nearness. A
modification of Galilei’s differential method (chapter VI., § 129,
and chapter XII., § 263) being adopted, the angular distance of _61
Cygni_ from two neighbouring stars, the faintness and immovability of
which suggested their great distance in space, was measured at frequent
intervals during a year. From the changes in these distances σ _a_, σ
_b_ (in fig. 85), the size of the small ellipse described by σ could be
calculated. The result, announced at the end of 1838, was that the star
had an annual parallax of about 1∕3″ (chapter VIII., § 161), _i.e._
that the star was at such distance that the greatest angular distance
of the earth from the sun viewed from the star (the angle S σ E in
fig. 86, where S is the sun and E the earth) was this insignificant
angle.[160] The result was confirmed, with slight alterations, by a
fresh investigation of Bessel’s in 1839-40, but later work seems to
shew that the parallax is a little less than 1∕2″.[161] With this
latter estimate, the apparent size of the earth’s path round the sun as
seen from the star is the same as that of a halfpenny at a distance
of rather more than three miles. In other words, the distance of the
star is about 400,000 times the distance of the sun, which is itself
about 93,000,000 miles. A mile is evidently a very small unit by which
to measure such a vast distance; and the practice of expressing such
distances by means of the time required by light to perform the journey
is often convenient. Travelling at the rate of 186,000 miles _per
second_ (§ 283), light takes rather more than six years to reach us
from _61 Cygni_.

279. Bessel’s solution of the great problem which had baffled
astronomers ever since the time of Coppernicus was immediately followed
by two others. Early in 1839 _Thomas Henderson_ (1798-1844) announced
a parallax of nearly 1″ for the bright star α _Centauri_ which he
had observed at the Cape, and in the following year _Friedrich Georg
Wilhelm Struve_ (1793-1864) obtained from observations made at Pulkowa
a parallax of 1∕4″ for _Vega_; later work has reduced these numbers to
3∕4″ and 1∕10″ respectively.

A number of other parallax determinations have subsequently been made.
An interesting variation in method was made by the late Professor
_Charles Pritchard_ (1808-1893) of Oxford by _photographing_ the star
to be examined and its companions, and subsequently measuring the
distances on the photograph, instead of measuring the angular distances
directly with a micrometer.

At the present time some 50 stars have been ascertained with some
reasonable degree of probability to have measurable, if rather
uncertain, parallaxes; α _Centauri_ still holds its own as the nearest
star, the light-journey from it being about four years. A considerable
number of other stars have been examined with negative or highly
uncertain results, indicating that their parallaxes are too small
to be measured with our present means, and that their distances are
correspondingly great.

280. A number of star catalogues and star maps—too numerous to mention
separately—have been constructed during this century, marking steady
progress in our knowledge of the position of the stars, and providing
fresh materials for ascertaining, by comparison of the state of the sky
at different epochs, such quantities as the proper motions of the stars
and the amount of precession. Among the most important is the great
catalogue of 324,198 stars in the northern hemisphere known as the Bonn
_Durchmusterung_, published in 1859-62 by Bessel’s pupil _Friedrich
Wilhelm August Argelander_ (1799-1875); this was extended (1875-85) so
as to include 133,659 stars in a portion of the southern hemisphere
by _Eduard Schönfeld_ (1828-1891); and more recently Dr. _Gill_ has
executed at the Cape photographic observations of the remainder of the
southern hemisphere, the reduction to the form of a catalogue (the
first instalment of which was published in 1896) having been performed
by Professor _Kapteyn_ of Groningen. The star places determined in
these catalogues do not profess to be the most accurate attainable,
and for many purposes it is important to know with the utmost accuracy
the positions of a smaller number of stars. The greatest undertaking
of this kind, set on foot by the German Astronomical Society in
1867, aims at the construction, by the co-operation of a number of
observatories, of catalogues of about 130,000 of the stars contained in
the “approximate” catalogues of Argelander and Schönfeld; nearly half
of the work has now been published.

The greatest scheme for a survey of the sky yet attempted is the
photographic chart, together with a less extensive catalogue to
be based on it, the construction of which was decided on at an
international congress held at Paris in 1887. The whole sky has been
divided between 18 observatories in all parts of the world, from
Helsingfors in the north to Melbourne in the south, and each of these
is now taking photographs with virtually identical instruments. It
is estimated that the complete chart, which is intended to include
stars of the 14th magnitude,[162] will contain about 20,000,000 stars,
2,000,000 of which will be catalogued also.

281. One other great problem—that of the distance of the sun—may
conveniently be discussed under the head of observational astronomy.

The transits of Venus (chapter X., §§ 202, 227) which occurred in
1874 and 1882 were both extensively observed, the old methods of
time-observation being supplemented by photography and by direct
micrometric measurements of the positions of Venus while transiting.

The method of finding the distance of the sun by means of observation
of Mars in opposition (chapter VIII., § 161) has been employed on
several occasions with considerable success, notably by Dr. Gill at
Ascension in 1877. A method originally used by Flamsteed, but revived
in 1857 by _Sir George Biddell Airy_ (1801-1892), the late Astronomer
Royal, was adopted on this occasion. For the determination of the
parallax of a planet observations have to be made from two different
positions at a known distance apart; commonly these are taken to be
at two different observatories, as far as possible removed from one
another in latitude. Airy pointed out that the same object could be
attained if only one observatory were used, but observations taken at
an interval of some hours, as the rotation of the earth on its axis
would in that time produce a known displacement of the observer’s
position and so provide the necessary base line. The apparent shift of
the planet’s position could be most easily ascertained by measuring
(with the micrometer) its distances from neighbouring fixed stars.
This method (known as the =diurnal method=) has the great advantage,
among others, of being simple in application, a single observer and
instrument being all that is needed.

The diurnal method has also been applied with great success to certain
of the minor planets (§ 294). Revolving as they do between Mars and
Jupiter, they are all farther off from us than the former; but there is
the compensating advantage that as a minor planet, unlike Mars, is, as
a rule, too small to shew any appreciable disc, its angular distance
from a neighbouring star is more easily measured. The employment of
the minor planets in this way was first suggested by Professor _Galle_
of Berlin in 1872, and recent observations of the minor planets
_Victoria_, _Sappho_, and _Iris_ in 1888-89, made at a number of
observatories under the general direction of Dr. Gill, have led to some
of the most satisfactory determinations of the sun’s distance.

282. It was known to the mathematical astronomers of the 18th century
that the distance of the sun could be obtained from a knowledge of
various perturbations of members of the solar system; and Laplace had
deduced a value of the solar parallax from lunar theory. Improvements
in gravitational astronomy and in observation of the planets and moon
during the present century have added considerably to the value of
these methods. A certain irregularity in the moon’s motion known as
the =parallactic inequality=, and another in the motion of the sun,
called the =lunar equation=, due to the displacement of the earth by
the attraction of the moon, alike depend on the ratio of the distances
of the sun and moon from the earth; if the amount of either of these
inequalities can be observed, the distance of the sun can therefore
be deduced, that of the moon being known with great accuracy. It was
by a virtual application of the first of these methods that Hansen (§
286) in 1854, in the course of an elaborate investigation of the lunar
theory, ascertained that the current value of the sun’s distance was
decidedly too large, and Leverrier (§ 288) confirmed the correction by
the second method in 1858.

Again, certain changes in the orbits of our two neighbours, Venus and
Mars, are known to depend upon the ratio of the masses of the sun and
earth, and can hence be connected, by gravitational principles, with
the quantity sought. Leverrier pointed out in 1861 that the motions of
Venus and of Mars, like that of the moon, were inconsistent with the
received estimate of the sun’s distance, and he subsequently worked out
the method more completely and deduced (1872) values of the parallax.
The displacements to be observed are very minute, and their accurate
determination is by no means easy, but they are both secular (chapter
XI., § 242), so that in the course of time they will be capable of very
exact measurement. Leverrier’s method, which is even now a valuable
one, must therefore almost inevitably outstrip all the others which
are at present known; it is difficult to imagine, for example, that
the transits of Venus due in 2004 and 2012 will have any value for the
purpose of the determination of the sun’s distance.

283. One other method, in two slightly different forms, has become
available during this century. The displacement of a star by aberration
(chapter X., § 210) depends upon the ratio of the velocity of light
to that of the earth in its orbit round the sun; and observations of
Jupiter’s satellites after the manner of Roemer (chapter VIII., § 162)
give the =light-equation=, or time occupied by light in travelling from
the sun to the earth. Either of these astronomical quantities—of which
aberration is the more accurately known—can be used to determine the
velocity of light when the dimensions of the solar system are known,
or _vice versa_. No independent method of determining the velocity
of light was known until 1849, when _Hippolyte Fizeau_ (1819-1896)
invented and successfully carried out a laboratory method.

New methods have been devised since, and three comparatively recent
series of experiments, by M. _Cornu_ in France (1874 and 1876) and by
Dr. _Michelson_ (1879) and Professor _Newcomb_ (1880-82) in the United
States, agreeing closely with one another, combine to fix the velocity
of light at very nearly 186,300 miles (299,800 kilometres) per second;
the solar parallax resulting from this by means of aberration is very
nearly 8″·8.[163]

284. Encke’s value of the sun’s parallax, 8″·571, deduced from
the transits of Venus (chapter X., § 227) in 1761 and 1769, and
published in 1835, corresponding to a distance of about 95,000,000
miles, was generally accepted till past the middle of the century.
Then the gravitational methods of Hansen and Leverrier, the earlier
determinations of the velocity of light, and the observations made
at the opposition of Mars in 1862, all pointed to a considerably
larger value of the parallax; a fresh examination of the 18th century
observations shewed that larger values than Encke’s could easily be
deduced from them; and for some time—from about 1860 onwards—a parallax
of nearly 8″·95, corresponding to a distance of rather more than
91,000,000 miles, was in common use. Various small errors in the new
methods were, however, detected, and the most probable value of the
parallax has again increased. Three of the most reliable methods, the
diurnal method as applied to Mars in 1877, the same applied to the
minor planets in 1888-89, and aberration, unite in giving values
not differing from 8″·80 by more than two or three hundredths of a
second. The results of the last transits of Venus, the publication and
discussion of which have been spread over a good many years, point to a
somewhat larger value of the parallax. Most astronomers appear to agree
that a parallax of 8″·8, corresponding to a distance of rather less
than 93,000,000 miles, represents fairly the available data.

285. The minute accuracy of modern observations is well illustrated
by the recent discovery of a variation in the latitude of several
observatories. Observations taken at Berlin in 1884-85 indicated a
minute variation in the latitude; special series of observations to
verify this were set on foot in several European observatories, and
subsequently at Honolulu and at Cordoba. A periodic alteration in
latitude amounting to about 1∕2″ emerged as the result. Latitude being
defined (chapter X., § 221) as the angle which the vertical at any
place makes with the equator, which is the same as the elevation of the
pole above the horizon, is consequently altered by any change in the
equator, and therefore by an alteration in the position of the earth’s
poles or the ends of the axis about which it rotates.

Dr. _S. C. Chandler_ succeeded (1891 and subsequently) in shewing
that the observations in question could be in great part explained by
supposing the earth’s axis to undergo a minute change of position in
such a way that either pole of the earth describes a circuit round its
mean position in about 427 days, never deviating more than some 30 feet
from it. It is well known from dynamical theory that a rotating body
such as the earth can be displaced in this manner, but that if the
earth were perfectly rigid the period should be 306 days instead of
427. The discrepancy between the two numbers has been ingeniously used
as a test of the extent to which the earth is capable of yielding—like
an elastic solid—to the various forces which tend to strain it.

286. All the great problems of gravitational astronomy have been
rediscussed since Laplace’s time, and further steps taken towards their
solution.

Laplace’s treatment of the lunar theory was first developed by _Marie
Charles Theodore Damoiseau_ (1768-1846), whose _Tables de la Lune_
(1824 and 1828) were for some time in general use.

Some special problems of both lunar and planetary theory were dealt
with by _Siméon Denis Poisson_ (1781-1840), who is, however, better
known as a writer on other branches of mathematical physics than as
an astronomer. A very elaborate and detailed theory of the moon,
investigated by the general methods of Laplace, was published by
_Giovanni Antonio Amadeo Plana_ (1781-1869) in 1832, but unaccompanied
by tables. A general treatment of both lunar and planetary theories,
the most complete that had appeared up to that time, by _Philippe
Gustave Doulcet de Pontécoulant_ (1795-1874), appeared in 1846, with
the title _Théorie Analytique du Système du Monde_; and an incomplete
lunar theory similar to his was published by _John William Lubbock_
(1803-1865) in 1830-34.

A great advance in lunar theory was made by _Peter Andreas Hansen_
(1795-1874) of Gotha, who published in 1838 and 1862-64 the treatises
commonly known respectively as the _Fundamenta_[164] and the
_Darlegung_,[165] and produced in 1857 tables of the moon’s motion
of such accuracy that the discrepancies between the tables and
observations in the century 1750-1850 were never greater than 1″
or 2″. These tables were at once used for the calculation of the
_Nautical Almanac_ and other periodicals of the same kind, and with
some modifications have remained in use up to the present day.

A completely new lunar theory—of great mathematical interest and of
equal complexity—was published by _Charles Delaunay_ (1816-1872) in
1860 and 1867. Unfortunately the author died before he was able to work
out the corresponding tables.

Professor Newcomb of Washington (§ 283) has rendered valuable
services to lunar theory—as to other branches of astronomy—by a
number of delicate and intricate calculations, the best known being
his comparison of Hansen’s tables with observation and consequent
corrections of the tables.

New methods of dealing with lunar theory were devised by the late
Professor _John Couch Adams_ of Cambridge (1819-1892), and similar
methods have been developed by Dr. _G. W. Hill_ of Washington; so
far they have not been worked out in detail in such a way as to be
available for the calculation of tables, and their interest seems to
be at present mathematical rather than practical; but the necessary
detailed work is now in progress, and these and allied methods may be
expected to lead to a considerable diminution of the present excessive
intricacy of lunar theory.

287. One special point in lunar theory may be worth mentioning. The
secular acceleration of the moon’s mean motion which had perplexed
astronomers since its first discovery by Halley (chapter X., § 201)
had, as we have seen (chapter XI., § 240), received an explanation in
1787 at the hands of Laplace. Adams, on going through the calculation,
found that some quantities omitted by Laplace as unimportant had
in reality a very sensible effect on the result, so that a certain
quantity expressing the rate of increase of the moon’s motion came out
to be between 5″ and 6″, instead of being about 10″, as Laplace had
found and as observation required. The correction was disputed at first
by several of the leading experts, but was confirmed independently
by Delaunay and is now accepted. The moon appears in consequence to
have a certain very minute increase in speed for which the theory of
gravitation affords no explanation. An ingenious though by no means
certain explanation was suggested by Delaunay in 1865. It had been
noticed by Kant that =tidal friction=—that is, the friction set up
between the solid earth and the ocean as the result of the tidal
motion of the latter—would have the effect of checking to some extent
the rotation of the earth; but as the effect seemed to be excessively
minute and incapable of precise calculation it was generally ignored.
An attempt to calculate its amount was, however, made in 1853 by
_William Ferrel_, who also pointed out that, as the period of the
earth’s rotation—the day—is our fundamental unit of time, a reduction
of the earth’s rate of rotation involves the lengthening of our unit
of time, and consequently produces an apparent increase of speed in
all other motions measured in terms of this unit. Delaunay, working
independently, arrived at like conclusions, and shewed that tidal
friction might thus be capable of producing just such an alteration
in the moon’s motion as had to be explained; if this explanation were
accepted the observed motion of the moon would give a measure of the
effect of tidal friction. The minuteness of the quantities involved
is shewn by the fact that an alteration in the earth’s rotation
equivalent to the lengthening of the day by 1∕10 second in 10,000 years
is sufficient to explain the acceleration in question. Moreover it
is by no means certain that the usual estimate of the amount of this
acceleration—based as it is in part on ancient eclipse observations—is
correct, and even then a part of it may conceivably be due to some
indirect effect of gravitation even more obscure than that detected by
Laplace, or to some other cause hitherto unsuspected.

288. Most of the writers on lunar theory already mentioned have also
made contributions to various parts of planetary theory, but some of
the most important advances in planetary theory made since the death of
Laplace have been due to the French mathematician _Urbain Jean Joseph
Leverrier_ (1811-1877), whose methods of determining the distance of
the sun have been already referred to (§ 282). His first important
astronomical paper (1839) was a discussion of the stability (chapter
XI., § 245) of the system formed by the sun and the three largest
and most distant planets then known, Jupiter, Saturn, and Uranus.
Subsequently he worked out afresh the theory of the motion of the sun
and of each of the principal planets, and constructed tables of them,
which at once superseded earlier ones, and are now used as the basis
of the chief planetary calculations in the _Nautical Almanac_ and most
other astronomical almanacs. Leverrier failed to obtain a satisfactory
agreement between observation and theory in the case of Mercury, a
planet which has always given great trouble to astronomers, and was
inclined to explain the discrepancies as due to the influence either
of a planet revolving between Mercury and the sun or of a number of
smaller bodies analogous to the minor planets (§ 294).

Researches of a more abstract character, connecting planetary theory
with some of the most recent advances in pure mathematics, have
been carried out by _Hugo Gyldén_ (1841-1896), while one of the most
eminent pure mathematicians of the day, M. _Henri Poincaré_ of Paris,
has recently turned his attention to astronomy, and is engaged in
investigations which, though they have at present but little bearing on
practical astronomy, seem likely to throw important light on some of
the general problems of celestial mechanics.

289. One memorable triumph of gravitational astronomy, the discovery of
Neptune, has been described so often and so fully elsewhere[166] that
a very brief account will suffice here. Soon after the discovery of
Uranus (chapter XII., § 253) it was found that the planet had evidently
been observed, though not recognised as a planet, as early as 1690, and
on several occasions afterwards.

When the first attempts were made to compute its orbit carefully, it
was found impossible satisfactorily to reconcile the earlier with the
later observations, and in Bouvard’s tables (chapter XI., § 247, note)
published in 1821 the earlier observations were rejected. But even
this drastic measure did not cure the evil; discrepancies between the
observed and calculated places soon appeared and increased year by
year. Several explanations were proposed, and more than one astronomer
threw out the suggestion that the irregularities might be due to the
attraction of a hitherto unknown planet. The first serious attempt to
deduce from the irregularities in the motion of Uranus the position
of this hypothetical body was made by Adams immediately after taking
his degree (1843). By October 1845 he had succeeded in constructing an
orbit for the new planet, and in assigning for it a position differing
(as we now know) by less than 2° (four times the diameter of the full
moon) from its actual position. No telescopic search for it was,
however, undertaken. Meanwhile, Leverrier had independently taken up
the inquiry, and by August 31st, 1846, he, like Adams, had succeeded
in determining the orbit and the position of the disturbing body. On
the 23rd of the following month Dr. Galle of the Berlin Observatory
received from Leverrier a request to search for it, and on the same
evening found close to the position given by Leverrier a strange body
shewing a small planetary disc, which was soon recognised as a new
planet, known now as Neptune.

It may be worth while noticing that the error in the motion of Uranus
which led to this remarkable discovery never exceeded 2′, a quantity
imperceptible to the ordinary eye; so that if two stars were side
by side in the sky, one in the true position of Uranus and one in
the calculated position as given by Bouvard’s tables, an observer of
ordinary eyesight would see one star only.

290. The lunar tables of Hansen and Professor Newcomb, and the
planetary and solar tables of Leverrier, Professor Newcomb, and
Dr. Hill, represent the motions of the bodies dealt with much more
accurately than the corresponding tables based on Laplace’s work, just
as these were in turn much more accurate than those of Euler, Clairaut,
and Halley. But the agreement between theory and observation is by no
means perfect, and the discrepancies are in many cases greater than
can be explained as being due to the necessary imperfections in our
observations.

The two most striking cases are perhaps those of Mercury and the moon.
Leverrier’s explanation of the irregularities of the former (§ 288) has
never been fully justified or generally accepted; and the position of
the moon as given in the _Nautical Almanac_ and in similar publications
is calculated by means of certain corrections to Hansen’s tables
which were deduced by Professor Newcomb from observation and have no
justification in the theory of gravitation.

291. The calculation of the paths of comets has become of some
importance during this century owing to the discovery of a number of
comets revolving round the sun in comparatively short periods. Halley’s
comet (chapter XI., § 231) reappeared duly in 1835, passing through
its perihelion within a few days of the times predicted by three
independent calculators; and it may be confidently expected again about
1910. Four other comets are now known which, like Halley’s, revolve
in elongated elliptic orbits, completing a revolution in between 70
and 80 years; two of these have been seen at two returns, that known
as Olbers’s comet in 1815 and 1887, and the Pons-Brooks comet in 1812
and 1884. Fourteen other comets with periods varying between 3-1∕3
years (Encke’s) and 14 years (Tuttle’s), have been seen at more than
one return; about a dozen more have periods estimated at less than
a century; and 20 or 30 others move in orbits that are decidedly
elliptic, though their periods are longer and consequently not known
with much certainty. Altogether the paths of about 230 or 240 comets
have been computed, though many are highly uncertain.

[Illustration: FIG. 87.—The path of Halley’s comet.]

292. In the theory of the tides the first important advance made after
the publication of the _Mécanique Céleste_ was the collection of actual
tidal observations on a large scale, their interpretation, and their
comparison with the results of theory. The pioneers in this direction
were Lubbock (§ 286), who presented a series of papers on the subject
to the Royal Society in 1830-37, and _William Whewell_ (1794-1866),
whose papers on the subject appeared between 1833 and 1851. Airy (§
281), then Astronomer Royal, also published in 1845 an important
treatise dealing with the whole subject, and discussing in detail the
theory of tides in bodies of water of limited extent and special form.
The analysis of tidal observations, a large number of which taken from
all parts of the world are now available, has subsequently been carried
much further by new methods due to Lord _Kelvin_ and Professor _G. H.
Darwin_. A large quantity of information is thus available as to the
way in which tides actually vary in different places and according to
different positions of the sun and moon.

Of late years a good deal of attention has been paid to the effect of
the attraction of the sun and moon in producing alterations—analogous
to oceanic tides—in the earth itself. No body is perfectly rigid, and
the forces in question must therefore produce some tidal effect. The
problem was first investigated by Lord Kelvin in 1863, subsequently
by Professor Darwin and others. Although definite numerical results
are hardly attainable as yet, the work so far carried out points to
the comparative smallness of these bodily tides and the consequent
great rigidity of the earth, a result of interest in connection with
geological inquiries into the nature of the interior of the earth.

Some speculations connected with tidal friction are referred to
elsewhere (§ 320).

293. The series of propositions as to the stability of the solar
system established by Lagrange and Laplace (chapter XI., §§ 244,
245), regarded as abstract propositions mathematically deducible from
certain definite assumptions, have been confirmed and extended by
later mathematicians such as Poisson and Leverrier; but their claim to
give information as to the condition of the actual solar system at an
indefinitely distant future time receives much less assent now than
formerly. The general trend of scientific thought has been towards the
fuller recognition of the merely approximate and probable character of
even the best ascertained portions of our knowledge; “exact,” “always,”
and “certain” are words which are disappearing from the scientific
vocabulary, except as convenient abbreviations. Propositions which
profess to be—or are commonly interpreted as being—“exact” and valid
throughout all future time are consequently regarded with considerable
distrust, unless they are clearly mere abstractions.

In the case of the particular propositions in question the progress of
astronomy and physics has thrown a good deal of emphasis on some of the
points in which the assumptions required by Lagrange and Laplace are
not satisfied by the actual solar system.

It was assumed for the purposes of the stability theorems that the
bodies of the solar system are perfectly rigid; in other words, the
motions relative to one another of the parts of any one body were
ignored. Both the ordinary tides of the ocean and the bodily tides to
which modern research has called attention were therefore left out of
account. Tidal friction, though at present very minute in amount (§
287), differs essentially from the perturbations which form the main
subject-matter of gravitational astronomy, inasmuch as its action is
irreversible. The stability theorems shewed in effect that the ordinary
perturbations produced effects which sooner or later compensated one
another, so that if a particular motion was accelerated at one time
it would be retarded at another; but this is not the case with tidal
friction. Tidal action between the earth and the moon, for example,
gradually lengthens both the day and the month, and increases the
distance between the earth and the moon. Solar tidal action has a
similar though smaller effect on the sun and earth. The effect in each
case—as far as we can measure it at all—seems to be minute almost
beyond imagination, but there is no compensating action tending at
any time to reverse the process. And on the whole the energy of the
bodies concerned is thereby lessened. Again, modern theories of light
and electricity require space to be filled with an “ether” capable of
transmitting certain waves; and although there is no direct evidence
that it in any way affects the motions of earth or planets, it is
difficult to imagine a medium so different from all known forms
of ordinary matter as to offer _no_ resistance to a body moving
through it. Such resistance would have the effect of slowly bringing
the members of the solar system nearer to the sun, and gradually
diminishing their times of revolution round it. This is again an
irreversible tendency for which we know of no compensation.

In fact, from the point of view which Lagrange and Laplace occupied,
the solar system appeared like a clock which, though not going
quite regularly, but occasionally gaining and occasionally losing,
nevertheless required no winding up; whereas modern research emphasises
the analogy to a clock which after all is running down, though at an
excessively slow rate. Modern study of the sun’s heat (§ 319) also
indicates an irreversible tendency towards the “running down” of the
solar system in another way.

294. Our account of modern descriptive astronomy may conveniently begin
with planetary discoveries.

The first day of the 19th century was marked by the discovery of a new
planet, known as Ceres. It was seen by _Giuseppe Piazzi_ (1746-1826)
as a strange star in a region of the sky which he was engaged in
mapping, and soon recognised by its motion as a planet. Its orbit—first
calculated by Gauss (§ 276)—shewed it to belong to the space between
Mars and Jupiter, which had been noted since the time of Kepler as
abnormally large. That a planet should be found in this region was
therefore no great surprise; but the discovery by _Heinrich Olbers_
(1758-1840), scarcely a year later (March 1802), of a second body
(_Pallas_), revolving at nearly the same distance from the sun, was
wholly unexpected, and revealed an entirely new planetary arrangement.
It was an obvious conjecture that if there was room for two planets
there was room for more, and two fresh discoveries (_Juno_ in 1804,
_Vesta_ in 1807) soon followed.

