A SHORT HISTORY 
 
 OF 
 
 ASTRONOMY 
 
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 gedacht, 
 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 
 I9IO 
 
UTra I- 
 "fc.A 
 
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 develop- 
 ment of astronomy is usually attributed. On the one 
 hand, it does not appear to me possible to form an in- 
 dependent 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 
 
vi Preface 
 
 the dogmatic form of a text-book. It has also seemed 
 advisable to lighten the book by omitting except in a very 
 few simple and important cases all accounts of astro- 
 nomical 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 im- 
 portant 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. 
 la particular, for these and for other reasons (stated more 
 fully at the beginning of chapter XHI.), I have -dealt in the 
 briefest possible way with the immense mass of observafcons 
 
Preface vii 
 
 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 astro- 
 nomers (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 witch- 
 craft 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 inci- 
 dent 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 
 
viii Preface 
 
 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 astro- 
 nomy 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 infor- 
 mation 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 
 
Preface ix 
 
 a further service of great value by drawing a consider- 
 able 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 photo- 
 graphs 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 Institute 
 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 
 
 i . Scope of astronomy . I 
 
 2-5. First notions : the motion of the sun : the motion 
 and phases of the moon : daily motion of the 
 stars i 
 
 6. Progress due to early civilised peoples : Egyptians, 
 
 Chinese, Indians, and Chaldaeans ... 3 
 
 7. The celestial sphere : its scientific value : apparent dis- 
 tance 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 
 
 IO-II. 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 (T), 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 . . . 1 5 
 
xii Contents 
 
 PAGE 
 
 1 6. 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 
 
 400A.D.), 19-54 21-75 
 
 19-20. Astronomy up to the time of Aristotle. The 
 Greek calendar : full and empty months : 
 the octaeteris : Melon'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 be- 
 lievers in the motion of the earth : Arist- 
 archus and Seleucus 25 
 
 25. Plato: uniform circular and spherical motions . 26 
 
 26. Eudoxus : representation of the celestial 
 
 motions by combinations of spheres : de- 
 scription of the constellations. x Callippus , 27 
 
 2 7~3- Aristotle ^his_spheres ; the phases of the moon ; 
 proofs that~the earttt iS""spherical : his 
 arguments against tfie motion of the earth : 
 relative distances of the celestial bodies : 
 other speculations : estimate of his astro- 
 nomical work 29 
 
 31-2. The early Alexandrine school : its rise : Arist- 
 archus: his_estimates ofthe^ distances of the 
 gun jind moon. Observations by Timochatis 
 and Anstyttus 34 
 
 33~4- Development of spherics : the Phenomena of 
 Euclid: the horizon t the zenith, poles of a 
 great circle, verticals, declination circles, the 
 meridian, celestial latitude and longitude, 
 right ascension and declination. Sun-dials . 36 
 
Contents xiii 
 
 PAGE 
 
 35. The division of the surface of the earth into 
 
 zones 37 
 
 36. Eratosthenes : his measurement of the earth : 
 
 and of the_flbHguit^ of the^ecliptic . . 39 
 
 37. Hipparehui : his life and chief contributions to 
 astronomy. Apollonius's representation of 
 the celestial motions by means of circles. 
 General account of the theory pf eccenfracs 
 anil-epicycles 40 
 
 3^-9- Hipparchus's representation of the motion of 
 the sun, by means of an ecgpi*ic-J3Jiag&, 
 perigee, line of apses, eccentricity : equation of 
 thelXn1r7r\^Q^ptcycir^^^& 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 con- 
 
 necting 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 5 1 
 
 43. Eclipses of the sun and moon : conjunction 
 and opposition: partial, total, and annular 
 eclipses : parallax $6 
 
 44. Delambre's estimate of Hipparchus 6l 
 
 45. The slow progress of astronomy after the time of Hip- 
 parchus : Pliny's proof that the earth is round : 
 new measurements of the earth by Posidonius . 6 1 
 
 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 ai.d Hypatia 73 
 
 54. Summary and estimate of Greek astronomy . . 74 
 
xiv Contents 
 
 CHAPTER III. 
 
 PAGE 
 
 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 
 
 6l. 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 cata- 
 logue .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 ofTivoli, 
 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 
 
Contents xv 
 
 PAGE 
 
 68. Purbach and Regiomontanus: influence of the 
 
 original Greek authors : the Niirnberg school : 
 Walther : employment of printing : conflict 
 between the views of Aristotle and of 
 Ptolemy : the celestial spheres of the Middle 
 Ages : t\\e firmament and theprimttm mobile 86 
 
 69. Lionardo da Vinci: tarthshine. Fracas for and 
 
 Apian : observations of comets. Nonius. 
 FerneCs 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: publi- 
 cation of the Commenlariolus : Rheticus and the 
 Prima Narratio : publication of the De Revo- 
 
 lutionibus 93 
 
 75. The central idea in the work of Coppernicus : 
 
 relation to earlier writers 99 
 
 76-9. Tne De Bevolutionibus. 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 no 
 
 84. The third book : precession . . . . .no 
 
 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 Ill 
 
 86-7. The fifth and sixth books : theory of the planets : 
 
 synodic and sidereal periods . . . .112 
 88. Explanation of the stationary points . . . 118 
 
xvi Contents 
 
 PAGE 
 
 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 I60I A.D.), 93~II2. . . . I2J-I44 
 
 93~4- The first reception of the De Revolutionibus : 
 
 Reinhold : the Prussian Tables . . . .125 
 95. Coppernicanism in England : Field, Records, 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 Burgi: the star catalogue : 
 Biirgi'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 Bar 136 
 
 106. Last years at Hveen : breach with the King . 138 
 
 107. Publication of the Astronomiae Instauratae 
 
 Mechanica and of the star catalogue : in- 
 vitation from the Emperor . . . .139 
 108. Life at Benatek : co-operation of Kepler : death 140 
 
 109. Fate of Tycho's instruments and observations . 141 
 
 1 10. Estimate of Tycho's work : the accuracy of his 
 observations : improvements in the art of 
 
 observing */ 141 
 
 III. 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 
 
Contents xvii 
 
 CHAPTER VI. 
 
 VAGE 
 
 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 
 
 1 16. 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 tele- 
 scope 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 im- 
 portance for the Coppernican controversy : 
 controversies .... 151 
 
 122. Appointment at the Tuscan court . . . 1 53 
 123. Observations of Saturn. Discovery of the 
 
 phases of Venus . . ... . . 154 
 
 124. Observations of sun-spots by Fabricius, Harriot, 
 
 Scheiner, and Galilei : the Macchie Solan : 
 proof that the spots were not planets : obser- 
 vations 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: // Saggiatore ...... 160 
 
 128. Dialogue on th^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 
 
 b 
 
xviii Contents 
 
 PAGE 
 
 130. Dynamical arguments in favour of the motion of 
 
 the earth : the First Law of Motion. The tides 1 65 
 
 131. The trial and condemnation. The thinly veiled 
 Coppernicanism of the Dialogue : the re- 
 markable 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 Cosmo- 
 graphicum . . . . . . . .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 
 
 6 143. 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 pro- 
 hibition : 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 
 
Contents xix 
 
 PAGE 
 
 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. Schemer's observations oifaculae 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 Auzoufs invention of the micro- 
 meter : PicarcCs telescopic " sights " . . . 202 
 
 156. Horrocks : extension of Kepler's theory to the 
 
 moon: observation of a transit of Venus . . 202 
 
 I 57~8. Huygens's rediscovery of the pendulum clock : 
 
 his theory of circular motion .... 203 
 
 159. Measurements of the earth by Snell, Norwood, and 
 
 Picard 204 
 
 1 60. The Paris Observatory : Domenico Cassini: his 
 discoveries of four.new satellites of Saturn : his 
 other work ........ 204 
 
 161. Richer s expedition to Cayenne : pendulum observa- 
 tions : observations of Mars in opposition : hori- 
 zontal parallax : annual or stellar parallax . . 205 
 
 162. Roemer and the velocity of light .... 208 
 
 $ 161. 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 . . 2IO 
 
 165. Early life, 1643 to 1665 210 
 
 166. Great productive period, 1665-87 . . . .211 
 
xx Contents 
 
 PAGE 
 
 167. Chief divisions of his work : aon'onomy, optics, pure 
 
 mathematics 21 1 
 
 1 68. 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 calcu- 
 lation of the motion of the moon : agreement 
 
 with observation 221 
 
 I75~6. Solution of the problem of elliptic motion : 
 
 Halleys 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 
 
 I79~8o. The Laws of Motion : the First Law : accelera- 
 tion 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 
 
Contents xxi 
 
 PAGE 
 
 191. Reception of the Principia 239 
 
 192. Third period of Newton's life, 1687-1727: Parlia- 
 mentary 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 workby Leibniz, by Lagrange, 
 
 and by himself 241 
 
 194. Comparison of his astronomical work with that of 
 his predecessors : " explanation " and " de- 
 scription " : conception of the material universe 
 as made up of bodies attracting one another 
 according to certain laws 242 
 
 195. Newton's scientific method : " Hypotheses nonfingo " 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 
 *97~8. Flamsteed : foundation of the Greenwich Ob- 
 servatory: his star catalogue .... 249 
 199. Halley : catalogue of Southern stars . . . 253 
 
 2OO. Halley 's comet . 253 
 
 2O I. 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 Green- 
 wich : 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 
 
 2l8. His observations : reduction . , . .271 
 
xxii Contents 
 
 PAGE 
 
 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 . . . -2,75 
 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. UAlembert'. his dynamics: precession and nuta- 
 tion : 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 . . . . 3 O1 
 
 237. Lagrange : his career : Berlin . and Paris : the 
 
 Mecanique Analytique 304 
 
 238. Laplace : his career : the Mecanique Celeste and the 
 Systeme du Monde : political appointments and 
 distinctions 36 
 
Contents xxiii 
 
 PAGE 
 
 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 Burg and Burck- 
 
 hardt 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. Tne magnitudes of some of the secular inequalities 318 
 
 247. Periodical inequalities: solar and planetary tables 
 
 based on the Me'canique Celeste . . . .3*8 
 
 248. Minor problems of gravitational astronomy : the 
 satellites : Saturn's ring : precession and nuta- 
 tion : figure of the earth : tides : comets : masses 
 of planets and satellites ..... 318 
 
 249. The solution of Newton's problem by the astro- 
 nomers 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 tele- 
 scope : 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 assump- 
 tion : its partial withdrawal. Employment of 
 
xxiv Contents 
 
 PAGE 
 
 brightness as a test of nearness : measurement 
 of brightness : " space-penetrating " power of a 
 telescope 332 
 
 259. Nebulae and star clusters : Herschel's great cata- 
 logues 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 find- 
 ing 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 a 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, observa- 
 tional, 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 : re- 
 
 discovery of the minor planet Ceres . . 358 
 
Contents xxv 
 
 PAGE 
 
 277. Bessel : his improvement in methods of re- 
 
 duction : his table of refraction : the Funda- 
 menta Nova and Tabulae Regiomontanae . 359 
 278. The para lax of 6 1 Cygni : its distance . . 360 
 
 279. Hendersons parallax of a Centauri and Struves 
 
 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 : Danwi- 
 seau,Poisson, Ponte'coulant, 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, Gylde'n, M. Poincare 370 
 
 289. The discovery of Neptune by Leverrier and Dr. 
 
 Galle : Adams's work 371 
 
 290. Lunar and planetary tables : outstanding dis- 
 
 crepancies 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, dis- 
 tribution, 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 atmo- 
 sphere 382 
 
xxvi Contents 
 
 PAGE 
 
 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 be- 
 tween sun-spots and terrestrial magnetism: 
 Carringtorfs observations of the motion and 
 distribution of spots: Wilson's theory of spots 385 
 
 299-300. Spectrum analysis: Newton, Wollaston, Fraun- 
 
 ho/er, 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 
 
 33- The constitution of the sun .... 392 
 
 304-5. Observations of comets: nucleus: theory of the 
 formation of their tails : their spectra : re- 
 lation 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 
 
 3 I S- Observations of variable stars .... 403 
 
 316. Stellar photometry: Pogsorfs light ratio: the 
 
 Oxford, Harvard, and Potsdam photometries 403 
 
 317. Structure of the sidereal system: relations of 
 
 stars and nebulae 405 
 
Contents xxvii 
 
 PAGE 
 
 318-20. Laplace's nebular hypothesis in the light of 
 later discoveries: the sun's heat: Helmholtats 
 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 1 1 
 
 5. The Great Bear To face p. 12 
 
 6. The apparent path of Jupiter 16 
 
 7. The apparent path of Mercury 17 
 
 8-1 1. 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 6b 
 
 32. Refraction by the atmosphere 63 
 
 xxix 
 
xxx List of Illustrations 
 
 FIG. PAOE 
 
 33. Parallax 68 
 
 34. Jupiter's epicycle and deferent 70 
 
 35. Theequant 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 ... M 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 f dee 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. 2OO 
 
 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 
 
List of Illustrations xxi.i 
 
 FIG. PAGE 
 
 bg. 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 Oygni and the two neighbouring stars used by Bessel ... 360 
 
 86. The parallax of 6 1 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 ... ... ... 3^2 
 
 94. The Apennines and the adjoining regions \ T r ^ c, 
 
 of the moon J 
 
 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 17 Argus ... 397 
 
 101. The orbit of Ursae ... ... ... ... 399 
 
 102. Spiral nebulae ... ... ... To face p. 400 
 
 103. The spectrum of /3 Aurigae ... ... ... fl 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). 
 
 i. 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. Cjn 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. 
 
History of Astronomy [CH. I. 
 
 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. u 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 fr.om 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 
 
$$ -tO The Beginnings of Astronomy 3 
 
 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 con- 
 venient 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 obser- 
 vations 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 astro- 
 nomical 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 observa- 
 tions 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. 
 
4 A Short History of Astronomy [CH. i 
 
 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, bear- 
 ing 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 * 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 oounded 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, f 
 
 Most of us at the present day have been taught in child- 
 hood 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. 
 
 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 
 
 * In our climate 2,000 is about the greatest number ever visible 
 at once, even to a keen-sighted person. 
 
 f Owing to the greater brightness of the stars overhead they 
 usually seem a little nearer than those near the horizon, and con- 
 sequently the visible portion of the celestial sphere appears to be 
 rather less than a half of a complete sphere. This is, however, of r,o 
 importance, and will for the future be ignoredt 
 
$ 7 ] The Celestial Sphere 5 
 
 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. i), 
 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 
 
 Q 
 
 FIG. I. The celestial sphere. 
 
 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 repre- 
 senting 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 
 
6 A Short History cf Astronomy [Cn. I. 
 
 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/, ^, 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 / 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 / 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 im- 
 portant 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.* 
 
 We might of course agree to regard the celestial sphere 
 as of a particular size, and then express the distance be- 
 tween 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 
 
 * A right angle is divided into ninety degrees (90), a degree into 
 sixty minutes (60'), and a minute into sixty seconds (60"). 
 
* 8] The Celestial Sphere : its Poles 7 
 
 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 abo'ut 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 trie sun, though this 
 
8 A Short History of Astronomy [Cn. i. 
 
 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 conse- 
 quence 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 o-, 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 o-', 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. 
 
 10. A slight familiarity with the stars is enough to shew 
 any one that the same stars are not always visible at ihc 
 
 f 
 
FIG. 2. The paths of circumpolar stars, shewing their move- 
 ment during seven hours. From a photograph by Mr. 
 H. Pain. The thickest line is the path of the pole star. 
 
 [To face p. 8. 
 
9, io] The Daily Motion of the Celestial Sphere 9 
 
 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 
 
 FIG. 3. The circles orthe celestial sphere. 
 
 unknown discoverer of this fact certainly made one of 
 ihe most important steps in early astronomy. 
 
 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 
 
to A Short History of Astronomy [CH. I. 
 
 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 
 
n] 
 
 The Annual Mo^on of the 
 
 NORTH POLE 
 
 months the sun is on the whole lower down in the sky than 
 in silmmer, and that in particular its midday height is less. 
 
 ii. 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'/, 
 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 1 100 B.C., and to have found 
 the remarkably accurate value 23 52' (cf. chapter n., 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 EC.) 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. 
 
 When the sun crosses the 
 equator the day is equal to 
 the night, and the times 
 when this occurs are con- 
 sequently known as the 
 equinoxes, the vernal equi- 
 nox occurring when the sun 
 crosses the equator from 
 south to north (about March 
 2ist), 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 eu.i in its path along the 
 
 SOUTH POLE 
 
 FIG. 4. The equator and the 
 ecliptic./' 
 
12 A Short History of Astronomy [CM. t. 
 
 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 con- 
 stellation 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 
 

 , X 
 
$; i2, i 3 ] The Constellations: the Zodiac 13 
 
 memory can follow them ; bears, lions, and fishes, large and 
 small, confuse all nomenclature." (Outlines of Astronomy^ 
 
 3 T -) 
 
 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 con- 
 stellation Berenice's Hair, named after an Egyptian queen 
 of the 3rd century B.C., is one of the few which com- 
 memorate a historical personage.* 
 
 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 <3ov, 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 
 ench " sign " were formed into a constellation, the " sign " 
 and the constellation each receiving the same name. Thus 
 
 * I have made no attempt either here or elsewhere to describe the 
 constellations and their positions, as I believe such verbal descrip- 
 tions 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. 
 
14 A Short History of Astronomy 
 
 arose twelve zodiacal constellations, the nam.s of which 
 have come down to us with unimportant changes from 
 early Greek times.* 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 n., 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 ( r ), 
 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 in., 64). 
 
 14. We have seen that the stars, as a whole, retain 
 invariable positions on the celestial sphere,t 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, \ 
 or wanderers, as distinguished from the fixed stars. 
 
 * The names, in the customary Latin forms, are : Aries, Taurus, 
 Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn us, 
 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. 
 
 f 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. 
 
 \ The custom of calling the sun and moon planets has now died 
 out, and the modern usage will be adopted henceforward in this 
 book. 
 
M 14, 15] The Planets \ 5 
 
 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 some- 
 times 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 retro- 
 grade motion only about 3 weeks (see figs. 6, 7). On the 
 whole the planets advance from west to j^ast and describe 
 circuits round the celestial sphere in periods which are 
 different for each planet. The explanation of these irregu 
 larities in the planetary motions was long one of the great 
 difficulties of astronomy. 
 .15. The idea that some of the heavenly bodies are 
 
i6 
 
 A Short History of Astronomy 
 
 [CH. I. 
 
 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 returnsTo' the 
 same place among the stars in about 29^ years, Jupiter in 
 12 years, Mars in 2 years, the sun in one year, Venus in 225 
 
 2 SOS. 
 
 2 303 
 
 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. 
 
 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 
 
> 16] The Measurement of Time 17 
 
 annual journey, being nevet more than about 47 and 29 
 respectively distant from it, on either side ; while the other 
 planets are not thus restricted in their motions. 
 
 1 6. 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 wiih it light and heat, 
 is the most obvious of astronomical facts, so the day is 
 
 I H. 40m. 20 in. XI H * nu 
 
 FIG. '/. The apparent path of Mercury from Aug. I to Oct. 3, 
 1898. The dates printed in capital letters shew the positions 
 of the sun ; the other dates shew those of Mercury. 
 
 the simplest unit of time.* 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 
 
 * 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, vvx^fJ^pov. 
 
 2 
 
 20m. 
 
1 8 A Short History of Astronomy [CH. I. 
 
 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 there- 
 fore 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 1 2 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! days, and as a period 
 consisting of an exact number of days was obviously con- 
 venient 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 n., 19). 
 
 Both Chaldaeans and Egyptians appear to have known 
 that the year consisted of about 365^ 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 
 na'tion 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, i5th, and 22nd hours 
 of the first day, the 5th, i2th, and igth of the second day, 
 and so on ; Jupiter ruled the 2nd, 9th, i6th, and 23rd 
 hours of the first day, and. subsequently the ist hour of 
 
17] The Measurement of Time: Eclipses 19 
 
 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.* 
 
 ^vi;. 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 con- 
 tributions 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 
 
 * Compare the French : Mardi, Mercredi, Jeudi, Vendredi ; or 
 better still the Italian : Martedi, Mercoledi, Giovedi, Venerdi. 
 
2o A Short History of Astronomy [CH. I., 18 
 
 be illustrated in modern times by the eclipses of the sun 
 which took place on July i8th, 1860, on July 29th, 1878, 
 and on August pth, 1896 ; but the first was visible in 
 Southern Europe, the second in North America, and the 
 third in Northern Europe and Asia. 
 
 1 8. 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 in- 
 fluence 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.* 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 in., 56, and chapter v., 
 99> I00 )- 
 
 *. See, for example, Old Moore's or Zadkiels Almanack. 
 
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^ 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! 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 
 
22 A Short History of Astronomy [Ca. 11. 
 
 resulted, of which Aristophanes makes the Moon complain 
 in his play The Clouds, acted in 423 B.C. : 
 
 44 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 
 
 , I in every fourth period of 19 years a "full" month for one of 
 n I 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 publish- 
 ing from time to time tables (TrapaTn/y/xara) 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 
 
2022] The Greek and Roman Calendars 23 
 
 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 consider- 
 ably 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 politi- 
 cal and commercial favouritism, the priests allowed the 
 calendar to fall into a state of great confusion, so that, 
 as Voltaire remarked, " les generaux remains triomphaient 
 toujours, mais ils ne savaient pas quel jour ils triom- 
 phaient." 
 
 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, #n extra day being added to Feb- 
 ruary every fourth year (our leap-year), so that the average 
 length of the year would be 365! 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 con- 
 venient 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 i6th 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 
 
24 A Short History of Astronomy [CH. n. 
 
 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 n 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 
 ^L Pythagoras* 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 
 
 * 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 
 e century or beginning of the next. 
 
M 23, 24] The Pythagoreans 25 
 
 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 VIL). 
 
 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 in- 
 visible 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 (avrt^wi/), which brought 
 the number of moving bodies up to ten, a sacred Pytha- 
 gorean number. The suggestion of such an important 
 idea as that of the motion of the earth, an idea so 
 
26 A Short History of Astronomy [Cn. n. 
 
 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) especi- 
 ally 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, Heraditus, 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.* 
 
 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 enrth 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. Unfor- 
 tunately 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 
 
 * Theophrastus was born about half a century, Plutarch nearly 
 five centuries, later than Plato. 
 
$25,26] Aristarchns : Plato 27 
 
 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.* 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 
 
 * Republic, VII. 529, 530. 
 
28 A Short History of Astronomy [Cn. u. 
 
 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 com- 
 bination 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 con- 
 jectured 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, i 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 
 
$$ 2 7j 28] Eudoxus : Aristotle 29 
 
 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 observa- 
 tions 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 observa- 
 tions 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 inci- 
 dental 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.* 
 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. 
 
 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 
 
 * 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. 
 
A Short History of Astronomy 
 
 [Cn. II. 
 
 phases are those which would be assumed by a spherical 
 
 tbody 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 
 
 FIG. 8. The phases of the moon. 
 
 (s), earth (E), and moon (M), A B c D representing on a 
 much enlarged scale a section of the moon itself, the 
 portion DAB which is turned away from the sun is dark, 
 while the portion ADC, 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 i, 2, 3, 4, 5, 6, 7, 8 in 
 fig. 10, its appearances are those represented by the cor- 
 responding numbers in fig. n, the moon thus passing 
 
 FIG. 9. The phases 
 of the moon. 
 
$ * 9 ] Aristotle : the Phases of the Moon 3 1 
 
 through the familiar phases of crescent, half full, gibbous, 
 full moon, and gibbous, half full, crescent again.* 
 
 C C> 
 
 DIRECTION 
 
 OF THE SUN 
 
 c e< 
 
 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 
 
 COOO3 
 
 1234567 
 FIG. II. The phases of the me on. 
 
 us, that a spherical form is appropriate to bodies moving as 
 the heavenly bodies appear to do. 
 
 29. His proofs that the earth is spherical are more in- 
 teresting. 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 
 
 * 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. 
 
32 A Short History of Astronomy [Ca. n. 
 
 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 dif- 
 ferent 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 com- 
 pared 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 
 
 B "^v^ v .5 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 
 FIG. 12. The curvature of at A must be in a different direc- 
 the earth. tion 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. 
 
 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 revo- 
 lution 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 
 
$ so] Aristotle 33 
 
 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 xin., 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 neigh- 
 bourhood, 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 astro- 
 nomical 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 subse- 
 quent research was shewing to be untenable. The advice 
 which he gives to his readers at the beginning of his ex- 
 position of the planetary motions, to compare his views 
 with those which they arrived at themselves or met with 
 
34 ^ Short History of Astronomy [Ca. II. 
 
 elsewhere, might with advantage have been noted and 
 followed by many of the so-called Aristotelians of the 
 Middle Ages and of the Renaissance.* 
 s 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 Phila- 
 delphos, 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. 
 