[Illustration: FIG. 88.—Photographic trail of a minor planet.]

The new bodies were very much smaller than any of the other planets,
and, so far from readily shewing a planetary disc like their neighbours
Mars and Jupiter, were barely distinguishable in appearance from fixed
stars, except in the most powerful telescopes of the time; hence the
name =asteroid= (suggested by William Herschel) or =minor planet= has
been generally employed to distinguish them from the other planets.
Herschel attempted to measure their size, and estimated the diameter
of the largest at under 200 miles (that of Mercury, the smallest of
the ordinary planets, being 3000), but the problem was in reality
too difficult even for his unrivalled powers of observation. The minor
planets were also found to be remarkable for the great inclination
and eccentricity of some of the orbits; the path of Pallas, for
example, makes an angle of 35° with the ecliptic, and its eccentricity
is 1∕4, so that its least distance from the sun is not much more
than half its greatest distance. These characteristics suggested to
Olbers that the minor planets were in reality fragments of a primeval
planet of moderate dimensions which had been blown to pieces, and the
theory, which fitted most of the facts then known, was received with
great favour in an age when “catastrophes” were still in fashion as
scientific explanations.

The four minor planets named were for nearly 40 years the only ones
known; then a fifth was discovered in 1845 by _Karl Ludwig Hencke_
(1793-1866) after 15 years, of search. Two more were found in 1847,
another in 1848, and the number has gone on steadily increasing
ever since. The process of discovery has been very much facilitated
by improvements in star maps, and latterly by the introduction of
photography. In this last method, first used by Dr. _Max Wolf_ of
Heidelberg in 1891, a photographic plate is exposed for some hours;
any planet present in the region of the sky photographed, having moved
sensibly relatively to the stars in this period, is thus detected by
the trail which its image leaves on the plate. The annexed figure shews
(near the centre) the trail of the minor planet _Svea_, discovered by
Dr. Wolf on March 21st, 1892.

At the end of 1897 no less than 432 minor planets were known, of which
92 had been discovered by a single observer, M. _Charlois_ of Nice, and
only nine less by Professor _Palisa_ of Vienna.

The paths of the minor planets practically occupy the whole region
between the paths of Mars and Jupiter, though few are near the
boundaries; no orbit is more inclined to the ecliptic than that of
Pallas, and the eccentricities range from almost zero up to about 1∕3.

Fig. 89 shews the orbits of the first two minor planets discovered, as
well as of No. 323 (_Brucia_), which comes nearest to the sun, and of
No. 361 (not yet named), which goes farthest from it. All the orbits
are described in the standard, or west to east, direction. The most
interesting characteristic in the distribution of the minor planets,
first noted in 1866 by _Daniel Kirkwood_ (1815-1895) is the existence
of comparatively clear spaces in the regions where the disturbing
action of Jupiter would by Lagrange’s principle (chapter XI., § 243)
be most effective: for instance, at a distance from the sun about
five-eighths that of Jupiter, a planet would by Kepler’s law revolve
exactly _twice_ as fast as Jupiter; and accordingly there is a gap
among the minor planets at about this distance.

[Illustration: FIG. 89.—Paths of minor planets.]

Estimates of the sizes and masses of the minor planets are still very
uncertain. The first direct measurement of any of the discs which
seem reliable are those of Professor _E. E. Barnard_, made at the Lick
Observatory in 1894 and 1895; according to these the three largest
minor planets, Ceres, Pallas, and Vesta, have diameters of nearly 500
miles, about 300 and about 250 miles respectively. Their sizes compared
with the moon are shewn on the diagram (fig. 90). An alternative
method—the only one available except for a few of the very largest of
the minor planets—is to measure the amount of light received, and hence
to deduce the size, on the assumption that the reflective power is the
same as that of some known planet. This method gives diameters of about
300 miles for the brightest and of about a dozen miles for the faintest
known.

[Illustration: FIG. 90.—Comparative sizes of three minor planets and
the moon.]

Leverrier calculated from the perturbations of Mars that the total
mass of all known or unknown bodies between Mars and Jupiter could not
exceed a fourth that of the earth; but such knowledge of the sizes as
we can derive from light-observations seems to indicate that the total
mass of those at present known is many hundred times less than this
limit.

295. Neptune and the minor planets are the only planets which have been
discovered during this century, but several satellites have been added
to our system.

[Illustration: FIG. 91.—Saturn and its system.]

Barely a fortnight after the discovery of Neptune (1846) a satellite
was detected by _William Lassell_ (1799-1880) at Liverpool. Like the
satellites of Uranus, this revolves round its primary from east to
west—that is, in the direction contrary to that of all the other known
motions of the solar system (certain long-period comets not being
counted).

[Illustration: FIG. 92.—Mars and its satellites.]

Two years later (September 16th, 1848) _William Cranch Bond_
(1789-1859) discovered, at the Harvard College Observatory, an,
eighth satellite of Saturn, called _Hyperion_, which was detected
independently by Lassell two days afterwards. In the following year
Bond discovered that Saturn was accompanied by a third comparatively
dark ring-now commonly known as the =crape ring=—lying immediately
inside the bright rings (see fig. 95); and the discovery was made
independently a fortnight later by _William Rutter Dawes_ (1799-1868)
in England. Lassell discovered in 1851 two new satellites of Uranus,
making a total of four belonging to that planet. The next discoveries
were those of two satellites of Mars, known as _Deimos_ and _Phobos_,
by Professor _Asaph Hall_ of Washington on August 11th and 17th, 1877.
These are remarkable chiefly for their close proximity to Mars and
their extremely rapid motion, the nearer one revolving more rapidly
than Mars rotates, so that to the Martians it must rise in the west
and set in the east. Lastly, Jupiter’s system received an addition
after nearly three centuries by Professor Barnard’s discovery at the
Lick Observatory (September 9th, 1892) of an extremely faint fifth
satellite, a good deal nearer to Jupiter than the nearest of Galilei’s
satellites (chapter VI., § 121).

[Illustration: FIG. 93.—Jupiter and its satellites.]

296. The surfaces of the various planets and satellites have been
watched with the utmost care by an army of observers, but the
observations have to a large extent remained without satisfactory
interpretation, and little is known of the structure or physical
condition of the bodies concerned.

[Illustration: FIG. 94.—The Apennines and adjoining regions of the
moon. From a photograph taken at the Paris Observatory.]

Astronomers are naturally most familiar with the surface of our
nearest neighbour, the moon. The visible half has been elaborately
mapped, and the heights of the chief mountain ranges measured by means
of their shadows. Modern knowledge has done much to dispel the view,
held by the earlier telescopists and shared to some extent even by
Herschel, that the moon closely resembles the earth and is suitable for
inhabitants like ourselves. The dark spaces which were once taken to
be seas and still bear that name are evidently covered with dry rock;
and the craters with which the moon is covered are all—with one or
two doubtful exceptions—extinct; the long dark lines known as =rills=
and formerly taken for river-beds have clearly no water in them. The
question of a lunar atmosphere is more difficult: if there is air its
density must be very small, some hundredfold less than that of our
atmosphere at the surface of the earth; but with this restriction
there seems to be no bar to the existence of a lunar atmosphere
of considerable extent, and it is difficult to explain certain
observations without assuming the existence of some atmosphere.

297. Mars, being the nearest of the superior planets, is the most
favourably situated for observation. The chief markings on its
surface—provisionally interpreted as being land and water—are fairly
permanent and therefore recognisable; several tolerably consistent maps
of the surface have been constructed; and by observation of certain
striking features the rotation period has been determined to a fraction
of a second. Signor _Schiaparelli_ of Milan detected at the opposition
of 1877 a number of intersecting dark lines generally known as
=canals=, and as the result of observations made during the opposition
of 1881-82 announced that certain of them appeared doubled, two nearly
parallel lines being then seen instead of one. These remarkable
observations have been to a great extent confirmed by other observers,
but remain unexplained.

The visible surfaces of Jupiter and Saturn appear to be layers of
clouds; the low density of each planet (1·3 and ·7 respectively, that
of water being 1 and of the earth 5·5), the rapid changes on the
surface, and other facts indicate that these planets are to a great
extent in a fluid condition, and have a high temperature at a very
moderate distance below the visible surface. The surface markings are
in each case definite enough for the rotation periods to be fixed with
some accuracy; though it is clear in the case of Jupiter, and probably
also in that of Saturn, that—as with the sun (§ 298)—different parts of
the surface move at different rates.

Laplace had shewn that Saturn’s ring (or rings) could not be, as it
appeared, a uniform solid body; he rashly inferred—without any complete
investigation—that it might be an irregularly weighted solid body. The
first important advance was made by _James Clerk Maxwell_ (1831-1879),
best known as a writer on electricity and other branches of physics.
Maxwell shewed (1857) that the rings could neither be continuous solid
bodies nor liquid, but that all the important dynamical conditions
would be satisfied if they were made up of a very large number of
small solid bodies revolving independently round the sun.[167] The
theory thus suggested on mathematical grounds has received a good deal
of support from telescopic evidence. The rings thus bear to Saturn
a relation having some analogy to that which the minor planets bear
to the sun; and Kirkwood pointed out in 1867 that Cassini’s division
between the two main rings can be explained by the perturbations due to
certain of the satellites, just as the corresponding gaps in the minor
planets can be explained by the action of Jupiter (§ 294).

The great distance of Uranus and Neptune naturally makes the study
of them difficult, and next to nothing is known of the appearance or
constitution of either; their rotation periods are wholly uncertain.

[Illustration: FIG. 95.—Saturn and its rings. From a drawing by
Professor Barnard.]

Mercury and Venus, being inferior planets, are never very far from
the sun in the sky, and therefore also extremely difficult to observe
satisfactorily. Various bright and dark markings on their surfaces have
been recorded, but different observers give very different accounts of
them. The rotation periods are also very uncertain, though a good many
astronomers support the view put forward by Sig. Schiaparelli, in 1882
and 1890 for Mercury and Venus respectively, that each rotates in a
time equal to its period of revolution round the sun, and thus always
turns the same face towards the sun. Such a motion—which is analogous
to that of the moon round the earth and of Japetus round Saturn
(chapter XII., § 267)—could be easily explained as the result of tidal
action at some past time when the planets were to a great extent fluid.

[Illustration: FIG. 96.—A group of sun-spots. From a photograph taken
by M. Janssen at Meudon on April 1st, 1894.]

298. Telescopic study of the surface of the sun during the century
has resulted in an immense accumulation of detailed knowledge of
peculiarities of the various markings on the surface. The most
interesting results of a general nature are connected with the
distribution and periodicity of sun-spots. The earliest telescopists
had noticed that the number of spots visible on the sun varied from
time to time, but no law of variation was established till 1851, when
_Heinrich Schwabe_ of Dessau (1789-1875) published in Humboldt’s
_Cosmos_ the results of observations of sun-spots carried out during
the preceding quarter of a century, shewing that the number of spots
visible increased and decreased in a tolerably regular way in a period
of about ten years.

Earlier records and later observations have confirmed the general
result, the period being now estimated as slightly over 11 years on
the average, though subject to considerable fluctuations. A year later
(1852) three independent investigators, Sir _Edward Sabine_ (1788-1883)
in England, _Rudolf Wolf_ (1816-1893) and _Alfred Gautier_ (1793-1881)
in Switzerland, called attention to the remarkable similarity
between the periodic variations of sun-spots and of various magnetic
disturbances on the earth. Not only is the period the same, but it
almost invariably happens that when spots are most numerous on the
sun magnetic disturbances are most noticeable on the earth, and that
similarly the times of scarcity of the two sets of phenomena coincide.
This wholly unexpected and hitherto quite unexplained relationship
has been confirmed by the occurrence on several occasions of decided
magnetic disturbances simultaneously with rapid changes on the surface
of the sun.

A long series of observations of the position of spots on the sun
undertaken by _Richard Christopher Carrington_ (1826-1875) led to the
first clear recognition of the difference in the rate of rotation
of the different parts of the surface of the sun, the period of
rotation being fixed (1859) at about 25 days at the equator, and two
and a half days longer half-way between the equator and the poles;
while in addition spots were seen to have also independent “proper
motions.” Carrington also established (1858) the scarcity of spots in
the immediate neighbourhood of the equator, and confirmed statistically
their prevalence in the adjacent regions, and their great scarcity more
than about 35° from the equator; and noticed further certain regular
changes in the distribution of spots on the sun in the course of the
11-year cycle.

Wilson’s theory (chapter XII., § 268) that spots are depressions
was confirmed by an extensive series of photographs taken at Kew in
1858-72, shewing a large preponderance of cases of the perspective
effect noticed by him; but, on the other hand, Mr. _F. Howlett_,
who has watched the sun for some 35 years and made several thousand
drawings of spots, considers (1894) that his observations are decidedly
against Wilson’s theory. Other observers are divided in opinion.

299. =Spectrum analysis=, which has played such an important part in
recent astronomical work, is essentially a method of ascertaining the
nature of a body by a process of sifting or analysing into different
components the light received from it.

It was first clearly established by Newton, in 1665-66 (chapter IX., §
168), that ordinary white light, such as sunlight, is composite, and
that by passing a beam of sunlight—with proper precautions—through a
glass prism it can be decomposed into light of different colours; if
the beam so decomposed is received on a screen, it produces a band of
colours known as a =spectrum=, red being at one end and violet at the
other.

[Illustration: FIG. 97.—Fraunhofer’s map of the solar spectrum. (The
red end of the spectrum is on the left, the violet on the right.)]

Now according to modern theories light consists essentially of a series
of disturbances or waves transmitted at extremely short but regular
intervals from the luminous object to the eye, the medium through
which the disturbances travel being called =ether=. The most important
characteristic distinguishing different kinds of light is the interval
of time or space between one wave and the next, which is generally
expressed by means of =wave-length=, or the distance between any point
of one wave and the corresponding point of the next. Differences in
wave-length shew themselves most readily as differences of colour;
so that light of a particular colour found at a particular part of
the spectrum has a definite wave-length. At the extreme violet end of
the spectrum, for example, the wave-length is about fifteen millionths
of an inch, at the red end it is about twice as great; from which it
follows (§ 283), from the known velocity of light, that when we look
at the red end of a spectrum about 400 billion waves of light enter
the eye per second, and twice that number when we look at the other
end. Newton’s experiment thus shews that a prism sorts out light of a
composite nature according to the wave-length of the different kinds
of light present. The same thing can be done by substituting for the
prism a so-called =diffraction-grating=, and this is for many purposes
superseding the prism. In general it is necessary, to ensure purity
in the spectrum and to make it large enough, to admit light through a
narrow slit, and to use certain lenses in combination with one or more
prisms or a grating; and the arrangement is such that the spectrum is
not thrown on to a screen, but either viewed directly by the eye or
photographed. The whole apparatus is known as a =spectroscope=.

The solar spectrum appeared to Newton as a continuous band of colours;
but in 1802 _William Hyde Wollaston_ (1766-1828) observed certain dark
lines running across the spectrum, which he took to be the boundaries
of the natural colours. A few years later (1814-15) the great Munich
optician _Joseph Fraunhofer_ (1787-1826) examined the sun’s spectrum
much more carefully, and discovered about 600 such dark lines, the
positions of 324 of which he mapped (see fig. 97). These dark lines are
accordingly known as =Fraunhofer lines=: for purposes of identification
Fraunhofer attached certain letters of the alphabet to a few of the
most conspicuous; the rest are now generally known by the wave-length
of the corresponding kind of light.

It was also gradually discovered that dark bands could be produced
artificially in spectra by passing light through various coloured
substances; and that, on the other hand, the spectra of certain flames
were crossed by various _bright_ lines.

Several attempts were made to explain and to connect these various
observations, but the first satisfactory and tolerably complete
explanation was given in 1859 by _Gustav Robert Kirchhoff_ (1824-1887)
of Heidelberg, who at first worked in co-operation with the chemist
Bunsen.

Kirchhoff shewed that a luminous solid or liquid—or, as we now know, a
highly compressed gas—gives a continuous spectrum; whereas a substance
in the gaseous state gives a spectrum consisting of bright lines
(with or without a faint continuous spectrum), and these bright lines
depend on the particular substance and are characteristic of it.
Consequently the presence of a particular substance in the form of gas
in a hot body can be inferred from the presence of its characteristic
lines in the spectrum of the light. The _dark_ lines in the solar
spectrum were explained by the fundamental principle—often known as
Kirchhoff’s law—that a body’s capacity for stopping or absorbing light
of a particular wave-length is proportional to its power, under like
conditions, of giving out the same light. If, in particular, light from
a luminous solid or liquid body, giving a continuous spectrum, passes
through a gas, the gas absorbs light of the same wave-length as that
which it itself gives out: if the gas gives out more light of these
particular wave-lengths than it absorbs, then the spectrum is crossed
by the corresponding bright lines; but if it absorbs more than it gives
out, then there is a deficiency of light of these wave-lengths and the
corresponding parts of the spectrum appear dark—that is, the spectrum
is crossed by dark lines in the same position as the bright lines in
the spectrum of the gas alone. Whether the gas absorbs more or less
than it gives out is essentially a question of temperature, so that
if light from a hot solid or liquid passes through a gas at a higher
temperature a spectrum crossed by bright lines is the result, whereas
if the gas is cooler than the body behind it dark lines are seen in the
spectrum.

300. The presence of the Fraunhofer lines in the spectrum of the sun
shews that sunlight comes from a hot solid or liquid body (or from a
highly compressed gas), and that it has passed through cooler gases
which have absorbed light of the wave-lengths corresponding to the dark
lines. These gases must be either round the sun or in our atmosphere:
and it is not difficult to shew that, although some of the Fraunhofer
lines are due to our atmosphere, the majority cannot be, and are
therefore caused by gases in the atmosphere of the sun.

For example, the metal sodium when vaporised gives a spectrum
characterised by two nearly coincident bright lines in the yellow
part of the spectrum; these agree in position with a pair of dark
lines (known as D) in the spectrum of the sun (see fig. 97); Kirchhoff
inferred therefore that the atmosphere of the sun contains sodium. By
comparison of the dark lines in the spectrum of the sun with the bright
lines in the spectra of metals and other substances, their presence
or absence in the solar atmosphere can accordingly be ascertained. In
the case of iron—which has an extremely complicated spectrum—Kirchhoff
succeeded in identifying 60 lines (since increased to more than 2,000)
in its spectrum with dark lines in the spectrum of the sun. Some
half-dozen other known elements were also identified by Kirchhoff in
the sun.

The inquiry into solar chemistry thus started has since been prosecuted
with great zeal. Improved methods and increased care have led to the
construction of a series of maps of the solar spectrum, beginning
with Kirchhoff’s own, published in 1861-62, of constantly increasing
complexity and accuracy. Knowledge of the spectra of the metals has
also been greatly extended. At the present time between 30 and 40
elements have been identified in the sun, the most interesting besides
those already mentioned being hydrogen, calcium, magnesium, and carbon.

The first spectroscopic work on the sun dealt only with the light
received from the sun as a whole, but it was soon seen that by throwing
an image of the sun on to the slit of the spectroscope by means of a
telescope the spectrum of a particular part of the sun’s surface, such
as a spot or a facula, could be obtained; and an immense number of
observations of this character have been made.

301. Observations of total eclipses of the sun have shewn that the
bright surface of the sun as we ordinarily see it is not the whole,
but that outside this there is an envelope of some kind too faint to
be seen ordinarily but becoming visible when the intense light of
the sun itself is cut off by the moon. A white halo of considerable
extent round the eclipsed sun, now called the =corona=, is referred to
by Plutarch, and discussed by Kepler (chapter VII., § 145) Several
18th century astronomers noticed a red streak along some portion of
the common edge of the sun and moon, and red spots or clouds here
and there (cf. chapter X., § 205). But little serious attention was
given to the subject till after the total solar eclipse of 1842.
Observations made then and at the two following eclipses of 1851 and
1860, in the latter of which years photography was for the first time
effectively employed, made it evident that the red streak represented
a continuous envelope of some kind surrounding the sun, to which the
name of =chromosphere= has been given, and that the red objects,
generally known as =prominences=, were in general projecting parts of
the chromosphere, though sometimes detached from it. At the eclipse of
1868 the spectrum of the prominences and the chromosphere was obtained,
and found to be one of bright lines, shewing that they consisted of
gas. Immediately afterwards M. _Janssen_, who was one of the observers
of the eclipse, and Sir _J. Norman Lockyer_ independently devised a
method whereby it was possible to get the spectrum of a prominence at
the edge of the sun’s disc in ordinary daylight, without waiting for an
eclipse; and a modification introduced by Sir _William Huggins_ in the
following year (1869) enabled the form of a prominence to be observed
spectroscopically. Recently (1892) Professor _G. E. Hale_ of Chicago
has succeeded in obtaining by a photographic process a representation
of the whole of the chromosphere and prominences, while the same method
gives also photographs of faculae (chapter VIII., § 153) on the visible
surface of the sun.

The most important lines ordinarily present in the spectrum of the
chromosphere are those of hydrogen, two lines (H and K) which have
been identified with some difficulty as belonging to calcium, and a
yellow line the substance producing which, known as =helium=, has only
recently (1895) been discovered on the earth. But the chromosphere when
disturbed and many of the prominences give spectra containing a number
of other lines.

[Illustration: FIG. 98.—The total solar eclipse of August 29th, 1886.
From a drawing based on photographs by Dr. Schuster and Mr. Maunder.]

The corona was for some time regarded as of the nature of an optical
illusion produced in the atmosphere. That it is, at any rate in great
part, an actual appendage of the sun was first established in 1869 by
the American astronomers Professor _Harkness_ and Professor _C.
A. Young_, who discovered a bright line—of unknown origin[168]—in
its spectrum, thus shewing that it consists in part of glowing gas.
Subsequent spectroscopic work shews that its light is partly reflected
sunlight.

The corona has been carefully studied at every solar eclipse during
the last 30 years, both with the spectroscope and with the telescope,
supplemented by photography, and a number of ingenious theories of its
constitution have been propounded; but our present knowledge of its
nature hardly goes beyond Professor Young’s description of it as “an
inconceivably attenuated cloud of gas, fog, and dust, surrounding the
sun, formed and shaped by solar forces.”

302. The spectroscope also gives information as to certain motions
taking place on the sun. It was pointed out in 1842 by _Christian
Doppler_ (1803-1853), though in an imperfect and partly erroneous way,
that if a luminous body is approaching the observer, or _vice versa_,
the waves of light are as it were crowded together and reach the eye at
shorter intervals than if the body were at rest, and that the character
of the light is thereby changed. The colour and the position in the
spectrum both depend on the interval between one wave and the next, so
that if a body giving out light of a particular wave-length, _e.g._
the blue light corresponding to the F line of hydrogen, is approaching
the observer rapidly, the line in the spectrum appears slightly on
one side of its usual position, being displaced towards the violet
end of the spectrum; whereas if the body is receding the line is, in
the same way, displaced in the opposite direction. This result is
usually known as =Doppler’s principle=. The effect produced can easily
be expressed numerically. If, for example, the body is approaching
with a speed equal to 1∕1000 of light, then 1001 waves enter the eye
or the spectroscope in the same time in which there would otherwise
only be 1000; and there is in consequence a virtual shortening of the
wave-length in the ratio of 1001 to 1000. So that if it is found that a
line in the spectrum of a body is displaced from its ordinary position
in such a way that its wave-length is apparently decreased by 1∕1000
part, it may be inferred that the body is approaching with the speed
just named, or about 186 miles per second, and if the wave-length
appears increased by the same amount (the line being displaced towards
the red end of the spectrum) the body is receding at the same rate.

Some of the earliest observations of the prominences by Sir J. N.
Lockyer (1868), and of spots and other features of the sun by the same
and other observers, shewed displacements and distortions of the lines
in the spectrum, which were soon seen to be capable of interpretation
by this method, and pointed to the existence of violent disturbances in
the atmosphere of the sun, velocities as great as 300 miles per second
being not unknown. The method has received an interesting confirmation
from observations of the spectrum of opposite edges of the sun’s
disc, of which one is approaching and the other receding owing to the
rotation of the sun. Professor _Dunér_ of Upsala has by this process
ascertained (1887-89) the rate of rotation of the surface of the sun
beyond the regions where spots exist, and therefore outside the limits
of observations such as Carrington’s (§ 298).

303. The spectroscope tells us that the atmosphere of the sun contains
iron and other metals in the form of vapour; and the photosphere,
which gives the continuous part of the solar spectrum, is certainly
hotter. Moreover everything that we know of the way in which heat is
communicated from one part of a body to another shews that the outer
regions of the sun, from which heat and light are radiating on a very
large scale, must be the coolest parts, and that the temperature in
all probability rises very rapidly towards the interior. These facts,
coupled with the low density of the sun (about a fourth that of the
earth) and the violently disturbed condition of the surface, indicate
that the bulk of the interior of the sun is an intensely hot and highly
compressed mass of gas. Outside this come in order, their respective
boundaries and mutual relations being, however, very uncertain,
first the photosphere, generally regarded as a cloud-layer, then the
=reversing stratum= which produces most of the Fraunhofer lines, then
the chromosphere and prominences, and finally the corona. Sun-spots,
faculae, and prominences have been explained in a variety of
different ways as joint results of solar disturbances of various kinds;
but no detailed theory that has been given explains satisfactorily more
than a fraction of the observed facts or commands more than a very
limited amount of assent among astronomical experts.

[Illustration: FIG. 99.—The great comet of 1882 (ii) on November 7th.
From a photograph by Dr. Gill.]

304. More than 200 comets have been seen during the present century;
not only have the motions of most of them been observed and their
orbits computed (§ 291), but in a large number of cases the appearance
and structure of the comet have been carefully observed telescopically,
while latterly spectrum analysis and photography have also been
employed.