 32. Among the earlier members of the Alexandrine 
 school were Aristarchus of Samos, Aristyllus, and Timo- 
 charis, three nearly contemporary astronomers belonging 
 
 FIG. 13. The method of Aristarchus for comparing the distances 
 of the sun and moon. 
 
 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 
 
 * See, for example, the account of Galilei's controversies, in 
 chapter vi. 
 
$ si, aa] Aristarchus 35 
 
 known ; its shape is therefore completely determined, and 
 the ratio of its sides EM, 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 1 8 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), he also found 
 that the diameter of the moon was about -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 |), and his conclusions seem to contradict 
 one another. He appears also to have believed the dis- 
 tance 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 
 
A Short History of Astronomy 
 
 [Cn. IL 
 
 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. 
 
 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 indi- 
 vidual steps of which it is difficult to trace. We have, 
 
 FIG. 14. The equator and the ecliptic. 
 
 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-n), 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,* the zenith,f or 
 
 * 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. 
 
 f The word "zenith " is Arabic, not Greek : cf. chapter in., 64. 
 
** 33-351 Spherics 37 
 
 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 r N 
 at right angles in N, and another great circle (a declination 
 circle) cutting the equator at M, and if T 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 r N, N s, which are called the celestial 
 longitude and latitude respectively, or by the arcs r M, M s, 
 called respectively the right ascension and declination.* 
 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 con- 
 struction 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 gradua- 
 tion 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 correspond- 
 ing division of the earth into zones. We have already 
 seen that the height of the pole above the horizon varies in 
 
 * Most of these names are not Greek, but of later origin. 
 
A Short History of Astronomy 
 
 [CH. II. 
 
 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 per- 
 manently invisible, and the south pole would appear above 
 the horizon. 
 
 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, there- 
 fore, 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^. Consequently 
 at places at which the. height of the pole is less than 23! 
 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 recog- 
 nised 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 |, its distance from the pole is 66^; therefore in 
 regions so far north that the height p K of the north pole 
 
 FIG. 15. The equator, the horizon, 
 and the meridian. 
 
3 6] The Measurement of the Earth 39 
 
 is more than 66 1, 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. 
 
 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 
 
 FIG. 1 6. The measurement ot the earth. 
 
 that at the summer solstice the angular distance of the 
 sun from the zenith at Alexandria was at midday J^th 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 ^th 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 i to 50. The distance between Alexandria and Syene 
 
40 A Short History of Astronomy ecu. 11 
 
 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 Jo another interpretation * the 
 result is less than i 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 
 f of a right angle, or 23 51', the error in which is only 
 about 7'. 
 
 37. An immense advance in astronomy was made by 
 ffipparchuSy 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 astro- 
 nomers 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 inter- 
 mediate 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 mathe- 
 
 * That of M. Paul Tannery : Recherches sur VHistoire de V Astro- 
 nomic Ancienne, chap. v. 
 
5 37, 38] Hipparchus 41 
 
 matics,* 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 com- 
 binations of uniform circular motions than by the revolv- 
 ing spheres of Eudoxus and his school ( 26) is generally 
 attributed to the great Alexandrine mathematician Apol- 
 lonius 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 
 
 * Trigonometry. 
 
42 A Short Plistory of Astronomy [Cn. It. 
 
 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 observa- 
 tions 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.* 
 
 ,/The Greeks, however, had not the requisite algebraical 
 knowledge for such a method of representation, and Hip- 
 parchus, like his predecessors, made use of a geometrical 
 
 * 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 
 I 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 re- 
 quirements, supplemented if desired by the corresponding figures 
 for fractions of seconds; but the mathematician represents the same 
 
$ 39) Hipparchus 43 
 
 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 ACS 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 neces- 
 sary 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 
 
 facts more simply and in a way more satisfactory to the mind by the 
 formula s = 16 t-, where s denotes the number of feet fallen, and 
 / 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 / = nt + 2 e sin nt, where / is the 
 distance from a fixed point in the orbit, t the time, and n t e certain 
 numerical quantities. 
 
44 -A Short History of Astronomy [Ca. 11. 
 
 times of year by quantities so small that they might fairly 
 be attributed to errors of observation. 
 
 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 
 
 FIG. 17. The eccentric. 
 
 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 
 
39] 
 
 Hipparchus 
 
 45 
 
 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 
 
 FIG. 1 8. The position of the sun's apogee. 
 
 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. 
 
 /In the case of the particular problem of the deter- 
 mination 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 
 
46 A Short History of Astronomy [Cn. n. 
 
 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 ( n). By means of his own observa- 
 tions, and of others made by his predecessors, he ascer- 
 tained 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^ days, the 
 length of the year being 365^ days. As the sun moves 
 in each season through the same angular distance, a rign't 
 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. J} If, ' therefore, in fig. t8, 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 PEA 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 ^ T . 
 
 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, 
 
Hipparchus 
 
 47 
 
 the motion of the sun can be represented so accurately 
 that the error never exceeds about i', 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 
 \\hereas in fact the angular diameter of the sun varies by 
 about ^th part of itself, or by about i' 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. 
 - Hipparchus saw that the motion of the sun could equally 
 well be represented by the other device suggested by 
 Apollonius, ( the epi- 
 cycle, 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 J 
 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 FIG. 19. The epicycle and the deferent, 
 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 com- 
 plicated than that of the sun has always presented difficulties 
 to astronomers,* and Hipparchus required for it a more 
 elaborate construction. Some further description of the 
 
 * At the present time there is still a small discrepancy between the 
 observed and calculated places of the moon. See chapter xin., 290. 
 
48 A Short History of Astronomy [CH. n. 
 
 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! days) is spoken of as a lunation or 
 vjsynodic monthy; 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 periqd (about 27 
 days 8 hours) is known as thevsidereal month.) Mgain, 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., in), 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, Vthe draconitic 
 month,^jto 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 
 
 * The name is interesting as a remnant of a very early supersti- 
 tion. 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 8 <Q still used to denote the two nodes 
 are supposed to represent the head and tail of the dragon. 
 
41 Hipparchus 49 
 
 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 ade- 
 quately 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 dis- 
 crepancies 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. 
 
 4 1. /In the case of the planets Hipparchus found so 
 small a supply of satisfactory observations by his prede- 
 cessors, 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. 
 
 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 
 
 4 
 
5 
 
 A Short History of Astronomy 
 
 [CH. II. 
 
 it .with the known angular dia- 
 meters of the sun and moon, 
 he obtained, by a simple cal- 
 culation,* a relation between 
 the distances of the sun and 
 moon, which gives either when 
 
 * 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 AD = n . (radius of moon), n 
 being 19 according to Aristarchus, 
 G K QM : n . (radius of moon) GK 
 
 : ; i : 
 n . (radius oi moon) GK 
 
 = n G K n QM 
 
 .'. radius of moon + radius of 
 shadow 
 
 = +~) (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 = f radius of moon; 
 .'. radius of moon 
 
 = I'T C 1 + ~) (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 (i + -) (radius of the earth), 
 which is roughly Hipparchus's 
 result, if n be any fairly large 
 
 FIG. 2O. The eclipse method 
 of connecting the distances 
 of the sun and moon. 
 
$ 40 Hipparchus 51 
 
 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 sub- 
 stantially unchanged for many centuries (chapter vin., 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 
 i, 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 con- 
 stellations, 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 in., 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 
 
52 A Short History of Astronomy [CH. n. 
 
 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 ppsition and that the change had been caused 
 
 N 
 FIG. 21.- The increase of the longitude of a star. 
 
 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. 
 
 The argument may be made clearer by figures. In 
 fig. 21 let TM denote the ecliptic, TN the equator, s a 
 star as seen by Timocharis, s M a great circle drawn per- 
 pendicular to the ecliptic. Then s M is the latitude, TM 
 the longitude. Let s' denote the star as seen by Hipparchus ; 
 
Juis 53 
 
 then he found that s' M was equal to the former s M, 
 but that TM' was greater than. the former TM, or that M' 
 
 M 
 
 FIG. 22. The movement of the equator. 
 
 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 T , say from T to T ' 
 
 (fig. 22), i.e. by a motion of 
 
 the equator from TN to 
 
 T'N', its inclination N' T'M 
 
 remaining equal to its former 
 
 amount N T M. The general 
 
 effect of this change is shewn 
 
 in a different way in fig. 23, 
 
 where T T' =~ * 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) FIG. 23. The precession of the 
 
 the same in the time of equinoxes. 
 
 Hipparchus, r, === being the 
 
 earlier positions of the two equinoctial points, and r', t 
 
 the later positions. 
 
54 A Short History of Astronomy [CH. n. 
 
 The annual motion r r ' 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". 
 
 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 
 
 7 
 FIG. 24. The precession of the equinoxes. 
 
 otherwise would. From this fact is derived the name pre- 
 cession 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 r T' denote the motion of the equinoctial point 
 during a tropical year, then the sun after starting from the 
 
* 42] Hipparchus 55 
 
 equinoctial point at T arrives at the end of a tropical 
 year at the new equinoctial point at T'; but the sidereal 
 year is only complete when the sun has furiher described 
 the arc r'r and returned to its original starting-point T. 
 Hence, taking the modern estimate 50" of the arc T T', 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 , days or about 20 
 
 1296000 
 
 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 astrono- 
 mers 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 \ 
 days for the length of the year had to be diminished 
 by g-J^th 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 observa- 
 tions, and concludes that the time of a solstice may be 
 erroneous to the extent of about f day, while that of an 
 equinox may be expected to be within \ day of the truth. 
 In the illustration given, this would indicate a possible 
 error of i| 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 
 
5 6 A Short History of Astronomy [CM. n. 
 
 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! 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 
 
43l 
 
 Hipparchus 
 
 57 
 
 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. 
 
 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 
 
 FIG. 25. The earth's shadow. 
 
 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 tim^, and hence upon her 
 position with respect to the nodes of her orbit ( 40). If 
 conjunction takes place when the sun and moon happen 
 
 M 
 
 S ECLIPTIC 5' 
 
 FIG. 26. The ecliptic and the moon's path. 
 
 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 eclipje 
 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 | ; consequently when their 
 discs just touch, as in fig. 27, the distance between their 
 centres is also about J. If then at conjunction the dis- 
 tance between their centres is less than this amount, an 
 
A Short History of Astronomv 
 
 [Cn. II. 
 
 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 i. 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 
 
 FIG. 27. The sun 
 and moon. 
 
 FIG. 28. Partial eclipse of 
 the moon. 
 
 FIG. 29. Total eclipse of 
 the moon. 
 
 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. 
 
 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 
 
$ 43] Hipparchus "59 
 
 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. 
 
 Thus eclipses take place if, and only if, 
 the distance of the moon from a node at 
 the time of conjunction or opposition lies FlG 30 . Annular 
 within certain limits approximately known ; eclipse of the 
 and the problem of predicting eclipses sun. 
 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, con- 
 junction (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 com- 
 pletely 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 
 
60 A Short History of Astronomy [CH. n. 
 
 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 
 
 M 
 
 FIG. 3i.-j-Parallax 
 
 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 BMC 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 i and 
 could not be neglected. 
 
44, 45] Hipparchus 6 1 
 
 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 Hippar- 
 chus 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 : 
 
 11 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."* 
 
 45. For nearly three centuries after the death of Hippar- 
 chus, 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 sur- 
 face 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 
 
 * Histoire de I 'Astronomic Anctenne, Vol. I., p. 185. 
 
.62 A Short History of Astronomy [CH. n. 
 
 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) tne 
 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 
 , . /yfjf 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* 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 
 
 * The chief MS. bears the title fieydXr] (rvvraZts, or great composi- 
 tion ^though the author refers to his book elsewhere as fj.adr)fj.aTiKr) 
 o-Wats (mathematical composition). The Arabian translators, either 
 through admiration or carelessness, converted jmeydXr), great, into 
 HeyLffT-rj, greatest, and hence it became known by the Arabs as 
 Al Magisti, whence the Latin Almagestum and our Almagest. 
 
* 46, 47] 
 
 Ptolemy 
 
 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 
 
 FIG. 32. Refraction by the atmosphere. 
 
 refraction more than the upper, so that a contraction of 
 the vertical diameter results, the horizontal contraction 
 being much less.* 
 
 47. The Almagest is avowedly based largely on the work 
 of earlier astronomers, and in particular on that of Hippar- 
 chus, for whom Ptolemy continually expresses the greatest 
 admiration and respect. Many of its contents have there- 
 fore 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, 
 
 * 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. 
 
64 A Short History of Astronomy [Cn. n. 
 
 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,* and a mathematical tablet of 
 considerable accuracy and extent. But the most interest- 
 ing 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 180,00* 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 pro- 
 bable than any others with which the author was acquainted, 
 on which to base mathematical calculations which should 
 explain observed phenomena.* His attitude is thus 
 
 * In spherical trigonometry. 
 
 f A table of chords (or double sines of half-angles) for every ^ 
 from o to 1 80. 
 
 \ His procedure may be compared with that of a political 
 economist of the school of Ricardo, who, in order to establish some 
 
4 8] The Almagest 65 
 
 essentially different from that either, of the early Greeks, 
 such as Pythagoras, or of the controversialists of the i6th 
 and early lyth 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 immo- 
 bility 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 import- 
 ance 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 
 
 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 ol the qualities of the "economic man" 
 cannot of course be regarded as his deliberate and final estimate 
 of human nature. 
 
66 A Short History of Astronomy [On. II, 
 
 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 obser- 
 vation, 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.* 
 
 * 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 + b sin (2 0), where a, b are two numerical 
 quantities, in round numbers 6 and 1, 6 is the angular distance of 
 the moon from perigee, and is the angular distance from the sun. 
 At conjunction and opposition is O or 180, and the two terms 
 reduce to (a 6) sin 6. 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 e + sin (2 0)], or 2 b sin cos (0-0), 
 
 which vanishes at conjunction or opposition, but reduces at the 
 quadratures to 2 b sin 0, which again vanishes if the moon is at apogee 
 or perigee (0 = o or 180), but has its greatest value half-way 
 between, when = 90. Ptolemy's construction gave rise also to 
 a still smaller term of the type, 
 
 c sin 2 [cos (204 0) + 2 cos (2 0)], 
 
 which, it will be observed, vanishes at quadratures as well as at 
 conjunction and opposition. 
 
* 49] The Almagest 67 
 
 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 epi- 
 cycles 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 sometimt-S-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.* 
 
 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 st en 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. 
 
 * Here, as elsewhere, I have given no detailed account of astro- 
 nomical 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. 
 
68 A Short History of Astronomy [CH. n. 
 
 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 
 
 M 
 
 FIG. 33. Parallax. 
 
 ' ?" ; 
 
 about ^ of the true number, as indicated by modern 
 work (chapter xiii^ 284). 
 
 The sixth book is devoted to eclipses, and contains no 
 substantial additions to the work cf 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. More- 
 over, Ptolemy professes to deduce from a comparison of 
 "his observations with those of Hipparchus and others the 
 (erroneous) value 56" for the precession, which Hipparchus 
 had given as the least possible value, and which Ptolemy 
 regards as his final estimate. But an examination of 
 
5$ 5, si] The Almagest 69 
 
 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 contribu- 
 tions 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 inter- 
 preted 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,) and to the 
 west (as at J 2 ) of the fictitious planet y; and the extent of 
 the oscillation on each side, and the interval between suc- 
 cessive appearances in the extreme positions (j b J 2 ) on either 
 side, can be made right by choosing appropriately the size 
 
A Short History of Astronomy 
 
 [CH. II. 
 
 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, ar 
 j l and J 2 the motion as seen from the earth is nearly the 
 same as that of./; while at J 4 the two motions are in 
 
 opposite directions, and the 
 size and motion of the epi- 
 cycle 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 there- 
 fore 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 ji and 
 J 2 the planet appears to 
 be moving from west to 
 
 east, and at J< in the opposite direction, and sudden 
 changes of motion do not occur in astronomy, there must 
 be a position between jj 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. 
 
 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. 
 
 FIG. 34. Jupiter's epicycle 
 and deferent. 
 
Si] 
 
 The Almagest 
 
 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. 
 
 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, how- 
 ever, 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 
 
 FIG. 35. 1 hi: equant. 
 
72 A Short History of Astronomy [Cn. it. 
 
 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 re- 
 present 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 i4th and i5th centuries ought to have 
 objected (as some of them actually did), to this innova- 
 tion of Ptolemy's. But there seems no good reason fo* 
 attributing this a priori attitude to the later scientific Greek 
 astronomers (cf. also 38, 47).* 
 
 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 
 vine 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 re- 
 spectively) beyond the sun in that order. As Venus and 
 
 * The advantage derived from the use of the equant can be made 
 clearer by a mathematical comparison with the elliptic motion in- 
 troduced by Kepler. In elliptic motion the angular motion and 
 distance are represented approximately by the formulae nt + 2e sin nt 
 a (i cos nt} respectively; the corresponding formulae given by 
 the use of the simple eccentric are nt + e' sin nt, a (i e' cos nt). 
 To make the angular motions agree we must therefore take e' 2e, 
 but to make the distances agree we must take e' = e; the two con 
 ditions are therefore inconsistent. But by the introduction of an 
 equant the formulae become nt + 2e' sin nt, a (l e' cos nf), 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. 
 
* 52, ss] Ptolemy and his Successors 73 
 
 Mercury accompany the sun, and may therefore be regarded 
 as on the average performing their revolutions in a y^ar, 
 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 accom- 
 plished and original mathematician.* 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 HI., 59). The 
 onlv 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.f 
 
 * De Morgan classes him as a geometer with Archimedes, Euclid, 
 and Apollonius, the three great geometers of antiquity. 
 
 f The legend that the books in the library served for six months as 
 
74 -4 Short History of Astronomy [CH. 11. 
 
 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 Pytha- 
 goras 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 VIL, 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 in- 
 vented. 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 
 
 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. 
 
54] Estimate of Grtek Astronomy 7$ 
 
 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 im- 
 portant 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 Copper- 
 nicus 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 publica- 
 tion 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 yth 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 
 
 76 
 
CH. in., $$ 55, 5 o The Bagdad School 77 
 
 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,* 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. More- 
 over 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 
 
 * 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 
 Ihe text, bear strong marks of having been based on Greek writings. 
 
78 A Short History of Astronomy [Cu. ill, 
 
 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 (P-873) and 
 by his son Ishak ben Honein (P-Qio 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 observa- 
 tions 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 import- 
 ance 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 
 
$$ 57 6o] The Bagdad School: Albc.tcgnins 79 
 
 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 n., 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., u) and of precession. He wrote also a 
 treatise on astronomy which contained improved tables 
 of the sun and moon, and included his most notable dis- 
 covery 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 n., 
 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.* 
 
 60. The last of the Bagdad astronomers was Abul Wafa 
 
 
 
 * He introduced into trigonometry the use of sines, and made also 
 some little use of tangents, without apparently realising their im- 
 portance : he also used some new formulas for the solution of 
 spherical triangles. 
 
80 A Short History of Astronomy [Cn. ill . 
 
 (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 Waia 
 here detected an inequality rediscovered by Tycho Brahe 
 (chapter v., in) and known as the variation, but it 
 is equally likely that he was merely restating Ptolemy's 
 prosneusis (chapter n., 48).* 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 (?-ioo8), 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 
 
 * 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. 
 
I 
 *$ 61, 62] The Spanish School : Nassir Eddin 6 A 
 
 the same school are due some improvements in instru- 
 ments 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 remark- 
 able for their size and careful construction, and were 
 probably better than any used in Europe in the time of 
 Coppernicws, 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 IL, 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 
 
 6 
 
82 A Short History of Astronomy [CH, in. 
 
 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 11., 42) 
 was fixed at 51", or within about i" 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 J3egh (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 improve- 
 ments in mathematical methods.* Among the most 
 important of their services to mathematics, and hence to 
 astronomy, must be counted the introduction, from India, 
 
 * For example, the practice of treating the trigonometrical functions 
 as algebraic quantities to be manipulated by formulae, not merely 
 as geometrical lines. 
 
$ 6365] Ulugh Begh : Esti/nate of Arab Astronomy 83 
 
 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 sim- 
 plification was thereby introduced into arithmetical work.* 
 More important than the actual original contributions of 
 the Arabs to astronomy was the service that they perform jJ 
 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 cdfnmon 
 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 break- 
 up 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. 
 
 * 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, 
 
84 A Short History of Astronomy [Cn. ill. 
 
 Charlemagne not only founded a higher school at his 
 own court, but was also successful in urging the ecclesi- 
 astical 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 loth 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 n., 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 i2th 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 1116, and translated Alba- 
 tegnius's Astronomy ( 59), as well as other astronomical 
 books. At about the same time Euclid's Elemwts^ 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 i3th 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. 
 
$ 66, 6 7 ] The Revival of Astronomy in the West 85 
 
 Aristotle's writings on logic had been preserved in 
 Latin translations from classical times, and were already 
 much esteemed by the scholars of the nth and i2th 
 centuries. His other writings were first met with in Arabic 
 versions, and were translated into Latin during the end 
 of the 1 2th and during the i3th centuries; in one or two 
 cases translations were also made from the original Greek. 
 The influence of Aristotle over mediaeval 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 volu- 
 minous encyclopaedia 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 1 3th century belong also several of the great 
 scholars, such as Albertus Magnus, Roger Bacon, and 
 Cecco cTAscoli (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 
 
86 A Short Historv of Astronomy [Cn. HI. 
 
 of learning. He inveighs warmly against excessive adher- 
 e*nce to authority, especially to that of Aristotle, whose 
 books he wishes burnt, and speaks strongly of the import- 
 ance 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 sur- 
 veyed 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 Mundiv^ 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 iyth 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 i5th century which succeeded 
 in making some additions to knowledge, not in themselves 
 of first-rate importance, but significant of the greater inde- 
 pendence 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 
 
68] Sacrobosco, Purbach, Regiomontanus 87 
 
 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 Miiller of Konigsberg (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 astro- 
 Comical 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 
 I Purbach's Epitome of Astronomy, made some observations, 
 lectured, wrote a mathematical treatise * 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 
 
 * On trigonometry. He reintroduced the stne, 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 mathe- 
 matics was a table of sines calculated for every minute from o to 9 
 
88 A Short History of Astronomy [CH. in. 
 
 turned his attention from Greek to fighting, and Regiomon- 
 tanus moved once more, settling this time in Niirnberg, 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 Niirnberg 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 
 Niirnberg 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 j 
 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 
 
68] 
 
 The Celestial Spheres 
 
 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 arrange- 
 ment. 
 
 C3l 
 
 FIG. 36. The celestial spheres. From Apian's Costnograplita. 
 
 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 Hippar- 
 chus and Ptolemy used. These were mere geometrical 
 
90 A Short History of Astronomy [Cn. ill 
 
 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 Niirnberg press were issued also a suc- 
 cession 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.* 
 
 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 (chap- 
 ter 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 n., 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 Niirnberg astronomical school of some 
 kind lasted into the lyth century. 
 
 69. A few minor discoveries in astronomy belong to this 
 or to a slightly later period and may conveniently be dealt 
 with here. 
 
 Lionardo 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 
 
 * That of " lunar distances." 
 
69] Regiomontnnus and Others 91 
 
 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 v 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 obser- 
 vations 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 xin., 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.* 
 
 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 Feme I (1497-1558), 
 who arrived at a result the error in which (less than i per 
 cent.) was far less than could reasonably have been ex- 
 pected from the rough methods employed. 
 
 The life of Regiomontanus overlapped that of Copper- 
 nicus 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. 
 
 * He did not invent the measuring instrument called the vernier, 
 often attributed to him, but something quite different and of very 
 inferior value. 
 
CHAPTER IV. 
 
 COPPERNICUS. 
 
 " But in this our age, one rara witte (seeing the continuall errors 
 that from time to time more and more continually, have been dis- 
 covered, 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 i5th century, the stimulus to the study of the Greek 
 authors, due in part to the scholars who were driven west- 
 wards 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 
 
 92 
 
CH IV., $ 70, 7 i] The Revival of Learning 93 
 
 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 independ- 
 ently 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 be- 
 tween Ptolemy and Aristotle, partly in connection with 
 the various astronomical tables which, though on sub- 
 stantially 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 grefet astronomer whose work has next to be 
 considered. 
 
 71. Nicholas Coppernic or Coppernicus* was born on 
 February iQth, 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 
 
 * 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. 
 
94 A Short History of Astronomy [CH. iv. 
 
 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 \\.io 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 Bradzewski, 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 in., 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 favour- 
 able 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. 
 
 During his absence he was appointed (about 1497) to 
 
COPPERNICUS. 
 
 [To face p. 94. 
 
$ 72] Life of Coppernicus 95 
 
 a canonry at Frauenburg, and at some uncertain date 
 he 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, 
 rind 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 alto- 
 gether 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 memo- 
 randum 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 
 
96 A Short History of Astronomy [CH. iv. 
 
 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 in., 62, 63), 
 and not even as good as those which he could have pro- 
 cured if he had wished from the workshops of Niirnberg ; 
 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; Copper- 
 nicus 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 n.. 
 