Independent lines of inquiry point to the extremely unsubstantial
character of a comet, with the possible exception of the bright central
part or =nucleus=, which is nearly always present. More than once,
as in 1767 (chapter XI., § 248), a comet has passed close to some
member of the solar system, and has never been ascertained to affect
its motion. The mass of a comet is therefore very small, but its bulk
or volume, on the other hand, is in general very great, the tail
often being millions of miles in length; so that the density must be
extremely small. Again, stars have often been observed shining through
a comet’s tail (as shewn in fig. 99), and even through the head at no
great distance from the nucleus, their brightness being only slightly,
if at all, affected. Twice at least (1819, 1861) the earth has passed
through a comet’s tail, but we were so little affected that the fact
was only discovered by calculations made after the event. The early
observation (chapter III., § 69) that a comet’s tail points away from
the sun has been abundantly verified; and from this it follows that
very rapid changes in the position of the tail must occur in some
cases. For example, the comet of 1843 passed very close to the sun at
such a rate that in about two hours it had passed from one side of the
sun to the opposite; it was then much too near the sun to be seen, but
if it followed the ordinary law its tail, which was unusually long,
must have entirely reversed its direction within this short time. It
is difficult to avoid the inference that the tail is not a permanent
part of the comet, but is a stream of matter driven off from it in
some way by the action of the sun, and in this respect comparable with
the smoke issuing from a chimney. This view is confirmed by the fact
that the tail is only developed when the comet approaches the sun,
a comet when at a great distance from the sun appearing usually as
an indistinct patch of nebulous light, with perhaps a brighter spot
representing the nucleus. Again, if the tail be formed by an outpouring
of matter from the comet, which only takes place when the comet is
near the sun, the more often a comet approaches the sun the more must
it waste away; and we find accordingly that the short-period comets,
which return to the neighbourhood of the sun at frequent intervals (§
291), are inconspicuous bodies. The same theory is supported by the
shape of the tail. In some cases it is straight, but more commonly it
is curved to some extent, and the curvature is then always _backwards_
in relation to the comet’s motion. Now by ordinary dynamical principles
matter shot off from the head of the comet while it is revolving round
the sun would tend, as it were, to lag behind more and more the farther
it receded from the head, and an apparent backward curvature of the
tail—less or greater according to the speed with which the particles
forming the tail were repelled—would be the result. Variations in
curvature of the tails of different comets, and the existence of two
or more differently curved tails of the same comet, are thus readily
explained by supposing them made of different materials, repelled from
the comet’s head at different speeds.

The first application of the spectroscope to the study of comets was
made in 1864 by _Giambattista Donati_ (1826-1873), best known as the
discoverer of the magnificent comet of 1858. A spectrum of three bright
bands, wider than the ordinary “lines,” was obtained, but they were
not then identified. Four years later Sir William Huggins obtained a
similar spectrum, and identified it with that of a compound of carbon
and hydrogen. Nearly every comet examined since then has shewn in its
spectrum bright bands indicating the presence of the same or some
other hydrocarbon, but in a few cases other substances have also been
detected. A comet is therefore in part at least self-luminous, and some
of the light which it sends us is that of a glowing gas. It also shines
to a considerable extent by reflected sunlight; there is nearly always
a continuous spectrum, and in a few cases—first in 1881—the spectrum
has been distinct enough to shew the Fraunhofer lines crossing it. But
the continuous spectrum seems also to be due in part to solid or liquid
matter in the comet itself, which is hot enough to be self-luminous.

305. The work of the last 30 or 40 years has established a remarkable
relation between comets and the minute bodies which are seen in the
form of =meteors= or =shooting stars=. Only a few of the more important
links in the chain of evidence can, however, be mentioned. Showers of
shooting stars, the occurrence of which has been known from quite early
times, have been shewn to be due to the passage of the earth through a
swarm of bodies revolving in elliptic orbits round the sun. The paths
of four such swarms were ascertained with some precision in 1866-67,
and found in each case to agree closely with the paths of known comets.
And since then a considerable number of other cases of resemblance or
identity between the paths of meteor swarms and of comets have been
detected. One of the four comets just referred to, known as Biela’s,
with a period of between six and seven years, was duly seen on several
successive returns, but in 1845-46 was observed first to become
somewhat distorted in shape, and afterwards to have divided into two
distinct comets; at the next return (1852) the pair were again seen;
but since then nothing has been seen of either portion. At the end of
November in each year the earth almost crosses the path of this comet,
and on two occasions (1872, and 1885) it did so nearly at the time when
the comet was due at the same spot; if, as seemed likely, the comet had
gone to pieces since its last appearance, there seemed a good chance of
falling in with some of its remains, and this expectation was fulfilled
by the occurrence on both occasions of a meteor shower much more
brilliant than that usually observed at the same date.

Biela’s comet is not the only comet which has shewn signs of breaking
up; Brooks’s comet of 1889, which is probably identical with Lexell’s
(chapter XI., § 248), was found to be accompanied by three smaller
companions; as this comet has more than once passed extremely close to
Jupiter, a plausible explanation of its breaking up is at once given
in the attractive force of the planet. Moreover certain systems of
comets, the members of which revolve in the same orbit but separated by
considerable intervals of time, have also been discovered. Tebbutt’s
comet of 1881 moves in practically the same path as one seen in 1807,
and the great comet of 1880, the great comet of 1882 (shewn in fig.
99), and a third which appeared in 1887, all move in paths closely
resembling that of the comet of 1843, while that of 1668 is more
doubtfully connected with the same system. And it is difficult to avoid
regarding the members of a system as fragments of an earlier comet,
which has passed through the stages in which we have actually seen the
comets of Biela and Brooks.

Evidence of such different kinds points to an intimate connection
between comets and meteors, though it is perhaps still premature to
state confidently that meteors are fragments of decayed comets, or that
conversely comets are swarms of meteors.

306. Each of the great problems of sidereal astronomy which Herschel
formulated and attempted to solve has been elaborately studied by the
astronomers of the 19th century. The multiplication of observatories,
improvements in telescopes, and the introduction of photography—to
mention only three obvious factors of progress—have added enormously
to the extent and accuracy of our knowledge of the stars, while the
invention of spectrum analysis has thrown an entirely new light on
several important problems.

William Herschel’s most direct successor was his son _John Frederick
William_ (1792-1871), who was not only an astronomer, but also made
contributions of importance to pure mathematics, to physics, to the
nascent art of photography, and to the philosophy of scientific
discovery. He began his astronomical career about 1816 by re-measuring,
first alone, then in conjunction with _James South_ (1785-1867), a
number of his father’s double stars. The first result of this work was
a catalogue, with detailed measurements, of some hundred double and
multiple stars (published in 1824), which formed a valuable third term
of comparison with his father’s observations of 1781-82 and 1802-03,
and confirmed in several cases the slow motions of revolution the
beginnings of which had been observed before. A great survey of nebulae
followed, resulting in a catalogue (1833) of about 2500, of which
some 500 were new and 2000 were his father’s, a few being due to other
observers; incidentally more than 3000 pairs of stars close enough
together to be worth recording as double stars were observed.

[Illustration]

307. Then followed his well-known expedition to the Cape of Good Hope
(1833-1838), where he “swept” the southern skies in very much the
same way in which his father had explored the regions visible in our
latitude. Some 1200 double and multiple stars, and a rather larger
number of new nebulae, were discovered and studied, while about 500
known nebulae were re-observed; star-gauging on William Herschel’s
lines was also carried out on an extensive scale. A number of special
observations of interest were made almost incidentally during this
survey: the remarkable variable star η _Argus_ and the nebula
surrounding it (a modern photograph of which is reproduced in fig.
100), the wonderful collections of nebulae clusters and stars, known as
the _Nubeculae_ or _Magellanic Clouds_, and Halley’s comet were studied
in turn; and the two faintest satellites of Saturn then known (chapter
XII., § 255) were seen again for the first time since the death of
their discoverer.

An important investigation of a somewhat different character—that
of the amount of heat received from the sun—was also carried out
(1837) during Herschel’s residence at the Cape; and the result agreed
satisfactorily with that of an independent inquiry made at the same
time in France by _Claude Servais Mathias Pouillet_ (1791-1868). In
both cases the heat received on a given area of the earth in a given
time from direct sunshine was measured; and allowance being made for
the heat stopped in the atmosphere as the sun’s rays passed through it,
an estimate was formed of the total amount of heat received annually by
the earth from the sun, and hence of the total amount radiated by the
sun in all directions, an insignificant fraction of which (one part in
2,000,000,000) is alone intercepted by the earth. But the allowance for
the heat intercepted in our atmosphere was necessarily uncertain, and
later work, in particular that of Dr. _S. P. Langley_ in 1880-81, shews
that it was very much under-estimated by both Herschel and Pouillet.
According to Herschel’s results, the heat received annually from the
sun—including that intercepted in the atmosphere—would be sufficient
to melt a shell of ice 120 feet thick covering the whole earth;
according to Dr. Langley, the thickness would be about 160 feet.[169]

308. With his return to England in 1838 Herschel’s career as an
observer came to an end; but the working out of the results of his
Cape observations, the arrangement and cataloguing of his own and his
father’s discoveries, provided occupation for many years. A magnificent
volume on the _Results of Astronomical Observations made during
the years 1834-8 at the Cape of Good Hope_ appeared in 1847; and a
catalogue of all known nebulae and clusters, amounting to 5,079, was
presented to the Royal Society in 1864, while a corresponding catalogue
of more than 10,000 double and multiple stars was never finished,
though the materials collected for it were published posthumously in
1879. John Herschel’s great catalogue of nebulae has since been revised
and enlarged by Dr. _Dreyer_, the result being a list of 7,840 nebulae
and clusters known up to the end of 1887; and a supplementary list
of discoveries made in 1888-94 published by the same writer contains
1,529 entries, so that the total number now known is between 9,000
and 10,000, of which more than half have been discovered by the two
Herschels.

309. Double stars have been discovered and studied by a number of
astronomers besides the Herschels. One of the most indefatigable
workers at this subject was the elder Struve (§ 279), who was
successively director of the two Russian observatories of Dorpat and
Pulkowa. He observed altogether some 2,640 double and multiple stars,
measuring in each case with care the length and direction of the line
joining the two components, and noting other peculiarities, such as
contrasts in colour between the members of a pair. He paid attention
only to double stars the two components of which were not more than
32″ apart, thus rejecting a good many which William Herschel would
have noticed; as the number of known doubles rapidly increased, it was
clearly necessary to concentrate attention on those which might with
some reasonable degree of probability turn out to be genuine binaries
(chapter XII., § 264).

In addition to a number of minor papers Struve published three separate
books on the subject in 1827, 1837, and 1852.[170] A comparison of
his own earlier and later observations, and of both with Herschel’s
earlier ones, shewed about 100 cases of change of relative positions of
two members of a pair, which indicated more or less clearly a motion
of revolution, and further results of a like character have been
obtained from a comparison of Struve’s observations with those of later
observers.

[Illustration:
 FIG. 101.—The orbit of ξ _Ursae_, shewing the relative positions
 of the two components at various times between 1781 and 1897, (The
 observations of 1781 and 1802 were only enough to determine the
 direction of the line joining the two components, not its length.)]

William Herschel’s observations of binary systems (chapter XII., § 264)
only sufficed to shew that a motion of revolution of some kind appeared
to be taking place; it was an obvious conjecture that the two members
of a pair attracted one another according to the law of gravitation,
so that the motion of revolution was to some extent analogous to that
of a planet round the sun; if this were the case, then each star of a
pair should describe an ellipse (or conceivably some other conic) round
the other, or each round the common centre of gravity, in accordance
with Kepler’s laws, and the apparent path as seen on the sky should be
of this nature but in general foreshortened by being projected on to
the celestial sphere. The first attempt to shew that this was actually
the case was made by ξ _Ursae_, which was found to be revolving in a
period of about 60 years.

Many thousand double stars have been discovered by the Herschels,
Struve, and a number of other observers, including several living
astronomers, among whom Professor _S. W. Burnham_ of Chicago, who
has discovered some 1300, holds a leading place. Among these stars
there are about 300 which we have fair reason to regard as binary,
but not more than 40 or 50 of the orbits can be regarded as at all
satisfactorily known. One of the most satisfactory is that of Savary’s
star ξ _Ursae_, which is shewn in fig. 101. Apart from the binaries
discovered by the spectroscopic method (§ 314), which form to some
extent a distinct class, the periods of revolution which have been
computed range between about ten years and several centuries, the
longer periods being for the most part decidedly uncertain.

310. William Herschel’s telescopes represented for some time the utmost
that could be done in the construction of reflectors; the first advance
was made by Lord _Rosse_ (1800-1867), who—after a number of less
successful experiments—finally constructed (1845), at Parsonstown in
Ireland, a reflecting telescope nearly 60 feet in length, with a mirror
which was six feet across, and had consequently a “light-grasp” more
than double that of Herschel’s greatest telescope. Lord Rosse used the
new instrument in the first instance to re-examine a number of known
nebulae, and in the course of the next few years discovered a variety
of new features, notably the spiral form of certain nebulae (fig. 102),
and the resolution into apparent star clusters of a number of nebulae
which Herschel had been unable to resolve and had accordingly
put into “the shining fluid” class (chapter XII., § 260). This last
discovery, being exactly analogous to Herschel’s experience when he
first began to examine nebulae hitherto only observed with inferior
telescopes, naturally led to a revival of the view that nebulae
are indistinguishable from clusters of stars, though many of the
arguments from probability urged by Herschel and others were in reality
unaffected by the new discoveries.

[Illustration: FIG. 102.—Spiral nebulae. From drawings by Lord Rosse.]

311. The question of the status of nebulae in its simplest form may
be said to have been settled by the first application of spectrum
analysis. Fraunhofer (§ 299) had seen as early as 1823 that stars had
spectra characterised like that of the sun by dark lines, and more
complete investigations made soon after Kirchhoff’s discoveries by
several astronomers, in particular by Sir William Huggins and by the
eminent Jesuit astronomer _Angelo Secchi_ (1818-1878), confirmed this
result as regards nearly all stars observed.

The first spectrum of a nebula was obtained by Sir William Huggins in
1864, and was seen to consist of three _bright_ lines; by 1868 he had
examined 70, and found in about one-third of the cases, including that
of the Orion nebula, a similar spectrum of bright lines. In these cases
therefore the luminous part of the nebula is gaseous, and Herschel’s
suggestion of a “shining fluid” was confirmed in the most satisfactory
way. In nearly all cases three bright lines are seen, one of which is a
hydrogen line, while the other two have not been identified, and in the
case of a few of the brighter nebulae some other lines have also been
seen. On the other hand, a considerable number of nebulae, including
many of those which appear capable of telescopic resolution into
star clusters, give a continuous spectrum, so that there is no clear
spectroscopic evidence to distinguish them from clusters of stars,
since the dark lines seen usually in the spectra of the latter could
hardly be expected to be visible in the case of such faint objects as
nebulae.

312. Stars have been classified, first by Secchi (1863), afterwards in
slightly different ways by others, according to the general arrangement
of the dark lines in their spectra; and some attempts have been made to
base on these differences inferences as to the relative “ages,” or at
any rate the stages of development, of different stars.

Many of the dark lines in the spectra of stars have been identified,
first by Sir William Huggins in 1864, with the lines of known
terrestrial elements, such as hydrogen, iron, sodium, calcium; so that
a certain identity between the materials of which our own earth is made
and that of bodies so remote as the fixed stars is thus established.

In addition to the classes of stars already mentioned, the spectroscope
has shewn the existence of an extremely interesting if rather
perplexing class of stars, falling into several subdivisions, which
seem to form a connecting link between ordinary stars and nebulae,
for, though indistinguishable telescopically from ordinary stars,
their spectra shew _bright_ lines either periodically or regularly. A
good many stars of this class are variable, and several “new” stars
which have appeared and faded away of late years have shewn similar
characteristics.

313. The first application to the fixed stars of the spectroscopic
method (§ 302) of determining motion towards or away from the observer
was made by Sir William Huggins in 1868. A minute displacement from its
usual position of a dark hydrogen line (F) in the spectrum of Sirius
was detected, and interpreted as shewing that the star was receding
from the solar system at a considerable speed. A number of other stars
were similarly observed in the following year, and the work has been
taken up since by a number of other observers, notably at Potsdam under
the direction of Professor _H. C. Vogel_, and at Greenwich.

[Illustration: FIG. 103.—The spectrum of β _Aurigae_, shewing the K
line single and double. From a photograph taken at Harvard.]

314. A very remarkable application of this method to binary stars has
recently been made. If two stars are revolving round one another, their
motions towards and away from the earth are changing regularly and
are different; hence, if the light from both stars is received in the
spectroscope, two spectra are formed—one for each star—the lines of
which shift regularly relatively to one another. If a particular line,
say the F line, common to the spectra of both stars, is observed when
both stars are moving towards (or away from) the earth at the same
rate—which happens twice in each revolution—only one line is seen; but
when they are moving differently, if the spectroscope be powerful
enough to detect the minute quantity involved, the line will appear
doubled, one component being due to one star and one to the other.
A periodic doubling of this kind was detected at the end of 1889 by
Professor _E. C. Pickering_ of Harvard in the case of ζ _Ursae_, which
was thus for the first time shewn to be binary, and found to have the
remarkably short period of only 104 days. This discovery was followed
almost immediately by Professor Vogel’s detection of a periodical shift
in the position of the dark lines in the spectrum of the variable
star Algol (chapter XII., § 266); but as in this case no doubling of
the lines can be seen, the inference is that the companion star is
nearly or quite dark, so that as the two revolve round one another the
spectrum of the bright star shifts in the manner observed. Thus the
eclipse-theory of Algol’s variability received a striking verification.

A number of other cases of both classes of spectroscopic binary stars
(as they may conveniently be called) have since been discovered. The
upper part of fig. 103 shews the doubling of one of the lines in the
spectrum of the double star β _Aurigae_; and the lower part shews the
corresponding part of the spectrum at a time when the line appeared
single.

315. Variable stars of different kinds have received a good deal of
attention during this century, particularly during the last few years.
About 400 stars are now clearly recognised as variable, while in a
large number of other cases variability of light has been suspected;
except, however, in a few cases, like that of Algol, the causes of
variability are still extremely obscure.

316. The study of the relative brightness of stars—a branch of
astronomy now generally known as stellar =photometry=—has also been
carried on extensively during the century and has now been put on
a scientific basis. The traditional classification of stars into
magnitudes, according to their brightness, was almost wholly arbitrary,
and decidedly uncertain. As soon as exact quantitative comparisons of
stars of different brightness began to be carried out on a considerable
scale, the need of a more precise system of classification became felt.
John Herschel was one of the pioneers in this direction; he suggested
a scale capable of precise expression, and agreeing roughly, at any
rate as far as naked-eye stars are concerned, with the current usages;
while at the Cape he measured carefully the light of a large number
of bright stars and classified them on this principle. According to
the scale now generally adopted, first suggested in 1856 by _Norman
Robert Pogson_ (1829-1891), the light of a star of any magnitude bears
a fixed ratio (which is taken to be 2·512 ...) to that of a star of
the next magnitude. The number is so chosen that a star of the sixth
magnitude—thus defined—is 100 times fainter than one of the first
magnitude.[171] Stars of intermediate brightness have magnitudes
expressed by fractions which can be at once calculated (according to a
simple mathematical rule) when the ratio of the light received from the
star to that received from a standard star has been observed.[172]

Most of the great star catalogues (§ 280) have included estimates of
the magnitudes of stars. The most extensive and accurate series of
measurements of star brightness have been those executed at Harvard
and at Oxford under the superintendence of Professor E. C. Pickering
and the late Professor Pritchard respectively. Both catalogues deal
with stars visible to the naked eye; the Harvard catalogue (published
in 1884) comprises 4,260 stars between the North Pole and 30° southern
declination, and the _Uranometria Nova Oxoniensis_ (1885), as it is
called, only goes 10° south of the equator and includes 2,784 stars.
Portions of more extensive catalogues dealing with fainter stars, in
progress at Harvard and at Potsdam, have also been published.

[Illustration: FIG. 104.—The Milky Way near the cluster in Perseus.
From a photograph by Professor Barnard.]

317. The great problem to which Herschel gave so much attention, that
of the general arrangement of the stars and the structure of the
system, if any, formed by them and the nebulae, has been affected in a
variety of ways by the additions which have been made to our knowledge
of the stars. But so far are we from any satisfactory solution of
the problem that no modern theory can fairly claim to represent the
facts now known to us as well as Herschel’s earlier theory fitted the
much scantier stock which he had at his command. In this as in so
many cases an increase of knowledge has shewn the insufficiency of a
previously accepted theory, but has not provided a successor. Detailed
study of the form of the Milky Way (cf. fig. 104) and of its relation
to the general body of stars has shewn the inadequacy of any simple
arrangement of stars to represent its appearance; William Herschel’s
cloven grindstone, the ring which his son was inclined to substitute
for it as the result of his Cape studies, and the more complicated
forms which later writers have suggested, alike fail to account for
its peculiarities. Again, such evidence as we have of the distance of
the stars, when compared with their brightness, shews that there are
large variations in their actual sizes as well as in their apparent
sizes, and thus tells against the assumption of a certain uniformity
which underlay much of Herschel’s work. The “island universe” theory
of nebulae, partially abandoned by Herschel after 1791 (chapter XII.,
§ 260), but brought into credit again by Lord Rosse’s discoveries
(§ 310), scarcely survived the spectroscopic proof of the gaseous
character of certain nebulae. Other evidence has pointed clearly to
intimate relations between nebulae and stars generally; Herschel’s
observation that nebulae are densest in regions farthest from the Milky
Way has been abundantly verified—as far as irresoluble nebulae are
concerned—while obvious star clusters shew an equally clear preference
for the neighbourhood of the Milky Way. In many cases again individual
stars or groups seen on the sky in or near a nebula have been clearly
shewn, either by their arrangement or in some cases by peculiarities of
their spectra, to be really connected with the nebula, and not merely
to be accidentally in the same direction. Stars which have bright
lines in their spectra (§ 312) form another link connecting nebulae
with stars.

A good many converging lines of evidence thus point to a greater
variety in the arrangement, size, and structure of the bodies with
which the telescope makes us acquainted than seemed probable when
sidereal astronomy was first seriously studied; they also indicate the
probability that these bodies should be regarded as belonging to a
single system, even if it be of almost inconceivable complexity, rather
than to a number of perfectly distinct systems of a simpler type.

318. Laplace’s nebular hypothesis (chapter XI., § 250) was published a
little more than a century ago (1796), and has been greatly affected by
progress in various departments of astronomical knowledge. Subsequent
discoveries of planets and satellites (§§ 294, 295) have marred to some
extent the uniformity and symmetry of the motions of the solar system
on which Laplace laid so much stress; but it is not impossible to give
reasonable explanations of the backward motions of the satellites
of the two most distant planets, and of the large eccentricity and
inclination of the paths of some of the minor planets, while apart
from these exceptions the number of bodies the motions of which have
the characteristics which Laplace pointed out has been considerably
increased. The case for some sort of common origin of the bodies of the
solar system has perhaps in this way gained as much as it has lost.
Again, the telescopic evidence which Herschel adduced (chapter XII., §
261) in favour of the existence of certain processes of condensation in
nebulae has been strengthened by later evidence of a similar character,
and by the various pieces of evidence already referred to which connect
nebulae with single stars and with clusters. The differences in the
spectra of stars also receive their most satisfactory explanation as
representing different stages of condensation of bodies of the same
general character.

319. An entirely new contribution to the problem has resulted from
certain discoveries as to the nature of heat, culminating in the
recognition (about 1840-50) of heat as only one form of what physicists
now call =energy=, which manifests itself also in the motion of
bodies, in the separation of bodies which attract one another, as
well as in various electrical, chemical, and other ways. With this
discovery was closely connected the general theory known as the
=conservation of energy=, according to which energy, though capable
of many transformations, can neither be increased nor decreased in
quantity. A body which, like the sun, is giving out heat and light is
accordingly thereby losing energy, and is like a machine doing work;
either then it is receiving energy from some other source to compensate
this loss or its store of energy is diminishing. But a body which goes
on indefinitely giving out heat and light without having its store of
energy replenished is exactly analogous to a machine which goes on
working indefinitely without any motive power to drive it; and both are
alike impossible.

The results obtained by John Herschel and Pouillet in 1836 (§ 307)
called attention to the enormous expenditure of the sun in the form of
heat, and astronomers thus had to face the problem of explaining how
the sun was able to go on radiating heat and light in this way. Neither
in the few thousand years of the past covered by historic records, nor
in the enormously great periods of which geologists and biologists take
account, is there any evidence of any important permanent alteration in
the amount of heat and light received annually by the earth from the
sun. Any theory of the sun’s heat must therefore be able to account
for the continual expenditure of heat at something like the present
rate for an immense period of time. The obvious explanation of the sun
as a furnace deriving its heat from combustion is found to be totally
inadequate when put to the test of figures, as the sun could in this
way be kept going at most for a few thousand years. The explanation
now generally accepted was first given by the great German physicist
_Hermann von Helmholtz_ (1821-1894) in a popular lecture in 1854. The
sun possesses an immense store of energy in the form of the mutual
gravitation of its parts; if from any cause it shrinks, a certain
amount of gravitational energy is necessarily lost and takes some
other form. In the shrinkage of the sun we have therefore a possible
source of energy. The precise amount of energy liberated by a definite
amount of shrinkage of the sun depends upon the internal distribution
of density in the sun, which is uncertain, but making any reasonable
assumption as to this we find that the amount of shrinking required to
supply the sun’s expenditure of heat would only diminish the diameter
by a few hundred feet annually, and would therefore be imperceptible
with our present telescopic power for centuries, while no earlier
records of the sun’s size are accurate enough to shew it. It is easy to
calculate on the same principles the amount of energy liberated by a
body like the sun in shrinking from an indefinitely diffused condition
to its present state, and from its present state to one of assigned
greater density; the result being that we can in this way account for
an expenditure of sun-heat at the present rate for a period to be
counted in millions of years in either past or future time, while if
the rate of expenditure was less in the remote past or becomes less in
the future the time is extended to a corresponding extent.

No other cause that has been suggested is competent to account for
more than a small fraction of the actual heat-expenditure of the sun;
the gravitational theory satisfies all the requirements of astronomy
proper, and goes at any rate some way towards meeting the demands of
biology and geology.

If then we accept it as provisionally established, we are led to the
conclusion that the sun was in the past larger and less condensed than
now, and by going sufficiently far back into the past we find it in a
condition not unlike the primitive nebula which Laplace presupposed,
with the exception that it need not have been hot.

320. A new light has been thrown on the possible development of the
earth and moon by Professor G. H. Darwin’s study of the effects of
tidal friction (cf. § 287 and §§ 292, 293). Since the tides increase
the length of the day and month and gradually repel the moon from
the earth, it follows that in the past the moon was nearer to the
earth than now, and that tidal action was consequently much greater.
Following out this clue. Professor Darwin found, by a series of
elaborate calculations published in 1879-81, strong evidence of a past
time when the moon was close to the earth, revolving round it in the
same time in which the earth rotated on its axis, which was then a
little over two hours. The two bodies, in fact, were moving as if they
were connected; it is difficult to avoid the probable inference that
at an earlier stage the two really were one, and that the moon is in
reality a fragment of the earth driven off from it by the too-rapid
spinning of the earth, or otherwise.

Professor Darwin has also examined the possibility of explaining in
a similar way the formation of the satellites of the other planets
and of the planets themselves from the sun, but the circumstances
of the moon-earth system turn out to be exceptional, and tidal
influence has been less effective in other cases, though it gives a
satisfactory explanation of certain peculiarities of the planets and
their satellites. More recently (1892) Dr. _See_ has applied a somewhat
similar line of reasoning to explain by means of tidal action the
development of double stars from an earlier nebulous condition.

Speaking generally, we may say that the outcome of the 19th century
study of the problem of the early history of the solar system has
been to discredit the details of Laplace’s hypothesis in a variety of
ways, but to establish on a firmer basis the general view that the
solar system has been formed by some process of condensation out of an
earlier very diffused mass bearing a general resemblance to one of the
nebulae which the telescope shews us, and that stars other than the
sun are not unlikely to have been formed in a somewhat similar way;
and, further, the theory of tidal friction supplements this general
but vague theory, by giving a rational account of a process which
seems to have been the predominant factor in the development of the
system formed by our own earth and moon, and to have had at any rate an
important influence in a number of other cases.



AUTHORITIES AND BOOKS FOR STUDENTS.


I. GENERAL.

I have made great use throughout of R. Wolf’s _Geschichte der
Astronomie_, and of the six volumes of Delambre’s _Histoire de
l’Astronomie_ (_Ancienne_, 2 vols.; _du Moyen Age_, 1 vol.; _Moderne_,
2 vols.; _du Dixhuitième Siècle_, 1 vol.). I shall subsequently
refer to these books simply as _Wolf_ and _Delambre_ respectively.
I have used less often the astronomical sections of Whewell’s
_History of the Inductive Sciences_ (referred to as _Whewell_), and
I am indebted—chiefly for dates and references—to the histories of
mathematics written respectively by Marie, W. W. R. Ball, and Cajori,
to Poggendorff’s _Handwörterbuch der Exacten Wissenschaften_, and
to articles in various biographical dictionaries, encyclopaedias,
and scientific journals. Of general treatises on astronomy Newcomb’s
_Popular Astronomy_, Young’s _General Astronomy_, and Proctor’s _Old
and New Astronomy_ have been the most useful for my purposes.

It is difficult to make a selection among the very large number of
books on astronomy which are adapted to the general reader. For
students who wish for an introductory account of astronomy the
Astronomer Royal’s _Primer of Astronomy_ may be recommended; Young’s
_Elements of Astronomy_ is a little more advanced, and Sir R. S. Ball’s
_Story of the Heavens_, Newcomb’s _Popular Astronomy_, and Proctor’s
_Old and New Astronomy_ enter into the subject in much greater detail.
Young’s _General Astronomy_ may also be recommended to those who are
not afraid of a little mathematics. There are also three modern English
books dealing generally with the history of astronomy, in all of which
the biographical element is much more prominent than in this book: viz.
Sir R. S. Ball’s _Great Astronomers_, Lodge’s _Pioneers of Science_,
and Morton’s _Heroes of Science: Astronomers_.


II. SPECIAL PERIODS.


_Chapters I. and II._—In addition to the general histories quoted
above—especially _Wolf_—I have made most use of Tannery’s _Recherches
sur l’Histoire de l’Astronomie Ancienne_ and of several biographical
articles (chiefly by De Morgan) in Smith’s _Dictionary of Classical
Biography and Mythology_. Ideler’s _Chronologische Untersuchungen_,
Hankel’s _Geschichte der Mathematik im Alterthum und Mittelalter_, G.
C. Lewis’s _Astronomy of the Ancients_, and Epping & Strassmaier’s
_Astronomisches aus Babylon_ have also been used to some extent.
Unfortunately my attention was only called to Susemihl’s _Geschichte
der Griechischen Litteratur in der Alexandriner Zeit_ when most of my
book was in proof, and I have consequently been able to make but little
use of it.

I have in general made no attempt to consult the original Greek
authorities, but I have made some use of translations of Aristarchus,
of the _Almagest_, and of the astronomical writings of Plato and
Aristotle.


_Chapter III._—The account of Eastern astronomy is based chiefly on
Delambre, and on Hankel’s _Geschichte der Mathematik im Alterthum und
Mittelalter_; to a less extent on Whewell. For the West I have made
more use of Whewell, and have borrowed biographical material for the
English writers from the _Dictionary of National Biography_. I have
also consulted a good many of the original astronomical books referred
to in the latter part of the chapter.

I know of no accessible book in English to which to refer students
except _Whewell_.


_Chapter IV._—For biographical material, for information as to the
minor writings, and as to the history of the publication of the _De
Revolutionibus_ I have used little but Prowe’s elaborate _Nicolaus
Coppernicus_, and the documents printed in it. My account of the _De
Revolutionibus_ is taken from the book itself. The portrait is taken
from Dandeleau’s engraving of a picture in Lalande’s possession. I have
not been able to discover any portrait which was clearly made during
Coppernicus’s lifetime, but the close resemblance between several
portraits dating from the 17th century and Dandeleau’s seems to shew
that the latter is substantially authentic.

There is a readable account of Coppernicus, as well as of several other
astronomers, in Bertrand’s _Fondateurs de l’Astronomie Moderne_; but I
have not used the book as an authority.


_Chapter V._—For the life of Tycho I have relied chiefly on Dreyer’s
_Tycho Brahe_, which has also been used as a guide to his scientific
work; but I have made constant reference to the original writings:
I have also made some use of Gassendi’s _Vita Tychonis Brahe_. The
portrait is a reproduction of a picture in the possession of Dr.
Crompton of Manchester, described by him in the _Memoirs of the
Manchester Literary and Philosophical Society_, Vol. VI., Ser. III. For
minor Continental writers I have used chiefly _Wolf_ and _Delambre_,
and for English writers, _Whewell_, various articles by De Morgan
quoted by him, and articles in the _Dictionary of National Biography_.

Students will find in Dreyer’s book all that they are likely to want to
know about Tycho.


_Chapter VI._—For Galilei’s life I have used chiefly Karl von Gebler’s
_Galilei und die Römische Curie_, partly in the original German form
and partly in the later English edition (translated by Mrs. Sturge).
For the disputed questions connected with the trial I have relied as
far as possible on the original documents preserved in the Vatican,
which have been published by von Gebler and independently by L’Épinois
in _Les Pièces du Procès de Galilée_: in the latter book some of the
most important documents are reproduced in facsimile. For personal
characteristics I have used the charming _Private Life of Galileo,
compiled chiefly from his correspondence and that of his daughter
Marie Céleste_. I have also read with great interest the estimate of
Galilei’s work contained in H. Martin’s _Galilée_, and have probably
borrowed from it to some extent. What I have said about Galilei’s
scientific work has been based almost entirely on study of his own
books, either in the original or in translation: I have used freely the
translations of the _Dialogue on the Two Chief Systems of the World_
and of the _Letter to the Grand Duchess Christine_ by Salusbury, that
of the _Two New Sciences_ by Weston (as well as that by Salusbury), and
that of the _Sidereal Messenger_ by Carlos. I have also made some use
of various controversial tracts written by enemies of Galilei, which
are to be found (together with his comments on them) in the magnificent
national edition of his works now in course of publication; and of the
critical account of Galilei’s contributions to dynamics contained in
Mach’s _Geschichte der Mechanik_.

_Wolf_ and _Delambre_ have only been used to a very small extent in
this chapter, chiefly for the minor writers who are referred to.

The portrait is a reproduction of one by Sustermans in the Uffizi
Gallery.

There is an excellent popular account of Galilei’s life and work in the
_Lives of Eminent Persons_ published by the Society for the Diffusion
of Useful Knowledge; students who want fuller accounts of Galilei’s
life should read Gebler’s book and the _Private Life_, which have
been already quoted, and are strongly recommended to read at any rate
parts of the _Dialogue on the Two Chief Systems of the World_, either
in the original or in the picturesque old translation by Salusbury:
there is also a modern German version of this book, as well as of the
_Two New Sciences_, in Ostwald’s series of _Klassiker der exakten
Wissenschaften_.


_Chapter VII._—For Kepler’s life I have used chiefly _Wolf_ and the
life—or rather biographical material—given by Frisch in the last
volume of his edition of Kepler’s works, also to a small extent
Breitschwerdt’s _Johann Keppler_. For Kepler’s scientific discoveries
I have used chiefly his own writings, but I am indebted to some extent
to _Wolf_ and _Delambre_, especially for information with regard to his
minor works. The portrait is a reproduction of one by Nordling given in
Frisch’s edition.

The _Lives of Eminent Persons_, already referred to, also contains an
excellent popular account of Kepler’s life and work.


_Chapter VIII._—I have used chiefly _Wolf_ and _Delambre_; for the
English writers Gascoigne and Horrocks I have used _Whewell_ and
articles in the _Dict. Nat. Biog._ What I have said about the work of
Huygens is taken directly from the books of his which are quoted in
the text; and for special points I have consulted the _Principia_ of
Descartes, and a very few of Cassini’s extensive writings.

There is no obvious book to recommend to students.


_Chapter IX._—For the external events of Newton’s life I have relied
chiefly on Brewster’s _Memoirs of Sir Isaac Newton_; and for the
history of the growth of his ideas on the subject of gravitation I have
made extensive use of W. W. R. Ball’s _Essay on Newton’s Principia_,
and of the original documents contained in it. I have also made some
use of the articles on Newton in the _Encyclopaedia Britannica_
and the _Dictionary of National Biography_; as well as of Rigaud’s
_Correspondence of Scientific Men of the Seventeenth Century_, of
Edleston’s _Correspondence of Sir Isaac Newton and Prof. Cotes_, and
of Baily’s _Account of the Rev^{d.} John Flamsteed_. The portrait is a
reproduction of one by Kneller.

Students are recommended to read Brewster’s book, quoted above, or
the abridged _Life of Sir Isaac Newton_ by the same author. The Laws
of Motion are discussed in most modern textbooks of dynamics; the
best treatment that I am acquainted with of the various difficulties
connected with them is in an article by W. H. Macaulay in the _Bulletin
of the American Mathematical Society_, Ser. II., Vol. III., No. 10,
July 1897.


_Chapter X._—For Flamsteed I have used chiefly Baily’s _Account of the
Rev^{d.} John Flamsteed_; for Bradley little but the _Miscellaneous
Works and Correspondence of the Rev. James Bradley_ (edited by
Rigaud), from which the portrait has been taken. My account of Halley’s
work is based to a considerable extent on his own writings; there is a
good deal of biographical information about him in the books already
quoted in connection with Newton and Flamsteed, and there is a useful
article on him in the _Dictionary of National Biography_. I have made a
good deal of use in this chapter of _Wolf_ and _Delambre_, especially
in dealing with Continental astronomers; and for special parts of
the subject I have used Grant’s _History of Physical Astronomy_,
Todhunter’s _History of the Mathematical Theories of Attraction and the
Figure of the Earth_, and Poynting’s _Density of the Earth_.


_Chapter XI._—Most of the biographical material has been taken from
_Wolf_ from articles in various encyclopaedias and biographical
dictionaries, chiefly French, and from Delambre’s _Eloge_ of Lagrange.
The two portraits are taken respectively from Serret’s edition of the
_Oeuvres de Lagrange_ and from the Academy’s edition of the _Oeuvres
Complètes de Laplace_. Gautier’s _Essai Historique sur le Problème des
Trois Corps_ and Grant’s _History of Physical Astronomy_ have been the
books most used for my account of the scientific contributions of the
various astronomers dealt with; I have also consulted various modern
treatises on gravitational astronomy, especially Tisserand’s _Mécanique
Céleste_, Brown’s _Lunar Theory_, and to a less extent Cheyne’s
_Planetary Theory_ and Airy’s _Gravitation_. For special points I
have used Todhunter’s _History_, already referred to. Of the original
writings I have made a good deal of use of Laplace’s _Mécanique
Céleste_ as well as of his _Système du Monde_; I have also consulted
a certain number of his other writings and of those of Lagrange and
Clairaut; but have made no systematic study of them.

Students who wish to know more about gravitational astronomy but
have little knowledge of mathematics should try to read Airy’s
_Gravitation_; Herschel’s _Outlines of Astronomy_ and Grant’s
_History_ (quoted above) also deal with the subject without employing
mathematics, and are tolerably intelligible.


_Chapter XII._—The account of Herschel’s career is taken chiefly from
Mrs. John Herschel’s _Memoir of Caroline Herschel_, from Miss A. M.
Clerke’s _The Herschels and Modern Astronomy_, from the _Popular
History of Astronomy in the Nineteenth Century_ by the same author,
and from Holden’s _Sir William Herschel, his Life and Works_. The last
three books and the _Synopsis and Subject Index to the Writings of Sir
William Herschel_ by Holden & Hastings have been my chief guides to
Herschel’s long series of papers; but nearly every thing that I have
said about his chief pieces of work is based on his own writings. I
have made also some little use of Grant’s _History_ (already quoted),
of _Wolf_, and of Miss Clerke’s _System of the Stars_.

Students are recommended to read any or all of the first four books
named above; the _Memoir_ gives a charming picture of Herschel’s
personal life and especially of his relations with his sister. There is
also a good critical account of Herschel’s work on sidereal astronomy
in Proctor’s _Old and New Astronomy_.


_Chapter XIII._—Except in the articles dealing with gravitational
astronomy I have constantly used Miss Clerke’s _History_ (already
quoted), a book which students are strongly recommended to read; and in
dealing with the first half of the century I have been helped a good
deal by Grant’s _History_. But for the most part the materials for the
chapter have been drawn from a great number of sources—consisting very
largely of the original writings of the astronomers referred to—which
it would be difficult and hardly worth while to enumerate; for the
lives of astronomers (especially of English ones), as well as for
recent astronomical history generally, I have been much helped by the
obituary notices and the reports on the progress of astronomy which
appear annually in the _Monthly Notices_ of the Royal Astronomical
Society.

I add the names of a few books which deal with special parts of modern
astronomy in a non-technical way:—

 _The Sun_, C. A. Young; _The Sun_, R. A. Proctor; _The Story of the
 Sun_, R. S. Ball; _The Sun’s Place in Nature_, J. N. Lockyer.

 _The Moon_, E. Neison; _The Moon_, T. G. Elger.

 _Saturn and its System_, R. A. Proctor.

 _Mars_, Percival Lowell.

 _The World of Comets_, A. Guillemin (a well-illustrated but uncritical
 book, now rather out of date); _Remarkable Comets_, W. T. Lynn (a very
 small book full of useful information); _The Great Meteoritic Shower
 of November_, W. F. Denning.

 _The Tides and Kindred Phenomena in the Solar System_, G. H. Darwin.

 _Remarkable Eclipses_, W. T. Lynn (of the same character as his book
 on Comets.)

 _The System of the Stars_, A. M. Clerke.

 _Spectrum Analysis_, H. Schellen; _Spectrum Analysis_, H. E. Roscoe.



INDEX OF NAMES.

 _Roman figures refer to the chapters, Arabic to the articles. The
 numbers given in brackets after the name of an astronomer are the
 dates of birth and death. All dates are_ A.D. _unless otherwise
 stated. In cases in which an authors name occurs in several articles,
 the numbers of the articles in which the principal account of him or
 of his work is given are printed in clarendon type thus_: =286=. _The
 names of living astronomers are italicised._


  Abul Wafa. _See_ Wafa

  Adams (1819-1892), XIII. 286, 287, 289

  Adelard. _See_ Athelard

  Airy (1801-1892), X. 227 _n_; XIII. =281=, =282=

  Albategnius (?-929), II. 53; III. =59=, 66, 68 _n_; IV. 84, 85

  Albert (of Prussia), V. 94

  Albertus Magnus (13th cent.), III. 67

  Alcuin (735-804), III. 65

  Alembert, d’. _See_ D’Alembert

  Alexander, II. 31

  Alfonso X. (1223-1284), III. =66=, 68; V. 94

  Al Mamun, III. 57, 69

  Al Mansur, III. 56

  Al Rasid, III. 56

  Alva, VII. 135

  Anaxagoras (499 B.C.?-427 B.C.?), I. 17

  Anaximander (610 B.C.-546 B.C.?), I. 11

  Apian (1495-1552), III. =69=; V. 97; VII. 146

  Apollonius (latter half of 3rd cent. B.C.), II. =38=, 39, 45, 51,
    52 _n_; X. 205

  Arago, XII. 254

  Archimedes, II. 52 _n_; III. 62

  Argelander (1799-1875), XIII. 280

  Aristarchus (earlier part of 3rd cent. B.C.), II. =24=, =32=, 41,
    42, 54; IV. 75

  Aristophanes, II. 19

  Aristotle (384 B.C.-322 B.C.), II. 24, =27-30=, 31, 47, 51, 52;
    III. 56, 66, 67, 68; IV. 70, 77; V. 100; VI. 116, 121, 125, 134;
    VIII. 163

  Aristyllus (earlier part of 3rd cent. B.C.), II. =32=, 42

  Arzachel (_fl._ 1080), III. =61=, 66

  d’Ascoli, Cecco (13th cent.), III. 67

  Athelard (beginning of 12th cent.), III. 66

  Auzout (?-1691), VIII. =155=, 160; X. 198


  Bacon, Francis (1561-1627), VI. 134; VIII. 163

  Bacon, Roger (1214?-1294), III. =67=; VI. 118

  Bailly, XI. 237

  _Ball_, XIII. 278 _n_

  Bär, Reymers (Ursus) (?-1600), V. 105

  Barberini (Urban VIII.), VI. 125, 127, 131, 132

  _Barnard_, XIII. 294, 295

  Baronius, VI. 125

  Barrow, Isaac, IX. 166

  Bayer, XII. 266

  Bede, III. 65

  Begh, Ulugh. _See_ Ulugh Begh

  Bellarmine, VI. 126

  Bentley, IX. 191

  Berenice, I. 12

  Bernouilli, Daniel (1700-1782), XI. 230

  Bernouilli, James (1654-1705), XI. 230

  Bernouilli, John (1667-1748), XI. 229, 230

  Bessel (1784-1846), X. 198 _n_, 218; XIII. 272, =277-278=, 279, 280

  Bille, V. 99

  Bliss (1700-1764), X. 219

  Bond, William Cranch (1789-1859), XIII. 295

  Borelli (1608-1679), IX. 170

  Bouguer (1698-1758), X. 219, =221=

  Boullian. _See_ Bullialdus

  Bouvard, XI. 247 _n_; XIII. 289

  Bradley (1693-1762), X. 198, =206-218=, 219, 222-226; XI. 233;
    XII. 257, 258, 263, 264; XIII. 272, 273, 275, 277

  Brahe, Tycho (1546-1601), III. 60, 62; V. 97, 98 _n_, =99-112=;
    VI. 113, 117, 127; VII. 136-139, 141 _n_, 142, 145, 146, 148;
    VIII. 152, 153, 162; IX. 190; X. 198, 203, 225; XII. 257; XIII. 275

  Brudzewski, IV. 71

  Bruno, VI. 132

  Bullialdus (1605-1694), XII. 266

  Bunsen, XIII. 299

  Burckhardt (1773-1825), XI. 241

  Bürg (1766-1834), XI. 241

  Bürgi (1552-1632), V. =97=, =98=; VIII. 157

  Burney, Miss, XII. 260

  _Burnham_, XIII. 309


  Caccini, VI, 125

  Caesar, II. 21; III. 67

  Callippus (4th cent. B.C.), II. =20=, =26=, 27

  Capella, Martianus (5th or 6th cent. A.D.), IV. 75

  Carlyle, XI. 232

  Carrington (1826-1875), XIII. 298, 302

  Cartesius. _See_ Descartes

  Cassini, Count (1748-1845), X. 220

  Cassini, Giovanni Domenico (1625-1712), VIII. =160=, =161=; IX. 187;
    X. 216, =220=, 221, 223; XII. 253, 267; XIII. 297

  Cassini, Jacques (1677-1756), X. =220=, =221=, 222

  Cassini de Thury (1714-1784), X. 220

  Castelli, VI. 125

  Catherine II., X. 227; XI. 230, 232

  Cavendish (1731-1810), X. 219

  Cecco d’Ascoli (1257?-1327), III. 67

  _Chandler_, XIII. 285

  Charlemagne, III. 65

  Charles II., X. 197, 223

  _Charlois_, XIII. 294

  Christian (of Denmark), V. 106

  Christine (the Grand Duchess), VI. 125

  Clairaut (1713-1765), XI. 229, 230, =231=, 232, =233-235=, 237, 239,
    248; XIII. 290

  Clement VII., IV. 73

  _Clerke_, XIII. 289 _n_

  Clifford, IX. 173 _n_

  Colombe, Ludovico delle, VI. 119 _n_

  Columbus, III. 68

  La Condamine (1701-1774), X. 219, =221=

  Conti, VI. 125

  Cook, X. 227

  Coppernicus (1473-1543), II. 24, 41 _n_, 54; III. 55, 62, 69;
    =IV.= _passim_; V. 93-97, 100, 105, 110, 111; VI. 117, 126, 127,
    129; VII. 139, 150; IX. 186, 194; XII. 257; XIII. 279

  _Cornu_, XIII. 283

  Cosmo de Medici, VI. 121

  Cotes (1682-1716), IX. 192

  Crosthwait, X. 198

  Cusa, Nicholas of (1401-1464), IV. 75


  D’Alembert (1717-1783), X. 215; XI. 229, 230, 231 _n_, =232-235=,
    237-239, 248

  Damoiseau (1768-1846), XIII. 286

  Dante, III. 67; VI. 119 _n_

  _Darwin_, XIII. 292, 320

  Da Vinci. _See_ Vinci

  Dawes (1799-1868), XIII. 295

  Delambre, II. 44; X. 218; XI. 247 _n_; XIII. 272

  Delaunay (1816-1872), XIII 286, 287

  De Morgan, II. 52 _n_

  Descartes (1596-1650), VIII. 163

  Diderot, XI. 232

  Digges, Leonard (?-1571?), VI. 118

  Digges, Thomas (?-1595), V. 95

  Donati (1826-1873), XIII. 304

  Doppler (1803-1853), XIII. 302

  _Dreyer_, XIII. 308

  _Dunér_, XIII. 302


  Ecphantus (5th or 6th cent. B.C.), II. 24

  Eddin, Nassir (1201-1273), III. =62=, 68; IV. 73

  Encke (1791-1865), X. 227; XIII. 284

  Eratosthenes (276 B.C.-195 or 196 B.C.), II. =33=, 45, 54; X. 221

  Euclid (_fl._ 300 B.C.), II. 33, 52 _n_; III. 62, 66; VI. 115; IX. 165

  Eudoxus (409 B.C.?-356 B.C.?), II. =26=, 27, 38, 42, 51

  Euler (1707-1783), X. 215, 226; XI. 229, =230=, 231 _n_, =233-236=,
    237, 239, 242, 243; XIII. 290


  Fabricius, John (1587-1615?), VI. 124

  Ferdinand (the Emperor), VII. 137, 147

  Fernel (1497-1558), III. 69

  Ferrel, XIII. 287

  Field (1525?-1587), V. 95

  Fizeau (1819-1896), XIII. 283

  Flamsteed (1646-1720), IX. 192; X. =197=, =198=, 199, 204, 207 _n_,
    218, 225; XII. 257; XIII. 275, 281

  Fracastor (1483-1553), III. =69=; IV. 89; VII. 146

  Fraunhofer (1787-1826), XIII. =299=, 311

  Frederick II. (of Denmark), V. 101, 102, 106

  Frederick II. (the Emperor), III. 66

  Frederick II. (of Prussia), XI. 230, 232, 237


  Galen, II. 20; III. 56; VI. 116

  Galilei, Galileo (1564-1642), II. 30 _n_, 47; IV. 73; V. 96, 98 _n_;
    =VI.= _passim_; VII. 135, 136, 138, 145, 151; VIII. 152-154, 157,
    163; IX. 165, 168, 170, 171, 173, 179, 180, 186, 190, 195; X. 216;
    XII. 253, 257, 263, 268; XIII. 278, 295

  Galilei, Vincenzo, VI. 113

  _Galle_, XIII. 281, 289

  Gascoigne (1612?-1644), VIII. =155=, 156; X. 198

  Gauss (1777-1855), XIII. =275=, =276=, 294

  Gautier (1793-1881), XIII. 298

  Genghis Khan, III. 62

  George III, XII. 254-256

  Gerbert (?-1003), III. 66

  Gherardo of Cremona (1114-1187), III. 66

  Gibbon, II. 53 _n_

  Giese, IV. 74

  Gilbert (1540-1603), VII. 150

  _Gill_, XIII. 280, 281

  _Glaisher_, XIII. 289 _n_

  Godin (1704-1760), X. 221

  Goodricke (1764-1786), XII. 266

  Grant, XIII, 289 _n_

  Grassi, VI. 127

  Gregory, James (1638-1675), IX. =168=, 169; X. 202

  Gregory XIII., II. 22

  Gyldén (1841-1896), XIII. 288


  Hainzel, V. 99

  _Hale_, XIII. 301

  Halifax, John. _See_ Sacrobosco

  Halifax (Marquis of), IX. 191

  _Hall_, XIII. 283 _n_, 295

  Halley (1656-1742), VIII. 156; IX. =176=, =177=, 192 _n_; X. 198,
    =199-205=, 206, 216, 223, 224, 227; XI. 231, 233, 235, 243;
    XII. 265; XIII. 287, 290