$ 73l Life of Copper nicus 97 
 
 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.* 
 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 attrac t notice outside astronomical circles. 
 About 1531 he had the distinction of being ridiculed on 
 the stage at some popular performance in the neighbour- 
 hood ; 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 interest- 
 ing 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. 
 
 * Nullo demutn loco itteptior est quam . . . ubt nim s pueriliter 
 hallucinatur : Nowhere is he more foolish than . . where he suffers 
 from delusions of too childish a character. 
 
 7 
 
gS A Short History of Astronomy [CH. IV. 
 
 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.* Born in 
 1514, he studied astronomy under Schoner at Niirnberg, 
 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.f 
 
 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 com- 
 plete 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 Niirnberg. 
 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 
 
 * 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 (Rhsetia). 
 
 f The Commcntariolus 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. 
 
* 74, 7s] Publication of the " De Revolutionibus " 99 
 
 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 
 Prindpia (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 Pytha- 
 goreans, 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 n., 
 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 AD. 
 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 
 
ioo A Short History of Astronomy [CH. IV. 
 
 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 funda- 
 mental 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 how- 
 ever 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 prin- 
 
$ 76, 77] The Motion f jf' the Ea*-tv 101 
 
 ciples or postulates, but on account of their novelty takes 
 a little more trouble than his predecessor (cf. chapter n., 
 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 reason- 
 ing, 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 com- 
 pounded of such motions, is, as might be expected, sup- 
 ported 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 con- 
 clusive. 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, 
 
io2 X Short 'Hhtvry.of Astronomy [CH. iv. 
 
 no motion is perceived, as between the object seen and the 
 observer." * 
 
 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. 
 
 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 
 
 FIG. 37. Relative motion. 
 
 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, 
 
 * Omnts enim quce videtur secundum locum mutatio, aut est propter 
 locum mutatio, aut est propter spedatce rei motum, aut videntis, aut 
 certe disparem utriusque mutationem. Nam inter mota cequaliter 
 ad eadem non percipitur motus, inter rent visam dico, et mdentem (De 
 Rev., I. v.). 
 
 I have tried to remove some of the crabbedness of the original 
 passage by translating freely. 
 
$ 78] Relative Motion 103 
 
 at the corresponding times, the sun remaining at rest at s, 
 exactly the same effect is produced on the eye, provided 
 that the lines as, ^s, rs, */s are, as in the figure, equal in 
 length and parallel in direction to E A, E B, E c, E D re- 
 spectively. 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 abed. 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 #), yet each circle as a whole is described, 
 
 6 
 FIG. 38. The relative motion of the sun and moon. 
 
 as indicated by the arrow-heads, 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. 
 
 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, terraeque urbesque recedunt." 
 
 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 
 
io 4 
 
 A Short History of Astronomy 
 
 [Cri. IV. 
 
 judging the direction of an object ta'Ken oy 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 sup- 
 pose 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 ce- 
 lestial 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 
 
 FL-. 39. The daily rotation of 
 the earth. 
 
M 79, s.\i The Motion of the Earth 105 
 
 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.* 
 
 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 
 
 * 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. 
 
 
io6 A Short History of Astronomy [CH. iv. 
 
 earth Coppernicus discusses the chief objections which had 
 been urged by Ptolemy. To the objection that if the earth 
 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 im- 
 perceptible as compared with the distance of the stars. 
 
 8 1. He goes on to argue that the chief irregularity in the 
 motion of the planets, in virtue of which they move back- 
 wards at intervals (chapter I., 14, and chapter IL, 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 
 
$ 8i] The Arrangement vf the Solar System 107 
 
 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 *), Saturn most 
 slowly (in about 30 years). On the Coppernican system 
 
 FIG. 40. The solar system according to Coppernicus. From the 
 De Revolutionibus. 
 
 the moon alone still revolved round the earth, being the 
 only celestial body the status of which was substantially 
 
 * 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. 
 
io8 
 
 A Short History of Astronomy 
 
 [CH. IV. 
 
 unchanged ; and thus Coppernicus was able to give the 
 accompanying diagram of the solar system (fig. 40), repre- 
 senting his view of its general arrangement (though not of 
 the right proportions of the different parts) and of the 
 various motions. 
 
 82. The effect of the motion of the earth round the sun 
 on the length of the day and other seasonal effects is 
 
 FIG. 41. Coppernican explanation of the seasons. From the 
 De Revolutionibus. 
 
 discussed in some detail, and illustrated by diagram^ which 
 are here reproduced.* 
 
 In fig. 41 A, B, c, D represent the centre of the earth in four 
 positions, occupied by it about December 23rd, March 2ist, 
 June 22nd, and September 22nd respectively (i.e. at the 
 
 * Fig. 42 has been slightly altered, so as to make it agree with 
 % .41- 
 
$ 82] The Seasons, according to Coppernicus 109 
 
 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, 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 
 
 Paries Bores. 
 
 Partes Auftrinae. 
 
 FIG. 42. Coppernican explanation of the seasons. From the 
 De Revolutionibus. 
 
 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 1 2 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 
 
no A Short History of Astronomy [CH. iv. 
 
 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.* 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 elabprate discussion 
 of the precession of the equinoxes (chapter IL, 42). From 
 a comparison of results obtained by Timocharis, by later 
 Greek astronomers, and by Albategnius, Coppernicus ififers 
 that the amount of precession has varied, but that its 
 average value is 5o"'2 annually (almost exactly the true 
 value), and accepts accordingly Tabit ben Korra's unhappy 
 suggestion of the trepidation (chapter in., 58). An 
 examination of the data used by Coppernicus shews that 
 the erroneous or fraudulent observations of Ptolemy 
 (chapter n., 50) are chiefly responsible for the perpetua- 
 tion of this mistake. 
 
 * Coppernicus, instead of giving longitudes as measured from the 
 first point of Aries (or vernal equinoctial point, chapter i., 'n, 13), 
 which moves on account of precession, measured the longitudes from 
 a standard fixed star (a Arietis) not far from this point, 
 
* 8385] Precession 1 1 1 
 
 Of much more interest than the detailed discussion of tre- 
 pidation 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, en 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 5o"*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 ^ T instead of ^), 
 fixes the position of the apogee and perigee (chapter n., 39), 
 or rather of the equivalent aphelion and peiihelion (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 in., 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 
 
ii2 A Short History of Astronomy [CH. iv. 
 
 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 11., 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 n., 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 n., 48) or Abul 
 Wafa's inequality (chapter in., 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.* 
 
 In discussing the distances and sizes of the sun and 
 moon Coppernicus follows Ptolemy closely (chapter n., 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 explana- 
 tion of the motion could with little difficulty be rearranged 
 so as to fit the ideas of Coppernicus. We have seen 
 (chapter 11., 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 
 
 * According to the theory of Coppernicus, the diameter of the 
 moon when greatest was about -J greater than its average amount; 
 modern observations make this fraction about ^3. Or, to put it other- 
 wise, 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. 
 
86] 
 
 Motion of the Planets 
 
 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 
 
 FIG. 43. The orbits of Venus and of the earth. 
 
 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. 
 
 In discussing the time of revolution of a planet a dis- 
 tinction has to be made, as in the case of the moon (chap- 
 ter ii., 40), between the synodic and sidereal periods of 
 revolution. Venus, for example, is seen as an evening star 
 
 8 
 
A Short History of Astronomy 
 
 [Ca. IV. 
 
 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 stln ; 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 posiiion relatively to the stars, as seen either from 
 the sun or from the earth. If at first Venus and the 
 
 FIG. 44. The synodic and sidereal periods of Venus. 
 
 earth are at v,, E I 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 h we performed an exact revolution round the 
 sun. It is important to know how long the line s v l takes 
 to return to the same position, i.e. how long Venus takes 
 to return to the same position with respect to the stars, 
 
$ 8 7 ] The Motion of the Planets 115 
 
 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 x |ff days as far 
 as the earth in 365 days, i.e. Venus completes a revolution 
 in 584 x Iff 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 t (fig. 44) such that v x E, touches the circle in 
 which Venus moves, and the angle E, v, s is then (by 
 a known property of a circle) a right angle. The angle 
 s E! v t being observed, the shape of the triangle s E, v t 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 387. 
 
 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 
 
n6 
 
 A Short History of Astronomy 
 
 [CH. IV. 
 
 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 peculi- 
 arity is that the radius of the epicycle of the planet, 
 / j, is always parallel to the line E s joining the earth 
 and sun, and consequent^ performs a complete re- 
 volution 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 * 
 that the line j' j is equal 
 and parallel to E/, that 
 consequently j describes 
 a circle round j' just as 
 j round E. Hence the 
 motion of the planet can 
 equally well be repre- 
 sented 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 
 ]' of the epicycle to be actually the sun. Ptolemy's epicycle 
 and deferent are therefore capable of being replaced, with- 
 put affecting the position of the planet in the sky, by a 
 * Euclid, I. 33. 
 
 FIG. 45. The epicycle of Jupiter. 
 
* 8 7 J 
 
 The Mot 'urn of the Planets 
 
 117 
 
 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. 
 
 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 n., 43). The 
 
 FIG. 46. The relative sizes of the orbits ot the earth and of a 
 superior planet. 
 
 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.* Coppernicus thus obtained very fairly accurate 
 
 * If p be the synodic period of a phmet (in years), and s the 
 
 sidereal period, then we evidently have - + I = - for an inferior 
 
 II PS 
 
 planet, and I - - = - for a superior planet. 
 
1 1 8 A Short History of Astronomy [Cn. IV. 
 
 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 i|, 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 Mer- 
 cury 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, and M , 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, 
 
88J 
 
 Stationary Points 
 
 119 
 
 or from west to east. Hence between M, and M its motion 
 has changed from direct to retrograde, and therefore at 
 some intermediate point, say ;//, (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 
 ;;/., between M and M a (about Sept. 13 in fig. 7). 
 
 In the case of a superior planet, say Jupiter, the argument 
 
 FIG. 47. The stationary points of Mercury. 
 
 is nearly the same. When in opposition at j (as on 
 Mar. 26 in fig. 6), Jupiter moves more slowly than the 
 earih, 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, or j (in which the earth appears 
 to the observer on Jupiter to be at its greatest distance 
 
I2O 
 
 A Short History of Astronomy 
 
 [CH. IV. 
 
 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 
 
 FIG. 48. The stationary points of Jupiter. 
 
 Jupiter appears to be also from right to left, or from west 
 to east. Hence, as before, between j, and j and between 
 j and J 2 there must be points / j\ (Jan. 24 and May 27, 
 in fig. 6) at which Jupiter appears for the moment to be 
 stationary. 
 
 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 
 
* 8 9 , 90] The Motion of the Planets j 2 1 
 
 that the stationary points must exist, but shews how to 
 calculate their exact positions. 
 
 89. So far the theory of the planets has cnly been 
 sketched very roughly, in order to bring into prominence 
 the essential differences between the Coppernican and the 
 Ptolemaic explanations of their motions, and n.; 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. Copper- 
 nicus, 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 dis- 
 cussed 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 in., 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 
 
i2z A Short History of Astronomy [Cn. iv. 
 
 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 
 
 E E' S 
 
 FIG. 49. The alteration in a planet's apparent position due to an 
 alteration in the earth's distance from the sun. 
 
 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 obser- 
 vations 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 representa- 
 tion of the motion of the planets and some other geometrical 
 scheme. 
 
 91. The merits of Coppernicus are so great, and the part 
 
$*9*i9*] The Use of Epieydes by Coppernicus 123 
 
 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.* 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 Copper- 
 nicus as to that of his predecessors; and it was Kepler 
 (chapter vn.), 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 n., 43). As such a displacement 
 had never been observed, Coppernicus explained the 
 apparent contradiction by supposing the 'fixed stars so 
 
 * 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 mathe- 
 matics, 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 vn.). 
 
124 
 
 A Short History of Astronomy CH. i v , 92 
 
 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 
 
 FIG. 50. Stellar parallax. 
 
 the sun is, the angle E s' E' would be less than 1 2', 
 a quantity which the instruments of the time were barely 
 capable of detecting.* But more accurate observations 
 of the fixed stars might be expected to throw further light 
 on this problem. 
 
 * 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. 
 
CHAPTER V. 
 
 THE RECEPTION OF THE COPPERNICAN THEORY AND THE 
 PROGRESS OF OBSERVATION. 
 
 11 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 kno.wledge 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 calcu- 
 lating 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 
 
 125 
 
126 A Short History of Astronomy [Cn. v. 
 
 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 Tabulcz Prutenicce^ or Prussian Tables. 
 Reinhold revised most of the calculations made by Copper- 
 nicus, 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 in., 66), and gradually came more and 
 more into favour, until superseded three-quarters of a cen- 
 tury 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 
 
94,95] The Reception of the Coppernican Ideas 127 
 
 tables naturally had great weight in inducing the astro- 
 nomical 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 (?-i595), in his Alae siveScalae Mathe- 
 maticae (1573), an astronomical treatise of no great import- 
 ance, 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 i7th 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 
 
128 A Short History of Astronomy [Ca. V. 
 
 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, de- 
 veloped during the latter half of the i6th 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 in., 64), 
 was already familiar, the other two (decimal fractions and 
 logarithms) were suggested in the i6th century and were 
 in working order early in the iyth century. 
 
 97. The first important set of observations taken after 
 the death of Regiomontanus and Walther (chapter in., 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 
 
$$ 97, 98] The Cassel Observatory 129 
 
 him to devote most of his energy to the duties of govern 
 ment, 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 J 'oost Biirgi 
 (m 1579). Rothmann, of whose life extremely little is 
 known, appears to have been a mathematician and theo- 
 retical 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. 
 Biirgi (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.* 
 
 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 (r) 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 observa- 
 tions 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 Biirgi, and in particular by his discovery that a clock 
 could be regulated by a pendulum, a discovery which he 
 
 * There is little doubt that he invented what were substantially 
 legarithms independently of Napier, but, with characteristic inability 
 or unwillingness to proclaim his discoveries, allowed the invention 
 to die with him. 
 
 9 
 
130 A Short History of Astronomy [Cn. v. 
 
 appears to have taken no steps to publish, and which had 
 in consequence to be made again independently before it 
 received general recognition.* 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 i59ot 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) Brake. 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, accom- 
 panied, 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 in., 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, 
 
 * A similar discovery was in fact made twice again, by Galilei 
 (chapter vi., 114)- and by Huygens (chapter vin., 157). 
 
 f He obtained leave of absence to pay a visit to Tycho Brahe 
 and never returned to Cassel. He must have died between 15^9 
 and 1608. 
 
of Tycho Brake 131 
 
 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 ic 
 had any perceptible parallax (chapter n., 43, 49). No 
 parallax could be definitely detected, and he deduced accord 
 ingly 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 unchange- 
 
132 A Short History of Astronomy [CH. v., $101 
 
 able, 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 ulti- 
 mately 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 sub- 
 sequently made tlje 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 Landgiave, 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 Regens- 
 burg, where he obtained a copy of the, Commentariolus of 
 Coppernicus (chapter iv., 73), and finally came home 
 by way of Saalfeld and Wittenberg. 
 
 1 01. 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 
 
134 A Short History of Astronomy [CH. v. 
 
 derived from the rents of the island by an annual payment 
 cf 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. 
 
 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.* 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 obliga- 
 tions, 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 
 
 * He even did not forget to provide one of the most necessary 
 parts of a mediaeval castle, a prison ! 
 
io2, io 3 ] Life at Hveen 135 
 
 Hveen his tenants complained of work which he illegally 
 f reed 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 
 hold 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 
 
136 A Short History of Astronomy [Cn. V. 
 
 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 pub- 
 lished 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. 
 
 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 
 
jo 4 , io S ] TyMs System of the World 
 
 137 
 
 Coppernican system introduced, but was unable to answer 
 two of the serious objections ; he regarded any motion of 
 
 K 
 
 H 
 
 FIG. 52. Tycho's system of the world. From his book on the 
 comet of 1577. 
 
 " the sluggish and heavy earth " as contrary to " physical 
 principles," and he objected to the great distance of the 
 
138 A Short History of Astronomy [CH V. 
 
 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.* Biblical difficul- 
 ties t 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 Bar or Ursus (?-i6oo), 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 hesita- 
 tion 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, how- 
 ever, 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. 
 
 1 06. 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 IT, 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 
 
 * It would be interesting to know what use he assigned to the 
 (presumably) still vaster space beyond the stars. 
 
 j- 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 thrt the 
 apparent diameters of sun and moon are very nearly equal ! 
 
TYCHO BRAKE. 
 
 [To face p. 139. 
 
W 106, i 7 ] Last Years at Hveen 139 
 
 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 re- 
 garded the heavy expenditure at Hveen as an extravagance, 
 and it is alse 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 
 
140 A Short History of Astronomy tCa. V. 
 
 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 
 
 1 08. 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 VIL, 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 re- 
 lations 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 
 
$ xos no] Tycho's Last Years 141 
 
 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 charac- 
 teristic 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 (IVefrustra 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 instru- 
 ments 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. 
 
 no. 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 charac- 
 teristics as an observer before referring to special dis- 
 coveries. 
 
 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 con- 
 tinuously (cf. chapter HI., 57), but they had too little 
 
142 A Short History of Astronomy [Cn. v. 
 
 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 HI., 68), 
 if he had lived, might probably have to a consider- 
 able 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 observa- 
 tions 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 i', 
 and in only one case as great as 2' (this error being chiefly 
 due to refraction (chapter n., 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 obser- 
 vations did not exceed i' 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,* 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 
 * By transversals. 
 
$ in] Estimate of Tycho' s Work 143 
 
 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 ascer- 
 tain 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 IL, 43, 49), 
 though the amount of this correction was uncertain. In 
 cases where special accuracy was required, Tycho accord- 
 ingly 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 acci- 
 dental sources of error in the separate observations should 
 as far as possible neutralise one another. 
 
 in. Almost every astronomical quantity of importance 
 was re-determined and generally corrected by him. The 
 annual motion of the sun's apogee relative to r , for example, 
 which Coppernicus had estimated at 24", Tycho fixed ut 
 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 i', 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 n., 41). 
 
 In the theory of the moon Tycho made several important 
 discoveries. He found that the irregularities in its move- 
 ment were not fully represented by the equation of the 
 centre and the evection (chapter n., 39, 48), but that 
 
144 -^ Short History of Astronomy [CH. v., 112 
 
 there was a further irregularity which vanished at opposition 
 and conjunction as well as at quadratures, but in inter- 
 mediate positions of the moon might be as great as 40'. 
 This irregularity, known as the variation, was, as has been 
 already mentioned (chapter in., 60), very possibly dis- 
 covered 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, how- 
 ever, 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 n., 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 in., 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 Galilee." 
 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 chrono- 
 logical 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 them- 
 selves 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 con- 
 spicuous, 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 
 
 I4>; IO 
 
146 A Short History of Astronomy [CH. vi. 
 
 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 scien- 
 tific 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 con- 
 tinued 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 
 
ii4ii6] The Pendulum : Falling Bodies 147 
 
 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,* 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 i6th century of such simple experiments. 
 The mediaeval tradition of carrying out scientific investiga- 
 tion 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 
 
 * On an instrument which he had invented, called the hydrostatic 
 balance. 
 
148 A Short History of Astronomy [CH. vi. 
 
 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 col- 
 leagues 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 co Id 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 ap- 
 pointment 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 
 o have been very great and very varied ; in addition to 
 jiving his regular lectures, to audiences which rapidly in- 
 creased, 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. 
 
tt 117, 118] First Astronomical Discoveries 149 
 
 No record exists of the exact time at which he first 
 adopted the astronomical views of Coppernicus, hut 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 con- 
 stellation Serpentarius, and was shewn by him to be at 
 any rate more distant than the planets, a result confirm^ 
 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. 
 
 1 1 8. By this time Galilei had become famous through- 
 out 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 HI., 67) had claimed to have de- 
 vised 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 (?-i6i9), a spectacle-maker of Middle- 
 burg, 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 informa- 
 tion 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 
 
150 A Short History of Astronomy [Cn. vi. 
 
 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 mathe- 
 matician 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 pub- 
 lished 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.* 
 
 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 precisiori, he calculated from observations of this nature 
 the height of some of the more conspicuous lunar moun- 
 
 * 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.;, 
 i i which the poet and Beatrice expound two different " explanations " 
 cf the spots on the moon. 
 
FIG. 53. One of Galilei's drawings of the moon. From the 
 
 Sidereus Nuncius. [To face p. 150. 
 
M 119121] Observations of the Moon 151 
 
 tains, 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 viu., 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 * attempted 
 to explain away the apparent contradiction between the old 
 theory and the new observations by the ingenious sugges- 
 tion 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 sub- 
 stance, at least ten times as high as any which he had 
 observed. 
 
 1 20. 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 Praesepe (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 
 
 * Ludovico delle Colombe in a tract Contra II Moto della Terra, 
 which is reprinted in the national edition of Galilei's works, Vol. III. 
 
152 A Short History of Astronojny [CH. vi. 
 
 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 i3th, and the motions of all four were soon recog- 
 nised by Galilei as being motions of revolution round 
 Jupiter as a centre. With characteristic thoroughness he 
 
 Ori. * * O * Occ 
 
 FIG. 54. Jupiter and its satellites as seen on Jan. 7, 1610. 
 From the Sidereus Nuncius. 
 
 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. 
 
 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 them- 
 selves 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 
 
122] The Satellites of Jupiter 153 
 
 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 objec- 
 tionable 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 satel- 
 lites 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. How- 
 ever, 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 xin., 295). 
 
 122. The reputation acquired oy Galilei by the publica- 
 tion of the Messenger enabled him to bring to a satisfactory 
 issue negotiations \\hich 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 
 
154 A Short History of Astronomy fCn. vi. 
 
 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 vm., 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 viu., 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, deriv- 
 ing its light from the sun ; so that its similarity to the earth 
 was thereby made more evident. 
 
 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,* 
 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 announce- 
 ment 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, 
 
 * 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. 
 

 F G 
 
 r 
 
 H 
 
 FIG. 55. Sun-spots. From Galilei's Macchie Soiari. 
 
 'T> face >- 154. 
 
$$123,124] Phases of Venus: Sun-spots 155 
 
 according to which celestial bodies were perfect and un- 
 changeable. 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,* 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 nis 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 
 
 * Historia c Dituostrasioni intorno alle Macchie Solan. 
 
156 A Short History of Astronomy [Cn. vi 
 
 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 
 DOE, 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 do' ted 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 , 
 measured by the angle d o e. Moreover, it required only 
 a simple calculation, performed by Galilei in several cases, 
 
 FIG. 56. Galilei's proof that sun-spots are not planets. 
 
 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 on^ hand, 
 no sort of reason could be given, and which, on the other, 
 was entirely out of harmony with the uniformity to which 
 mediaeval astronomy clung so firmly. 
 
 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 
 
$ las] Sun-spots 157 
 
 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 con- 
 siderably 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 consider- 
 ably 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 
 \\ho 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 betn alive. 
 
 Jt 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 1 61 1, 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 contro- 
 versy as to the relative validity in scientific matters of 
 
158 A Short History of Astronomy [CH. vi. 
 
 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 j 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?"* and in 1615 
 Galilei was secretly denounced to the Inquisition on the 
 strength of the letter to Cistelli 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 
 
 * Acts i. ii. The pun is not quite so bad in its Latin form : Viri 
 Gali'act, etc. 
 
$ 126] The First Condemnation of Galilei 159 
 
 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, 
 
 tc That the intention of the Holy Ghost is to teach us not 
 
 ,j how the heavens go, but how to go to heaven," * 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." t 
 
 1 26. 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 
 
 * Spiritni sanclo nienfcin Juisse nos docerc, quo tnodo ad Coeluin 
 catiir^ iion mttcm, quomodo Coeliim gradiaiitr. 
 
 f From the translation by Salusbury, in Vol. I. of his Mathematical 
 Collections, 
 
160 A Short History of Astronomy [CH. vi. 
 
 abandon the said opinion," which the Cardinal did.* 
 Immediately afterwards a decree was issued condemning 
 the opinions in question and placing on the well-known 
 Index of Prohibited Books three books containing Copper- 
 nican 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 after- 
 wards, not to be relied upon to keep time accurately during 
 
 * 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. 
 
$ i2 7 ] The Problem of Longitude : Comets 161 
 
 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 // 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." 
 
 II 
 
1 62 A Short History of Astronomy [Cn. VI. 
 
 Although in scientific importance the Saggiatorc 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 
 
$$128,129] The Dialogue on the Tivo Chief Systems 163 
 
 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 re- 
 written 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 read- 
 able, 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-con- 
 tradiction. 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 
 
1 64 <A Short History of Astronomy [CH. VI. 
 
 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 n., 43, 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 ; * 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 direc- 
 tion), of which one might be presumed from its greater 
 
 * 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. 
 