  Hansen (1795-1874), XIII. 282, 284, =286=, 290

  _Harkness_, XIII. 301

  Harriot (1560-1621), VI. 118, 124

  Harrison, X. 226

  Harun al Rasid, III. 56

  Helmholtz (1821-1894), XIII. 319

  Hencke (1793-1866), XIII. 294

  Henderson (1798-1844), XIII. 279

  Heraclitus (5th cent. B.C.), 11, 24

  Herschel, Alexander, XII. 251

  Herschel, Caroline (1750-1848), XII. 251, 254-256, 260

  Herschel, John (1792-1871), I. 12; X. 221; XI. 242; XII. 256;
    XIII. 289 _n_, =306-308=, 309, 316, 317, 319

  Herschel, William (1738-1822), IX. 168; X. 223, 227; XI. 250;
    =XII.= _passim_; XIII. 272, 273, 294, 296, 306-311, 317, 318

  Hesiod, 11, 19, 20

  Hevel (1611-1687), VIII. =153=; X. 198; XII. 268

  Hicetas (6th or 5th cent. B.C.), II. 24; IV. 75

  _Hill_, XI. 233 _n_; XIII. =286=, 290

  Hipparchus (2nd cent. B.C.), I. 13; II. 25, 27, 31, 32, =37-44=,
    45, 47-52, 54; III 63, 68; IV. 73, 84; V. 111; VII. 145

  Hippocrates, III. 56

  Holwarda (1618-1651), XII. 266

  Holywood. _See_ Sacrobosco

  Honein ben Ishak (?-873), III. 56

  Hooke (1635-1703), IX. =174=, 176; X. 207, 212

  Horky, VI. 121

  Horrocks (1617?-1641), VIII. =156=; IX. 183; X. 204

  _Howlett_, XIII. 298

  _Huggins_, XIII. 301, 304, 311-313

  Hulagu Khan, III. 62

  Humboldt, XIII. 298

  Hutton (1737-1823), X. 219

  Huygens (1629-1695), V. 98 _n_; VI. 123; VIII. =154=, =155=, =157=,
    =158=; IX. 170-172, 191

  Hypatia (?-415), II. 53


  Ibn Yunos. _See_ Yunos

  Ishak ben Honein (?-910 or 911), III. 56


  James I., V. 102; VII. 147

  James II., IX. 192

  _Janssen_, XIII. 301, 307 _n_

  Joachim. _See_ Rheticus


  Kaas, V. 106

  Kant, XI. 250; XII. 258, 260; XIII. 287

  _Kapteyn_, XIII. 280

  _Kelvin_, XIII. 292

  Kepler (1571-1630), II. 23, 51 _n_, 54; IV. 91; V. 94, 100, 104,
    108-110; VI. 113, 121, 130, 132; =VII.= _passim_; VIII. 152, 156,
    160; IX. 168-170, 172, 175, 176, 190, 194, 195; X. 202, 205, 220;
    XI. 228, 244; XIII. 294, 301, 309

  Kirchhoff (1824-1887), XIII. =299=, =300=, 311

  Kirkwood (1815-1895), XIII. 294, 297

  Koppernigk, IV. 71

  Korra, Tabit ben. _See_ Tabit


  Lacaille (1713-1762), X. =222-224=, 225, 227; XI. 230, 233, 235;
    XII. 257, 259

  La Condamine (1701-1774), X. 219, =221=

  Lagrange (1736-1813), IX. 193; XI. 229, 231 _n_, 233 _n_, 236, =237=,
    238-240, =242-245=, 247, 248; XII. 251; XIII. 293, 294

  Lalande (1732-1807), XI. =235=, 241, 247 _n_; XII. 265

  Lambert (1728-1777), XI. 243; XII. 265

  Lami, IX. 180 _n_

  Landgrave of Hesse. _See_ William IV.

  _Langley_, XIII. 307

  Lansberg (1561-1632), VIII. 156

  Laplace (1749-1827), XI. 229, 231 _n_, =238-248=, =250=; XII. 251,
    256; XIII. 272, 273, 282, 286-288, 290, 293, 297, 318-320

  Lassell (1799-1880), XII. 267; XIII. =295=

  Lavoisier, XI. 237

  Legendre (1752-1833), XIII. =275=, 276

  Leibniz, IX. 191, 193

  Lemaire, XII. 255

  Leverrier (1811-1877), XIII. 282, 284, =288=, =289=, 290, 293, 294

  Lexell (1740-1784), XII. 253

  Lindenau, XI. 247 _n_

  Lionardo da Vinci. _See_ Vinci

  Lippersheim (?-1619), VI. 118

  Locke, IX. 191

  _Lockyer_, XIII. 301, 302

  _Loewy_, XIII. 283 _n_

  Louis XIV., VIII. 160

  Louis XVI., XI. 237

  Louville (1671-1732), XI. 229

  Lubbock (1803-1865), XIII. 286, 292

  Luther, IV. 73; V. 93


  Machin (?-1751), X. 214

  Maclaurin (1698-1746), X. =196=; XI. 230, 231; XII. 251

  Maestlin, VII. 135

  Maraldi, X. 220

  Marius (1570-1624), VI. =118=; VII. 145

  Martianus Capella. _See_ Capella

  Maskelyne (1732-1811), X. =219=; XII. 254, 265

  Mason (1730-1787), X. 226; XI. 241

  Matthias (the Emperor), VII. 143, 147

  Maupertuis (1698-1759), X. 213, =221=; XI. 229, 231

  Maxwell (1831-1879), XIII. 297

  Mayer, Tobias (1723-1762), X. 217, =225=, =226=; XI. 233, 241;
    XII. 265

  Melanchthon, IV. 73, 74; V. 93

  Messier (1730-1817), XII. 259, 260

  Meton (460 B.C.?-?), II. 20

  Michel Angelo, VI. 113

  Michell, John (1724-1793), X. 219; XII. 263, 264

  _Michelson_, XIII. 283

  Molyneux (1689-1728), X. 207

  Montanari (1632-1687), XII. 266

  Müller. _See_ Regiomontanus


  Napier, V. 97 _n_

  Napoleon I., XI. 238; XII. 256

  Napoleon, Lucien, XI. 238 _n_

  Nassir Eddin. _See_ Eddin, Nassir

  _Newcomb_, X. 227 _n_; XIII. =283=, =286=, 290

  Newton (1643-1727). II. 54; IV. 75; VI. 130, 133, 134; VII. 144, 150;
    VIII. 152; =IX.= _passim_; X. 196-200, 211, 213, 215-217, 219, 221;
    XI. 228, 229, 231-235, 238, 249; XII. 257; XIII. 273, 299

  Niccolini, VI. 132

  Nicholas of Cusa (1401-1464), IV. 75

  Nonius (1492-1577), III. 69

  Norwood (1590?-1675), VIII. =159=; IX. 173

  Numa, II. 21

  Nunez. _See_ Nonius

  _Nyrén_, XIII. 283 _n_


  Olbers (1758-1840), XIII. 294

  Orange, Prince of, V. 107

  Osiander, IV. 74; V. 93


  _Palisa_, XIII. 294

  Palitzsch (1723-1788), XI. 231

  Pemberton, IX. 192

  Philolaus (5th cent. B.C.), II. =24=; IV. 75

  Piazzi (1746-1826), XIII. 294

  Picard (1620-1682), VIII. =155=, =157=, =159-161=, 162; IX. 174;
    X. 196, 198, 221

  _Pickering_, XIII. 314, 316

  Plana (1781-1869), XIII. 286

  Plato (428 B.C.?-347 B.C.?), II. =24=, =25=, 26, 51; IV. 70

  Plato of Tivoli (_fl._ 1116), III. 66

  Pliny (23 A.D.-79 A.D.), II. 45

  Plutarch, II. 24; XIII. 301

  Pogson (1829-1891), XIII. 316

  _Poincaré_, XIII. 288

  Poisson (1781-1840), XIII. 286, 293

  Pontécoulant (1795-1874), XIII. 286

  Porta, VI. 118

  Posidonius (135 B.C.?-51 B.C.?), II 45, 47

  Pouillet (1791-1868), XIII. 307, 319

  Pound, X. 206, 216

  Prévost (1751-1839), XII. 265

  Pritchard (1808-1893), XIII. 278 _n_, =279=, 316

  Ptolemy, Claudius (_fl._ 140 A.D.), II. 25, 27, 32, 37, =46-52=, 53,
    54; III. 55, 57, 59-63, 68; IV. 70, 76, 80, 83-87, 89, 91; V. 94,
    105; VI. 121, 129, 134; VII. 145; VIII. 161; IX. 194; X. 205;
    XI. 236

  Ptolemy Philadelphus, II. 31

  Purbach (1423-1461), III. =68=; IV. 71

  Pythagoras (6th cent. B.C.), I. 11, 14; II. =23=, 28, 47, 51, 54


  Recorde (1510-1558), V. 95

  Regiomontanus (1436-1476), III. =68=, 69; IV. 70, 71; V. 97, 110

  Reimarus. _See_ Bär

  Reinhold (1511-1553), V. =93-96=; VII. 139

  Reymers. _See_ Bär

  Rheticus (1514-1576), IV. =73=, =74=; V. =93=, =94=, 96

  Ricardo, II. 47 _n_

  Riccioli (1598-1671), VIII. 153

  Richer (?-1696), VIII. =161=; IX. 180, 187; X. 199, 221

  Rigaud. X. 206 _n_

  Roemer (1644-1710), VIII. =162=; X. 198, 210, 216, 220, 225; XIII. 283

  Rosse (1800-1867), XIII. 310, 317

  Rothmann (_fl._ 1580), V. =97=, =98=, 106

  Rudolph II. (the Emperor), V. 106-108; VII. 138, 142, 143


  Sabine (1788-1883), XIII. 298

  Sacrobosco (?-1256?), III. =67=, 68

  St. Pierre, X. 197

  Savary (1797-1841), XIII. 309

  Scheiner (1575-1650), VI. =124=, 125; VII. =138=; VIII. =153=;
    XII. 268

  _Schiaparelli_, XIII. 297

  Schomberg, IV. 73

  Schoner, IV. 74

  Schönfeld (1828-1891), XIII. 280

  Schroeter (1745-1816), XII. 267, =271=

  Schwabe (1789-1875), XIII. 298

  Secchi (1818-1878), XIII. 311, 312

  _See_, XIII. 320

  Seleucus (2nd cent, B.C.), II. 24

  Shakespeare, VI. 113

  Sharp (1651-1742), X. 198

  Slusius, IX. 169

  Smith, XII. 251

  Snell (1591-1626), VIII. 159; IX. 173

  Sosigenes (_fl._ 45 B.C.), II. 21

  South (1785-1867), XIII. 306

  Struve, F. G. W. (1793-1864), XIII. 279, 309

  _Struve, O._, XIII. 283 _n_

  Svanberg (1771-1851), X. 221

  Sylvester II. _See_ Gerbert


  Tabit ben Korra (836-901), III. =56=, =58=, 68; IV. 84

  Tamerlane, III. 63

  _Tannery_, II. 36 _n_

  Thales (640 B.C.?-546 B.C.?), II. =23=; III. 55

  Theon (_fl._ 365 A.D.), II. 53

  Theophrastus, II. 24

  Theophylactus, IV. 72

  Thomson, T., X. 208 _n_

  _Thomson, William._ _See_ _Kelvin_

  Thury, Cassini de. _See_ Cassini de Thury

  Timocharis (beginning of 3rd cent. B.C.). II. =32=, 42; IV. 84

  Tycho Brahe, _See_ Brahe


  Ulugh Begh (1394-1449), III. =63=, 68; IV. 73

  Urban VIII. (Barberini), VI. 125, 127, 131, 132

  Ursus. _See_ Bär


  Varignon, IX. 180 _n_

  Vinci, Lionardo da (1452-1519), III. 69

  _Vogel_, XIII. 313, 314

  Voltaire, II. 21; XI, 229


  Wafa, Abul (939 or 940-998), III. =60=, 68 _n_; IV. 85; V. 111

  Wallenstein, VII. 149

  Walther (1430-1504), III. 68; V. 97

  Wargentin (1717-1783), X. 216

  Watzelrode, IV. 71

  Wefa. _See_ Wafa

  Welser, VI. 124

  Whewell (1794-1866), XIII. 292

  William IV. (Landgrave of Hesse) (1532-1592), V. =97=, =98=, 100,
    105, 106, 110

  Wilson (1714-1786), XII. =268=; XIII. 298

  _Wolf, Max_, XIII. 294

  Wolf, Rudolf (1816-1893), XIII. 298

  Wollaston (1766-1828), XIII. 299

  Wren (1632-1723), IX. 174, 176

  Wright, Thomas (1711-1786), XII. 258, 265


  _Young_, XIII. 301

  Yunos, Ibn (?-1008), III. =60=, 62, 68 _n_


  von Zach, XI. 247 _n_



GENERAL INDEX.

 _Roman figures refer to the chapters, Arabic to the articles. When
 several articles are given under one heading the numbers of the most
 important are printed in clarendon type, thus_: =207=. _The names of
 books are printed in italics._

  Aberration, X. 206, =207-211=, 212, 213, 216, 218; XII. 263;
    XIII. 277, 283, 284

  Académie Française, XI. 232, 238

  Academy of Berlin, XI. 230, 237

  Academy of St. Petersburg, XI. 230, 233

  Academy of Sciences (of Paris), X. 221, 223; XI. 229-233, 235-237

  Academy of Turin, XI. 237, 238

  Acceleration, VI. =133=; IX. =171=, 172, 173, 179, 180, 185, 195;
    X. 223

  _Ad Vitellionem Paralipomena_ (of Kepler), VII. 138

  _Alae sive Scalae_ (of Digges), V. 95

  Aldebaran, III. 64; XIII. 316 _n_

  Alexandrine school, II. 21, 31-33, 36-38, 45, 53

  _Alfonsine Tables_, III. =66=, 68; V. 94, 96, 99

  Algol, XII. 266; XIII. 314, 315

  _Almagest_ (of Ptolemy), II. =46-52=; III. 55, 56, 58, 60, 62, 66, 68;
    IV. 75, 76, 83

  _Almagest_ (of Abul Wafa), III. 60

  _Almagest, New_ (of Kepler), VII. 148

  _Almagest, New_ (of Riccioli), VIII. 153

  _Almanac, Nautical_, X. 218; XIII. 286, 288, 290

  Almanacks, I. 18 _n_; II. 20, 38; III. 64, 68; V. 94, 95, 100;
    VII. 136; X. 218, 224; XIII. 286, 288, 290

  Altair, III. 64; XIII. 316 _n_

  Analysis, analytical methods, X. =196=; XI. 234

  Angles, measurement of, I. 7

  Annual equation, V. =111=; VII. 145

  Annual motion of the earth. _See_ Earth, revolution of

  Annual motion of the sun. _See_ Sun, motion of

  Annual parallax. _See_ Parallax, stellar

  Annular eclipse, II. =43=; VII. 145

  Anomalistic month, II. 40

  Ἀντιχθών, II. 24

  Apex of solar motion, XII. 265

  Aphelion, IV. 85

  Apogee, II. =39=, 40, 48; III. 58, 59; IV. 85; V. 111; IX. 184 XI. 233

  Apparent distance, I. 7

  Apple, Newton’s, IX. 170

  Apse, apse-line, II. =39=, 40, 48; IV. 85; IX. 183; XI. 235, 236, 242,
    246

  Arabic numerals, III. 64; V. 96

  Arabs, Arab astronomy, II. 46, 53; III. =56-61=, =64=, 66, 68; V. 110;
    VIII. 159; XII. 266

  Arctic regions, II. 35

  Arcturus, XII. 258

  Aries, first point of (♈), I. =13=; II. 33, 42; IV. 83 _n_; V. 98,
    III; X. 198

  Asteroids. _See_ Minor planets

  Astrolabe, II. 49; III. 66

  Astrology, I. 16, =18=; III. 56; V. 99, 100; VII. 136, 149, 151

  _Astronomiae Fundamenta_ (of Lacaille), X. 224

  _Astronomiae Instauratae Mechanica_ (of Tycho), V. 107

  _Astronomiae Instauratae Progvmnasmata_ (of Tycho), V. 104

  Astronomical Society, German, XIII. 280

  Astronomical Society, Royal, XII. 256, 263

  _Astronomicum Caesareum_ (of Apian), V. 97

  _Astronomisches Jahrbuch_, XII. 253

  Astronomy, divisions of, XIII. 272

  Astronomy, descriptive, XIII. =272=, 273, 294

  Astronomy, gravitational, X. =196=; XIII. 272, 273, 286

  Astronomy, observational, XIII. 272, 273

  Astronomy, origin of, I. =2=

  Astronomy, scope of, I. 1

  Attraction. _See_ Gravitation

  Autumnal equinox, I. II. _See also_ Equinox

  Axioms. _See_ Laws of Motion

  Axis (of an ellipse), XI. 236, 244, 245


  Babylonians. _See_ Chaldaeans

  Bagdad, III. 56, 57, 60, 68

  Belts of Jupiter, XII. 267

  Betelgeux, III. 64

  Biela’s comet, XIII. 305

  Binary stars. _See_ Stars, double and multiple

  Bode’s _Astronomisches Jahrbuch_, XII. 253

  Bodily tides, XIII. 292, 293

  Brightness of stars. _See_ Stars, brightness of

  Brooks’s comet, XIII. 305

  Brucia, XIII. 294

  Bureau des Longitudes, XI. 238


  Cairo, III. 60

  Calculus of Variations, XI. 237 _n_

  Calendar, Greek, II. =19=, =20=, 21

  Calendar, Gregorian, II. =22=; IX. 165 _n_; X. 217

  Calendar, Julian, II. =21=, 22; III. 68; IV. 73; IX. 165 _n_;
    X. 197 _n_, 217

  Calendar, Mahometan, III. 56

  Calendar, Roman, II. 21

  Caliphs, III. 56, 57, 69

  Canals of Mars, XIII. 297

  Cartesianism, VIII. 163; IX. 191, 195; XI. 229

  Castor, XII. 263, 264, 266

  Catalogues of stars. _See_ Star-catalogues

  Cavendish experiment, X. 219

  Celestial latitude. _See_ Latitude, celestial

  Celestial longitude. _See_ Longitude, celestial

  Celestial sphere, I. =7=, =8=, 9-11, 13, 14; II. 24, 26, 33, 38-40,
    45, 47, 51; III. 64, 67; IV. 78, 80, 83, 88; V. 98, 105; VI. 129;
    VIII. 157; X. 198, 207, 208, 214; XII. 257; XIII. 272

  Celestial sphere, daily motion of. _See_ Daily motion

  Celestial spheres, II. 23, 26, 27, 38, 51, 54; III. 62, =68=; V. 100,
    103

  Centre of gravity, IX. 186

  Centrifugal force, VIII. 158 _n_

  Ceres, XIII. 276, 294

  Chaldaeans (Babylonians), Chaldaean astronomy, I. 6, 11, 12, =16-18=;
    II. 30, 34, 40, 51, 54; III. 56

  Chart, photographic, XIII. 280

  Chemistry of the sun, XIII. =299-301=, 303

  Chinese astronomy, I. 6, 11

  Chords, II. 47 _n_

  Chromosphere, X. 205; XIII. =301=, 303

  Chronometer, X. 226

  Circular motion, II. 25, 26, 38, 51; IV. 76; VII. 139; VIII. =158=;
    IX. =171-173=. _See also_ Eccentric, Epicycle

  Circumpolar stars, I. =9=; II. 35

  Clock. _See_ Pendulum clock _and_ Chronometer

  Clustering power, XII. 261. _See also_ Condensation of nebulae

  _Coelum Australe Stelliferum_ (of Lacaille), X. 223

  Collimation error, X. 225 _n_

  Comets, I. 1; II. 30; III. 68, =69=; V. 100, 103-105; VI. 127, 129;
    VII. 144, =146=; VIII. 153; IX. =190=, 192; X. =200=, 205, 217,
    224; XI. 228, =231=, 243, 248, 250; XII. 253, 256, 259;
    XIII. 272, =291=, =304=, =305=, 307

  Comet, Biela’s, XIII. 305

  Comet, Brooks’s, XIII. 305

  Comet, Encke’s, XIII. 291

  Comet, Halley’s, VII. 146; X. =200=, 205; XI. =231=, 232; XIII. 291,
    307

  Comet, Lexell’s, XI. 248; XIII. 305

  Comet, Olbers’s, XIII. 291

  Comet, Pons-Brooks, XIII. 291

  Comet, Tebbutt’s, XIII. 305

  Comet, Tuttle’s, XIII. 291

  _Cometographia_ (of Hevel), VIII. 153

  _Commentaries on the Motions of Mars_ (of Kepler), VII. 135 _n_,
    139, 141, 150 _n_

  _Commentariolus_ (of Coppernicus), IV. =37=; V. 100

  _Compleat System of Optics_ (of Smith), XII. 251

  Complete induction, IX. 195

  Condensation of nebulae, XI. 250; XII. 261; XIII. 318

  Conic, conic section, VII. =140= _n_; XIII. 309

  Conjunction, II. =43=, 48 _n_; III. 60; V. 110, III; X. 227

  Conservation of energy, XIII. 319

  Constant of aberration, X. =209=, 210; XIII. 283 _n_

  Constellations, I. =12=, 13; II. 20, =26=, 34, =42=; X. =223=

  Construction of the heavens, XII. 257. _See also_ Sidereal system,
    structure of

  Corona, VII. 145; X. 205; XIII. =301=, 303

  Counter-earth, II. 24

  Crape-ring, XII. 267; XIII. 295

  Craters (on the moon), VIII. 153; XIII. 296

  Curvature of the earth. _See_ Earth, shape of


  Daily Motion (of the celestial sphere), I. 5, =8=, =9=, 10; II. 23,
    24, 26, 33, 38, 39, 46, 47; III. 67, 68; IV. =78=, =80=, 83;
    V. 98, 105; VI. 129; VIII. 157

  D’Alembert’s principle, XI. 232, 237 _n_

  Damascus, III. 57

  _Darlegung_ (of Hansen), XIII. 286

  Day, I. 4, II. 16; II. 47; XIII. 287, 293, 320

  Day-and-night, I. 16 _n_

  Day-hour, I. 16

  Declination, II. =33=, 35, 39; X. 213, 218; XIII. 276

  Declination circle, II. 33

  _De Coelo_ (of Aristotle), II. 27

  Deductive method, inverse, IX. 195

  Deferent, II. =39=, 48, 51; III. 68; IV. 86, 87

  Degree, I. 7

  Deimos, XIII. 295

  _De Magnete_ (of Gilbert), VII. 150

  _De Motu_ (of Newton), IX. 177, 191

  _De Mundi aetherei_ (of Tycho), V. 104

  _De Nova Stella_ (of Tycho), V. 100

  _De Revolutionibus_ (of Coppernicus), II. 41 _n_; IV. =74-92=;
    V. 93, 94; VI. 126

  _De Saturni Luna_ (of Huygens), VIII. 154

  Descriptive astronomy, XIII. =272=, 273, 294

  Deviation error, X. 225 _n_

  _Dialogue on the Two Chief Systems_ (of Galilei), VI. 124 _n_,
    =128-132=, 133

  Differential method of parallax, VI. =129=; XII. 263; XIII. 278

  Diffraction-grating, XIII. 299

  Dione, VIII. 160

  _Dioptrice_ (of Kepler), VII. 138

  Direct motion, I. 14

  Disturbing force. _See_ Perturbations

  Diurnal method of parallax, XIII. =281=, 284

  Doctrine of the sphere. _See_ Spherics

  Doppler’s principle, XIII. =302=, 313, 314

  Double hour, I. 16

  Double-star method of parallax. _See_ Differential method of parallax

  Double stars. _See_ Stars, double and multiple

  Draconitic month, II. =40=, 43

  _Durchmusterung_, XIII. 280

  Dynamics, VI. 133, 134; IX. 179, 180; XI. 230, 232, 237

  _Dynamique, Traité de_ (of D’Alembert), XI. 232


  Earth, I. 1, 15, 17; II. 28, 29, 32, 39, 41, 43, 47, 49, 51; III. 66,
    69; IV. 80, 86; VI. 117, 121, 133; VII. 136 _n_, 144, 145, 150;
    VIII. 153, 154; IX. 173, 174, 179-182, 184, 186, 195; XI. 228, 245;
    XIII. 285, 287, 292, 293, 297, 320. _See also_ the following
    headings

  Earth, density, mass of, IX. 180, 185, 189; X. =219=; XI. 235;
    XIII. 282, 294

  Earth, motion of, II. 24, 32, 47; IV. =73=, 76, =77=; V. 96. 97,
    =105=; VI. 121, 125-127, =129-132=; VIII. 161, 162; IX. 186, 194;
    XII. 257. _See also_ Earth, revolution of _and_ rotation of

  Earth, revolution of, annual motion of, II. 23, =24=, 28 _n_, =30=,
    47; IV. =75=, =77=, =79-82=, =85-88=, 89, 90, =92=; V. 111;
    VI. 119, 126, =129=, 131, 133; VII. 139, 142, 146; VIII. 161;
    IX. 172, 183; X. 207-210, 212, 227; XI. 235, 236, 240; XII. 263;
    XIII. 278, 282, 283

  Earth, rigidity of, XIII. 285, 292

  Earth, rotation of, daily motion of, II. 23, =24=, 28 _n_; IV. =75=,
    =78=, 79 _n_, =80=, 84; V. =105=; VI. 124, 126, =129=, =130=;
    IX. 174, 194; X. 206, 207, 213; XIII. 281, 285, =287=, 320

  Earth, shape of, II. =23=, =29=, =35=, =45=, =47=, 54; IV. 76; VIII.
    =161=; IX. =187=, 188; X. 196, 213, 215, 220, =221=, 222, 223; XI.
    229, =231=, 237, 248

  Earth, size of, II. =36=, 41, 45, 47, 49; III. 57, 69; IV. 85;
    VII. 145; VIII. =159=, 161; IX. 173, 174; X. =221=, 222, 223

  Earth, zones of, II. =35=, 47

  Earthshine, III. 69

  Easter, rule for fixing, II. 20

  Eccentric, II. 37, =39=, =40=, 41, =48=, =51=; III. 59; IV. =85=, =89=,
    =91=; VII. 139, 150

  Eccentricity, II. =39=; IV. 85; VII. =140= _n_; XI. 228, 236, 240,
    244-246, 250; XIII. 294, 318

  Eccentricity fund, XI. 245

  Eclipses, I. 11, 15, =17=; II. 29, 32, 40-42, =43=, 47-49, 54;
    III. 57, 68; IV. 76, 85; V. 110; VI. 127; VII. 145, 148: VIII. 162;
    X. 201, 205, 210, 216, 227; XI. 240; XIII. 287, 301

  Eclipses, annular, II. =43=; VII. 145

  Eclipses, partial, II. 43

  Eclipses, total, II. =43=; VII. 145; X. 205; XIII. =301=

  Ecliptic, I. =11=, 13, 14; II. 26, 33, 35, 36, 38, 40, 42, 51; III.
    58, 59, 68; IV. 80, 82-84, 87, 89; V. 111; VIII. 154; X. 203, 209,
    213, 214, 227; XI. 235, 236, 244, 246, 250

  Ecliptic, obliquity of, I. =11=; II. 35, 36, 42; III. 59, 68; IV. 83,
    84; XI. 235, 236

  École Normale, XI. 237, 238

  École Polytechnique, XI. 237

  Egyptians, Egyptian astronomy, I. 6, 11, 12, 16; II. 23, 26, 30, 45;
    IV. 75

  _Elements_ (of Euclid), III. 62, 66; IX. 165

  Elements (of an orbit), XI. =236=, 240, 242, 244, 246; XIII. 275, 276

  Elements, variation of. _See_ Variation of elements

  Ellipse, II. 51 _n_; III. 66; VII. =140=, 141; IX. 175, 176, 190, 194;
    X. 200, 209, 214; XI. 228, 236, 242, 244; XIII. 276, 278, 309

  Ellipticity, X. 221

  Empty month, II. =19=, 20

  Empyrean, III. 68

  Enceladus, XII. 255

  Encke’s comet, XIII. 291

  Encyclopaedia, the French, XI. 232

  Energy, XIII. 319

  Ephemerides. _See_ Almanacks

  _Ephemerides_ (of Regiomontanus), III. 68

  Epicycle, II. 37, =39=, 41, 45, =48=, =51=, 54; III. 68; IV. =85-87=,
    =89-91=; VII. 139, 150; VIII. 163; IX. 170, 194

  _Epitome_ (of Kepler), VI. 132; VII. 144, =145=

  _Epitome_ (of Purbach), III. 68

  Equant, II. =51=; III. 62; IV. 85, 89, 91; VII. 139, 150

  Equation of the centre, II. =39=, 48; III. 60; V. 111

  Equator, I. =9=, 10, 11; II. 33, 35, 39, 42; IV. 82, 84; V. 98;
    VI. 129, 133; IX. 187; X. 207, 220, 221; XIII. 285

  Equator, motion of. _See_ Precession

  Equinoctial points, I. =11=, 13; II. 42. _See also_ Aries, first
    point of

  Equinoxes, I. =11=; II. 39, 42

  Equinoxes, precession of. _See_ Precession

  _Essai philosophique_ (of Laplace), XI. 238

  Ether, XIII. 293, 299

  Evection, II. =48=, 52; III. 60; IV. 85; V. 111; VII. 145

  Evening star, I. 14. _See also_ Venus

  _Exposition du Système du Monde_ (of Laplace), XI. =238=, 242 _n_,
    =250=


  Faculae, VIII. =153=; XIII. 300, 301, 303

  Figure of the earth. _See_ Earth, shape of

  Firmament, III. 68

  First point of Aries, Libra. _See_ Aries, Libra, first point of

  Fixed stars, I. 14. _See_ Stars

  Fluxions, IX. 169, 191; X. 196

  _Fluxions_ (of Maclaurin), XII. 251

  Focus, VII. =140=, 141; IX. 175; XI. 236

  Force, VI. 130; IX. =180=, 181

  Fraunhofer lines, XIII. =299=, =300=, 303, 304

  Front-view construction. _See_ Herschelian telescopes

  Full month, II. =19=, 20

  Full moon. _See_ Moon, phases of

  _Fundamenta_ (of Hansen), XIII. 286

  _Fundamenta Astronomiae_ (of Bessel), X. 218; XIII. =277=

  Funds of eccentricity, inclination, XI. 245


  Galactic circle, XII. 258, 260

  Galaxy. _See_ Milky Way

  Gauges, gauging. _See_ Star-gauging

  Georgium Sidus. _See_ Uranus

  Gravitation, gravity, II. 38 _n_; VII. 150; VIII. 158, 161;
    =IX.= _passim_; X. 196, 201, 213, 215, 219, 220, 223, 226;
    =XI.= _passim_; XII. 264; XIII. 282, 284, 286-293, 309, 319

  Gravitational astronomy, X. =196=; XIII. 272, 273, 286

  Gravity, variation of, VIII. =161=; IX. 180; X. 199, 217, 221, 223;
    XI. =231=

  Great Bear, I. 12; XII. 266

  Great circle, I. =11=; II. 33, 42; IV. 82, 84

  Gregorian Calendar. _See_ Calendar, Gregorian

  Grindstone theory, XII. =258=; XIII. 317


  _Hakemite Tables_, III. =60=, 62

  Halley’s comet, VII. 146; X. =200=, 205; XI. =231=, 232; XIII. 291, 307

  _Harmonics_ (of Smith), XII. 251

  _Harmony of the World_ (of Kepler), VII. 144

  Helium, XIII. 301

  Herschelian telescope, XII. 255, 256

  _Historia Coelestis_ (of Flamsteed), X. 198

  Holy Office. _See_ Inquisition

  Horizon, I. 3, 9; II. 29, =33=, 35, 39, 46; VIII. 161; XIII. 285

  Horoscopes, V. 99

  Hour, I. 16

  Hydrostatic balance, VI. 115 _n_

  Hyperbola, IX. 190; XI. 236 _n_

  Hyperion, XIII. 295


  _Ilkhanic Tables_, III. 62

  _Il Saggiatore_ (of Galilei), VI. 127

  Inclination, III. 58; IV. 89; XI. =228=, =244=, 245, 246, 250;
    XIII. 294, 318

  Inclination fund, XI. 245

  _Index of Prohibited Books_, VI. 126, 132; VII. 145

  Indians, Indian astronomy, I. 6; III. 56, 64

  Induction, complete, IX. 195

  Inequalities, long, XI. 243

  Inequalities, periodic, XI. =242=, 243, 245, 247

  Inequalities, secular, XI. =242=, 243-247; XIII. 282. _See also_
    Perturbations

  Inequality, parallactic, XIII. 282

  Inferior planets, I. =15=; IV. 87, 88. _See also_ Mercury, Venus

  Inquisition (Holy Office), VI. 126, 132, 133

  Institute of France, XI. 241

  Inverse deductive method, IX. 195

  Inverse square, law of, IX. =172-176=, =181=, 195; XI. 233. _See also_
    Gravitation

  Ionian school, II. 23

  Iris, XIII. 281

  Irradiation, VI. 129

  Island universe theory, XII. =260=; XIII. 317


  Japetus, VIII. 160; XII. 267; XIII. 297

  Julian Calendar. _See_ Calendar, Julian

  Juno, XIII. 294

  Jupiter, I. 14-16; 11. 25, 51; IV. 81, 87, 88; V. 98, 99; VI. 121,
    127; VII. 136 n, 142, 144, 145, 150; VIII. 154, 156, 162; IX. 172,
    181, 183, 185-187; X. 204, 216; XI. 228, 231, 235, 236, 243-245,
    248; XII. =267=; XIII. 281, 288, 294, =297=, 305. _See also_ the
    following headings

  Jupiter, belts of, XII. 267

  Jupiter, mass of, IX. 183, 185

  Jupiter, rotation of, VIII. =160=; IX. 187; XIII. =297=

  Jupiter, satellites of, II. 43; VI. =121=, =127=, 129, 133; VII.
    =145=, 150; VIII. 160, 162; IX. 170, 184, 185; X. 210, =216=;
    XI. 228, =248=; XII. 267; XIII. 283, =295=, 297

  Jupiter’s satellites, mass of, XI. 248


  Kepler’s Laws, VII. =141=, =144=, 145, 151; VIII. 152; IX. 169, 172,
    175, 176, 186, 194, 195; X. 220; XI. 244; XIII. 294, 309


  Latitude (celestial), II. =33=, 42, 43; III. 63; IV. 89

  Latitude (terrestrial), III. 68, 69; IV. 73; X. =221=; XIII. 285

  Latitude, variation of, XIII. 285

  Law of gravitation. _See_ Gravitation

  Laws of motion, VI. =130=, =133=; VIII. 152, 163; IX. 171, =179-181=,
    183, 186, 194, 195; XI. 232

  Leap-year, I. 17; II. =21=, 22

  Least squares, XIII. =275=, 276

  _Letter to the Grand Duchess_ (of Galilei), VI. 125

  Level error, X. 225 _n_

  Lexell’s comet, XI. 248; XIII. 305

  Libra, first point of (♎), I. =13=; II. 42

  Librations of the moon, VI. =133=; X. 226; XI. =237=, 239

  _Libros del Saber_, III. 66

  Light-equation, XIII. 283

  Light, motion of, velocity of, VIII. =162=; X. 208-211, 216, 220;
    XIII. 278, 279, =283=, 302. _See also_ Aberration

  Logarithms, V. 96, 97 _n_

  Long inequalities, XI. 243

  Longitude (celestial), II. =33=, 39, 42, 43; III. 63; IV. 87; VII. 139

  Longitude (terrestrial), III. 68; VI. =127=, 133; VII. 150; X. 197,
    216, =226=

  Longitudes, Bureau des, XI. 238

  Lunar distances, III. 68 _n_

  Lunar eclipses. _See_ Eclipses

  Lunar equation, XIII. 282

  Lunar theory, II. =48=, 51; V. 111; VII. 145; VIII. 156; IX. =184=,
    192; X. =226=; XI. =228=, 230, 231, =233=, =234=, =240=, =241=,
    242, 248; XIII. 282, =286=, =287=, 288, 290. _See also_ Moon,
    motion of

  Lunation, II. =40=. _See also_ Month, synodic


  _Macchie Solari_ (of Galilei), VI. =124=, 125

  Magellanic clouds, XIII. 307

  Magnetism, VII. 150; XIII. 276, =298=

  _Magnitudes and Distances of the Sun and Moon_ (of Aristarchus), II. 32

  Magnitudes of stars, II. 42; XII. 266; XII. 280, 316. _See also_
     Stars, brightness of

  Mars, I. 14-16; II. 25, 26, 30, 51; III. 68; IV. 81, 87; V. 108; VI.
    129; VII. 136 _n_, 139-142, 144, 145; VIII. 154, 161; IX. 181, 183,
    185; X. 223, 227; XI. 235, 245; XII. 267; XIII. 281, 282, 284, 294,
     295, =297=. _See also_ the following headings

  Mars, canals of, XIII. 297

  Mars, mass of, XI. 248

  Mars, opposition of, VIII. =161=; XIII. =281=, 284, 297

  Mars, rotation of, VIII. 160; XIII. 295, 297

  Mars, satellites of, XIII. 295

  Mass, IX. =180=, 181, 185

  Mass of the earth, sun, Venus.... _See_ Earth, Sun, Venus ... mass of

  _Mécanique Analytique_ (of Lagrange), XI. 237

  _Mécanique Celeste_ (of Laplace), XI. =238=, 241, 247, 249, 250;
    XIII. 292

  Medicean Planets. _See_ Jupiter, satellites of

  Meraga, III. 62

  Mercury, I. 14-16; II. 25, 26, 45, 47, 51; III. 66; IV. 73, 75, 81,
    86-89; VI. 121, 124; VII. 136 _n_, 139, 142, 144; IX. 185; XIII.
    288, 290, 294, =297=. _See also_ the following headings

  Mercury, mass of, XI. 248

  Mercury, phases of, VI. 129

  Mercury, rotation of, XIII. 297

  Mercury, transit of, X. 199

  Meridian, II. =33=, 39; III. 57; VI. 127; VIII. 157; X. 207, 218, 221

  _Meteorologica_ (of Aristotle), II. 27

  Meteors, XIII. 305

  Meton’s cycle, II. 20

  Metric system, XI. 237

  Micrometer, VIII. =155=; XIII. 279, 281

  Milky Way, II. 30, 33; VI. 120; XII. =258=, =260-262=; XIII. =317=

  Mimas, XII. 255

  Minor planets, XI. 250 _n_; XIII. 276, 281, 284, 288, =294=, 295, 297,
    318

  Minor planets, mass of, XIII. 294

  Minute (angle), I. 7

  Mira, XII. 266

  Mongols, Mongol astronomy, III. 62

  Month, I. 4, 16; II. 19-21, 40, 48; IX. 173; XI. 240; XIII. 293, 320.
   _See also_ the following headings

  Month, anomalistic, II. 40

  Month, draconitic, II. =40=, 43

  Month, empty, II. =19=, 20

  Month, full, II. =19=, 20

  Month, lunar, I. =16=; II. 19, 20, 40

  Month, sidereal, II. 40

  Month, synodic, II. =40=, 43

  Moon, I. =1=, 4, 5, 11, 13-16; II. 19-21, 25, 28, 30, 32, 39, 43;
    III. 68, 69; IV. 81, 86; V. 104, 105 _n_; VI. =119=, 121, 123, 129,
    130, 133; VII. 145, 150; VIII. =153=; IX. 169, 180, 181, 188, 189;
    X. 198, 204, 213, 215, =226=; XI 228, 235; XII. 256, 257, =271=;
    XIII. 272, 292, 293, =296=, 297, 301, 320. _See also_ the following
    headings

  Moon, angular or apparent size of, II. 32, =41=, =43=, 46 _n_, 48;
    IV. 73, 85, 90; V. 105 _n_

  Moon, apparent flattening of, II. 46

  Moon, atmosphere of, XIII. 296

  Moon, distance of, I. 15; II. 24, 25, 30, =32=, =41=, 43, 45, 48,
    =49=, 51; IV. 85, 90; V. 100, 103; IX. 173 185; X. 223; XIII. 293,
    320

  Moon, eclipses of. _See_ Eclipses

  Moon, librations of, VI. =133=; X. 226; XI. =237=, 239

  Moon, map of, X. 226; XIII. 296

  Moon, mass of, IX. 188, =189= XI. 235

  Moon, motion of, I. =4=, 8, 13, 15, 17; II. 20, 24-26, 28, 37, 39,
    =40=, 43, 47, =48=, 51; III. 60; IV. 73, 81, 85, 89, 90; V. 111;
    VI. 133; VII. =145=, 150; VIII. 156; IX. 169, =173=, =174=, 179,
    =184=, 185, 189, 194, 195; X. =201=, 204, 213, =226=; XI. 235,
    237, 248; XIII. 287, 290, 297, 320. _See also_ Lunar theory

  Moon, origin of, XIII. 320

  Moon, parallax of, II. =43=, =49=; IV. 85. _Cf. also_ Moon, distance
    of

  Moon, phases of, I. =4=, 16, 17; II. 19, 20, 23, =28=, 43, 48;
    III. 68, 69; VI. 123

  Moon, rotation of, X. 226; XI. 248; XII. 267; XIII. 297

  Moon, shape of, II. 23, 28, 46; VI. 119; XI. 237

  Moon, size of, II. =32=, 41; IV. 85

  Moon, tables of. _See_ Tables, lunar

  Moons. _See_ Satellites

  Morning star, I. 14. _See also_ Venus

  Morocco, III. 61

  Motion, laws of. _See_ Laws of motion

  Multiple stars. _See_ Stars, double and multiple

  Mural quadrant, X. 218, 225 _n_

  Music of the spheres, II. =23=; VII. 144

  _Mysterium Cosmographicum_ (of Kepler), V. 108; VII. 136, 144


  Nadir, III. 64

  _Nautical Almanac._ _See_ Almanac, Nautical

  Nebula in Argus, XIII. 307

  Nebula in Orion, XII. 252, 259, 260; XIII. 311

  Nebulae, X. 223; XI. 250; XII. 252, 256, =259-261=; XIII. =306-308=,
    =310=, =311=, =317=, =318=, 319, 320

  Nebulae, spiral, XIII. 310

  Nebular hypothesis, XI. =250=; XIII. 318-320

  Nebulous stars, X. 223; XII. 260, 261

  Neptune, XIII. =289=, 295, =297=

  Neptune, satellite of, XIII. 295

  _New Almagest_ (of Kepler), VII. 148

  =New Almagest= (of Riccioli), VIII. 153

  New moon. _See_ Moon, phases of

  New stars. _See_ Stars, new

  New Style (N.S.), II. 22. _See also_ Calendar, Gregorian

  Newton’s problem, XI. =228=, 229, 249

  Newtonian telescope, IX. =168=; XII. 252, 253, 256

  Night-hour, I. 16

  Node, II. =40=, 43; V. 111; IX. 184; X. 213, 214; XI. =236=, 246

  Nubeculae, XIII. 307

  Nucleus (of a comet), XIII. 304

  Nürnberg school, III. =68=; IV. 73

  Nutation, X. 206, 207, =213-215=, 216, 218; XI. 232, 248; XII. 263

  Νυχθήμερον, I. 16 _n_


  Oberon, XII. 255

  Obliquity of the ecliptic. _See_ Ecliptic, obliquity of

  Observational astronomy, XIII. 272, 273

  Occultations, I. =15=; II. 30

  Octaeteris, II. 19

  Olbers’s comet, XIII. 291

  _Old Moore’s Almanack_, I. 18 _n_

  Old Style (O.S.). _See_ Calendar, Julian

  Opposition, II. =43=, 48 _n_; III. 60; IV. 87, 88; V. 111; VIII. 161;
    XIII. 281, 284, 297

  Opposition of Mars, VIII. =161=; XIII. =281=, 284, 297

  Optical double stars, XII. 264

  _Optics_ (of Gregory), X. 202

  _Optics_ (of Newton), IX. 192

  _Optics_ (of Ptolemy), II. 46

  _Optics_ (of Smith), XII. 251

  _Opus Majus_, _Minus_, _Tertium_ (of Bacon), III. 67

  _Opuscules Mathématiques_ (of D’Alembert), XI. 233

  Orion, nebula in, XII. 252, 259, 260; XIII. 311

  _Oscillatorium Horologium_ (of Huygens), VIII. =158=; IX. 171


  Pallas, XIII. 294

  Parabola, IX. =190=; XI. 236 _n_; XIII. 276

  Parallactic inequality, XIII. 282

  Parallax, II. =43=, =49=; IV. 85, =92=; V. 98, 100, 110; VI. =129=;
    VII. 145; VIII. =161=; X. 207, 212, 223, 227; XII. 257, 258, 263,
    264; XIII. 272, 278, 279, 281-284

  Parallax, annual, VIII. 161. _See also_ Parallax, stellar

  Parallax, horizontal, VIII. 161

  Parallax of the moon. _See_ Moon, parallax of

  Parallax of the sun. _See_ Sun, parallax of

  Parallax, stellar, IV. =92=; V. 100; VI. =129=; VIII. =161=; X. 207,
    =212=; XII. 257, 258, 263, 264; XIII. 272, =278=, =279=

  Parallelogram of forces, IX. 180 _n_

  Parameters, variation of, XI. 233 _n_. _See also_ Variation of elements

  Παραπήγματα II. 20

  Partial eclipses, II. 43

  Pendulum, pendulum clock, V. =98=; VI. =114=; VIII. =157=, 158, =161=;
    IX. 180, 187; X. 199, 217, 221, 223; XI. 231. _See also_ Gravity,
    variation of

  _Pendulum Clock_ (of Huygens), VIII. =158=; IX. 171

  Penumbra (of a sun-spot), VI. =124=; XII. 268

  Perigee, II. =39=, 40, 48; IV. 85. _See also_ Apse, apse-line

  Perihelion, IV. =85=; XI. 231. _See also_ Apse, apse-line

  Periodic inequalities. _See_ Inequalities, periodic

  Perturbations, VIII. 156; IX. =183=, =184=; X. 200, 204, 224, 227;
    =XI.= _passim_; XIII. 282, 293, 294, 297

  Phases of the moon. _See_ Moon, phases of

  _Phenomena_ (of Euclid), II. 33

  Phobos, XIII. 295

  Photography, XIII. 274, 279-281, 294, 298, 299, 301, 306

  Photometry, XIII. 316. _See also_ Stars, brightness of

  Photosphere, XII. =268=; XIII. 303

  Physical double stars, XII. 264. _See also_ Stars, double and multiple

  Planetary tables. _See_ Tables, planetary

  Planetary theory, II. 51, 52, 54; III. 68; IV. 86-90; XI. =228=, 230,
    231, 233, =235=, =236=, =242-247=, 248; XIII. 286, =288-290=, 293,
    _See also_ Planets, motion of

  Planets, I. 13, =14=, 15, 16; II. 23-27, 30, 32, 51; III. 68; IV. 81;
    V. 104, 105, 110, 112; VI. 119, 121; VII. 136, 144; VIII. =154=,
    155; X. 200; XI. 228, 250; XII. 253, 255, 257, =267=, 271; XIII.
    272, 275, 276, 281, 282, 294-296, =297=, 318, 320. _See also_ the
    following headings, _and_ the several planets Mercury, Venus, etc.

  Planets, discoveries of, XII. =253=, 254, 255, 267; XIII. =289=,
    =294=, 295, 318

  Planets, distances of, I. 15; II. 30, 51; IV. 81, 86, 87; VI. 117;
    VII. 136, 144; IX. 169, 172, 173

  Planets, inferior, I. =15=; IV. 87, 88. _See also_ Mercury, Venus

  Planets, masses of, IX. 185; XI. 245, 248; XIII. 294. _See also_ under
    the several planets

  Planets, minor. _See_ Minor planets

  Planets, motion of, I. 13, =14=, =15=; II. 23-25, =26=, =27=, 30, =41=,
    45, 47, =51=, 52; III. 62, 68; IV. =81=, =86-90=, 92; V. 100, 104,
    =105=, 112; VI. 119, 121, 129; VII. =139-142=, =144=, 145, 150, 151;
    VIII. 152, 156; IX. 169, =170=, =172-177=, =181=, =183=, 194;
    X. 199, 204; XI. 228, 229, 245, =250=; XIII. 275, 276, 282, 294.
    _See also_ Planetary theory

  Planets, rotation of, VIII. =160=; IX. 187; XI. 228, 250; XII. 267;
    XIII. =297=

  Planets, satellites of. _See_ Satellites

  Planets, stationary points of, I. =14=; II. 51; IV. =88=

  Planets, superior, I. =15=; IV. 87, 88. _See also_ Mars, Jupiter, etc.