$ 13] The Dialogue on the Two Chief Systems 165 
 
 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 
 
 E 
 
 FIG. 57. The differential method of parallax. 
 
 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 xin., 278). 
 
 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 appl'ei 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 
 
1 66 A Short History of Astronomy [CH. VI. 
 
 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 abstrac- 
 tion 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, 
 t.g. t 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 mast- 
 head 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 
 
130] The Dialogue on the Two Chief Systems 167 
 
 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 mast-head 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.* 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 earch, 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 
 
 * 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. 
 
1 68 A Short History of Astronomy [CH. vi. 
 
 correctly with a problem of which the vastly more powerful 
 resources of modern science can only give an imperfect 
 solution (cf. chapter XL, 248, and chapter xm., 292). 
 
 131. The book as a whole was in effect, though not in 
 form, a powerful indeed unanswerable plea for Copper- 
 nicanism. 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 illformed 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 
 
$$ isi, 132] Galileos Trial 169 
 
 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." * 
 
 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 exciied ; 
 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 evi- 
 dently frightened by the summons, and tried to avoid com- 
 pliance 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 admoni- 
 tion 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 
 
 * From the translation by Salusbury, in Vol. I. of his Mathematical 
 Collections, 
 
170 A Short History of Astronomy [CH. vi. 
 
 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 2ist he was examined under threat of 
 torture,* 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 con- 
 finement at a country-house near Rome belonging to the 
 Grand Duke, and Galilei moved there on June 24th. t 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. 
 
 * The official minute is : Et ei dido quod dicat veritatem, alias 
 devenietur ad torturam. 
 
 f The three days June 21-24 are 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. 
 
[To face p. 171. 
 
f 132] The Second Condemnation of Galilei 171 
 
 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 iyth century; 
 during Galilei's life-time (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 con- 
 demnation was seriously meant by the Pope, the Holy 
 Office, and others, to believe that in some senses at least 
 his vie\vs must be wrong, although, as a matter of observa- 
 tion and pure reason, he was unable to see how or why. 
 In fact, like many other excellent people, he kept water- 
 tight 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 vn., 145), 
 which had been put on the Index shortly afterwards. It 
 
172 A Short History of Astronomy [CH. vi., $ 133 
 
 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 Lite 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. 
 
 It had long been well known that as the moon t!e>cribes 
 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 
 
174 A Short History of Astronomy [CH. VI. 
 
 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 CQ 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 1 is invisible and Q appears to be in 
 the centre of the disc. Thus in the course of the day 
 the portion a Q 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 altera- 
 tions 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, Mathe- 
 matical 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 
 
$ i33] Libration; the Two New Sciences 175 
 
 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,* 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 dis- 
 turbed 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.t 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 y 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 
 
 * Equivalent to portions of the subject now called dynamics or 
 (more correctly) kinematics and kinetics. 
 
 f 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. 
 
176 A Short History of Astronomy [Cn. VI. 
 
 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 Ley den 
 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 in- 
 cidents of his persecution, formed conjointly a contribution 
 to the Coppernican controversy which was in effect 
 decisive. Astronomical text-books still continued to give 
 side by side accounts of the Ptolemaic and of the Copper- 
 nican 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 Copper- 
 nican 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 
 
$ i34] Estimate of Galilei's Work 177 
 
 astronomy he was building on foundations laid by pre- 
 vious generations, in dynamics it was no question of im- 
 proving 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 
 a\ 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 super- 
 structure which later investigators added. 
 
 It is customary to associate with our countryman Francis 
 Bacon (1561-1627) the reform in methods of scientific 
 discover)' 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 
 
 12 
 
178 A Short History of Astronomy [CH. vi., $ i 34 
 
 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.'' * 
 
 * Two New Sciences, translated by Western, p. 255. 
 
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 persever- 
 ance and ingenuity with which he renewed his attack upon the 
 riddles of nature." 
 
 JEVONS. 
 
 135. JOHN KEPLER, or Keppler,* was born in 1571, seven 
 years after Galilei, at Weil in Wiirtemberg ; his parents were 
 in reduced circumstances, though his father had some claims 
 to noble descent. Though Weil itself was predominantly 
 Koman 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 n 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 
 
 * 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), 
 t|as the form Kepler, while the dedication of the same book is signed 
 Ke| pier. 
 
 179 
 
180 A Short History of Astronomy [CH. vii. 
 
 manual work at home he was sen* 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 Tubingen, then one of the great centres of 
 Protestant theology. 
 
 In 1588 he obta'njd the B.A. degree, and in the following 
 year entered the philosophical faculty at Tubingen. 
 
 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 stifdy of theology. 
 
 136. In 1594, however, the Protestant Estates of Styria 
 applied to Tubingen for a lecturer on mathematics (in- 
 cluding 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 Wiirtemberg. 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 washed 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 
 
*$ 136, 13?] Kepler s Rarlv Astronomical M'ork 181 
 
 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.* 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 U?iiverse, 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 
 
 * 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 t'ie 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. 
 
1 82 A Short History of Astronomy [Cn. vn 
 
 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 ha'l 
 hopes of converting the astronomer. Kepler's hearers had, 
 however, mostly been scattered by the persecution, it be- 
 came difficult to ensure regular payment of his stipend, 
 and the rising tide of Catholicism made his position in- 
 creasingly 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 w..s 
 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 pos- 
 session, but as he had little taste or skill for observing, the 
 loss was probably not great ; fortunately, after some diffi- 
 culties 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 ex- 
 ample, 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 11., 46, and chapter v., no). A 
 later optical treatise (the Dioptrice of 1611) contained a 
 
KEPLER. 
 
 [To face p. 183. 
 
* i38, 139] Optical Work: Study of Mars 183 
 
 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 instru- 
 ment (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. 
 
 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 Copper- 
 nican 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 evi- 
 dently at fault. 
 
 The solution of the problem was clearly more likely 
 to be found by the study of a planet in which the de 4 - 
 viations 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 un- 
 wearied 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 
 
184 A Short History of Astronomy [Cn. vn. 
 
 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 simpli- 
 fication was accordingly effected when Kepler succeeded, 
 by an ingenious combination of observations taken at suit- 
 able 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'.* 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 : 
 
 11 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." f 
 
 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 in- 
 clined to believe in an egg-shaped oval, larger at one end than 
 at the other, but soon had to abandon this idea. Finally 
 
 * Two stars 4' apart only just appear distinct to the naked eye of 
 a person with average keenness of sight. 
 
 f Commentaries on the Motions of Mars, Part II., end of chapter xix. 
 
$ 140] The Discovery of the Elliptic Mot ion of Mars 185 
 
 he tried the simplest known oval curve, the ellipse,* 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 plnnet 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. 6ot 
 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 
 
 * 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. 
 
 Thus if, in the figure, s and H are the foci, and P, Q are any two 
 
 FIG. 59. An ellipse. 
 
 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. 
 
 f 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 \ in the figure, that of Mars being fa. 
 
1 86 A Short History of Astronomy [Ca. vn. 
 
 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. 
 
 FIG. 60. Kepler's second law. 
 
 141. Thus were established for the case of Mars the two im- 
 portant results generally known as Kepler's frst 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* 
 
 142. Although the two laws of planetary motion just 
 given were only fully established for the case of Mars, 
 
 * Astronomia Nova 00-10X07777-0? seu Physica Coelestis, tradita Com- 
 mentariis de Motibus Stellae Martis. Ex Observationibus G. V. 
 Tychonis Brahc. 
 
141, 142] 
 
 Kepler's First and Second Laws 
 
 187 
 
 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 
 
 FIG. 61. Diagram used by Kepler to establish his laws of planetary 
 motion. From the Commentaries on Mars. 
 
 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 
 
1 88 A Short History of Astronomy [Cn vu 
 
 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, how- 
 ever, he decided to stay with Rudolph. 
 
 In the same year Kepler lost his wife, who had long 
 been in weak bodily and mentalhealth. 
 
 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 Better 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 house- 
 hold led to the publication of a pamphlet, of some mathe- 
 matical interest, dealing with the proper way of measuring 
 the contents of a cask with curved sides.* 
 
 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^ and a treatise on Comets. 
 
 * It contains the germs of the method of infinitesimals, 
 t Harmonices Mundi Libri V. 
 
143, 144] 
 
 Kepler s Third Law 
 
 189 
 
 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 specula- 
 tive and mystic temperament led him constantly to search 
 for relations between the various numerical quantities occur- 
 ring 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 dis- 
 covered 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 con- 
 veniently 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, i, i'88, 1 1*86, 29-457, and their distances from the 
 sun are respectively -387, 723, i, 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 j 
 periodic l 
 time J 
 
 5 8 
 
 378 
 
 I 
 
 3'54 
 
 1407 
 
 867-7 
 
 Cube oH 
 mean j- 
 distance J 
 
 5 8 
 
 378 
 
 I 
 
 3-54 
 
 140-8 
 
 867-9 
 
 Here it will be seen tha.t the .two series of numbers, in the 
 
190 A Short History of Astronomy [Cn. vn. 
 
 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 after- 
 wards 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."* 
 
 It should be noticed further that Kepler's law requires 
 no knowledge of the actual distances of the several planets 
 
 
 ** 
 
 Saturnus Jupiter Mars fere Terra 
 
 f 
 
 Venus *"* Mercunus Hie locum habetctiam> 
 
 FIG. 62. The " music of the spheres," according to Kepler. 
 From the Harmony of the World. 
 
 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 
 
 * 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 in. 
 
Ms] The Harmony of the " II 'or Id and the Epitome 191 
 
 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 (miserid) 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 dis- 
 coveries. Such a text-book 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 olher 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 n., 48) 
 and variation (chapter in., 60; chapter v., in) being 
 fully dealt with, though the "annual equation" which 
 Tycho had just begun to recognise at the end of his life 
 (chapter v., in) 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 
 
192 A Short History of Astronomy [CH. vii. 
 
 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 n., 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 there- 
 fore 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 n., 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 
 
1^6] Kepler s Epitome and his Book on Comets 193 
 
 which has attracted so much attention in the last half- 
 century (chapter xm., 3 01 )- 
 
 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 in., 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 views as to the nature of light, is a curiously 
 correct anticipation of modern theories of comets' tails 
 (chapter xm., 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." 
 
194 -A Short History of Astronomy [CH. vil. 
 
 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 attach- 
 ment 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 L, 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 perse- 
 cution of the Protestants at Gratz ( 137) and who had 
 few scientific interests. Kepler was, however, after some 
 delay, confirmed in his appointment as Imperial Mathe- 
 matician. 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. 
 
$$ i47 is J The Rudolphine Tables 195 
 
 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 i5th (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 men- 
 tioned 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 
 
196 
 
 A Short History of Astronomy [CH. vii. 
 
 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 yords, 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 there- 
 fore a conflict be- 
 tween 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." * 
 FIG. 63. Kepler's idea of gravity. , 
 
 From the Epitome. Ihe annexed 
 
 diagram is given 
 
 by Kepler in illustration of this rather confused and vague 
 theory. 
 
 He believed also in a more general " gravity," which he 
 defined f 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 
 
 * Epitome, Book IV., Part 2. 
 
 f Introduction to the Commentaries on the Motions of Mars. 
 
$ i 5 i] Estimate of Kepler's Work 197 
 
 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 import- 
 ance 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 con- 
 temporaries 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 specula- 
 tion 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 im- 
 pressed 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. 
 
 ".Andgr.owjthc kfty 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 observa- 
 tion 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 under- 
 lying 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. 
 
 198 
 
CH. viii., $$ 152-154] Telescapic Discoveries 199 
 
 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 ms 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. 
 Revel'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 Ricdoli (1598-1671) in his bulky treatise 
 on astronomy called the New Almagest (165 1). 
 
 Hevel, who was an indefatigable worker, published two 
 large books on comets, Prodromus Cometicus (1654) and 
 C0metogra#Aia(i66&), containing the first systematic account 
 of all recorded comets. He constructed also a catalogue 
 cf 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, 
 
2oo A Short History of Astronomy [CH. vin., $ 154 
 
 Christiaan Huygens (1629-1695), a native of the Hague. 
 Huygens possessed remarkable ability, both practical and 
 theoretical, in several different directions, and his contribu- 
 tions 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 XIL, 253, 255 ; chapter xin., 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 dis- 
 covery of the satellite was published, De Saturni Luna 
 Observatio Nova (1656); and three years afterwards (1659) 
 the larger Systema Saturnium appeared, in which the in- 
 terpretation 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 pre- 
 sented 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 pre- 
 sents the appearance of two arms projecting from the body 
 
FIG. 64. Saturn's ring, as drawn by Huygens. From the 
 Systeitia Saltirninin. 
 
 FIG. 65. Saturn, with the ring seen edge-wise. From the 
 
 Systewa Satttrnium. [To face p. 200 
 
2O2 A Short History of Asironciny [CH. vm. 
 
 of Saturn; farther off still the ring appeals wider and the 
 opening becomes visible; and about seven years before 
 and after i 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 son e of them were to the discovery of the ring. 
 
 155. To our countryman William Gascoigne (1612 ?-i644) 
 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 Marstcn 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 * with which he was able to measure 
 angles of a few seconds. Nothing was known of his dis- 
 coveries at the time, and it was left for Huygens to invent 
 independently a micrometer of an inferior kind (1658), and 
 for Adrien Auzout (?-i69i) 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 y^TZ Picard (1620-1682). 
 
 156. With Gascoigne should be mentioned his friend 
 Jeremiah Horrocks (1617 ?-i 641), who was an- enthusiastic 
 admirer of Kepler and had made a considerable improve- 
 ment 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 
 
 * Substantially the filar micrometer of modern astronomy 
 
FIG. 67. Earlv drawings of Saturn. From the Systema Saturnium. 
 
 [To face p. 202. 
 
* 155158] GaScoigilt) Ilorro.ks, Huygens 203 
 
 of Philips von Lansberg (1561-1632), which Hurrocks 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 disturb- 
 ing 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 (chap er x., 
 204 ; chapter XL, 243). 
 
 157. Another of Huygens's discoveries revolutionised the 
 art of exact astronomical observation. This was the inven- 
 tion of the pendulum-clock (made 1656, patented in 1657). 
 It has been already mentioned how the same discovery 
 was made by Biirgi, 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.* Henceforward 
 it was possible to take reasonably accurate time-observa- 
 tions, 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 
 
 * 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. 
 
2O4 A Short History of Astronomy [Cn. viil. 
 
 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 * (or, in other 
 words, the rate of falling of a heavy body) afforded a prac- 
 tical 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,t is 
 of fundamental importance in accounting for the planetary 
 motions by gravitation. 
 
 159. During the iyth century also the first measurements 
 of the earth were made which were a definite advance on 
 those of the Greeks and Arabs (chapter n., 36, 45, 
 and chapter in., 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 ?-i675) measured the distance from London 
 tc 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. 
 
 1 60. 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 
 
 * In modern notation : time of oscillation = 2 IT ^ljg> 
 f I.e. he obtained the familiar formula v*/r, and several equivalent 
 forms for centrifugal force*. 
 
$ 159-161] Measurements of the Earth : Cassini 205 
 
 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 Obser- 
 vatory, 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 dis- 
 tinctly 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 vn., 138), and issued in 1693 a fr esn 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. 
 
 1 6 1. It was probably at the suggestion of Picard or Cassini 
 that one of their fellow astronomers, John Richer (?-i696), 
 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 by Cassini, Picard, 
 
206 
 
 A Short History of Astronomy [Cu. vin. 
 
 M 
 
 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 IL, 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 iden- 
 tical with that long used in 
 the case of the moon (chap- 
 ter IL, 49). One observer 
 is, say, at Paris (P, in fig. 69), 
 and observes the direction in 
 
 FIG. 68. Mars in opposition. which Mars appears, i.e. the 
 
 direction of the line P M ; the 
 
 other at Cayenne (c) observes similarly the direction of 
 the line CM. The line CP, joining Paris and Cayenne, is 
 known geographically ; the shape of the triangle c P M and 
 
 FIG. 69. The parallax of a planet. 
 
 the length of one of its sides being thus known, the 
 lengths of the other sides are readily calculated. 
 
 The result of an investigation of this sort is often most 
 conveniently expressed by means of a certain angle, from 
 
i6x] Parallax 207 
 
 vvhic.i the distance in terms of the radius of the earth, and 
 hence in miles, can readily bj deJucad when desired. 
 
 The parallax of a heavenly boJy such as the moon, the 
 sun, or a planet, being in the first instance defined generally 
 (chapter IL, 43) as the angb (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 xin., 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 xin., 284), being 
 
208 A Short History of Astronomy [Cn. vm. 
 
 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 i7th century, and although 
 it was generally believed to be seriously wrong, most 
 corrections of it had been purely conjectural (chapter vii., 
 
 I45)- 
 
 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 Itoemer (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 con- 
 sequently 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 Rene 
 Descartes* (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 
 
 * Also frequently referred to by the Latin name Cartesius. 
 
$$ i6 2 , 163] The Velocity of Ligfit : Descartes 209 
 
 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 con- 
 tributed 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;* 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 
 earlv in life, but ultimately became so marked that, after 
 
 * 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 u 
 22). 
 
CH. ix., 164-168] Newton's Early Life 211 
 
 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 under- 
 graduate, but is reported to have soon abandoned it as 
 being " a trifling book," in favour of more advanced reading. 
 In January 1665 he graduated in the ordinary course as 
 Bachelor of Arts. 
 
 1 66. 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 1m t >ry of his life is almost exclusively the history 
 of his succ.es ,ive discoveries. 
 
 167. His scientific work falls into three main groups, 
 astronomy (including dynamics), optics, and pure mathe- 
 matics. 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. 
 
 1 68. 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, how- 
 ever, 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 
 
212 A Short History of Astronomy [Cn. IX. 
 
 of Galilei and Kepler. Telescopes on this principle, differ- 
 ing however in some important particulars from Newton's, 
 had already been described in 1663 by James Gregory 
 (1638-1675), with whose i Itas 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 astrcno nical discoveries were made 
 with them before the time of William Herschel (chapter XIL), 
 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 xin., 
 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 
 
$169,170] First Discoveries : Gravity 213 
 
 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." * 
 
 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 VIL, 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 
 
 * From a MS. among the Portsmouth Papers, quoted in the Preface 
 to the Catalogue of the Portsmouth Papers. 
 
214 
 
 A Short History of Astronomy 
 
 [Cn. IX. 
 
 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 VIIL, 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. 
 
 171. Viewed as a purely general question, apart from 
 its astronomical applications, the problem may be said to 
 
 be to examine under 
 what condiiions 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. Conse- 
 quently by Galilei's First 
 Law (chapter vi., 
 . 130, 133), if left to 
 itself and uninfluenced by any other body, it would con- 
 tinue to move with the same speed and in the same 
 direction, i.e. along the line A a, and consecjuently 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 
 
 FIG. 70. Motion in a circle. 
 
$$ i 7 i, 172] Motion in a Circle 215 
 
 constantly to draw the body towards o, or counteracting 
 the tendency which it has, in virtu'e 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 accele- 
 ration of 32 feet per second per second. A further in- 
 vestigation 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 ( = J 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 VIIL, 
 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 
 
2i6 A Short History of Astronomy [CH. ix. 
 
 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 to- 
 wards 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 vn., 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 conse- 
 quently 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 dis- 
 tances, 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 
 
$ i 7 3] The Law of the Inverse Square 217 
 
 the sun ; i.e. at twice the distance it is \ as great, at three 
 times the distance \ as great, at ten times the distance -j-J^ 
 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 ^ 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 x 5 to 12 x 12, 
 or as 25 to 144. The accelerations of Jupiter and of the 
 earth towards the sun are therefore as 25-4-5 to 144, 
 or as 5 to 144 ; hence Jupiter's acceleration towards the 
 sun is about -^ that of the earth, and if we had taken 
 more accurate figures this fraction would have come out 
 more nearly ^V 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 
 ^ith 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 down- 
 wards, 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 o\vn centre to bodies near its surface. Newton noticed 
 that this power extended at any rate to the tops of moun- 
 tains, 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 
 
218 A Short History of Astronomy [CH. ix. 
 
 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 x 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 
 
$ 1 73 ] Extension of Gravity to the Moon 219 
 
 distance of the m;)on (which is 60 times the radius of the 
 
 earth, or 20,000,000 feet); that is, it is 3>3 - ^- 
 
 60 x 2^,000,000' 
 
 which reduces to about TTTr . 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 -. or between 
 
 no 
 
 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 BDE. 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." * 
 
 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 2oth, 1686) 
 
 * W. K. Clifford, Aims and Instruments of Scientific Thought. 
 
220 
 
 A Short History of Astronomy 
 
 [CH IX. 
 
 he refers to the calculation as not having been made accu- 
 rately 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 VIIL, 
 159) ; it is known that even at a later stage he was unable 
 
 FIG. 71. The moon as a projectile. 
 
 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 dis- 
 satisfied with his results as to regard his hypothesis as 
 
$ 174, i7s] The Motion of the Moon> and of the Planets 221 
 
 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 Royul Society, founded in 1662. Robert Hooke (1635- 
 1703), who claimed credit for most of the scientific dis- 
 coveries 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 vni., 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 
 
222 A Short History of Astronomy [Cn. ix. 
 
 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 (chap- 
 ter 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, tne 
 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.* 
 Later in the year Halley visited Newton at Cambridge 
 and learnt from him the answer. Newton had, character- 
 istically 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 
 
 * It is interesting to read that Wren offered a prize of 405. to 
 whichever of the other two should solve this the central problem of 
 the solar system. 
 
H 176-179] Elliptic Motions the Prindpia 223 
 
 to the Royal Society, a tract called Propositioned de Motu, 
 the 1 1 propositions of which contained the results already 
 mentioned and some others relating to the motion of 
 bodies under attraction to a centre. Although the pro- 
 positions 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 Philosophic^ Naturalis Prindpia Mathematica. 
 
 178. The Prindpia, as it is commonly called, consists of 
 three books in addition to introductory matter: the first 
 book de ,1s generally with problems of the motion of bodies, 
 solved for ihe 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 
 
224 A Short History of Astronomy [Cn. ix. 
 
 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 accelera- 
 tion (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 accelera- 
 tion 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. 
 
 1 80. 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 m 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 
 
$ 180] Newtorfs Laws of Motion : Mass 225 
 
 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 lea*den 
 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 par- 
 ticular definition of mass, but the really important step 
 v.as 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 measure- 
 ment of the action exerted by one body on another the pro- 
 duct 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 
 
 IS 
 
226 A Short History of Astronomy [Cn. ix. 
 
 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 com- 
 pared 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 VIIL, 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 x 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.* 
 
 -VA 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 
 
 * The familiar parallelogram offerees, of which earlier writers had 
 had indistinct ideas, was clearly stated and proved in the intro- 
 duction to the Princtpia, and was, by a curious coincidence, published 
 also in the same year by Varignon and Lamt. 
 
i8i] Mass: Action and Reaction 227 
 
 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,* its acceleration is enor- 
 mously 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 
 from the earths 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 pro- 
 portional 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 
 
 * It is between 13 and 14 billion billion pounds. Sec chapter x. 
 219. 
 
228 A Short History of Astronomy [Cn. IX. 
 
 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 pro- 
 portional 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* 
 
 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 
 
 * As far as I know Newton gives no short statement of the law 
 in a perfectly complete and general form ; separate parts of it arc 
 given in different passages of the Principia. 
 
$ i8a, 183] Universal Gravitation 229 
 
 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 
 
230 A Short History of Astronomy [CH. ix. 
 
 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 in., 
 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 vin., 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 n., 40, 48 ; chapter v., in), had already 
 been discovered by observation. Newton was able to 
 shew that the disturbing action of the sun would neces- 
 sarily 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;* and he also dis- 
 covered 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. 
 
 * 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. 
 
M 184186] Universal Gravitation 231 
 
 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 earttu 
 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 x J. x 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 corre- 
 sponding 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 em- 
 ployment 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. 
 
 1 86. 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 
 
232 A Short History of Astronomy [CH. ix. 
 
 the case of the large planets Jupiter and Saturn (cf. chap- 
 ter vn., 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 con- 
 demnation 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 them- 
 selves obedient to the papal decrees on the subject.* 
 
 * 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 
 
* is?] Relativ: .'\fotion .- fhe Snap / the Earth 233 
 
 187. The variation of the tin,c of oscillation of a 
 pendulum in different parts c" .ne earth, discovered by 
 Richer in 1672 (chapter vm $161), indicated that the 
 earth was probably not a sy iere. Newton pointed out 
 that this departure from the >pherical form was a conse- 
 quence 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 
 (BO a \ in fig. 72), and then found the condition that these 
 two columns of water OB, 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 
 
 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 obseived motions of the bodies of the solar system. 
 
A Short History of Astronomy 
 
 [CH. IX. 
 
 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\ is about 
 -2^ 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 ^-g- 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 
 
 FIG. 72. The spheroidal form of the earth. 
 
 the length #A is made, for. the sake of distinctness, nearly 
 30 times as great as it should be. 
 
 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. 
 
 1 88. The discovery of the form of the earth led to 
 an explanation of the precession of the equinoxes, a 
 phenomenon which had been discovered i, 800 years before 
 
$ 1 8?, 189] The Shape of the Earth : Precession 235 
 
 (chapter n., 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.* 
 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 pro- 
 duced, on the water nearest to the moon is greater than 
 
 * 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. 
 
236 A Short History of Astronomy [Cn. ix. 
 
 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 achieve- 
 ment. Newton's theory of the tides was based on certain 
 hypotheses which had to be made in order to render the 
 
190] Tides and Comets 237 
 
 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 vn., 
 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, recog- 
 nised them as belonging to the celestial bodies, were un- 
 able 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 Prindpia, 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, 
 
238 A Short History of Astronomy [CH. IX. 
 
 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 mathemati- 
 cally 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 Prindpia the motions of a 
 number of other comets were investigated with a similar 
 
 FIG. 73. An elongated ellipse and a parabola. 
 
 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 
 Prindpia. 
 
 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 xin., 304), that 
 
i9l Comets : Reception of the Prindpia 239 
 
 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 Prindpia 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 
 Prindpia, 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, in- 
 volving 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 VIH., 
 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 differen- 
 tial method (out of which the differential and integral 
 calculus have developed) grew in intensity and fresh com- 
 batants 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 XL, 229). In our 
 country the case was different ; not only was the Prindpia 
 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 
 
240 A Short History of Astronomy [CH. ix. 
 
 to grasp Newton's general ideas, even though the details 
 of his mathematics were out of their range. It was more- 
 over soon discovered that his scientific ideas could be 
 used with advantage as theological arguments- 
 
 192. One unfortunate result of the great success of the 
 Prindpia was that Newton was changed from a quiet 
 Cambridge professor, with abundant leisure and a slender 
 income, into a public character, with a continually increas- 
 ing portion of his time devoted to public business of one 
 sort or another. 
 
 Just before the publication of the Prindpia 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.* 
 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 occa- 
 sional 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 Prindpia; and in 1691 there was 
 already some talk of a new edition of the Prindpia, 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. 
 
 Soon after recovering from this illness he made some 
 
 * He once told Halley in despair that the lunar theory ' made 
 his head ache and kept him awake so often that he would think ol 
 it no more." 
 
[To face p. 240. 
 
i92, 193] J\?;i'/Wj Later Life 241 
 
 preparations for a new edition of the Prindpia, besides 
 going on with the lunar theory, but the work was again 
 interrupted in 1695, when he received the valuable appoint- 
 ment of Warden to the Mint, from which he was promoted 
 to the Mastership four years later. He had, in conse- 
 quence, 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 suc- 
 cessor Lagrange (chapter XL, 237), "Newton was the 
 
 16 
 
242 A Short History of Astronomy [CH. ix. 
 
 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 sea- 
 shore, 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 differ- 
 ence 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 epi- 
 cycles ; 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 par- 
 ticular 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, correspond- 
 ing to certain epicycles, existed, he would have replied that 
 it was because of certain motions cf the earth, from which 
 
iQ 4 ] Explanation and Description 243 
 
 these (apparent) planetary motions could be de'duced 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 th..i 
 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 specu- 
 lating 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 neces- 
 sary 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 
 mother. 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 
 
244 d Short History of Astronomy [CH. ix. 
 
 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 satis- 
 factory 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 depart- 
 ments 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. 
 
$ i95] Newton's Scientific Method 245 
 
 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 gtneral 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 
 perhups been somewhat obscured by his famous remark, 
 Hypotheses nonfingo (I do not invent hypotheses), dissociated 
 from its context. The words occur in the conclusion of 
 the Printipia, 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 phenomei a 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 couM 1>- 4 
 
246 A Short History of Astronomy [On. ix., $ 195 
 
 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 pro- 
 cess 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 xin., 290). 
 
CHAPTER X. 
 
 OBSERVATIONAL ASTRONOMY IN THE l8TH 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 country- 
 men contributed next to nothing throughout the i8th 
 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 Prindpia. 
 
 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 Prindpia 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 differ- 
 ences between them to say that in the geometrical treatment 
 of an astronomical problem each step of the reasoning is 
 
 247 
 
248 A Short History of Astronomy [CH. x. 
 
 expressed in such a way as to be capable of being inter- 
 preted 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 con- 
 . ducted 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 Prindpia, 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 
 1 8th century, and with the exception of some admirable 
 work by Colin Madaurin (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 
 
j 197] Continental Analysis and English Observation 249 
 
 gravitation beyond the point at which it was left in the 
 Prindpia. 
 
 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 Observa- 
 tory of Paris, at which Picard and others had done such 
 admirable work (chapter vni., 160-2), produced little of 
 real importance for nearly a century afterwards, and a lar.:e 
 part of the best observing work of the i8th 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.* Unfor- 
 tunately 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 tho e 
 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 ne was appointed a member of a Committee to repo r t 
 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 
 
 * December 3ist, 1719, according to the unreformed calendar (O.S*) 
 then in use in England. 
 
250 A Short History of Astronomy [Cn. x. 
 
 office of Astronomer Royal, with a salary of ^100 a year, 
 and the warrant for building an Observatory at Greenwich 
 was signed on June i2th, 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 vin., 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 assist- 
 ant. 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 Govern- 
 ment 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 improve- 
 ments in practical astronomy, the best known being his 
 method of finding the position of the first point of Aries 
 (chapter n,, 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 vin., 157), 
 and, as soon as he could get the necessary instruments, 
 he regularly used the telescopic sights of Gascoigne and 
 Auzout (chapter vin., 155), instead of making naked-eye 
 
$198] Flamsteed 251 
 
 observations. Thus while Hevel (chapter vin., 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 observa- 
 tions ; 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 ful- 
 filled 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 observa- 
 tions 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 
 
252 A Short History of Astronomy [CH. x. 
 
 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 per- 
 manently valuable part was a catalogue of the places of 
 nearly 3,000 stars.* 
 
 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 t that, whereas 
 Tycho's determinations of the positions of the stars were 
 on the average about i' in error, the corresponding errors 
 in Flamsteed's case were about ID". 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 ; 
 but by unflagging industry and scrupulous care he succeeded 
 in bequeathing to his successors an immense treasure of 
 
 * The apparent number is 2,935, Dut 12 of these are duplicates. 
 
 f By Bessel (chapter xin., 277). 
 
 j 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." 
 
i 99 , 2r ] Flamsteed' s Observations : Halley 253 
 
 observations, executed with all the accuracy that his in- 
 strumental 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. 
 
 Bom in 1656, ten year,s 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 observa- 
 tion, 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 vin., 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 wLh 
 comets, particularly in the later editions. In 1705 he 
 published a Synopsis of Comet ary Astronomy in which no I 
 less than 24 cometary orbits were calculated. Struck by * 
 
254 ^ Short History of Astronomy [CH. x. 
 
 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 appear- 
 ances 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 XL, 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 XL, 240, and 
 chapter XHI., 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 
 
$1 301204] If alky 255 
 
 his observation of the transit of Mercifry 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 dis- 
 appearance, 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 T \-^ 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 un- 
 naturally 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 stand- 
 ing astronomical problem of improving the tables of the 
 moon and planets, particularly the former. He made 
 
256 A Short History of Astronomy [CH. X. 
 
 observations of the moon as early as 1 683, 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 vin., 
 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 com- 
 pleted 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 XL, 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 n., 38) and the star catalogue of Ptolemy 
 (chapter n., 50). He was also one of the first of modern 
 
$ 2c S , 206] Hattey 257 
 
 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 VH., 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 xin., 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 dis- 
 coveries, 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 
 
258 A Short History of Astronomy [CH. X. 
 
 biographer * 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 pro- 
 fession." 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 Astro- 
 nomy 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 Obser- 
 vatory 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. 
 
 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 
 
 * Rigaud, in the memoirs prefixed to Bradley's Miscellaneous 
 Works. 
 
{To face p. 258. 
 
$ 207] Bradley; Aberration 259 
 
 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 obser- 
 vation 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 (y 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 posi- 
 tion in which it would be seen in the field of view of 
 the telescope. The first observations were taken on 
 December i4th, 1725 (N.S.), and by December 28th 
 Bradley believed that he had already noticed a slight dis- 
 placement of the star towards the south. This motion 
 was clearly verified on January ist, and was then observed 
 to continue ; in the following March the star reached its 
 extreme southern position, and then began to move north- 
 wards 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 
 
260 A Short History of Astronomy [CH. x. 
 
 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 i st a star * 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 firSv 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 y 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 y 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 y 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 altera- 
 tion 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 ques- 
 tion, although amply established, still remained quite 
 unexplained. 
 
 By this time Bradley had mounted an instrument of his 
 
 * A telescopic star named 37 Camelopardi in Flamsteed's 
 catalogue. 
 
$ 2c8] Aberration 26 1 
 
 own at Wansted, so arranged that it was possible to observe 
 through it the motions of stars other than y 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 pheno- 
 menon 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,* ' 
 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 men- 
 tioned proceeded from the progressive motion of light and the 
 earth's annual motion in its orbit. For I perceived that, if light 
 \vas 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 
 
 * 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 UQ allusion to this 
 
262 
 
 A Short History of Astronomy 
 
 [Cn. X. 
 
 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 sup- 
 posed 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 DEC, 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. . . . 
 
 " 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 pari- 
 
 D A B 
 
 FIG. 74. The aberra 
 
 tion of light. From bus) greater or less, according to the 
 Bradley's paper in different proportion between the velocity 
 the Phil Trans. o f light and tnat of the eye. So that if 
 we could suppose that light was propa- 
 gated 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." 
 
209] 
 
 Aberration 
 
 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 wpuld 
 believe that there was a slight head-wind blowing the rain 
 towards hirr. 
 
 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 displace- 
 ment from its true position is represented by 
 the angle B c A, which depends on the pro- 
 portion 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 u, 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* 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, dif- 
 ferent 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) 
 * It is k sin CAB, where k is the constant of aberration. 
 
 IB' 
 
 A B 
 
 FIG. 75. The 
 aberration of 
 light. 
 
264 
 
 A Short History of Astronomy 
 
 [CH. X. 
 
 FIG. 76. The aberrational ellipse. 
 
 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 direc- 
 tion 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 inter- 
 mediate times it 
 undergoes its least 
 displacement, as at 
 B and B'. The 
 greatest displace- 
 ]A ment s A, or half of 
 A A', which is the 
 same for all stars, 
 is known as the con- 
 stant of aberration, 
 and was fixed by 
 Bradley at between 
 20" and 2o|", the 
 value at present accepted being 20" '4 7. The least displace- 
 ment, 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. 
 
 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 u minutes (chapter vin., 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 
 
M 210-213] Aberration 265 
 
 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 i8th 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 ex- 
 planation 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 y Dra- 
 conis 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 |", 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 
 
266 A Short History of Astronomy [CH. x. 
 
 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 declina- 
 tion 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 there- 
 fore 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 
 
$ 2i 3 ] Nutation 267 
 
 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." 
 
 in his discussion of precession (chapter ix., 188; 
 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 ot 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 
 
268 
 
 A Short History of Astronomy 
 
 [C'H. X. 
 
 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 gf his theory, and 
 only published his results definitely at the beginning of 
 1748. 
 
 214. Bradley's observations established the existence of 
 certain alterations in the positions of various stars, which 
 
 FIG. 77. Precession and nutation. 
 
 could be accounted for by supposing that, on the one 
 hand, the distance of the pole from the ecliptic fluctu- 
 ated, 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 mathe- 
 maticians 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 
 
$ 2i 4 -2i6] Nutation 269 
 
 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.* 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 XL, 232) published a treatise t 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 du_e to t|iejittraction 
 o the moon on the protuberant parts ^)f the ^arth round 
 trie v ~equator (cf. chapter ix., 187), while Newton's ex- 
 planation of precession was confirmed by the same piece 
 of work. Euler (chapter XL, 236) published soon after- 
 wards 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 t 
 of minute irregularities in the motion of the earth's axis. 
 
 216. Although aberration and nutation have been dis- 
 cussed first, as being the most important of Bradley's 
 
 * 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 i8"'42 and 13"'75, so that his ellipse 
 was decidedly less flat than it should have been. 
 
 t Recherches sur la precession des equinoxes et siir la nutation de 
 I axe de la terre. 
 
270 A Short History of Astronomy [C'n. x. 
 
 discoveries, other investigations were carried out by him 
 before or during the same time. 
 
 The earliest important piece of work which he accom- 
 plished 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 VIH., 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 Viihelm Wargentin (1717-1783) had in- 
 dependently 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 observa- 
 tion had revealed to him, though he was well aware of the 
 importance of such an undertaking, and definitely expressed 
 the hope " that some geometer,* 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 
 
 * The word "geometer" was formerly used, as "geometre" still 
 is in French, in the wider sense in which " mathematician " is now 
 customary. 
 
$217,218] Bradley V Minor Work: his Observations 271 
 
 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 IL, 22). 
 
 218. It remains to give some account of the magnificent 
 series of observations carried out during Bradley's adminis- 
 tration 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 con- 
 sisted 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 tUl 1818 
 that the publication of Bessel's Fundamenta Astronomiae 
 (chapter xin., 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 
 
272 A Short History of Astronomy [Cn. x. 
 
 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 cor- 
 rections which have the effect of allowing for these various 
 sources of error. This process of reduction, as it is techni- 
 cally 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 
 
, ai9] Bradley s Observations 273 
 
 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 com- 
 paring 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 
 
 18 
 
274 -^ Short History of Astronomy [LH. x. 
 
 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 extra- 
 ordinary insight, estimated that the density of the earth 
 was between five and six times as great as that of water.* 
 
 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 deflec- 
 tions of the plumb-line, which indicated an attraction by 
 Chimborazo, near which they were working ; but the obser- 
 vations 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 attrac- 
 tions. 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 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 \ater fa/in 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^ times that of water. 
 
 * Principia, Book III., proposition 10. 
 
220, a i] The Density of the Earth 275 
 
 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|, thus verifying in a most striking 
 way both Newton's conjecture and Cavendish's original 
 experiment. 
 
 \Vith this value of the density the mass of the earth is 
 a little more than 13 billion billion pounds, or more 
 pi ' isely 13,136,000,000,000,000,000,000,000 Ibs. 
 
 2^0. While Greenwich was furnishing the astronomical 
 wond 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. Un- 
 fortunately 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, 
 
276 A Short History of Astronomy [CH. x. 
 
 in which the Cassinis had a considerable share, were made 
 during the i8th 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 vin., 
 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 cham- 
 pioned 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 i 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.* 
 
 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 
 
 * It is important for the purposes of this discussion to notice that 
 the vertical io not the linf drawn from the centre of the earth to the 
 place of observation. 
 
221] 
 
 The Shape of the Earth 
 
 277 
 
 latitudes differing by 10 ; so that two consecutive verticals 
 are inclined in every case at an angle of 10. 
 
 If, as in fig. 78, the shape of the earth is drawn in accord- 
 ance with Newton's views, the figure shews at once that 
 the arcs A A,, A, A^ 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 
 
 ; FiG. 78. The varying curvature of the earth. 
 
 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 n., 36) with some made 
 in Europe (chapter vm., 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 in- 
 dicated by a comparison of different portions of an extensive 
 
278 A Short History of Astronomy [CH. x. 
 
 French arc, about 9 in length, extending from Dunkirk 
 to the Pyrenees, which was measured under the super- 
 intendence 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 Conda- 
 mine (1701-1774), and Louis Godin (1704-1760), with 
 whom two Spanish naval officers also co-operated. 
 
 The expedition started in 1735, ar >d> 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 i 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.* 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 p, of fig. 78 
 divided by the greater of them. From comparison of the 
 three arcs just mentioned several very different values of the 
 
 * 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 miK s 
 
222] The Shape of the Earth 279 
 
 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 Jons Svanberg ([771-1851) 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 
 1 8th and igth 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 ^J^, 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 i8th century in the domain of 
 exact observation. France, however, produced during this 
 period one great observing astronomer who actually accom- 
 plished 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 intro- 
 duced to Jacques Cassini, and appointed one of the 
 assistants at the Piris 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 ap- 
 pointed (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 w r hole 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 
 
280 A Short History of Astronomy [Ca. X. 
 
 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* 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 xn., 
 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 ex- 
 tending 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 vm., 161). These observations ( f 
 Mars compared with some made in Europe by Bradley an- 1 
 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 vui., 161), though 
 obtained by more reliable processes. 
 
 A large number of observations of the moon, of which 
 
 * 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. 
 
$$ 22 3 , 224] Lacaille 281 
 
 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, not- 
 withstanding 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 observa- 
 tions 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 i^ 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 
 
282 A Short History of Astronomy [-;;. X. 
 
 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 Gottingen 
 in 1 75 1, 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,* 
 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. t 
 
 He improved Lacaille's tables of the sun, and made a 
 catalogue of 998 zodiacal stars, published posthumously in 
 T 775 ') 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. 
 
 * A mural quadrant. , 
 
 f The ordinary approximate theory of the collimation error, level 
 error, and deviation error of a transit, as given in text-books of 
 spherical and practical astronomy, is substantially his. 
 
FIG. 79. Tobias Mayer's map of the moon. 
 
 [To face p. 282. 
 
* 225, 226] Tobias Mayer 283 
 
 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 i8th century. A funher 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 XL, 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 improve- 
 ment 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, re- 
 ported 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 * 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, 
 
 * The title-page is dated 1767 ; but it is known not to have been 
 actually published till three years later. 
 
284 A Short History of Astronomy [Cu. X. 
 
 which rendered practicable an entirely different method 
 of finding the longitude (chapter vi., 127). 
 
 227. The astronomers of the i8th century had two 
 opportunities of utilising a transit of Venus for the deter- 
 mination 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! and of 12 1 1 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 vin., 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 i8th 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 
 
$227] Transits of Venus 285 
 
 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.* 
 
 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 pre- 
 vented 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 vin., 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"*57i, corresponding to a distance of 95,370,000 miles, 
 
 * For a more detailed discussion of the transit of Venus, see Airy's 
 Popular Astronomy and Newcomb's Potular Astronomy, 
 
286 A Short History of Astrcnomy [Cn. x., $ 2 7 
 
 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 i8th century and 
 the earlier part of the iQth century. As, however, his 
 work was essentially different from that of almost all the 
 astronomers of the i8th century, and gave a powerful 
 impulse to a department of astronomy hitherto almost 
 ignored, it is convenient to postpone to a later chapter (xn.) 
 the discussion of his work 
 
CHAPTER XI. 
 
 GRAVITATIONAL ASTRONOMY IN THE l8TH 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 Mecanique Celeste. 
 
 228. THE solar system, as it was known at the beginning 
 of the 1 8th 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, how- 
 ever, 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 considera- 
 tion, most of the bodies of the system were known from 
 
 287 
 
288 A Short History of Astronomy [Cn. xi. 
 
 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 proba- 
 bility, 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.* 
 
 The problem which presented itself, and which may con- 
 veniently 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 con- 
 venient 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 
 
 * 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. 
 
$ 228] Newton's Problem : the Problem of Three Bodies 289 
 
 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 
 1, initially 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 mathe- 
 matical methods of the early i8th 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 
 
 I 9 
 
290 A Short History of Astronomy [CH. XL 
 
 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. * 
 
 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 v 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 sub- 
 stantially 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 philo- 
 sophy. 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 
 
 * The arithmetical processes of working out, figure by figure, a 
 non-terminating decimal or a square root are simple cases of successive 
 approximation. 
 
$$229,230] Development of Gravitational Astronomy 291 
 
 earth on Newtonian lines, and the appearance six years later 
 of Voltaire's extremely readable Elements de la Philosophic 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 
 1 9th 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 
 1 8th 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. 
 
 If 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 fames Bernouilli (1654-1705), the first of 
 a famous family of mathematicians. Leonhard Euler him- 
 self 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 dis- 
 tinguished mathematicianjike 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 
 
292 A Short History of Astronomy [Cn. XI. 
 
 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 cxne 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 ; * 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 
 
 * "C'est que je viens d'un pays oil, quand on parle, on est pendu." 
 
23i] Euler and Clairaut 293 
 
 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 accelera- 
 tion due to gravity, which are in satisfactory agreement with 
 the results of pendulum experiments. 
 
 Although the subject has since been more elaborately 
 
2 9 4 
 
 A Short History of Astronomy 
 
 [CH. XI. 
 
 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 con- 
 tained 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. 
 
 Halley's comet (chapter x., 200) was "due" about 
 
 \ 
 
 1898 
 
 FIG. 80. The path of Halley's comet. 
 
 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 
 
$232] Hallefs Comet: D'Alembert 295 
 
 perihelion (the point of its orbit nearest the sun, i> in fig. 80) 
 about April i3th 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. Un- 
 fortunately 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.* 
 
 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 Traite de Dynamique (1743), whic'i 
 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, 
 
 * 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. 
 
296 A Short History of Astronomy tea. xi. 
 
 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 Academic Franfaise* 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 mathe- 
 matics, but published a number of books on philosophical, 
 literary, and political subjects ; t as secretary of the 
 Academy he also wrote obituary notices (eloges] 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, use- 
 ful 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 
 
 * 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. 
 
 f t E.g. Melanges de Philosophic, de FHistoire, et de Litte'ra'ure ; 
 Elements de Philosophic ; Sur la Destruction des Jesiiites. 
 
$ 233] D^Alembert : Lunar Theory 297 
 
 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 ex- 
 tremely 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 gravi- 
 tation, 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.* 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 
 
 * I.e. he assumed a law of attraction represented by fj./r* + v/r*. 
 
298 A Short History of Astronomy [CH. XL 
 
 published in 1752, with the title Ti.corie 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 
 Theorie 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 differens 
 points importans du systeme du Monde. In 1756 he pub- 
 lished an improved set cf tables, and a few months afterward 
 a third volume of Recherches with some fresh developments 
 of the theory. The second volume of his Opuscules 
 Mathematiques (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 (Theorla Motuum Lunae) was 
 published in 1753, though it had been sent to the St. 
 Petersburg Academy a year or two earlier. In an appendix * 
 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 
 
 * 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. 
 
$1 234, 235] Lunar Theory 299 
 
 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 i|', 
 and would hence serve to determine the longitude to 
 within about j. 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 observa- 
 tion that his formulae cannot be regarded as true deductions 
 from gravitation in the same sense in which Clairaut's were. 
 Mathematically Euler's second jtheory is the most interest- 
 ing and was of the greatest importance as a basis for later 
 developments. The most modern lunar theory * 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 
 
 * That of the distinguished American astronomer Dr. G. W. Hill 
 (chapter xin., 286). 
 
300 A Short History of Astronomy [CH. xi. 
 
 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 i8th 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 perturba- 
 tions 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 per- 
 turbations 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 / T an( ^ tnat f Venus 
 that of the earth ; the first result was a considerable 
 
23 6] Planetary Theory 301 
 
 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.* 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 Jerome Le Francois Lalande 
 (1732-1807), who is chiefly known as an admirable popu- 
 lariser 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 treat- 
 ment of planetary perturbations in the second volume of 
 bis 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 satis- 
 factorily. 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 per- 
 turbations 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 
 
 * They give about 78 for the mass of Venus compared to that of 
 the earth. 
 
302 A Short History of Astronomy [Ca. XE. 
 
 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. 
 
 Euler extended this idea so as to represent any per- 
 turbation of a planet, whether experienced in the course 
 of one revolution or in a longer time, by means of changes 
 
$ 2s6j Variation of tfie Elements 303 
 
 in an elliptic orbit. For wherever a planet may be and 
 whatever (within certain limits *) 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 
 
 FIG. 81. A varying ellipse. 
 
 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 
 
 * The orbit might be a parabola or hyperbola, though this does 
 not occur in the case of any known planet 
 
304 A Short History of Astronomy [CH. xi. 
 
 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 over- 
 come 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 * 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., 
 '33) 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,! 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. 
 
 * On the Calculus of Variations. 
 
 f The establishment of the gene.-al equations of motion by 
 a combination of virtual velocities and D 1 Alemberf s principle. 
 
LAGRANGE. 
 
 [To face p. 35- 
 
2:?T Life of Lagmnge 305 
 
 Lagrange accepted this magnificently expressed invitation 
 and spent the next 21 years at Berlin. 
 
 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 Mecanique 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 melan- 
 choly 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 Mecanique 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 Com- 
 mission 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 govern- 
 ments which ruled France up to the time of his death ; 
 in 1793 he was specially exempted from a decree of banish- 
 ment directed against all foreigners ; subsequently he was 
 made professor of mathematics, first at the Ecole Normale 
 ( I 795) J and then at the 6cole Polytechnique (1797), the 
 last appointment being retained till his death in 1813. 
 During this period of his life he published, in addition 
 
 20 
 
306 A Short History of Astronomy [CH. xi. 
 
 to a large number of papers on astronomy and mathematics, 
 three important books on pure mathematics,* and at the 
 time of his death had not quite finished a second edition 
 of the Mecanique 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 de- 
 partments of mathematics. 
 