  Pleiades, VI. 120; XII. 260

  Poles (of a great circle), II. 33 _n_

  Poles (of the celestial sphere), I. =8=, 9, 10; II. 33, 35; IV. 78;
    VI. 129; X. 207, 214; XIII. 285

  Poles (of the earth), IV. 82; IX. 187; X. 220, 221; XIII. 285

  Pole-star, I. 8, 9

  Pollux, XII. 266

  Pons-Brooks comet, XIII. 291

  Postulates (of Ptolemy), II. 47

  Postulates (of Coppernicus), IV. 76

  Praesepe, XII. 260

  Precession (of the equinoxes), II. =42=, =50=; III. 58, 59, 62, 68;
    IV. 73, 83, =84=, 85; V. 104, 112; VI. 129; IX. =188=, 192;
    X. 213-215, 218, 221; XI. 228, =232=, 248; XIII. 277, 280

  _Prima Narratio_ (of Rheticus), IV. =74=; V. 94

  Primum Mobile, III. 68

  _Principia_ (of Descartes), VIII. 163

  _Principia_ (of Newton), IV. 75; VIII. 152; IX. 164, =177-192=, 195;
    X. 196, 199, 200, 213; XI. 229, 234, 235, 240

  _Principles of Philosophy_ (of Descartes), VIII. 163

  _Probabilités, Théorie Analytique des_ (of Laplace), XI. 238

  Problem of three bodies. _See_ Three bodies, problem of

  _Prodromus Cometicus_ (of Hevel), VIII. 153

  Prominences, XIII. =301=, 302, 303

  Proper motion (of stars), X. =203=, 225; XII. 257, =265=; XIII. 278, 280

  Prosneusis, II. =48=; III. 60; IV. 85

  _Prussian Tables_, V. =94=, 96, 97, 99; VII. 139

  Pythagoreans, II. 24; IV. 75


  Quadrant, V. 99; X. 218, 225 _n_

  Quadrature, II. =48=; III. 60; V. 111

  Quadrivium, III. 65


  _Recherches sur différens points_ (of D’Alembert), XI. =233=, 235

  _Recherches sur la précession_ (of D’Alembert), XI. 215

  Reduction of observations, X. 198, =218=; XIII. 277

  Reflecting telescopes, IX. =168=; XII. 251-255

  Refracting telescopes, IX. 168. _See also_ Telescopes

  Refraction, II. =46=; III. 68; V. 98, =110=; VII. =138=; VIII. 159,
    =160=; X. =217=, 218, 223; XIII. =277=

  Relative motion, principle of, IV. =77=; IX. 186 _n_

  Renaissance, IV. 70

  _Results of Astronomical Observations_ (of John Herschel), XIII. 308

  Retrograde motion, I. 14

  Reversing stratum, XIII. 303

  Reviews of the heavens, XII. 252, 253

  Revival of Learning, IV. 70

  Rhea, VIII. 160

  Rigel, III. 64

  Right ascension, II. =33=, 39; X. 198, 218; XIII. 276

  Rills, XIII. 296

  Rings of Saturn. _See_ Saturn, rings of

  Rotation of the celestial sphere. _See_ Daily motion

  Rotation of the earth, sun, Mars, etc. _See_ Earth, Sun, Mars, etc.,
    rotation of

  Royal Astronomical Society. _See_ Astronomical Society, Royal

  Royal Society, IX. 166, 174, 177, 191, 192; X. 201, 202, 206, 208;
    XII. 254, 256, 259, 263; XIII. 292, 308

  _Rudolphine Tables_, V. 94; VII. =148=, 151; VIII. 156

  Ruler, I. 16

  Running down of the solar system, XIII. 293, 319


  _Saggiatore_ (of Galilei), VI. 127

  Sappho, XIII. 281

  Saros, I. =17=; II. 43

  Satellites, VI. =121=, =127=, 129, 133; VII. 145, 150; VIII. =154=,
    =160=, 162; IX. 170, 183-185; X. 210, 216; XI. 228, 248; XII. 253,
    =255=, =267=; XIII. 272, 283, =295=, 296, 297, 318, 320. _See also_
    Jupiter, Saturn, etc., satellites of

  Satellites, direction of revolution of, XI. 250; XIII. 295, 318

  Satellites, rotation of, XI. 250; XII. 267; XIII. 297

  Saturn, I. 14-16; II. 25, 51; IV. 81, 87; V. 99; VI. 123; VII.
    136 _n_, 142, 144; VIII. 154, 156; IX. 183, 185, 186; X. 204;
    XI. 228, 231, 235, 236, 243-246; XII. 253, =267=; XIII. 288,
    =297=. _See also_ the following headings

  Saturn, mass of, IX. 185

  Saturn, rings of, VI. =123=; VIII. =154=, =160=; XI. 228, =248=;
    XII. 267; XIII. =295=, =297=

  Saturn, rotation of, XII. 267; XIII. 297

  Saturn, satellites of, VIII. =154=, =160=; IX. 184; XI. 228;
    XII. 253, =255=, 267; XIII. =295=, 297, 307

  Scientific method, II, 54; VI. =134=; IX. =195=

  Seas (on the moon), VI. 119; VIII. 153; XIII. 296

  Seasons, I. 3; II. =35=, 39; IV. =82=; XI. 245

  Second (angle), I. 7

  Secular acceleration of the moon’s mean motion, X. =201=;
    XI. 233, 234, =240=, 242; XIII. =287=

  Secular inequalities. _See_ Inequalities, secular

  _Selenographia_ (of Hevel), VIII. 153

  _Selenotopographische Fragmente_ (of Schroeter), XII. 271

  Sequences, method of, XII. 266

  Shadow of earth, moon. _See_ Eclipses

  “Shining-fluid” theory, XII. =260=; XIII. 310, 311

  Shooting stars. _See_ Meteors

  Short-period comets, XIII. 291

  Sidereal month, II. 40

  Sidereal period, IV. 86, 87

  Sidereal system, structure of, XII. 257, =258=, 259-262; XIII. =317=

  Sidereal year, II. 42

  _Sidereus Nuncius_ (of Galilei), VI. 119-122

  “Sights,” V. 110; VIII. =155=; X. 198

  Signs of the zodiac, I. 13

  Sine, II. 47 _n_; III. 59 _n_, 68 _n_

  Sirius, XIII. 316 _n_

  Solar eclipse. _See_ Eclipse

  Solar system, stability of, XI. 245; XIII. 288, 293

  Solstices, I. =11=; II. 36, 39, 42

  Solstitial points, I. 11

  Space-penetrating power, XII. 258

  Spanish astronomy, III. 61, 66

  Spectroscope, XIII. 299. _See also_ Spectrum analysis

  Spectrum, spectrum analysis, IX. 168; XIII. 273, =299-302=, 303,
    =304=, 306, 309, =311-314=, 317, 318

  _Sphaera Mundi_ (of Sacrobosco), III. 67

  Sphere, attraction of, IX. 173, =182=; XI. 228

  Sphere, celestial. _See_ Celestial sphere

  Sphere, doctrine of the. _See_ Spherics

  Spheres, celestial, crystal. _See_ Celestial spheres

  Spheres, music of the, II. =23=; VII. 144

  Spherical form of the earth, moon. _See_ Earth, Moon, shape of

  Spherics, II. =33=, 34

  Spica, II. 42

  Spiral nebulae, XIII. 310

  Stability of the solar system, XI. 245; XIII. 288, 293

  Stadium, II. 36, 45, 47

  Star-atlases, star-maps, I. 12 _n_; X. 198, 223; XII. 259, 266;
    XIII. 280, 294

  Star-catalogues, II. =32=, =42=, =50=; III. 62, =63=; IV. =83=;
    V. =98=, =107=, 110, 112; VIII. =153=; X. =198=, =199=, 205,
    =218=, =223-225=; XII. 257; XIII. =277=, =280=, 316

  Star-clusters, VI. 120; X. 223; XII. 258, =259=, =260=, 261; XIII.
    307, 308, 310, 311, 318

  Star-gauging, XII. =258=; XIII. 307

  Star-groups. _See_ Constellations

  Stars, I. 1, 5, 7-10, 12-15, 18; II. 20, 23-26, 29, 30, 32, 33, 39,
    40, 42, 45-47, 50; III. 56, 57, 62, 68; IV. 73, 78, 80, 86, 89, 92;
    V. 96-100, 104, 105, 110; VI. 120, 121, 129; VIII. 155, 157, 161;
    IX. 186 _n_; X. 198, 199, 203, 207-214, 218, 223; XI. 228; XII. 253,
    =257-266=, 267; XIII. 272, 277-280, 283, 304, =306-318=, 320.
   _See also_ the preceding and following headings

  Stars, binary. _See_ Stars, double and multiple

  Stars, brightness of, II. 42; XII. 258, 266; XIII. 278, 280, 316,
    317. _See also_ Stars, variable

  Stars, circumpolar, I. =9=; II. 35

  Stars, colours of, XII. 263; XIII. 309

  Stars, distances of, I. 7; II. 30, 32, 45, 47; IV. 80, =92=; V. 100;
    VI. 117, =129=; XI. 228; XII. 257, 258, 265, 266; XIII. =278=,
    =279=, 317. _See also_ Parallax, stellar

  Stars, distribution of, XII. 257, 258. _See also_ Sidereal system,
    structure of

  Stars, double and multiple, XII. 256, =263=, =264=; XIII. 306-308,
    =309=, =314=, 320

  Stars, magnitudes of, II. 42; XII. 266; XIII. 280, 316. _See also_
    Stars, brightness of

  Stars, motion of. _See_ Stars, proper motion of, _and_ Daily motion
   (of the celestial sphere)

  Stars, names of, I. 12, 13; III. 64

  Stars, nebulous, X. 223; XII. 260, 261

  Stars, new, II. 42; V. 100, 104; VI. 117, 129; VII. 138; XII. 266;
    XIII. 312

  Stars, number of, I. 7 _n_; XIII. 280

  Stars, parallax of. _See_ Parallax, stellar

  Stars, proper motion of, X. =203=, 225; XII. 257, =265=; XIII. 278,
    280

  Stars, rotation of, XII. 266

  Stars, spectra of, XIII. 311-314, 317

  Stars, system of. _See_ Sidereal system, structure of

  Stars, variable, XII. =266=, 269; XIII. 307, 312, =314=, =315=

  Stationary points, I. =14=; II. 51; IV. =88=

  Stjerneborg, V. 101

  Summer solstice, I. 11. _See also_ Solstices

  Sun, I. 1, 4, 10, 13, 14, 16; II. 21, 23-26, 28-30, 32, 35, 36, 40,
    43, 45, 48, 51; III. 68, 69; IV. 73, 75, 77, 79-82, 85-90, 92;
    V. 98, 103, 105, 110, 111; VI. 119, 121, 123, 124, 126, 127, 129,
    132; VII. 136, 139-141, 144-146, 150; VIII. 153, 154, 156;
    IX. 170, 172-175, 181, 183-186, 188-190, 194; X. 198, 200, 202,
    205, 210, 213, 223, 227; XI. 228, 235, 236, 240, 243, 245, 250;
    XII. 257, 265, =268=, =269=; XIII. 272, 278, 283, 288, 292-294,
    297, =298-303=, 304, 305, =307=, =319=, 320. _See also_ the
    following headings

  Sun, angular or apparent size of, II. =32=, 38, 39, =41=, 43, 46 _n_,
    48; IV. 73, 90; V. 105 _n_

  Sun, apparent flattening of, II. 46

  Sun, distance of, I. 15; II. 24, 25, 30, =32=, 38, =41=, 43, 45, 48,
    =49=, 51; IV. 81, =85=, 86, 87, 90, 92; V. 111; VII. 144, 145;
    VIII. 156, =161=: IX. 185, 188; X. 202, 205, 223, =227=; XI. 235;
    XIII. 278, =281-284=

  Sun, eclipses of. _See_ Eclipses

  Sun, heat of, XII. 268, 269; XIII. 303, =307=, =319=

  Sun, mass of, IX. 183, 184, =185=, 189; XI. 228; XIII. 282

  Sun, motion of, I. 3, 5, 8, =10=, =11=, 13, 15-17; II. 20, 21, 24-26,
    35, 37, =38=, =39=, 40, 42, 43, 47, 48, 51; III. 59; IV. 73, =77=,
    =79=, 85, 86, 87, 92; V. 104, =105=, 111; VI. 121, 126, 127, 132;
    VIII. 160; IX. =186=; X. 223; XI. 235; XII. =265=; XIII. 288

  Sun, parallax of, II. 43; V. 98, 110; VII. 145; VIII. =161=; X. 223,
    =227=; XIII. =281-284=. _See also_ Sun, distance of

  Sun, rotation of, VI. =124=; VII. 150; XI. 250; XIII. 297, =298=, 302

  Sun, size of, II. =32=; IV. 85; VII. 145; IX. 173; XIII. 319

  Sun, tables of. _See_ Tables, solar

  Sun-dials, II. 34

  Sun-spots, VI. =124=, 125; VIII. 153; XII. =268=, 269; XIII. =298=,
    300, 302, 303

  Superior planets, I. =15=; IV. 87, 88. _See also_ Mars, Jupiter, etc.

  Svea, XIII. 294

  Synodic month, II. =40=, 43

  Synodic period, IV. 86, 87

  _Synopsis of Cometary Astronomy_ (of Halley), X. 200

  _Systema Saturnium_ (of Huygens), VIII. 154

  _Système du Monde_ (of Laplace), XI. =238=, 242 _n_, =250=

  _Système du Monde_ (of Pontécoulant), XIII. 286


  _Table Talk_ (of Luther), IV. 73

  Tables, astronomical, III. 58, =60-63=, =66=, 68; IV. 70; V. =94=,
    96, 97, 99, 110; VII. 139, =148=; VIII. 156, =160=; X. =216=, 217;
    XIII. 277. _See also_ the following headings

  Tables, lunar, II. =48=; III. =59=; X. =204=, 216, 217, =226=; XI.
    =233=, 234, =241=; XIII. =286=, 290

  Tables, planetary, III. =63=; V. 108, 112; VII. 142, 143; X. =204=,
    216; XI. 235, =247=; XIII. =288=, 289, 290

  Tables, solar, III. =59=; IV. =85=; V. 111; VIII. 153; X. =224=, 225,
    226; XI. 235, =247=; XIII. 290

  Tables, Alfonsine, III. =66=, 68; V. 94, 96, 99

  Tables, Hakemite, III. =60=, 62

  Tables, Ilkhanic, III. 62

  Tables, Prussian, V. =94=, 96, 97, 99; VII. 139

  Tables, Rudolphine, V. 94; VII. =148=, 151; VIII. 156

  Tables, Toletan, III. =61=, 66

  _Tables de la Lune_ (of Damoiseau), XIII. 286

  _Tabulae Regiomontanae_ (of Bessel), XIII. 277

  Tangent, III. 59 _n_, 68 _n_

  Tartars, Tartar astronomy, III. 63

  Tebbutt’s comet, XIII, 305

  Telescope, III. 67; VI. =118-124=, 134; VII. =138=; VIII. =152-155=;
   IX. =168=; X. 207, 213, 218; XII. 251, =252-258=, 260, 262, =271=;
   XIII. =274=, 300, 301, 306, =310=, 317

  _Theoria Motus_ (of Gauss), XIII. 276

  _Theoria Motuum Lunae_ (of Euler), XI. 233

  _Théorie de la Lune_ (of Clairaut), XI. 233

  _Théorie ... des Probabilités_ (of Laplace), XI. 238

  _Théorie ... du Système du Monde_ (of Pontécoulant), XIII. 286

  _Theory of the Moon_ (of Mayer), X. 226

  _Theory of the Universe_ (of Wright), XII. 258

  Thetis, VIII. 160

  Three bodies, problem of, XI. =228=, =230-233=, 235

  Tidal friction, XIII. =287=, 292, 293, =320=

  Tides, VI. 130; VII. 150; IX. =189=; XI. 228-230, 235, =248=;
    XIII. =287=, =292=, 293, 297, =320=

  Time, measurement of, I. 4, 5, 16. _See also_ Calendar, Day, Hour,
     Month, Week, Year

  Titan, VIII. 154

  Titania, XII. 255

  _Toletan Tables_, III. =61=, 66

  Torrid zones, II. 35

  Total eclipse, II. =43=; VII. 145; X. 205; XIII. =301=. _See also_
    Eclipses

  Transit instrument, X. 218, 225 _n_

  Transit of Mercury, X. 199

  Transit of Venus, VIII. 156; X. =202=, 205, 224, =227=; XIII. =281=,
    282, 284

  Translations, III. 56, 58, 60, 62, 66, 68

  Transversals, V. 110 _n_

  Trepidation, III. =58=, 62, 68; IV. 84; V. 112

  Trigonometry, II. 37 _n_, 47 _n_; III. 59 _n_, 64 _n_, 68 _n_; IV. 74

  Trivium, III. 65

  Tropical year, II. 42

  Tuttle’s comet, XIII. 291

  Twilight, III. 69

  Twinkling of stars, II. 30

  _Two New Sciences_ (of Galilei), VI. =133=, 134 _n_; VIII. 152

  Tychonic system, V. =105=; VI. 127


  Umbra (of sun-spots), VI. =124=; XII. 268

  Uniform acceleration, VI. 133. _See also_ Acceleration

  Uraniborg, V. 101

  _Uranometria Nova Oxoniensis_, XIII. 316

  Uranus, XII. =253=, 254, 255, 267; XIII. 276, 288, 289, =297=

  Uranus, rotation of, XIII. 297

  Uranus, satellites of, XI. 250 _n_; XII. =255=, 267; XIII. 272, =295=


  Variable stars. _See_ Stars, variable

  Variation (of the moon), III. =60=; V. =111=; VII. 145

  Variation of elements or parameters, XI. 233 _n_, =236=, 245

  Variations, calculus of, XI. 237 _n_

  Vega, III. 64

  Venus, I. 14-16; II. 25, 26, 45, 47, 51; III. 68; IV. 75, 81, 86, 87;
    V. 98, 100, 103; VI. 121, 123; VII. 136 n, 139, 142, 144; VIII.
    =154=; IX. 181, 185; X. 223, 227; XI. 235, 245; XII. =267=, 271;
    XIII. 282, =297=. _See also_ the following headings

  Venus, mass of, XI. 235, 248

  Venus, phases of, VI. =123=, 129

  Venus, rotation of, VIII 160; XII. 267; XIII. 297

  Venus, transits of. _See_ Transits of Venus

  Vernal equinox, I. 11. _See also_ Equinoxes

  Vernier, III. 69 _n_

  Vertical, II. 33; X. 221; XIII. 285

  Vesta, XIII. 294

  Victoria, XIII. 281

  Virtual velocities, XI. 237 _n_

  Vortices, VIII. =163=; IX. 178, 195


  Wave, wave-length (of light) XIII. 299, 300, 302

  Weather, prediction of, II. 20; VII. 136

  Week, I. 16

  Weight, VI. 116, 130; IX. 180

  Weights and Measures, Commission on, XI. 237, 238

  _Whetstone of Witte_ (of Recorde), V. 95

  Winter solstice, I. 11. _See also_ Solstices


  Year, I. 3, 4, 16; II. 19-22, 42, 47; III. 66; V. 111

  Year, sidereal, II. 42

  Year, tropical, II. 42


  _Zadkiel’s Almanack_, I. 18 _n_

  Zenith, II. =33=, 35, 36, 46; III, 64; X. 221

  Zenith-sector, X. 206

  Zodiac, I. =13=; X. 224

  Zodiac, signs of the, I. 13

  Zodiacal constellations, I. 13

  Zones of the earth, II. =35=, 47



FOOTNOTES:

[1] In our climate 2,000 is about the greatest number ever visible at
once, even to a keen-sighted person.

[2] Owing to the greater brightness of the stars overhead they usually
seem a little nearer than those near the horizon, and consequently the
visible portion of the celestial sphere appears to be rather less than
a half of a complete sphere. This is, however, of no importance, and
will for the future be ignored.

[3] A right angle is divided into ninety degrees (90°), a degree into
sixty minutes (60′), and a minute into sixty seconds (60″).

[4] I have made no attempt either here or elsewhere to describe
the constellations and their positions, as I believe such verbal
descriptions to be almost useless. For a beginner who wishes to become
familiar with them the best plan is to get some better informed, friend
to point out a few of the more conspicuous ones, in different parts of
the sky. Others can then be readily added by means of a star-atlas, or
of the star-maps given in many textbooks.

[5] The names, in the customary Latin forms, are: Aries, Taurus,
Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus,
Aquarius, and Pisces; they are easily remembered by the doggerel
verses:—

    The Ram, the Bull, the Heavenly Twins,
    And next the Crab, the Lion shines,
      The Virgin and the Scales,
    The Scorpion, Archer, and He-Goat,
    The Man that bears the Watering-pot,
      And Fish with glittering tails.


[6] This statement leaves out of account small motions nearly or
quite invisible to the naked eye, some of which are among the most
interesting discoveries of telescopic astronomy; see, for example,
chapter X., §§ 207-215.

[7] The custom of calling the sun and moon planets has now died out,
and the modern usage will be adopted henceforward in this book.

[8] It may be noted that our word “day” (and the corresponding word in
other languages) is commonly used in two senses, either for the time
between sunrise and sunset (day as distinguished from night), or for
the whole period of 24 hours or day-and-night. The Greeks, however,
used for the latter a special word, νυχθήμερον.

[9] Compare the French: Mardi, Mercredi, Jeudi, Vendredi; or better
still the Italian: Martedi, Mercoledi, Giovedi, Venerdi.

[10] See, for example, _Old Moore’s_ or _Zadkiel’s Almanack_.

[11] We have little definite knowledge of his life. He was born in the
earlier part of the 6th century B.C., and died at the end of the same
century or beginning of the next.

[12] Theophrastus was born about half a century, Plutarch nearly five
centuries, later than Plato.

[13] _Republic_, VII. 529, 530.

[14] Confused, because the mechanical knowledge of the time was quite
unequal to giving any explanation of the way in which these spheres
acted on one another.

[15] I have introduced here the familiar explanation of the phases of
the moon, and the argument based on it for the spherical shape of the
moon, because, although probably known before Aristotle, there is,
as far as I know, no clear and definite statement of the matter in
any earlier writer, and after his time it becomes an accepted part of
Greek elementary astronomy. It may be noticed that the explanation is
unaffected either by the question of the rotation of the earth or by
that of its motion round the sun.

[16] See, for example, the account of Galilei’s controversies, in
chapter VI.

[17] The =poles= of a great circle on a sphere are the ends of a
diameter perpendicular to the plane of the great circle. Every point on
the great circle is at the same distance, 90°, from each pole.

[18] The _word_ “zenith” is Arabic, not Greek: cf. chapter III., § 64.

[19] Most of these names are not Greek, but of later origin.

[20] That of M. Paul Tannery: _Recherches sur l’Histoire de
l’Astronomie Ancienne_, chap. V.

[21] Trigonometry.

[22] The process may be worth illustrating by means of a simpler
problem. A heavy body, falling freely under gravity, is found (the
resistance of the air being allowed for) to fall about 16 feet in
1 second, 64 feet in 2 seconds, 144 feet in 3 seconds, 256 feet
in 4 seconds, 400 feet in 5 seconds, and so on. This series of
figures carried on as far as may be required would satisfy practical
requirements, supplemented if desired by the corresponding figures
for fractions of seconds; but the mathematician represents the same
facts more simply and in a way more satisfactory to the mind by the
formula s = 16 t^2, where s denotes the number of feet fallen,
and t the number of seconds. By giving t any assigned value, the
corresponding space fallen through is at once obtained. Similarly
the motion of the sun can be represented approximately by the more
complicated formula l = nt + 2 e sin nt, where l is the
distance from a fixed point in the orbit, t the time, and n, e
certain numerical quantities.

[23] At the present time there is still a small discrepancy between the
observed and calculated places of the moon. See chapter XIII., § 290.

[24] The name is interesting as a remnant of a very early superstition.
Eclipses, which always occur near the nodes, were at one time supposed
to be caused by a dragon which devoured the sun or moon. The symbols ☊
☋ still used to denote the two nodes are supposed to represent the head
and tail of the dragon.

[25] In the figure, which is taken from the _De Revolutionibus_ of
Coppernicus (chapter IV., § 85), let D, K, M represent respectively the
centres of the sun, earth, and moon, at the time of an eclipse of the
moon, and let S Q G, S R E denote the boundaries of the shadow-cone
cast by the earth; then Q R, drawn at right angles to the axis of the
cone, is the breadth of the shadow at the distance of the moon. We have
then at once from similar triangles

  G K - Q M : A D - G K :: M K : K D.

Hence if K D = _n_. M K and ∴ also A D = _n_. (radius of moon), _n_
being 19 according to Aristarchus, G K-Q M: _n_. (radius of moon)-G K

  :: 1 : _n_

  _n_ . (radius of moon) - G K

  = _n_ G K - _n_ Q M

∴ radius of moon + radius of shadow

  = (1 + 1∕_n_) (radius of earth).

By observation the angular radius of the shadow was found to be about
40′ and that of the moon to be 15′, so that

  radius of shadow = 8∕3 radius of moon;

  ∴ radius of moon

  = 3∕11 (1 + 1∕_n_) (radius of earth).

But the angular radius of the moon being 15′, its distance is
necessarily about 220 times its radius,

  and ∴ distance of the moon

  = 60 (1 + 1∕_n_) (radius of the earth),

which is roughly Hipparchus’s result, if _n_ be _any_ fairly large
number.

[26] _Histoire de l’Astronomie Ancienne_, Vol. I., p. 185.

[27] The chief MS. bears the title μεγάλη σύνταξις or
great composition though the author refers to his book elsewhere as
μαθηματικὴ σύνταξις (mathematical composition). The Arabian
translators, either through admiration or carelessness, converted
μεγάλη, great, into μεγίστη, greatest, and hence
it became known by the Arabs as _Al Magisti_, whence the Latin
_Almagestum_ and our _Almagest_.

[28] The better known apparent enlargement of the sun or moon when
rising or setting has nothing to do with refraction. It is an optical
illusion not very satisfactorily explained, but probably due to the
lesser brilliancy of the sun at the time.

[29] In spherical trigonometry.

[30] A table of chords (or double sines of half-angles) for every 1∕2°
from 0° to 180°.

[31] His procedure may be compared with that of a political economist
of the school of Ricardo, who, in order to establish some rough
explanation of economic phenomena, starts with certain simple
assumptions as to human nature, which at any rate are more plausible
than any other equally simple set, and deduces from them a number of
abstract conclusions, the applicability of which to real life has to be
considered in individual cases. But the perfunctory discussion which
such a writer gives of the qualities of the “economic man” cannot of
course be regarded as his deliberate and final estimate of human nature.

[32] The equation of the centre and the evection may be expressed
trigonometrically by two terms in the expression for the moon’s
longitude, _a sin_θ + _b sin_ (2φ-θ), where _a_, _b_ are two numerical
quantities, in round numbers 6° and 1°, θ is the angular distance of
the moon from perigee, and φ is the angular distance from the sun.
At conjunction and opposition φ is 0° or 180°, and the two terms reduce
to (_a_-_b_) _sin_θ. This would be the form in which the equation
of the centre would have presented itself to Hipparchus. Ptolemy’s
correction is therefore equivalent to adding on

  _b_ [_sin_θ + _sin_ (2φ - θ)], or 2 _b sin_φ _cos_ (φ-θ),

which vanishes at conjunction or opposition, but reduces at the
quadratures to 2 _b sin_θ, which again vanishes if the moon is at
apogee or perigee (θ = 0° or 180°), but has its greatest value half-way
between, when θ = 90°. Ptolemy’s construction gave rise also to a still
smaller term of the type,

  _c sin_ 2φ [_cos_ (2φ + θ) + 2 _cos_ (2φ - θ)],

which, it will be observed, vanishes at quadratures as well as at
conjunction and opposition.

[33] Here, as elsewhere, I have given no detailed account of
astronomical instruments, believing such descriptions to be in general
neither interesting nor intelligible to those who have not the actual
instruments before them, and to be of little use to those who have.