 Laplace's work on astronomy was to a great extent 
 incorporated in his Mecanique Celeste, 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 Systeme 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,t on which nearly all later developments of the 
 subject have been based, and in 1819 a more popular 
 Essai Philosophique on the same subject. 
 
 r * Theorie dcs Fonctions Analytiqucs (1797); Resolution dcs 
 Equations Numeriques ( \ 798) ; Lemons sur le Calcul dcs Fonctions 
 (1805). f Theorie Analytique des Probabi life's. 
 
\Tofacep. 307. 
 
$ 238] Life of Laplace 307 
 
 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 dis- 
 coverers. 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 Ecole 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)*; as a compensation he was made a member of 
 the newly created Senate. The third volume of the 
 Mecanique Celeste, 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 Academic 
 Fran$aise ; this distinction he seems to have owed in great 
 part to the literary excellence of the Systeme du Monde. 
 
 Notwithstanding these distractions he worked steadily 
 at mathematics and astronomy, and even after the com- 
 pletion of the Mecanique Celeste wrote a supplement to it 
 which was published after his death (1827). 
 
 His last words, " Ce que nous connaissons est peu de chose, 
 ie i 2 u: 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 pre- 
 sented during his career. 
 
 * 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. 
 
308 A Short History of Astronomy [CH. xi. 
 
 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 con- 
 tinued 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 com- 
 paratively 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 com- 
 pletely that he was inclined to discredit the early observa- 
 tions on which the existence of the phenomenon was 
 based. Laplace, after trying ordinary methods without 
 success, attempted to explain it by supposing that gravita- 
 tion 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 there- 
 fore 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 ir.oon would, in the course of a century, or in about 
 1,300 complete revolutions, gain about 10" (more exactly 
 io"'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, 
 
$ 239-241] Lun.ir Theory 309 
 
 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 01 
 importance in the solar system between theory and observa- 
 tion 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 xin., 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 Mecanique Celeste con- 
 tains 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 JBurg 
 (1766-1834) made considerable use of his formulae, 
 together with an immense number of Greenwich observa- 
 tions, 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 
 
310 A Short History of Astronomy [CH. XL 
 
 of the Mecanique Celeste. These were generally accepted 
 in lieu of Burg's, which had been in their turn an im- 
 provement on Mason's and Mayer's. 
 
 Later work on lunar theory may conveniently be regarded 
 as belonging to a new period of astronomy (chapter xin., 
 286). 
 
 242. Observation had shewn the existence of inequali- 
 ties 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 treatmer t 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 
 
$ 2 4 a] Periodic and Secular Inequalities 311 
 
 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 continu- 
 ally 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." * 
 
 " 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 
 ] er nanent, 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 
 
 * Outlines of Astronomy, 656. 
 
312 A Short History of Astronomy [CH. xi. 
 
 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." * 
 
 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 
 ( 2 35 2 36, 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 cf 
 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 HalJey 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 revo- 
 lution of two planets are exactly proportional to two whole 
 
 * Laplace, Systeme du Monde, 
 
243,244] Long Inequalities 313 
 
 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 con- 
 siderable effect.* 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, and 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 
 
 * If n, n' are the mean motions of the two planets, the expression 
 for the disturbing force contains terms of the type = CQS (n pn' p') t, 
 
 where/*, />' are integers, and the coefficient is of the order p t^ />' 
 in the eccentricities and inclinations. If now p and />' are such 
 that np r^f n' p' is small, the corresponding inequality has a perio 1 
 2 IT I (n p <^> n' p' '), and though its coefficient is of order/) f^/*', it 
 has the small factor np r*~s n p' (or its square) in the denominator and 
 may therefore be considerable. In the case of Jupiter and Saturn, 
 for example, n = 109257 in seconds of arc per annum, n' = 43,996; 
 5 n' 2n = 1,466; there is therefore an inequality of the third order, 
 
 with a period (in years) = = 900. 
 1,400 
 
314 A Short History of Astronomy [CM. xi. 
 
 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 ; * 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 men- 
 tioned) considered as small quantities ; and thus all t e 
 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 
 
 * 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. 
 
* 245] Stability of the- Solar System 3 1 5 
 
 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 dis- 
 cussion 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 fluctu- 
 ate 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, es- 
 tablished 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. 
 
316 A Short History of Astronomy [Cn. xi. 
 
 The second is precisely similar, save that eccentricity is 
 replaced by inclination.* 
 
 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 cf 
 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 ; conse- 
 quently the whole fund is small, and the share owned at 
 any time by any one planet must be small. t 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. 
 
 * S e*m */a = c, S tan~im \/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. 
 
 f 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. 
 
245] Stability of the Solar System 317 
 
 Stated biiefly, the results established by the two astro- 
 nomers were that the changes in axis, eccentricity, and 
 inclination of any planetary orbit are all permanently re- 
 stricted 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 in- 
 habited 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 mathe- 
 matical 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 re- 
 semblance. Apart from causes at present unforeseen, it is 
 therefore not unreasonable to expect that for a very con- 
 siderable 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 dis- 
 turbing causes, ignored or rejected by Laplace and Lagrange 
 on account of their insignificance, should not sooner or later 
 produce quite appreciable effects (cf. chapter xin., 293). 
 
318 A Short History of Astronomy [CH. xi. 
 
 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 TTy V^- 
 
 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 formulae given by Laplace and collected in the Mecanique 
 Celeste 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.* 
 
 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 
 
 * 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 (i770 f Lindenau for Venus, Mars, and Mercury (181013), 
 and of BoHvard for Jupiter, Saturn, and Uranus (1808 and 1821). 
 
$246-249] Planetary Theory : Minor Problems 319 
 
 a sort of miniature solar system but with several character- 
 istic 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 com- 
 [ letely 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 satis- 
 factorily 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 1 8th century, the results of which were summarised 
 in the Mecanique Celeste, it was shewn to be possible to 
 
320 A Short History of Astronomy [CH. xi. 
 
 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 Mecanique Celeste^ but as the inventor of the Nebular 
 Hypothesis, 
 
 This famous speculation was published (in 1796) in his 
 popular book the Systhne 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,* 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 
 
 * The motion of the satellites of Uranus (chapter xn., 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. 
 
$ 25 o] The Nebular Hypothesis 321 
 
 eccentricities of all the orbits were quite small, so that 
 they were nearly circular. 
 
 Comets, on the other hand, presented none of these pecu- 
 liarities ; 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 con- 
 necting link in these respects between comets and planets 
 or satellites.* 
 
 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 xn., 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, 
 
 * This statement again has to be modified in consequence of the 
 discoveries, beginning on January 1st, 1801, of the minor planets 
 (chapter xni., 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. 
 
 21 
 
322 A Short History of Astronomy iCn. XL, $250 
 
 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 (defiance) which every- 
 thing which is not a result of observation or of calculation 
 ought to inspire." * 
 
 * Systeme du Monde, Book V., chapter vi. 
 
CHAPTER XII. 
 
 HERSCHEL. 
 
 "Coelorum perrupit claustra." 
 
 HERSCHEL'S Epitaph. 
 
 251. Frederick William Herschel was born at Hanover on 
 November i5th, 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 ac- 
 cordingly 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 
 
 323 
 
324 A Short History of Astronomy [CH. XII. 
 
 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 de- 
 scribed ; 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, 
 arid 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 mathe- 
 matics, 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 
 
$$ 252, ass] fferschefs Early Life 325 
 
 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 suc- 
 ceeded 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 neces- 
 sary 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 knowr 
 
 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 i3th, 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 
 H Geminorum I perceived one that appeared visibly larger 
 than the rest ; being struck with its uncommon appearance I 
 compared it to H 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 
 
326 A Short History of Astronomy [CH. xii. 
 
 o dinary cometary orbit would in any way fit its motion, 
 ana wi.hin three or four months of its discovery it was 
 recogn.sed 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 vin., 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 memor- 
 able discovery is curiously illustrated by a discussion in 
 the leading astronomical journal (Bode's Astronomisches 
 JahrbucK] 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 dis- 
 covery 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. 
 
 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 
 
WILLIAM HERSCHEL. 
 
 [To face p. 327. 
 
$$ 254, ass] The Discovery of Uranus: Slough 327 
 
 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 oil il a e"te" fait le plus de decouvertes." 
 
 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 hence- 
 forward 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 3o-foot telescope ; the 
 discovery of Uranus and its consequences prevented the 
 renewal of the attempt for some time, but in 1785 he began 
 a 4o-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 construc- 
 tion, as it is called, was first tried with the 2o-foot, and led 
 to the discovery (January nth, 1787) of two satellites of 
 Uranus, Oberon and Titania ; it was henceforward regularly 
 
328 A Short History of Astronomy [Ca. xn. 
 
 employed. After several mishaps the 4o-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 i yth 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, 
 HerscheFs 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 
 
Life at Slough 
 
 329 
 
 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 
 
 FIG. 82. Herschel's forty-foot telescope. 
 
 welfare on these occasions, and it is perhaps rather a matter 
 of wonder that so few serious accidents occurred. 
 
 In addition to doing his real work Herschel had to 
 
33o A Short History of Astronomy [CH. xii. 
 
 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 Astro- 
 nomical Society. His last observation was taken almost at 
 the same time, and he died rather more than a year after- 
 wards (August 2ist, 1822), when he was nearly 84. 
 
 He left one son, John, who became an astronomer only 
 less distinguished than his father (chapter xin., 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. 
 
25?] fJerschfl's Astronomical Programme 331 
 
 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 lyth or in the i8th century would, 
 if asked, hive 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. More- 
 over 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 con- 
 siderable 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 interes*. 
 
 Herschel therefore struck out an entirely new path when 
 
532 A Short History of Astronomy [CH. XII. 
 
 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 dis- 
 tributed' 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 accumula- 
 tion 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. 
 
 In the absence of any a priori knowledge of the actual 
 
j --581 Star-gauging 333 
 
 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 2o-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* regions, 
 
 FIG. 83. Section of the sidtreal system. From Herschel's paper in 
 the Philosophical Transactions. 
 
 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 
 
 * 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. 
 
J34 A Short History of Astronomy [CH. xii. 
 
 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 conse- 
 quences 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 Her- 
 schel 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 gradu- 
 ally 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 
 
4 258] The Structure of tJie Sidereal System 335 
 
 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 neces- 
 sarily 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 reflect- 
 ing 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 
 
336 A Short History of Astronomy [CH. xn. 
 
 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 2o-foot telescope which he commonly 
 used, and the 4o-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 bright- 
 ness, 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 con- 
 clusions 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 maybe 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 instru- 
 mental appliances were on a far larger scale. In 1786 
 he presented to the Royal Society a catalogue of 1,000 
 
$ 259, 26 ] Nebulae and Star Clusters 337 
 
 new nebulae and clusters, three years later a second cata- 
 logue 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 extra- 
 ordinary 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 
 6 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 classifica- 
 tion 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 
 i.iany objects which to Messier were purely nebulous 
 appeared in his own great telescopes to be undoubted 
 clusters, though others still remained nebulous. Thus in 
 'lis 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," 
 
 22 
 
338 A Short History of Astronomy [Cn. xn 
 
 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 accord- 
 ingly 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 observa- 
 tion 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 some- 
 thing 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 
 
5 26i] Nebulae and Star Clusters 339 
 
 explanations Herschel chose the latter, considering the 
 nebulosity to be " a shining fluid, of a nature totally un- 
 known 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 distri- 
 bution 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 ii, capable 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 develop- 
 ment, some " clustering power " being at work converting 
 a diffused nebula into a brighter and more condensed 
 
34 d Short History of Astronomy [CH. xil. 
 
 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 luxu- 
 riant garden, which contains tbe 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 con- 
 dense 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 W T ay, 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 con- 
 vinced that this was not the case, but that there remained 
 cloudy portions which whether on account of their remote- 
 
; 262, 26j Condensation of Nzbulae : Double Stars 341 
 
 ness 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 
 rst. 
 
 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 
 d : scovery of double stars. He proposed to employ Galilei's 
 differential or double-star method (chapter vi., 129), i-i 
 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 Astro- 
 nomical 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 w^re 
 
342 A Short History of Astronomy [CH. xn. 
 
 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 argu- 
 ment 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, a and /?, rather less than 5" apart. Altogether there 
 are about 50 stars of the same order of brightness as a, and 
 400 like /?. 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 anothi r 
 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 i oo 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 opink n 
 that the odds in favour of a physical relation between il.c 
 
$ 264] Doubh Stars 343 
 
 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 Philo- 
 sophical Transactions in 1803 and 1804, which were fortu- 
 nately 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,* 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 xin., 
 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 
 
 * 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. 
 
344 -A Short History of Astronomy [CH. xii. 
 
 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 inade- 
 quate 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 
 
265] Motion of the Sun in Space 345 
 
 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. 
 
 L 
 
 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 X in the constella- 
 
346 A Short History of Astronomy [Cn. xn 
 
 tion 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.* This result was published in 1783, 
 and a few months later Pierre Prevost (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 con- 
 temporary astronomers with a large measure of distrust ; 
 jut 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 
 .ttempts 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 vari- 
 ability of the light of stars. Two remarkable cases of 
 variability had been for some time known. A star in the 
 Whale (o Ceti or Mir a) had been found to be at times 
 
 * 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) ; 2^ agreed in sign with 
 those which would result from the assumed motion of the sun. 
 
z66j I'ariabU Stars 347 
 
 invisible to the naked eye and at other times to be con- 
 spicuous ; a Dutch astronomer, Phocy tides Holwarda ( 1 6 1 8- 
 1651), first clearly recognised its variable character (1639), 
 and Ismail 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 paptr published 
 by Herschel in the Philosophical Transactions (1780). An 
 equally remarkable variable star is that known as Algol 
 (or ft 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 n., 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 i8th 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, 
 
34 8 -A Short History of Astronomy [CH. xii 
 
 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 Revised 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 pre- 
 pared 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 a 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 i8th century whether they were 
 observers or mathematicians he was a careful and success- 
 ful 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 
 
* 267 j Brightness of Stars : Planetary Observations 349 
 
 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 (chap- 
 ter 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 XIIL, 295), though he aid 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 loh. i6m. (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 io| 
 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 vin., 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 't 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 
 
350 A Short History of Astronomy CH. xn. 
 
 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 observa- 
 tions 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 nptch 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 
 
$ 263, z6 9 j Herschel' s Theory of the Sun 351 
 
 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 
 < r, in strictness of speaking, the only primary one of our 
 ^stem ; ... it is most probably also inhabited, like the rest 
 t 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 
 (ilasgow (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 observa- 
 tions 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 ex- 
 plained 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 of 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 xin., 298, 303). 
 
 269. Another suggestive idea of HerscheFs was the 
 analogy between the sun and a variable star, the known 
 
352 A Short History of Astronomy [CH. xn. 
 
 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 meteoro- 
 logical 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 con- 
 clusions. 
 
 270. In addition to carrying out the astronomical re- 
 searches already sketched, and a few others of less import- 
 ance, 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 im- 
 portant other considerations were, that though a great 
 number of his telescopes were supplied to other astro- 
 nomers, 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, wzsjohann Hieronymus 
 Schroetcr (1745-^16). 
 
 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 
 
$ 2 7 o, 271] Herschel and Schroeter 9 353 
 
 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 ot 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 mathe- 
 matics 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 fqr only one part of the astrono- 
 
 354 
 
CH. xiil., * 272, 273] Descriptive Astronomy 355 
 
 mical 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 re- 
 searches, 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 expert-. 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 
 
356 A Short History of Astronomy [CH. xm. 
 
 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 ac- 
 cordingly 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 
 
$$274,275] Descriptive Astronomy: Theory of Errors 357 
 
 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., 
 no), 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 Gottingen,* 
 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 condi- 
 tions, 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 
 
 * 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. 
 
358 A Short History of Astronomy [Cn. xin. 
 
 to deal with more complicated cases in which two or more 
 unknown quantities have to be determined from observa- 
 tions of different quantities, as, for example, when the 
 elements of the orbit of a planet (chapter XL, 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 favour- 
 able 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 prior i\ 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 affect- 
 ing astronomy. Gauss, on the other hand, was for nearly 
 half a century head of the observatory of Gottingen, 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 astrono- 
 mical observations, the best known being a method of 
 calculating the orbit of a planet from three complete 
 
2 7 6, 277 Legendre and Gauss 359 
 
 observations of its position, which was published in his 
 Thtorin Motus (1809). As we have seen (chapter XL, 
 236), the complete determination of a planet's orbit 
 depends on six independent elements : any complete ob- 
 servation of the planet's position in the sky, at any time, 
 gives two quantities, e.g. the right ascension and declination 
 (chapter n., 33) ; hence three complete observations 
 give six equations and are theoretically adequate to de- 
 termine 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 ob- 
 servations taken at particular times so chosen that certain 
 elements could be determined without any accurate know- 
 ledge of the others ; even Uranus had been under observa- 
 tion 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 ist, 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 Konigsberg. 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, 
 
360 A Short History of Astronomy [CH. xiii. 
 
 under die 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 con- 
 structing 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. 
 
 278. The most memorable of Bessel's special pieces of 
 
 a: 
 FIG. 85. 6 1 Cygniand the two neighbouring stars used by Bessel. 
 
 work was the first definite detection of the parallax of a 
 fixed star. He abandoned the test of brightness as an 
 
2 7 8] 
 
 The Parallax of b\ Cygni 
 
 indication of nearness, and selected a star (61 Cygni) 
 which was barely visible to the naked eye but was re- 
 markable 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 (chap- 
 ter vi., 129, and chapter XIL, 263) 
 being adopted, the angular distance 
 of 6 1 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 <r a, a- b 
 (in fig. 85), the size of the small ellipse 
 described by o- could be calculated. 
 The result, announced at the end of 
 1838, was that the star had an annual 
 parallax of about " (chapter VIIL, 
 161), i.e. that the star was at such 
 distance that the greatest angular dis- 
 tance of the earth from the sun viewed 
 from the star (the angle s a- E in fig. 86, 
 where s is the sun and E the earth) 
 was this insignificant angle.* The 
 result was confirmed, with slight altera- 
 tions, by a fresh investigation of 
 Bessel's in 1839-40, but later work FIG. 86. The parallax 
 seems to shew that the parallax is a of61 c >'S ni - 
 little less than ".t 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 
 
 * The figure has to be enormously exaggerated, the angle s <r E as 
 shewn there being about 10, and therefore about 100,000 times too 
 great. 
 
 f 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. 
 
362 A Short History of Astronomy [CH. xil 
 
 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 6 1 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 i" 
 for the bright star a 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 j* for Vega ; later work has reduced 
 these numbers to f " and T V respectively. 
 
 A number of other parallax determinations have subse- 
 quently 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 measur- 
 able, if rather uncertain, parallaxes ; a 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 know- 
 ledge 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 
 
* 279281] Parallax : Star Catalogues 363 
 
 the most important is the great catalogue of 324,198 stars 
 in the northern hemisphere known as the Bonn Durch 
 musterung, published in 1859-62 by Bessel's pupil Friedrich 
 M'ilhelm August Arge lander (1799-1875); this was extended 
 (1875-85) so as to include 133,659 stars in a portion of 
 the southern hemisphere by Eduard Schonfeld (\$>2$>-\'&y\) ; 
 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 Schonfeld ; 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,* 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, 
 
 * An average star of the I4th magnitude is 10,000 times fainter 
 than one of the 4th magnitude, \\hich again is about 150 times less 
 bright than Sirius. See 316. 
 
364 A Short History of Astronomy [CH. xm. 
 
 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 vni., 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, m ade 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 i8th century that the distance of the sun could be 
 obtained from a knowledge of various perturbations of 
 
282, a8 3 ] The Distance of the Sun 365 
 
 members of the solar system; and Laplace had deduced 
 a value of the solar parallax from lunar theory. Improve- 
 ments 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 paraUactic 
 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 neigh- 
 bours, 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 incon- 
 sistent 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 displace- 
 ments to be observed are very minute, and their accurate 
 determination is by no means easy, but they are both 
 secular (chapter XL, 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 displace- 
 ment of a star by aberration (chapter x., 210) depends 
 
366 A Short History of Astronomy [CH. xin. 
 
 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 vin., 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 com- 
 paratively 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.* 
 
 284. Encke's value of the sun's parallax, 8"'57i, 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 i8th 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, correspond- 
 ing 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 
 
 * Newcomb's velocity of light and Nyren's constant of aberration 
 (2o"'492i) give 8"794; Struve's constant of aberration (2O"'445), 
 Loewy's (2o"'447), and Hall's (2o"'454) each give 8"'8i, 
 
$$284286] Solar Parallax : Variation of Latitude 367 
 
 aberration, unite in giving values not differing from 8"'8o 
 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 |" 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 
 
368 A Short History of Astronomy *CH. xin. 
 
 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 Simeon Dents 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 unac- 
 companied 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 Pontecoulant 
 (1795-1874), appeared in 1846, with the title Theorie 
 Analytique du Systeme du Monde ; and an incomplete 
 lunar theory similar to his was published ^ 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 * and the Darlegung^C 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 i" 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 calcula- 
 tions, the best known being his comparison of Hansen's tables 
 with observation and consequent corrections of the tables. 
 
 * Fundamenta Nova Investigationis Orbitae Verae quant Luna 
 perlustrat. 
 
 f Darlegung der theoretischen Berechnung der in den Mondtafeln 
 angewandten Storungen. 
 
$ 287] Lunar Theory 369 
 
 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. \V. 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 XL, 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 
 ^he 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 
 IVilliam Ferret, 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 pro- 
 duces an apparent increase of speed in all other motions 
 
 24. 
 
370 A Short History of Astronomy [CH. xin. 
 
 measured in terms of this unit. Delaunay, working inde- 
 pendently, 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 T ^ second in 10,000 years 
 is sufficient to explain the acceleration in question. More- 
 over 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 men- 
 tioned 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 dis- 
 cussion of the stability (chapter XL, 245) of the system 
 formed by the sun and the three largest and most distant 
 planets then known, Jupiter, Saturn, and Uranus. Subse- 
 quently 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 agree- 
 ment 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 
 
* ass, 289] Planetary Theory 371 
 
 in pure mathematics, have been carried out by Hugo Gylden 
 (1841-1896), while one of the most eminent pure mathe- 
 maticians of the day, M. Henri Poincare 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 * that a very brief account will suffice 
 here. Soon after the discovery of Uranus (chapter xn., 
 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 XL, 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, how- 
 ever, undertaken. Meanwhile, Leverrier had independently 
 taken up the inquiry, and by August 3ist, 1846, he, like 
 Adams, had succeeded in determining the orbit and the 
 position of the disturbing body. On the 23rd of the follow- 
 
 * E.g. in Grant's History of Physical Astronomy, Herschel's Out- 
 lines 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. 
 
372 A Short History of Astronomy [CH. xm. 
 
 ing 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, Pro- 
 fessor Newcomb, and Dr. Hill, represent the motions of 
 the bodies dealt with much more accurately than the corre- 
 sponding 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 obser- 
 vation 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 Pfofessor Newcomb from 
 observation and have no justification in the theory of 
 gravitation. 
 
 291. The calculation of the paths of comets has be- 
 come 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 XL, 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 ; 
 
290292] 
 
 Orbits of Comets 
 
 373 
 
 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 1 8 1 2 and 1 884. Fourteen other comets with periods 
 varying between 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 
 
 1898 
 
 FIG. 87. The path of Halley's comet. 
 
 with much certainty. Altogether the paths of about 230 
 or 240 comets have been computed, though many are 
 highly uncertain. 
 
 292. In the theory of the tides the first important advance 
 made after the publication of the Mecanique Celeste 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 
 
374 ^ Short History of Astronomy [CH. xin. 
 
 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 fro-n 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 XL, 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 aoproximate 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. 
 
293] Tides: the Stability ^f the Solar System 375 
 
 Propositions which profess to be or are commonly inter- 
 preted 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 x 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 ?to 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 
 
376 A Short History of Astronomy [CH. xm 
 
 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 i9th 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 con- 
 jecture 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. 
 
 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 
 
.FiG. 88. Photographic trail of a minor planet. 
 
 [To face p. 377. 
 
$ 294] The Minor Planets 377 
 
 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 |, so that its least dis- 
 tance 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 Ludivig 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 
 2ist, 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. Chariots 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 . 
 
 Fig. 89 shews the orbits of the first two minor planets 
 discovered, as well as of No. 323 (Brucid), which comes 
 nearest to the sun, and of No. 361 (not yet named), 
 
373 
 
 A Short History of Astronomy 
 
 [CH. XIII. 
 
 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 
 
 FIG. 89. Paths of minor planets. 
 
 principle (chapter XL, 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. 
 
 Estimates of the sizes and masses of the minor planets 
 are still very uncertain. The first direct measurements 
 
=94] 
 
 The Minor Planets 
 
 379 
 
 of any of the discs which stem reliable are those of 
 Professor 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. Thd r 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 
 
 FIG. 90. Comparative sizes of three minor planets and the moon. 
 
 of the minor planets is to measure the amount of light re- 
 ceived, 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. 
 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 
 
380 A Short History of Astronomy [CH. xm. 
 
 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. 
 
 Barely a fortnight after the discovery of Neptune (1846) 
 
 \ 
 \ 
 
 JAPETUS 
 / HYPERION 
 
 FIG. 91. Saturn and its system. 
 
 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). 
 
 Two years later (September i6th, 1848) William Cranch 
 Bond (1789-1859) discovered, at the Harvard College 
 
$ 2 9 5] New Satellites 381 
 
 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 imme- 
 diately inside -the bright rings (see fig. 95); and the 
 discovery was made independently a fortnight later by 
 
 \ 
 
 \ 
 
 DE1MOS 
 
 
 
 FIG. 92. Mars and its satellites. 
 
 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 dis- 
 coveries were those of two satellites of Mars, known as 
 Deimos and Phobos, by Professor Asaph Hall of Washington 
 on August nth and lyth, 1877. These are remarkable 
 chiefly for their close proximity to Mars and their extremely 
 rapid motion, the nearer one revolving more rapidly than 
 
382 A Short History of Astronomy [CH. xin. 
 
 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 gth, 1892) of 
 
 
 
 \ 
 
 \ \ "-/ ')' /" ?'" i 
 
 . ' 
 
 \ \ v -- -' / / 
 
 \ '*----' / / ; 
 
 N s ' 
 
 ^*. .**' 
 
 ' / 
 
 \ 
 
 \ 
 
 \ / 
 
 X \ +*''' 
 
 * -** 
 
 **-... .** 
 
 FIG. 93. Jupiter and its satellites. 
 
 an extremely faint fifth satellite, a good deal nearer to Jupiter 
 than the nearest of Galilei's satellites (chapter vi., 121). 
 
 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. 
 
 Astronomers are naturally most familiar with the surface 
 
FIG. 94, The Apennines and adjoining regions ot the moon, trom a 
 
 photograph taken at the Paris Observatory. [To face p. 383. 
 
* 296, 2 9 7] Satellites of Jupiter : the Moon: Mars 383 
 
 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 
 nre evidently covered with dry rock ; and the craters with 
 which the moon is covered are all with one or two doubt- 
 ful 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 atmo- 
 sphere. 
 
 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. Signer 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 i 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 
 
384 A Short History of Astronomy [CH. XHI. 
 
 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.* The 
 theory thus suggested on mathematical grounds has re- 
 ceived 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. 
 
 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 astrono- 
 mers 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 
 * This had been suggested as a possibility by several earlier writers 
 
hfl 
 
 s 
 
 a 
 
 
 
 OJ 
 
 I 
 
o 
 a, 
 
 rt 
 
 I 
 
$ 29 3 1 Planetary Observations : Sun-spots 385 
 
 moon round the earth and of Japetus round Saturn 
 (chapter xn., 267) could be easily explained as the 
 result of tidal action at some past time when the planets 
 were to a great extent fluid- 
 
 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 Hein- 
 rich Schwabe of Dessau (1789-1 8 75) 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 n years on the average, though subject to 
 considerable fluctuations. A year later (1852) three inde- 
 pendent investigators, Sir Edward Sabine (1788-1883) in 
 England, Rudolf Wolf (1816-1893) and Alfred Gautier 
 (1793-1881) in Switzerland, called attention to the remark- 
 able 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 un- 
 explained 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 differ- 
 ence 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 
 
 25 
 
386 A Short History of Astronomy [Cn. xin. 
 
 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 i i-year cycle. 
 
 Wilson's theory (chapter xii., 268) that spots are de- 
 pressions 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 im- 
 portant 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 sun- 
 light, 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. 
 
 Now according to modern theories light consists essen- 
 tially 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 disturb- 
 ances 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 corre- 
 sponding point of the next. Differences in wave-length 
 shew themselves most readily as differences of colour ; so 
 
It 
 
 
 
$ 399] Spectrum Analysis 387 
 
 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 super 
 seding 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 spectro- 
 scope. 
 
 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 
 
388 A Short History of Astroncmy [CH. xin. 
 
 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 con- 
 tinuous 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 charac- 
 teristic 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 
 KirchhofFs 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 corre- 
 sponding 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 
 
300, 3 oi] Spectrum Analysis 38 
 
 atmosphere, the majority cannot be, and are therefor- 
 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 there- 
 fore 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 sub- 
 stances, their presence or absence in the solar atmosphere 
 can accordingly be ascertained. In the case of iron which 
 has an extremely complicated spectrum Kirchhoff suc- 
 ceeded 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 Kirchhoffs 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 
 
39^> A Short History of Astronomy [CH. Xlll. 
 
 Plutarch, and discussed by Kepler (chapter vn., 145) 
 Several i8th century astronomers noticed a red streak along 
 some portion of the common edge of the s-un 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 repre- 
 sentation of the whole of the chromosphere and prominences, 
 while the same method gives also photographs of faculae 
 (chapter VIIL, 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. 
 
 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 
 
FIG. 98. The total solar eclipse of August 29th, 1886. From a drawing 
 based on photographs by Dr. Schuster and Mr. Maunder. 
 
 \Toface p. 390. 
 
$302] Solar Spectroscopy : Dpppler>s Principle 391 
 
 Professor Harkness and Professor C. A. Young, who dis- 
 covered a bright line of unknown origin * 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, sur- 
 rounding 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 tha?:i 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 y^oo^ that f n g nt 5 then 1001 waves enter the eye or the 
 spectroscope in the same time in which there would other- 
 wise 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 
 
 * The discovery of a terrestrial substance with this line in its 
 spectrum has been announced while this book has been passing 
 through the press. 
 
392 A Short History of Astronomy [Cn. xin. 
 
 a way that its wave-length is apparently decreased by 
 J^-Q part, it may be inferred that the body is approach- 
 ing 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 dis- 
 placements 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 dis- 
 turbances 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 obser- 
 vations 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 Duner 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 com- 
 municated 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, how- 
 ever, 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 
 
tt 303, 304] Structure of the Sun : Comets 393 
 
 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. 
 
 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 un- 
 substantial 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 XL, 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 ob- 
 served 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 HI., 69) that a comet's tail points 
 away from the sun has been abundantly verified ; and from 
 this it follows that very rapid changes irr-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 
 
394 A Short History of Astronomy [CH. xiil 
 
 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 con- 
 
? 305] Comets and Meteors 395 
 
 tinuous 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 XL, 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 
 
396 A Short History of Astronomy [Cn. xiu. 
 
 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 
 1 88 1 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 i9th 
 century. The multiplication of observatories, improve- 
 ments 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 photo- 
 graphy, and to the philosophy of scientific discovery. He 
 began his astronomical career about 1816 by re-measuring, 
 first alone, then in conjunction with fames South (1785- 
 1867), a number of his father's double stars. The first 
 result of this work was a catalogue, with detailed measure- 
 ments, 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 
 
307 
 
 Tohn Herschel 397 
 
 (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. 
 
 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 v\ 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 xn., 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 allow- 
 ance 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 1 880-81, 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 
 
398 A Short History of Astronomy [CH. xili. 
 
 atmosphere would be sufficient to melt a shell of ice 
 1 20 feet thick covering the whole earth; according to 
 Dr. Langley, the thickness would be about 160 feet.* 
 
 308. With his return to England in 1838 HerschePs 
 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, amount- 
 ing 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. Jhn 
 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 io,oco, 
 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 
 
 * Observations made on Mont Blanc under the direction of 
 M. Janssen in 1897 indicate a slightly larger number than Dr. 
 Langley 's. 
 
308, 
 
 Double Stars 
 
 399 
 
 probability turn out to be genuine binaries (chapter xn., 
 264). 
 
 In addition to a number of minor papers Struve published 
 three separate books on the subject in 1827, 1837, and 1852.* 
 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 
 
 Scal 
 
 1876 
 
 FIG. 101. The orbit of 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.) 
 
 from a comparison of Struve's observations with those of 
 later observers. 
 
 William Herschel's observations of binary systems 
 (chapter XIL, 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 
 
 * Catalogus novus stellarum duplicium^ Stellarum duplicium et 
 multiplicium mensurac micrometricae, and Stellarum fixarum imprimis 
 duplicium et multiplicium positiones mediae pro epocha 1830. 
 
400 A Short History of Astronomy [Cn. xiil. 
 
 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 Felix 
 Savary (1797-1841) in 1827, the star being $ 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 Pro- 
 fessor 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 Ursae, which is shewn 
 in fig. 1 01. 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 ex- 
 periments 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 
 
FK.. 102. Spiral nebulae. From drawings by Lord Rosse. 
 
 [To face p 400. 
 
M 3IO 312] Double Stars and Nebulae 401 
 
 and had accordingly put into " the shining fluid " class 
 (chapter XIL, 260). This last discovery, being exactly 
 analogous to Herschel's experience when he first began to 
 examine nebulae hitherto only observed with inferior tele- 
 scopes, 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. 
 
 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 KirchhofFs discoveries by 
 several astronomers, in particular by Sir William Huggins 
 and by the eminent Jesuit astronomer Angela 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 
 
 26 
 
402 A Short History of Astronomy [Cn. xin. 
 
 differences inferences as to the relative " ages," or at any 
 rate the stages of developn. t, 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 in- 
 teresting if rather perplexing class of stars, falling into 
 several subdivisions, which seem to form a connecting 
 link between ordinary stars and nebulae, for, though in- 
 distinguishable 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 spectro- 
 scopic 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. 
 
 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 differ- 
 ent ; 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 
 
fi I- 
 
i 
 
W 313-316] Stellar Spectroscopy 403 
 
 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 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 Pro- 
 fessor Vogel's detection of a periodical shift in the position 
 of the dark lines in the spectrum of the variable star Algol 
 (chapter xn., 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 /? 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, how- 
 ever,, 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 photo- 
 metry 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 com- 
 parisons 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 
 
404 A Short History of Astronomy [CH. xm. 
 
 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 gener- 
 ally adopted, first suggested in 1856 by Norman Robert 
 Pogson (1829-1891), the ligit of a star of any magnitude 
 bears a fixed ratio (which is taken to be 2 '5 12...) 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.* 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. t 
 
 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 Urano- 
 metria Nova Oxoniensis (1885), as it is called, only goes 
 10 south of the equatorand includes 2,784 stars. Portions 
 of more extensive catalogues dealing with fainter stars, in 
 progress at Harvard and at Potsdam, have also been 
 published. 
 
 * I.e. 2-512... is chosen as being the number the logarithm of which 
 is -4, so that (2-512. ..) 5/2 = 10. 
 
 f 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 
 
 m- 1 
 \ b , whencem-i = - $ log L. 
 
 , 
 
 -5I2/ \ioo 2 
 
 A star brighter than Aldebaran has a magnitude less than I, while 
 the magnitude of Sinus, which is about nine times as bright as 
 Aldebaran, is a negative quantity, 1-4, according to the Harvard 
 photometry. 
 
FIG. 104. The Milky Way near the cluster in Perseus, 
 by Professor Barnard. 
 
 From a photograph 
 [To face P- 405- 
 
$ 3i?] Photometry : the Sidereal System 405 
 
 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 
 cm 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 
 tor 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 xn., 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, ekher by their arrangement 
 or in some cases by peculiarities of their spectra, to be really 
 connected with the nebula, and not merely to be accident- 
 ally in the same direction. Stars which have bright lines 
 
406 A Short History of Astronomy [CH. xm. 
 
 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 XL, 250) 
 was published a little more than a century ago (1796), and 
 has been greatly affected by progress in various depart- 
 ments 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 per- 
 haps in this way gained as much as it has lost. Again, the 
 telescopic evidence which Herschel adduced (chapter xn., 
 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 explana- 
 tion 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 
 
* 3 1 8, 319] The Evolution of the Solar System 407 
 
 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 
 l8 3 6 ( 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 gravita- 
 tional 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 
 
408 A Short History of Astronojny [CH. xm. 
 
 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 suffi- 
 ciently 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 
 
$ 320] The Evolution of the Solar System 409 
 
 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 
 1 9th 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 cf 
 other cases. 
 
AUTHORITIES AND BOOKS FOR STUDENTS. 
 
 I. GENERAL. 
 
 I HAVE made great use throughout of R. Wolfs Geschichte der 
 Astronomic, and of the six volumes of Delambre's Histoire 
 de V Astronomic (Ancienne, 2 vols. ; du Moyen Age, I vol. ; 
 Moderne, 2 vols. ; du Dixhuitieme Siecle, I vol.). I shall subse- 
 quently 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 Poggendorffs Handworterbuch 
 der Exacten Wissenschaften, and to articles in various bio- 
 graphical dictionaries, encyclopaedias, and scientific journals. 
 Of general treatises on astronomy Newcomb's Popular Astro- 
 nomy, 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 recom- 
 mended ; Young's Elements of Astronomyis 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. 
 
 411 
 
412 Authorities and Books for Students 
 
 II. SPECIAL PERIODS. 
 
 Chapters 1. and II. In addition to the general histories quoted 
 above especially Wolf- I have made most use of Tannery's 
 Recherches sur t'Histoire de I Astronomic Ancienne and of several 
 biographical articles (chiefly by De Morgan) in Smith's Dictionary 
 of Classical Biography and Mythology. Ideler's Chronologische 
 L'ntersuchungen, Hankel's Geschichte der Mathematik im Alter- 
 thum ^ind 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 Altcrthum und Mittclalter; 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 Rational 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 I7th 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 I 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 Brake, 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 
 
Authorities and Books for Students 413 
 
 Tychonis Brake. The portrait is a reproduction of a picture in 
 tne 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 
 t sed 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 Romische 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 
 j) reserved in the Vatican, which have been published by von 
 Gebler and independently by L'pinois in Les Pieces du Proces 
 de Galilee : in the latter book some of the most important docu- 
 ments 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 
 Celeste. I have also read with great interest the estimate of 
 Galilei's work contained in H. Martin's Galilee, 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 
 u-ed 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 
 c ontroversial 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 
 f >r 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 
 
414 Authorities 'and Books for Students 
 
 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 Wissen- 
 schaften. 
 
 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. 
 
 TheZ/z>,y 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 Diet. Nat. Biog. What I have 
 said abjout 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*- 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 
 text-books 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*- John Flamsteed; for Bradley little but the 
 Miscellaneous Works and Correspondence of the Rev. James 
 
Authorities and Books for Students 415 
 
 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 if 
 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 XL Most of the biographical material has been 
 taken from Wolf, from articles in various encyclopaedias and 
 biographical dictionaries, chiefly French, and from Delambre's 
 hloge 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 Completes de Laplace. 
 Gautier's Essai Historique sur le Probleme des Trots 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 
 Mecanique Celeste, 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 Mecanique Celeste as well as of his Systeme 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 everything that I 
 have said about his chief pieces of work is based on his own 
 
4i 6 Authorities and Books for Students 
 
 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 
 HerscheFs 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 gravita- 
 tional astronomy I have constantly used Miss Clerke's History 
 (already quoted), a book which students are strongly recom- 
 mended 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 Suns Place in Nature, J. N. 
 Lockyer. 
 
 The Moon, E. Neisori ; 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 in- 
 formation) ; 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 author s 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), xm. 286, 
 
 287, 289 
 
 Adelard. See Athelard 
 Airy (1801-1892), x. 227 ; 
 
 xm. 281, 292 
 Albategnius (7-929), n. 53 ; in. 
 
 59, 66, 68 n ; iv. 84, 85 
 Albert (of Prussia), v. 94 
 Albertus Magnus (i3th cent), 
 
 in. 67 
 
 Alcuin (735-804), in. 65 
 Alembert, d'. See D'Alembert 
 Alexander, n. 31 
 Alfonso X. (1223-1284), in. 66, 
 
 68 ; v. 94 
 
 Al Mamun, in. 57, 69 
 Al Mansur, in. 56 
 Al Rasid, in. 56 
 Alva, vn. 135 
 Anaxagoras (499 B.C. ?-427 
 
 B.C ?), i. 17 
 Anaximander (610 B.C.-546 
 
 B.C. ?), i. ii 
 Apian (1495-1552), in. 69; v. 
 
 97 ; vn. 146 
 
 Apollonius (latter half of 3rd 
 
 cent. B.C.), ii. 38, 39, 45 51. 
 
 52 n ; x. 205 
 Arago, xii. 254 
 Archimedes, n. 52 n ; in. 62 
 Argelander (1799-1875), xm. 
 
 280 
 Aqstarchus (earlier part of 3rd 
 
 cent. B.C.), n. 24, 32, 41, 42, 
 
 54; iv. 75 
 Aristophanes, ii. 19 
 Aristotle (384 B.C.-322 B.C.), 
 
 n. 24, 27-30. 31, 47, 5 1 . S 2 ; 
 
 in. 56, 66, 67, 68 ; iv. 70, 77 ; 
 
 v. 100 ; vi. 116, 121, 125, 
 
 134; vin. 163 
 Aristyllus (earlier part of 3rd 
 
 cent. B.C.), n. 32, 42 
 Arzachel (fl. *io8o), in. 61, 
 
 66 
 d'Ascoli, Cecco (i3th cent.), 
 
 in. 67 
 Athelard (beginning of I2th 
 
 cent.), in. 66 
 Auzout (7-1691), vin. 155, 
 
 160; x. 198 
 
 27 
 
4i8 
 
 Index of Names 
 
 [Roman figures refer to the chapters, Arabic to the articles.'] 
 
 Bacon, Francis (1561-1627), 
 
 vi. 134; vui. 163 
 Bacon, Roger (12147-1294), 
 
 in. 67 ; vi. 118 
 Bailly, xi. 237 
 Ball, xni. 278 n 
 Bar, Reymers (Ursus) (?-i6oo), 
 
 v. 105 
 Barberini (Urban VIII.), vi. 
 
 125, 127, 131, 132 
 Barnard, xm. 294, 295 
 Baronius, vi. 125 
 Barrow, Isaac, ix. 166 
 Bayer, xn. 266 
 Bede, in. 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 , 
 
 218; xm. 272, 277-278, 279, 
 
 280 
 
 Bille, v. 99 
 
 Bliss (1700-1764), x. 219 
 Bond, William Cranch (1789- 
 
 1859), xm. 295 
 Borelli (1608-1679), ix. 170 
 Bouguer (1698-1758), x. 219, 
 
 221 
 
 Boulliau. See Bullialdus 
 Bouvard, xi. 247 n ; xm. 289 
 Bradley (1693-1762), x. 198, 
 
 206-218, 219, 222-226; xi. 
 
 233; xn. 257,258, 263,264; 
 
 xm. 272, 273, 275, 277 
 Brahe, Tycho (1546-1601), in. 
 
 60, 62 ; v. 97, 98 , 99-112 ; 
 
 vi. 113, 117, 127 ; vii. 136- 
 
 139, 141 , 142, 145, 146, 148 ; 
 
 vui. 152, 153, 162; ix. 190; 
 
 x. 198, 203, 225; xn. 257; 
 
 xm. 275 
 
 Brudzewski, iv. 71 
 Bruno, vi. 132 
 
 Bullialdus (1605- 1 694), xn. 266 
 Bunsen, xm. 299 
 Burckhardt (1773-1825), xi. 
 
 241 
 
 Burg (1766-1834), xi. 241 
 Biirgi (1552-1632), v. 97, 98; 
 
 vui. 157 
 
 Burney, Miss, xn. 260 
 Burnham, xm. 309 
 
 Caccini, vi. 125 
 Caesar, n. 21 ; in. 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), xm. 
 
 298, 302 
 
 Cartesius. See Descartes 
 Cassini, Count (1748-1845), x. 
 
 220 
 Cassini, Giovanni Domenico 
 
 (1625-1712), vui. 160, 161; 
 
 ix. 187; x. 216, 220, 221, 
 
 223 ; xn. 253, 267 ; xm. 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 (12577-1327), 
 
 in. 67 
 
 Chandler, xm. 285 
 Charlemagne, in. 65 
 Charles II., x. 197, 223 
 Chariots, xm. 294 
 
Index of Names 
 
 419 
 
 [Kotnan figures refer to the chapters, Arabic to the articles^ 
 
 Christian (of Denmark), v. 106 
 Christine (the Grand Duchess), 
 
 vi. 125 
 Clairaut (1713-1765), XL 229, 
 
 230, 231, 232, 233-235, 237, 
 
 239, 248 ; XIIL 290 
 Clement VII., iv. 73 
 Clerke, xm. 289 n 
 Clifford, ix. 173 
 Colombe, Ludovico delle, vi. 
 
 119 n 
 
 Columbus, HI. 68 
 La Condamine (1701-1774), x. 
 
 219, 221 
 Conti, vi. 125 
 Cook, x. 227 
 Coppernicus (1473-1543), n. 
 
 24, 41 , 54; in. 55, 62, 69; 
 
 IV. passim ; v. 93-97, 100, 
 
 105, no, in; vi. 117, 126, 
 
 127, 129; vn. 139, 150; ix. 
 
 186, 194; xii. 257; xm. 279 
 Cornn, xm. 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; 
 
 XL 229, 230, 231 , 232-235, 
 
 237-239, 248 
 Damoiseau (1768-1846), xm. 
 
 286 
 
 Dante, in. 67 ; vi. 119 n 
 Darwin, XIIL 292, 320 
 Da Vinci. See Vinci 
 Dawes (1799-1868), xm. 295 
 Delambre, n. 44; x. 218; xi. 
 
 247 n ; xm. 272 
 Delaunay (1816-1872), xm. 
 
 286, 287 
 
 De Morgan, n. 52 n 
 Descartes (1596-1650), vin. 163 
 Diderot, xi. 232 
 
 Digges, Leonard (?-i57i ?), vi. 
 
 118 
 
 Digges, Thomas (?-i595) v. 95 
 Douati (1826-1873), xm - 34 
 Doppler (1803-1853), xm. 302 
 Dreyer, xm. 308 
 Duner, xm. 302 
 
 Ecphantus (5th or 6th cent. 
 
 B.C.), n. 24 
 Eddin, Nassir (1201-1273), -HI. 
 
 62, 68 ; iv. 73 
 Encke (1791-1865), x. 227; 
 
 xm. 284 
 Eratosthenes (276 B.C.-I95 or 
 
 196 B.C.), n. 36, 45. 54J x. 
 
 221 
 
 Euclid (fl. 300 B.C.), n. 33, 52 
 n\ HI. 62, 66; vi. 115; ix. 
 165 
 
 Eudoxus (409 B.C. ?-356 B.C. ?), 
 "26,27,38,42, 51 
 
 Euler(i7o7-i783),x. 215, 226; 
 xi. 229, 230, 231 , 233-236, 
 237, 239, 242, 243; xm. :o/> 
 
 Fabricius, John (1587-1615 ?), 
 
 vi. 124 
 Ferdinand (the Emperor), vn. 
 
 137, H7 
 
 Fernel (1497-1558), in. 69 
 Ferrel, xm. 287 
 Field (i 525 7-1587), v. 95 
 Fizeau (1819-1896), xm. 283 
 Flamsteed(i646-I72o), ix. 192; 
 
 x. 197, 198, 199, 204, 207 , 
 
 218, 225 ; xii. 257 ; xin. 275, 
 
 281 
 Fracastor (1483-1553), in. 69; 
 
 iv. 89; vii. 146 
 Fraunhofer (1787-1826), xm. 
 
 299,3H 
 
 Frederick II. (of Denmark), v. 
 101, 102, 106 
 
420 
 
 Index of Names 
 
 [Roman figures refer to the chapters, Arabic to the articles.] 
 
 Frederick II. (the Emperor), 
 
 in. 66 
 Frederick II. (of Prussia), XL 
 
 230, 232, 237 
 
 Galen, n. 20; HI. 56; vi. 116 
 
 Galilei, Galileo (1564-1642), 11. 
 30 , 47 ; iv. 73 ; v. 96, 98 n ; 
 VI. passim-, VH. 135, 136, 
 138, 145, 151 ; VHI. 152-154. 
 I57 163; ix. 165, 168, 170, 
 i?i, 173. J 79 1 80, 186, 190, 
 195; x. 216; xii. 253, 257, 
 263, 268 ; xiii. 278, 295 
 
 Galilei, Vincenzo, vi. 113 
 
 Galle, xiii. 281, 289 
 
 Gascoigne (16127-1644), vin. 
 155, 156; x. 198 
 
 Gauss (1777-1855), xiii. 275, 
 276, 294 
 
 Gautier (1793-1881), xiii. 298 
 
 Genghis Khan, HI.. 62 
 
 George III., xii. 254-256 
 
 Gerbert (7-1003), in. 66 
 
 Gherardo of Cremona (1114- 
 1187), in. 66 
 
 Gibbon, n. 53 n 
 
 Giese, iv. 74 
 
 Gilbert (1540-1603), vn. 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., n. 22 
 
 Gylden (1841-1896), xiii. 288 
 
 Hainzel, v. 99 
 Hcl , xiii. 301 
 Halifax, John. See Sacrobosco 
 
 Halifax (Marquis of), ix. 191 
 
 Hall, xiii. 283 n, 295 
 
 Halley (1656-1742), vin. 156; 
 ix. 176, 177, 192 ; 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, HI. 56 
 
 Helmholtz (1821-1894), xiii. 
 