[34] The advantage derived from the use of the equant can be made
clearer by a mathematical comparison with the elliptic motion
introduced by Kepler. In elliptic motion the angular motion and
distance are represented approximately by the formulae _nt_ + 2_e sin
nt_, _a_ (1 - _e cos nt_) respectively; the corresponding formulæ given
by the use of the simple eccentric are _nt + e′ sin nt_, _a_ (1 - _e′
cos nt_). To make the angular motions agree we must therefore take _e′_
= 2_e_, but to make the distances agree we must take _e′ = e_; the two
conditions are therefore inconsistent. But by the introduction of an
equant the formulæ become _nt_ + 2_e′ sin nt_, _a_ (1 - _e′ cos nt_),
and _both_ agree if we take _e′ = e_. Ptolemy’s lunar theory could have
been nearly freed from the serious difficulty already noticed (§ 48) if
he had used an equant to represent the chief inequality of the moon;
and his planetary theory would have been made accurate to the first
order of small quantities by the use of an equant both for the deferent
and the epicycle.

[35] De Morgan classes him as a geometer with Archimedes, Euclid, and
Apollonius, the three great geometers of antiquity.

[36] The legend that the books in the library served for six months as
fuel for the furnaces of the public baths is rejected by Gibbon and
others. One good reason for not accepting it is that by this time there
were probably very few books left to burn.

[37] The data as to Indian astronomy are so uncertain, and the evidence
of any important original contributions is so slight, that I have not
thought it worth while to enter into the subject in any detail. The
chief Indian treatises, including the one referred to in the text, bear
strong marks of having been based on Greek writings.

[38] He introduced into trigonometry the use of _sines_, and made also
some little use of _tangents_, without apparently realising their
importance: he also used some new formulæ for the solution of spherical
triangles.

[39] A prolonged but indecisive controversy has been carried on,
chiefly by French scholars, with regard to the relations of Ptolemy,
Abul Wafa, and Tycho in this matter.

[40] For example, the practice of treating the trigonometrical
functions as _algebraic_ quantities to be manipulated by formulæ, not
merely as geometrical lines.

[41] Any one who has not realised this may do so by performing with
Roman numerals the simple operation of multiplying by itself a number
such as MDCCCXCVIII.

[42] On trigonometry. He reintroduced the _sine_, which had been
forgotten; and made some use of the _tangent_, but like Albategnius (§
59 _n._) did not realise its importance, and thus remained behind Ibn
Yunos and Abul Wafa. An important contribution to mathematics was a
table of sines calculated for every minute from 0° to 90°.

[43] That of “lunar distances.”

[44] He did not invent the measuring instrument called the _vernier_,
often attributed to him, but something quite different and of very
inferior value.

[45] The name is spelled in a large number of different ways both by
Coppernicus and by his contemporaries. He himself usually wrote his
name Coppernic, and in learned productions commonly used the Latin form
Coppernicus. The spelling Copernicus is so much less commonly used by
him that I have thought it better to discard it, even at the risk of
appearing pedantic.

[46] _Nullo demum loco ineptior est quam ... ubi nim’s pueriliter
hallucinatur_: Nowhere is he more foolish than ... where he suffers
from delusions of too childish a character.

[47] His real name was Georg Joachim, that by which he is known having
been made up by himself from the Latin name of the district where he
was born (Rhætia).

[48] The _Commentariolus_ and the _Prima Narratio_ give most readers a
better idea of what Coppernicus did than his larger book, in which it
is comparatively difficult to disentangle his leading ideas from the
mass of calculations based on them.

[49] _Omnis enim quæ videtur secundum locum mutatio, aut est propter
locum mutatio, aut est propter spectatæ rei motum, aut videntis, aut
certe disparem utriusque mutationem. Nam inter mota æqualiter ad eadem
non percipitur motus, inter rem visam dico, et videntem_ (De Rev., I.
v.).

I have tried to remove some of the crabbedness of the original passage
by translating freely.

[50] To Coppernicus, as to many of his contemporaries, as well as to
the Greeks, the simplest form of a revolution of one body round another
was a motion in which the revolving body moved as if rigidly attached
to the central body. Thus in the case of the earth the second motion
was such that the axis of the earth remained inclined at a constant
angle to the line joining earth and sun, and therefore changed its
direction in space. In order then to make the axis retain a (nearly)
fixed direction in space, it was necessary to add a _third_ motion.

[51] In this preliminary discussion, as in fig. 40, Coppernicus gives
80 days; but in the more detailed treatment given in Book V. he
corrects this to 88 days.

[52] Fig. 42 has been slightly altered, so as to make it agree with
fig. 41.

[53] Coppernicus, instead of giving longitudes as measured from the
first point of Aries (or vernal equinoctial point, chapter I., §§ 11,
13), which moves on account of precession, measured the longitudes from
a standard fixed star (α _Arietis_) not far from this point.

[54] According to the theory of Coppernicus, the diameter of the moon
when greatest was about 1∕8 greater than its average amount; modern
observations make this fraction about 1∕13. Or, to put it otherwise,
the diameter of the moon when greatest ought to exceed its value when
least by about 8′ according to Coppernicus, and by about 5′ according
to modern observations.

[55] Euclid, I. 33.

[56] If P be the synodic period of a planet (in years), and S the
sidereal period, then we evidently have (1∕P) + 1 = 1∕S for an inferior
planet, and 1 - (1∕P) = 1∕S for a superior planet.

[57] Recent biographers have called attention to a cancelled passage in
the manuscript of the _De Revolutionibus_ in which Coppernicus shews
that an ellipse can be generated by a combination of circular motions.
The proposition is, however, only a piece of pure mathematics, and has
no relation to the motions of the planets round the sun. It cannot,
therefore, fairly be regarded as in any way an anticipation of the
ideas of Kepler (chapter VII.).

[58] It may be noticed that the differential method of parallax
(chapter VI., § 129), by which such a quantity as 12′ could have been
noticed, was put out of court by the general supposition, shared by
Coppernicus, that the stars were all at the same distance from us.

[59] There is little doubt that he invented what were substantially
logarithms independently of Napier, but, with characteristic inability
or unwillingness to proclaim his discoveries, allowed the invention to
die with him.

[60] A similar discovery was in fact made twice again, by Galilei
(chapter VI., § 114) and by Huygens (chapter VIII., § 157).

[61] He obtained leave of absence to pay a visit to Tycho Brahe and
never returned to Cassel. He must have died between 1599 and 1608.

[62] He even did not forget to provide one of the most necessary parts
of a mediæval castle, a prison!

[63] It would be interesting to know what use he assigned to the
(presumably) still vaster space _beyond_ the stars.

[64] Tycho makes in this connection the delightful remark that Moses
must have been a skilled astronomer, because he refers to the moon
as “the lesser light,” notwithstanding the fact that the apparent
diameters of sun and moon are very nearly equal!

[65] By transversals.

[66] On an instrument which he had invented, called the _hydrostatic
balance_.

[67] A fair idea of mediaeval views on the subject may be derived from
one of the most tedious Cantos in Dante’s great poem (_Paradiso_, II.),
in which the poet and Beatrice expound two different “explanations” of
the spots on the moon.

[68] _Ludovico delle Colombe_ in a tract _Contra Il Moto della Terra_,
which is reprinted in the national edition of Galilei’s works, Vol. III.

[69] In a letter of May 4th, 1612, he says that he has seen them for
eighteen months; in the _Dialogue on the Two Systems_ (III., p. 312,
in Salusbury’s translation) he says that he saw them while he still
lectured at Padua, _i.e._ presumably by September 1610, as he moved to
Florence in that month.

[70] _Historia e Dimostrazioni intorno alle Macchie Solari._

[71] Acts i. 11. The pun is not quite so bad in its Latin form: _Viri
Galilaci_, etc.

[72] _Spiritui sancto mentem fuisse nos docere, quo modo ad Coelum
eatur, non autem quomodo Coelum gradiatur._

[73] From the translation by Salusbury, in Vol. I. of his _Mathematical
Collections_.

[74] The only point of any importance in connection with Galilei’s
relations with the Inquisition on which there seems to be room for any
serious doubt is as to the stringency of this warning. It is probable
that Galilei was at the same time specifically forbidden to “hold,
teach, or defend in any way, whether verbally or in writing,” the
obnoxious doctrine.

[75] This is illustrated by the well-known optical illusion whereby a
white circle on a black background appears larger than an equal black
one on a white background. The apparent size of the hot filament in a
modern incandescent electric lamp is another good illustration.

[76] Actually, since the top of the tower is describing a slightly
larger circle than its foot, the stone is at first moving eastward
slightly faster than the foot of the tower, and therefore should reach
the ground slightly to the _east_ of it. This displacement is, however,
very minute, and can only be detected by more delicate experiments than
any devised by Galilei.

[77] From the translation by Salusbury, in Vol. I. of his _Mathematical
Collections_.

[78] The official minute is: _Et ei dicto quod dicat veritatem, alias
devenietur ad torturam_.

[79] The three days June 21-24 the only ones which Galilei _could_ have
spent in an actual prison, and there seems no reason to suppose that
they were spent elsewhere than in the comfortable rooms in which it is
known that he lived during most of April.

[80] Equivalent to portions of the subject now called _dynamics_ or
(more correctly) _kinematics_ and _kinetics_.

[81] He estimates that a body falls in a second a distance of 4
“bracchia,” equivalent to about 8 feet, the true distance being
slightly over 16.

[82] _Two New Sciences_, translated by Weston, p. 255.

[83] The astronomer appears to have used both spellings of his name
almost indifferently. For example, the title-page of his most important
book, the _Commentaries on the Motions of Mars_ (§ 141), has the form
Kepler, while the dedication of the same book is signed Keppler.

[84] The regular solids being taken in the order: cube, tetrahedron,
dodecahedron, icosahedron, octohedron, and of such magnitude that a
sphere can be circumscribed to each and at the same time inscribed in
the preceding solid of the series, then the radii of the six spheres so
obtained were shewn by Kepler to be approximately proportional to the
distances from the sun of the six planets Saturn, Jupiter, Mars, Earth,
Venus, and Mercury.

[85] Two stars 4′ apart only just appear distinct to the naked eye of a
person with average keenness of sight.

[86] _Commentaries on the Motions of Mars_, Part II., end of chapter
XIX.

[87] An ellipse is one of several curves, known as =conic sections=,
which can be formed by taking a section of a cone, and may also be
defined as a curve the sum of the distances of any point on which from
two fixed points inside it, known as the =foci=, is always the same.

[Illustration: FIG. 59.—An ellipse.]

Thus if, in the figure, S and H are the foci, and P, Q are _any_ two
points on the curve, then the distances S P, H P added together are
equal to the distances S Q, Q H added together, and each sum is equal
to the length A A′ of the ellipse. The ratio of the distance S H to the
length A A′ is known as the =eccentricity=, and is a convenient measure
of the extent to which the ellipse differs from a circle.

[88] The ellipse is _more_ elongated than the actual path of Mars, an
accurate drawing of which would be undistinguishable to the eye from a
circle. The eccentricity is 1∕3 in the figure, that of Mars being 1∕10.

[89] _Astronomia Nova_ αἰτιολογητος _seu Physica Coelestis,
tradita Commentariis de Motibus Stellae Martis._ _Ex Observationibus G.
V. Tychonis Brahe._

[90] It contains the germs of the method of infinitesimals.

[91] _Harmonices Mundi Libri V._

[92] There may be some interest in Kepler’s own statement of the
law: “Res est certissima exactissimaque, quod proportionis quae est
inter binorum quorumque planetarum tempora periodica, sit praecise
sesquialtera proportionis mediarum distantiarum, id est orbium
ipsorum.”—_Harmony of the World_, Book V., chapter III.

[93] _Epitome_, Book IV., Part 2.

[94] Introduction to the _Commentaries on the Motions of Mars_.

[95] Substantially the _filar micrometer_ of modern astronomy.

[96] Galilei, at the end of his life, appears to have thought of
contriving a pendulum with clockwork, but there is no satisfactory
evidence that he ever carried out the idea.

[97] In modern notation: time oπf oscillation = 2π√(_l_∕_g_).

[98] _I.e._ he obtained the familiar formula (_v^2_)∕_r_, and several
equivalent forms for _centrifugal force_.

[99] Also frequently referred to by the Latin name _Cartesius_.

[100] According to the unreformed calendar (O.S.) then in use in
England, the date was Christmas Day, 1642. To facilitate comparison
with events occurring out of England, I have used throughout this and
the following chapters the Gregorian Calendar (N.S.), which was at this
time adopted in a large part of the Continent (cf. chapter II., § 22).

[101] From a MS. among the Portsmouth Papers, quoted in the Preface to
the Catalogue of the Portsmouth Papers.

[102] W. K. Clifford, _Aims and Instruments of Scientific Thought_.

[103] It is interesting to read that Wren offered a prize of 40_s._ to
whichever of the other two should solve this the central problem of the
solar system.

[104] The familiar _parallelogram of forces_, of which earlier writers
had had indistinct ideas, was clearly stated and proved in the
introduction to the _Principia_, and was, by a curious coincidence,
published also in the same year by _Varignon_ and _Lami_.

[105] It is between 13 and 14 billion billion pounds. See chapter X. §
219.

[106] As far as I know Newton gives no short statement of the law in a
perfectly complete and general form; separate parts of it are given in
different passages of the _Principia_.

[107] It is commonly stated that Newton’s value of the motion of the
moon’s apses was only about half the true value. In a scholium of the
_Principia_ to prop. 35 of the third book, given in the first edition
but afterwards omitted, he estimated the annual motion at 40°, the
observed value being about 41°. In one of his unpublished papers,
contained in the Portsmouth collection, he arrived at 39° by a process
which he evidently regarded as not altogether satisfactory.

[108] Throughout the Coppernican controversy up to Newton’s time it had
been generally assumed, both by Coppernicans and by their opponents,
that there was some meaning in speaking of a body simply as being “at
rest” or “in motion,” without any reference to any other body. But all
that we can really observe is the motion of one body relative to one
or more others. Astronomical observation tells us, for example, of a
certain motion relative to one another of the earth and sun; and this
motion was expressed in two quite different ways by Ptolemy and by
Coppernicus. From a modern standpoint the question ultimately involved
was whether the motions of the various bodies of the solar system
relatively to the earth or relatively to the sun were the simpler to
express. If it is found convenient to express them—as Coppernicus and
Galilei did—in relation to the sun, some simplicity of statement is
gained by speaking of the sun as “fixed” and omitting the qualification
“relative to the sun” in speaking of any other body. The same motions
might have been expressed relatively to any other body chosen at will:
_e.g._ to one of the hands of a watch carried by a man walking up and
down on the deck of a ship on a rough sea; in this case it is clear
that the motions of the other bodies of the solar system relative to
this body would be excessively complicated; and it would therefore be
highly inconvenient though still possible to treat this particular body
as “fixed.”

A new aspect of the problem presents itself, however, when an
attempt—like Newton’s—is made to explain the motions of bodies of the
solar system as the result of forces exerted on one another by those
bodies. If, for example, we look at Newton’s First Law of Motion
(chapter VI., § 130), we see that it has no meaning, unless we know
what are the body or bodies relative to which the motion is being
expressed; a body at rest relatively to the earth is moving relatively
to the sun or to the fixed stars, and the applicability of the First
Law to it depends therefore on whether we are dealing with its motion
relatively to the earth or not. For most terrestrial motions it is
sufficient to regard the Laws of Motion as referring to motion relative
to the earth; or, in other words, we may for this purpose treat the
earth as “fixed.” But if we examine certain terrestrial motions more
exactly, we find that the Laws of Motion thus interpreted are not quite
true; but that we get a more accurate explanation of the observed
phenomena if we regard the Laws of Motion as referring to motion
relative to the centre of the sun and to lines drawn from it to the
stars; or, in other words, we treat the centre of the sun as a “fixed”
point and these lines as “fixed” directions. But again when we are
dealing with the solar system generally this interpretation is slightly
inaccurate, and we have to treat the centre of gravity of the solar
system instead of the sun as “fixed.”

From this point of view we may say that Newton’s object in the
_Principia_ was to shew that it was possible to choose a certain point
(the centre of gravity of the solar system) and certain directions
(lines joining this point to the fixed stars), as a base of reference,
such that all motions being treated as relative to this base, the Laws
of Motion and the law of gravitation afford a consistent explanation of
the observed motions of the bodies of the solar system.

[109] He estimated the annual precession due to the sun to be about
9″, and that due to the moon to be about four and a half times as
great, so that the total amount due to the two bodies came out about
50″, which agrees within a fraction of a second with the amount shewn
by observation; but we know now that the moon’s share is not much more
than twice that of the sun.

[110] He once told Halley in despair that the lunar theory “made his
head ache and kept him awake so often that he would think of it no
more.”

[111] December 31st, 1719, according to the unreformed calendar (O.S.)
then in use in England.

[112] The apparent number is 2,935, but 12 of these are duplicates.

[113] By Bessel (chapter XIII., § 277).

[114] The relation between the work of Flamsteed and that of Newton was
expressed with more correctness than good taste by the two astronomers
themselves, in the course of some quarrel about the lunar theory: “Sir
Isaac worked with the ore I had dug.” “If he dug the ore, I made the
gold ring.”

[115] Rigaud, in the memoirs prefixed to Bradley’s _Miscellaneous
Works_.

[116] A telescopic star named 37 Camelopardi in Flamsteed’s catalogue.

[117] The story is given in T. Thomson’s _History of the Royal
Society_, published more than 80 years afterwards (1812), but I have
not been able to find any earlier authority for it. Bradley’s own
account of his discovery gives a number of details, but has no allusion
to this incident.

[118] It is _k sin_ C A B, where _k_ is the constant of aberration.

[119] His observations as a matter of fact point to a value rather
greater than 18″, but he preferred to use round numbers. The figures
at present accepted are 18″·42 and 13″·75, so that his ellipse was
decidedly less flat than it should have been.

[120] _Recherches sur la précession des équinoxes et sur la nutation de
l’axe de la terre._

[121] The word “geometer” was formerly used, as “géomètre” still is in
French, in the wider sense in which “mathematician” is now customary.

[122] _Principia_, Book III., proposition 10.

[123] It is important for the purposes of this discussion to notice
that the vertical is _not_ the line drawn from the centre of the earth
to the place of observation.

[124] 69 miles is 364,320 feet, so that the two northern degrees were a
little more and the Peruvian are a little less than 69 miles.

[125] The remaining 8,000 stars were not “reduced” by Lacaille. The
whole number were first published in the “reduced” form by the British
Association in 1845.

[126] A mural quadrant.

[127] The ordinary approximate theory of the _collimation error_,
_level error_, and _deviation error_ of a transit, as given in
textbooks of spherical and practical astronomy, is substantially his.

[128] The title-page is dated 1767; but it is known not to have been
actually published till three years later.

[129] For a more detailed discussion of the transit of Venus, see
Airy’s _Popular Astronomy_ and Newcomb’s _Popular Astronomy_.

[130] _Some_ other influences are known—_e.g._ the sun’s heat causes
various motions of our air and water, and has a certain minute effect
on the earth’s rate of rotation, and presumably produces similar
effects on other bodies.

[131] The arithmetical processes of working out, figure by figure, a
non-terminating decimal or a square root are simple cases of successive
approximation.

[132] “C’est que je viens d’un pays où, quand on parle, on est pendu.”

[133] Longevity has been a remarkable characteristic of the great
mathematical astronomers: Newton died in his 85th year; Euler,
Lagrange, and Laplace lived to be more than 75, and D’Alembert was
almost 66 at his death.

[134] This body, which is primarily literary, has to be distinguished
from the much less famous Paris Academy of Sciences, constantly
referred to (often simply as the Academy) in this chapter and the
preceding.

[135] E.g. _Mélanges de Philosophie, de l’Histoire, et de Littérature_;
_Éléments de Philosophie_; _Sur la Destruction des Jésuites_.

[136] _I.e._ he assumed a law of attraction represented by μ∕_r^2_ +
ν∕_r^3_.

[137] This appendix is memorable as giving for the first time the
method of _variation of parameters_ which Lagrange afterwards developed
and used with such success.

[138] That of the distinguished American astronomer Dr. G. W. Hill
(chapter XIII., § 286).

[139] They give about ·78 for the mass of Venus compared to that of the
earth.

[140] The orbit might be a parabola or hyperbola, though this does not
occur in the case of any known planet.

[141] On the _Calculus of Variations_.

[142] The establishment of the general equations of motion by a
combination of _virtual velocities_ and _D’Alembert’s principle_.

[143] _Théorie des Fonctions Analytiques_ (1797); _Resolution des
Équations Numériques_ (1798); _Leçons sur le Calcul des Fonctions_
(1805).

[144] _Théorie Analytique des Probabilités._

[145] The fact that the post was then given by Napoleon to his brother
Lucien suggests some doubts as to the unprejudiced character of the
verdict of incompetence pronounced by Napoleon against Laplace.

[146] _Outlines of Astronomy_, § 656.

[147] Laplace, _Système du Monde_.

[148] If _n_, _n′_ are the mean motions of the two planets, the
expression for the disturbing force contains terms of the type =
_sin_(_np_ ± _n′p′_) _t_, _cos_ where _p_, _p′_ are integers, and
the coefficient is of the order _p_⁓_p′_ in the eccentricities and
inclinations. If now _p_ and _p′_ are such that _np_⁓_n′p′_ is small,
the corresponding inequality has a period 2π∕(_np_⁓_n′p′_), and
though its coefficient is of order _p_⁓_p′_, it has the small factor
_np_⁓_np′_ (or its square) in the denominator and may therefore be
considerable. In the case of Jupiter and Saturn, for example, _n_ =
109,257 in seconds of arc per annum, _n′_ = 43,996; 5_n′_ - 2_n_ =
1,466; there is therefore an inequality of the _third_ order, with a
period (in years) = 360°∕1,466″ = 900.

[149] This statement requires some qualification when perturbations are
taken into account. But the point is not very important, and is too
technical to be discussed.

[150] ∑_e^2m_√_a_ = _c_, ∑_tan^2im_√_a_ = _c′_, where _m_ is the mass
of any planet, _a_, _e_, _i_ are the semi-major axis, eccentricity, and
inclination of the orbit. The equation is true as far as squares of
small quantities, and therefore it is indifferent whether or not _tan
i_ is replaced as in the text by _i_.

[151] Nearly the whole of the “eccentricity fund” and of the
“inclination fund” of the solar system is shared between Jupiter
and Saturn. If Jupiter were to absorb the whole of each fund, the
eccentricity of its orbit would only be increased by about 25 per
cent., and the inclination to the ecliptic would not be doubled.

[152] Of tables based on Laplace’s work and published up to the time
of his death, the chief solar ones were those of _von Zach_ (1804) and
_Delambre_ (1806); and the chief planetary ones were those of _Lalande_
(1771), of _Lindenau_ for Venus, Mars, and Mercury (1810-13), and of
_Bouvard_ for Jupiter, Saturn, and Uranus (1808 and 1821).

[153] The motion of the satellites of Uranus (chapter XII., § 253, 255)
is in the opposite direction. When Laplace first published his theory
their motion was doubtful, and he does not appear to have thought it
worth while to notice the exception in later editions of his book.

[154] This statement again has to be modified in consequence of the
discoveries, beginning on January 1st, 1801, of the minor planets
(chapter XIII., § 294), many of which have orbits that are far more
eccentric than those of the other planets and are inclined to the
ecliptic at considerable angles.

[155] _Système du Monde_, Book V., chapter VI.

[156] In his paper of 1817 Herschel gives the number as 863, but a
reference to the original paper of 1785 shews that this must be a
printer’s error.

[157] The motion of Castor has become slower since Herschel’s time, and
the present estimate of the period is about 1,000 years, but it is by
no means certain.

[158] More precisely, counting motions in right ascension and in
declination separately, he had 27 observed motions to deal with (one
of the stars having no motion in declination); 22 agreed in sign with
those which would result from the assumed motion of the sun.

[159] The method was published by Legendre in 1806 and by Gauss in
1809, but it was invented and used by the latter more than 20 years
earlier.

[160] The figure has to be enormously exaggerated, the angle SσE as
shewn there being about 10°, and therefore about 100,000 times too
great.

[161] Sir R. S. Ball and the late Professor Pritchard (§ 279) have
obtained respectively ·47″ and ·43″; the mean of these, ·45″, may be
provisionally accepted as not very far from the truth.

[162] An average star of the 14th magnitude is 10,000 times fainter
than one of the 4th magnitude, which again is about 150 times less
bright than Sirius. See § 316.

[163] Newcomb’s velocity of light and Nyrén’s constant of aberration
(20″·4921) give 8″·794; Struve’s constant of aberration (20″·445),
Loewy’s (20″·447), and Hall’s (20″·454) each give 8″·81.

[164] _Fundamenta Nova Investigationis Orbitae Verae quam Luna
perlustrat._

[165] _Darlegung der theoretischen Berechnung der in den Mondtafeln
angewandten Störungen._

[166] _E.g._ in Grant’s _History of Physical Astronomy_, Herschel’s
_Outlines of Astronomy_, Miss Clerke’s _History of Astronomy in the
Nineteenth Century_, and the memoir by Dr. Glaisher prefixed to the
first volume of Adams’s _Collected Papers_.

[167] This had been suggested as a possibility by several earlier
writers.

[168] The discovery of a terrestrial substance with this line in its
spectrum has been announced while this book has been passing through
the press.

[169] Observations made on Mont Blanc under the direction of M. Janssen
in 1897 indicate a slightly larger number than Dr. Langley’s.

[170] _Catalogus novus stellarum duplicium_, _Stellarum duplicium et
multiplicium mensurae micrometricae_, and _Stellarum fixarum imprimis
duplicium et multiplicium positiones mediae pro epocha 1830_.

[171] _I.e._ 2·512... is chosen as being the number the logarithm of
which is ·4, so that (2·512...)^{5∕2} = 10.

[172] If L be the ratio of the light received from a star to that
received from a standard first magnitude star, such as Aldebaran or
Altair, then its magnitude _m_ is given by the formula

  L = (1∕2·512)^{m - 1} = (1∕100)^{(m - 1)∕5}, whence m - 1 = -5∕2log L.

A star brighter than Aldebaran has a magnitude less than 1, while the
magnitude of Sirius, which is about nine times as bright as Aldebaran,
is a _negative_ quantity,-1·4, according to the Harvard photometry.





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