 319 
 
 Hencke (1793-1866), xiii. 294 
 Henderson (1798-1844), xiii. 
 
 279 
 Heraclitus (5th cent. B.C.), n. 
 
 24 
 
 Herschel, Alexander, xn. 251 
 Herschel, Caroline ( 1 7 50- 1 848), 
 
 xn. 251, 254-256, 260 
 Herschel, John (1792-1871), 
 
 I. 12 ; x. 221 ; xi. 242; xn. 
 
 256; xiii. 289 , 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, n. 19, 20 
 Hevel (1611-1687), vin. 153; 
 
 x. 198 ; xn. 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; n. 25, 27, 31, 32, 37-44, 
 
 45, 47-52, 54J in- 6 3> 68; 
 
 iv. 73, 84; v. in ; VH. 145 
 Hippocrates, HI. 56 
 Holwarda (1618-165 1), xii. 266 
 Holywood. See Sacrobosco 
 
Index of Names 
 {Roman figures refer to the chapters, Arabic to the articles.} 
 
 42T 
 
 Honein ben Ishak (?-873), in. 
 
 56 
 Hooke (1635-1703), ix. 174, 
 
 176; x. 207, 212 
 Horky, vi. 121 
 Horrocks (1617 ?-i64i), vm. 
 
 156; ix. 183; x. 204 
 Hewlett, xin. 298 
 Huggins, xin. 301, 304, 311- 
 
 313 
 
 Hulagu Khan, in. 62 
 Humboldt, xin. 298 
 Hutton (1737-1823), x. 219 
 Huygens (1629-1695), v. 98 n ; 
 
 vi. 123 ; vm. 154, 155, 157, 
 
 158; ix. 170-172, 191 
 Hypatia(?-4i5), n. 53 
 
 Ibn Yunos. See Yunos 
 Ishak ben Honein (7-910 or 
 911), in. 56 
 
 James I., v. 102 ; vii. 147 
 James II., ix. 192 
 Janssen, xin. 301, 307 n 
 Joachim. See Rheticus 
 
 Kaas, v. 106 
 
 Kant, xi. 250; xii. 258, 260; 
 xin. 287 
 
 Kapteyn, xm. 280 
 
 Kelvin, xin. 292 
 
 Kepler (1571-1630), n. 23, 51 #, 
 54 ; iv. 91 ; v. 94, 100, 104, 
 108-110; vi. 113, 121, 130, 
 132 ; VII. passim ; vm. 152, 
 156, 160; ix. 168-170, 172, 
 175, 176, 190, 194, 195 ; x. 
 202, 205, 220 ; xi. 228, 244 ; 
 xin. 294, 301, 309 
 
 Kirchhoff (1824-1887), xin. 
 
 299,300,3" 
 
 Kirkwood (1815-1895), xin. 
 294, 297 
 
 Koppernigk, iv. 71 
 
 Korra, Tabit ben. See Tabit 
 
 Lacaille (1713-1762), x. 222- 
 
 224, 225, 227 ; xi. 230, 233, 
 
 235 ; xn. 257, 259 
 La Condamine (1701-1774), x. 
 
 219, 221 
 Lagrange(i736-i8i3), ix. 193 ; 
 
 xi. 229, 231 #, 233 , 236, 
 
 237, 238-240, 242-245, 247, 
 
 248; xn. 251 ; xm. 293, 294 
 Lalande (1732-1807), XL 235, 
 
 241, 247 n ; xn. 265 
 Lambert (1728-1777), xi. 243; 
 
 xn. 265 
 
 Lami, ix. 180 n 
 Landgrave of Hesse. See 
 
 William IV. 
 Lang ley, xm. 307 
 Lansberg(i56i-i632), vm. 156 
 Laplace (1749-1827), xi. 229, 
 31 n, 238-248, 250; xn. 
 
 251, 256; xin. 272, 273, 282, 
 
 286-288, 290, 293, 297, 318- 
 
 320 
 Lassell (1799-1880), xn. 267 ; 
 
 xm. 295 
 
 Lavoisier, xi. 237 
 Legendre 075 2 - l8 33). xm. 
 
 275, 276 
 
 Leibniz, ix. 191, 193 
 Lemaire, xn. 255 
 Leverrier ( 1 8 1 1 - 1 877), xin. 282, 
 
 284, 288, 289, 290, 293, 294 
 Lexell (1740-1784), xii. 253 
 Lindenau, xi. 247 n 
 Lionardo da Vinci. See Vinci 
 Lippersheim (7-1619), vi. 118 
 Locke, ix. 191 
 Lockyer^ xin. 301, 302 
 Loewy, xm. 283 n 
 Louis XIV., vm. 160 
 Louis XVI., xi. 237 
 Louville (1671-1732), xi. 229 
 
422 
 
 Index of Names 
 
 [Roman figured refer to the chapters, Arabic to the articles.'] 
 
 Lubbock (1803-1865), XHI. 286, 
 
 292 
 Luther, iv. 73 ; v. 93 
 
 Machin (7-1751), x. 214 
 Maclaurin (1698-1746), x. 196 ; 
 
 xi. 230, 231 ; xii. 251 
 Maestlin, vn. 135 
 Maraldi, x. 220 
 Marius (1570-1624), vi. 118; 
 
 vii. 145 
 Martianus Capella. See 
 
 Capella 
 Maskelyne(i732-i8ii),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), xm. 297 
 Mayer, Tobias (1723-1762), x. 
 
 217, 225, 226 ; xi. 233, 241 ; 
 
 xn. 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, xm. 283 
 Molyneux (1689-1728), x. 207 
 Montanari (1632-1687), xii. 266 
 Muller. 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 ; xm. 283,* 
 
 286,2 9 o 
 
 Newton (1643-1727), n. 54; 
 
 iv v 75; vi. 130, 133, 134; 
 
 vn. 144, 150; vin. 152; IX. 
 
 passim ; x. 196-200, 211, 
 
 213, 215-217, 219, 221 ; xi. 
 
 228, 229, 231-235, 238, 249; 
 
 xn. 257 ; xm. 273, 299 
 Niccolini, vi. 132- 
 Nicholas of Cusa (1401-1464), 
 
 iv. 75 
 
 Nonius (1492-1577), in. 69 
 Norwood (15907-1675), VIH. 
 
 159; ix. 173 
 Numa, ii. 21 
 Nunez. See Nonius 
 Nyre'n, xm. 283 n 
 
 Olbers (1758-1840), xm. 294 
 Orange, Prince of, v. 107 
 Osiander, iv. 74 ; v. 93 
 
 Palisa, xm. 294 
 
 Palitzsch (1723-1788), xi. 231 
 
 Pemberton, ix. 192 
 
 Philolaus (5th cent. B.C.), n. 
 
 24; iv. 75 
 
 Piazzi (1746-1826), xm. 294 
 Picard (1620-1682), vm. 155, 
 
 157, 159-161, 162; ix. 174; 
 
 X. 196, 198, 221 
 
 Pickering, xm. 314, 316 
 Plana (1781-1869), xm. 286 
 Plato (428 B.c.?-347 B.C.?), n. 
 
 24, 25, 26, 51 ; iv. 70 
 Plato of Tivoli (fi. 1116), in. 
 
 66 
 
 Pliny (23 A.D.-79 A.D.), n. 45 
 Plutarch, n. 24 ; xm. 301 
 Pogson (1829-1891), xm. 316 
 Poincare, xm. 288 
 Poisson (1781-1840), xm. 286, 
 
 2 93 
 Pontcoulant (1795-1874), xm. 
 
 286 
 Porta, vi. 118 
 
Index of 
 
 423 
 
 [Roman figures refer to the chapters, Arabic to the articles.] 
 
 Posidonius(i35 B.C. ?-5i B.C.?), 
 
 ". 45. 47 
 Pouillet (1791-1868), xin. 307, 
 
 3 ! 9 
 
 Pound, x. 206, 216 
 Provost (1751-1839), xn. 265 
 Pritchard (1808-1893), xin. 
 
 278 , 279, 316 
 Ptolemy, Claudius (fl. 140 A.D.), 
 
 ii. 25, 27, 32, 37, 46-52, 53, 
 
 54; in- 55, 57, 59-63, 68; 
 
 iv. 70, 76, 80, 83-87, 89, 91 ; 
 
 v. 94, 105 ; vi. 121, 129, 134; 
 
 vn. 145 ; viii. 161 ; ix. 194 ; 
 
 x. 205 ; xi. 236 
 Ptolemy Philadelphus, n. 31 
 Purbach (1423-1461), in. 68; 
 
 iv. 71 
 Pythagoras '(6th cent. B.C.), i. 
 
 n, 14; ii. 23, 28, 47, 51, 54 
 
 Recorde (1510-1558), V. 95 
 Regiomontanus (1436-1476), 
 
 111.68,69; iv. 70, 71; v. 97, 
 
 no 
 
 Reimarus. See Bar 
 Reinhold (1511-1553), v. 93- 
 
 96; vii. 139 
 Reymers. See Bar 
 Rheticus (1514-1576), iv. 73, 
 
 74 ; v. 93, 94, 96 
 Ricardo, n. 47 n 
 Riccioli (1598-1671), viii. 153 
 Richer (7-1696), viii. 161; ix. 
 
 180, 187 ; x. 199, 221 
 Rigaud, x. 206 n 
 Roemer (1644-1710), viii. 162 ; 
 
 x. 198, 210, 216, 220, 225 ; 
 
 xin. 283 
 Rosse (1800-1867), xni. 310, 
 
 317 
 Rothmann (fl. 1580), v. 97, 
 
 98, 106 
 Rudolph II. (the Emperor), v. 
 
 106-108; vii. 138, 142, 143 
 
 Sabine ( 1788- 1883), xin. 298 
 Sacrobosco (?-i256?), in. 67, 
 
 68 
 
 St. Pierre, x. 197 
 Savary (1797-1841), xin. 309 
 Scheiner (1575-1650), vi. 124, 
 
 125; vn. 138; viii. 163; 
 
 XII. 268 
 
 Schiaparelli, xin. 297 
 
 Schomberg, iv. 73 
 
 Schoner, iv. 74 
 
 Schonfeld (1828-1891), xin. 
 
 280 
 Schroeter( 1 745-1816), xn. 267, 
 
 271 
 
 Schwabe (1789-1875), xin. 298 
 Secchi (1818-1878), xni. 311, 
 
 312 
 
 See, xin. 320 
 
 Seleucus (2nd cent. B.C.), n. 24 
 Shakespeare, vi. 1 13 
 Sharp (1651-1742), x. 198 
 Slusius, ix. 169 
 Smith, xn. 251 
 Snell (1591-1626), viii. 159; 
 
 ix. 173 
 
 Sosigenes (fl. 45 B.C.), ii. 21 
 South (1785-1867), xni. 306 
 Struve, F. G. W. (1793-1864), 
 
 xni. 279, 309 
 Struve, O., xni. 283 n 
 Svanberg (1771-1851), x. 221 
 Sylvester II. See Gerbert 
 
 Tabit ben Korra (836-901), in. 
 
 56, 58, 68 ; iv. 84 
 Tamerlane, in. 63 
 Tannery, n. 36 n 
 Thales (640 B.C. ?~546 B.C. ?), 
 
 ii. 23; in. 55 
 Theon (fl. 365 A.D.), n. 53 
 Theophrastus, ii. 24 
 Theophylactus, iv. 72 
 Thomson, T., x. 208 n 
 Thomson, William. See Kelvin 
 
424 
 
 Index of Names 
 
 [Roman figures refer to the cliapters, Arabic to the articles ] 
 
 Thury, Cassini de. See Cassini 
 
 de Thury 
 Timocharis (beginning of 3rd 
 
 cent. B.C.). n. 32, 42 ; iv. 84 
 Tycho Brahe. See Brahe 
 
 Ulugh Begh (1394-1449), in. 
 
 63, 68 ; iv. 73 
 Urban VIII. (Barberini), vi. 
 
 125, 127, 131, 132 
 Ursus. See Bar 
 
 Varignon, ix. 180 n 
 
 Vinci, Lionardoda(i452-i5i9), 
 
 in. 69 
 
 Vogel, xni. 313, 314 
 Voltaire, 11.21 ; xi. 229 
 
 Wafa, Abul (939 or 940-998), 
 in. 60, 68 n ; iv. 85 ; v. 1 1 1 
 
 Wallenstein, VH. 149 
 
 Walther (1430-1504), in. 68; 
 v. 97 
 
 Wargentin (1717-1783), x. 216 
 
 Watzelrode, iv. 71 
 
 Wefa. See Wafa 
 
 Welser, vi. 124 
 
 Whevvell (1794-1866), xm. 292 
 
 William IV. (Landgrave of 
 
 Hesse) (1532-1592), v. 97, 
 
 98, ioo, 105, 106, 1 10 
 Wilson (1714-1786), xii. 268; 
 
 xni. 298 
 
 Wolf, Max, xni. 294 
 Wolf, Rudolf (1816-1893), xni. 
 
 298 
 Wollaston (1766-1828), xm. 
 
 299 
 
 Wren (1632-1723), ix. 174, 176 
 Wright, Thomas (1711-1786), 
 
 xn. 258, 265 
 
 Young, xm. 301 
 Yunos, Ibn ( ?-ioo8), in. 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, j 
 21*, 216, 218; xn. 263; xin. 
 277, 283, 284 
 
 Academic Francaise, 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. !79, l8o l8 5. 195; 
 
 x. 223 
 Ad Vitellionem Paralipomena (of 
 
 Kepler), vn. 138 
 
 Alaesive Scalae (of Dirges), v. 95 
 Aldebaran, in. 64 ; xm. 316 w 
 Alexandrine school, n. 21, 31-33, 
 
 36-38, 45, 53 
 A If on sine Tables, in. 66, 68; v. 
 
 94, 96, 99 
 
 Algol, xn. 266; xin. 314, 315 
 Almagest (of Ptolemy), 11. 46-52 ; 
 
 HI. 55, 56, 58, 60, 62, 66,68; 
 
 iv. 75, 76, 83 
 
 Almagest (of Abul Wafa), m. 60 
 Aln.ngest, New (of Kepler), vii. 
 
 148 
 Almagest, New (of Riccioli), vm. 
 
 153 
 
 425 
 
 Almanac , Nautical, x. 218; xin. 
 
 286, 288, 290 
 Almanacks, i. 18 ; n. 20, 38 ; in. 
 
 64, 68; v. 94, 95, 100; vii. 
 
 136; x. 218, 224; xin. 286, 
 
 288, 290 
 
 Altair, in. 64; xin. 316 n 
 Analysis, analytical methods, x. 
 
 196 ; xi. 234 
 
 Angles, measurement of, i. 7 
 Annual equation, v. Ill; vn. 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, n. 43; vn. 145 
 Anomalistic month, n. 40 
 'AvrtxQuv, ii. 24 
 Apex of solar motion, xu. 265 
 Aphelion, iv. 85 
 Apogee, n. 39, 40, 48 ; HI. 58, 59 ; 
 
 iv. 85; v. in; ix. 184- xi. 
 
 233 
 
 Apparent distance, i. 7 
 Apple, Newton's, ix. 170 
 Apse, apse-line, n. 39, 40, 48; iv. 
 
 85 ; ix. 183 ; xi. 235, 236, 242, 
 
 246 
 Arabic numerals, m. 64 ; v. 96 
 
426 
 
 General Index 
 
 [Roman figures refer to the chapters, Arabic to the articles J\ 
 
 Arabs, Arab astronomy, n. 46, 53; 
 
 HI. 56-61, 64, 66, 68; v. no; 
 
 viii. 159; xn. 266 
 Arctic regions, n. 35 
 Arcturus, xn. 258 
 Aries, first point of ( r ), i. 13 ; 
 
 ii. 33, 42; iv. 83 ; v. 98, 
 
 in ; x. 198 
 
 Asteroids. See Minor planets 
 Astrolabe, n. 49; in. 66 
 Astrology, i. 16, 18; in. 56; v. 
 
 99, 100 ; vii. 136, 149, 151 
 Astronomiae Fundamenta (of 
 
 Lacaille), x. 224 
 Astronomiae Instauratae Mecha- 
 
 nica (of Tycho), v. 107 
 Astronomiae Instauratae Progvm- 
 
 nasmata (of Tycho), v. 104 
 Astronomical Society, German, 
 
 xin. 280 
 Astronomical Society, Royal, xn. 
 
 256, 263 
 Astronomicum Caesareum (of 
 
 Apian), v. 97 
 
 Astronomisches Jahrbuch t xn. 253 
 Astronomy, divisions of, xin. 272 
 Astronomy, descriptive, xin. 272, 
 
 273. 294 
 Astronomy, gravitational, x. 196 ; 
 
 xin. 272, 273, 286 
 Astronomy, observational, xin. 
 
 272, 273 
 
 Astronomy, origin of, i. 2 
 Astronomy, scope of, i. i 
 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, HI. 56, 57, 60, 68 
 Btlts of Jupiter, xn. 267 
 Betelgeux, in. 64 
 Biela's comet, xin. 305 
 Binary stars. See Stars, double 
 and multiple 
 
 Bode's Astronomisches Jahrbuch, 
 
 xii. 253 
 
 Bodily tides, xin. 292, 293 
 Brightness of stars. See Stars, 
 
 brightness of 
 Brooks's comet, xin. 305 
 Brucia, xin. 294 
 Bureau des Longitudes, xi. 238., 
 
 Cairo, in. 60 
 
 Calculus of Variations, xi. 237 n 
 
 Calendar, Greek, ii. 19, 20, 21 
 
 Calendar, Gregorian, n. 22 ; IX. 
 165 n ; x. 217 
 
 Calendar, Julian, n. 21, 22 ; 111. 
 68; iv. 73; ix. 165 n ; x. 
 197 , 217 
 
 Calendar, Mahometan, in. 56 
 
 Calendar, Roman, n. 21 
 
 Caliphs, in. 56, 57, 69 
 
 Canals of Mars, xin. 297 
 
 Cartesianism, vin. 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 Longi- 
 tude, celestial 
 
 Celestial sphere, i. 7, 8, 9-11, 13, 
 14 ; n. 24, 26, 33, 38-40, 45, 47, 
 51 ; m. 64, 67; iv. 78, 80, 83, 
 88; v. 98, 105; vi. 129; vni. 
 157; x. 198, 207, 208, 214; xn. 
 257; xin. 272 
 
 Celestial sphere, daily motion of. 
 See Daily motion 
 
 Celestial spheres, n. 23, 26, 27, 38, 
 51, 54; in. 62, 68; v. 100, 103 
 
 Centre of gravity, ix. 1 86 
 
 Centrifugal force, vin. 158 
 
 Ceres, xin. 276, 294 
 
 Chaldaeans (Babylonians), Chal- 
 dacan astronomy, i. 6, 1 1, 12, 
 16-18; ii. 30, 34, 40, 51, 54; 
 in. 56 
 
General Index 
 
 427 
 
 \Romanfigures refer to the chapters, Arabic to the articles.] 
 
 Chart, photographic, xm. 280 
 Chemistry of the sun, xm. 299 
 
 301, 303 
 
 Chinese astronomy, i. 6, 1 1 
 Chords, ii. 47 
 Chromosphere, x. 205 ; xm. 301, 
 
 303 
 
 Chronometer, x. 226 
 Circular motion, 11. 25, 26, 38, 51 ; 
 
 iv. 76; vii. 139; vin. 158; ix. 
 
 171-173. See also Eccentric, 
 
 Epicycle 
 
 Circumpolar stars, i. 9; n. 35 
 Clock. See Pendulum clock and 
 
 Chronometer 
 Clustering power, xn. 261. See 
 
 also Condensation of nebulae 
 Coelum Australe Stelliferum (of 
 
 Lacaille), x. 223 
 Collimation error, x. 225 n , 
 Comets, i. i ; n. 30 ; in. 68, 69 ; v. 
 
 ioo, 103-105; vi. 127, 129; vii. 
 
 144, 146; vin. 153; ix. 190, 
 
 192 ; x. 200, 205, 217, 224; xi. 
 
 228, 231. 243, 248, 250 ; xn. 
 
 253, 256, 259; xm. 272, 291, 
 
 304, 305, 307 
 Comet, Biela's, xm. 305 
 Comet, Brooks's, xm. 305 
 Comet, Encke's, xm. 291 
 Comet, Halley's, vii. 146; x. 200, 
 
 205 ; xi. 231, 232 ; xm. 291, 307 
 Comet, Lexell's, xi. 248; xm. : 
 
 305 
 
 Comet, Olbers's, xm. 291 
 Comet, Pons-Brooks, xm. 291 
 Comet, Tebbutt's, xm. 305 
 Comet, Tuttle's, xm. 291 
 Cometographia (of Hevel), vm. 
 
 153 
 Commentaries on the Motions of \ 
 
 Mars(of Kepler), vii. 135 n, 139, 
 
 141, I50M 
 Comnientariolus (of Coppernicus), : 
 
 iv. 37 ; v. ioo 
 Compleat System of Optics (of 
 
 Smith), xii. 251 
 Complete induction, ix. 195 
 
 Condensation of nebulae, xi. 250; 
 
 xii. 261 ; xm. 318 
 Conic, conic section, vn. 140 n ; 
 
 xm. 309 
 Conjunction, n. 43, 48 n ; HI. 60 ; 
 
 v. 1 10, 1 1 1 ; x. 227 
 Conservation of energy, xm. 319 
 Constant of aberration, x. 209, 2 10; 
 
 xm. 283 n 
 Constellations, i. 12, 13; n. 20, 
 
 26, 34, 42 ; x. 223 
 Construction of the heavens, xn. 
 
 257. See also Sidereal system, 
 
 structure of 
 Corona, vn. 145 ; x. 205 ; xm. 
 
 301, 303 
 
 Counter-earth, n. 24 
 Crape-ring, xn. 267 ; xm. 295 
 Craters (on the moon), vm. 153 ; 
 
 xm. 296 
 Curvature of the earth. See 
 
 Earth, shape of 
 
 Daily Motion (of the celestial 
 sphere), i. 5, 8, 9, 10 ; n. 23, 24, 
 26, 33, 38, 39, 46, 47 J i". 67, 
 68; iv. 78, 80, 83 ; v. 98, 105 ; 
 vi. 129; vm. 157 
 
 D'Alembert's principle, xi. 232, 
 237 n 
 
 Damascus, HI. 57 
 
 Darlegung (of Hansen), xm. 286 
 
 Day, 1.4, 11,16; 11.47; xm. 287, 
 293, 320 
 
 Day-and-night, i. i6n 
 
 Day-hour, i. 16 
 
 Declination, n. 33, 35, 39 ; x. 213, 
 218 ; xm. 276 
 
 Declination circle, n. 33 
 
 De Coelo (of Aristotle), n. 27 
 
 Deductive method, inverse, ix. 
 
 195 
 Deferent, n. 39, 48, 51; m. 68 ; 
 
 iv. 86, 87 
 Degree, i. 7 
 Deimos, xm. 295 
 De Magnete (of Gilbert), vn. 
 
428 
 
 General Index 
 
 [Roman figures refer to the chapters, Arabic to the articles. ,] 
 
 De tfotu (of Newton), ix. 177, 
 
 191 
 De Mundi aetherei (of T^cho), 
 
 v. 104 
 
 De Nova Stella (ofTycho), v. 100 
 De Revolutionibus (of Copperni- 
 
 cus), ii. 41 n ; iv. 74-92; v. 93, 
 
 94; vi. 126 
 De Saturni Luna (of Huygens), 
 
 vni. 154 
 Descriptive astronomy, xin. 272, 
 
 273 294 
 
 Deviation error, x. 225 n 
 
 Dialogue on the Two Chief Sys- 
 tems (of Galilei), vi. 124^, 
 128-132, 133 
 
 Differential method of parallax, 
 vi. 129; xn. 263 ; xm. 278 
 
 Diffraction-grating, xin. 299 
 
 Dione, vin. 160 
 
 Dioptrice (of Kepler), vn. 138 
 
 Direct motion, i. 14 
 
 Disturbing force. See Perturba- 
 tions 
 
 Diurnal method of parallax, xin. 
 281, 284 
 
 Doctrine of the sphere. See 
 Spherics 
 
 Doppler's principle, xm. 302, 
 
 313, 3H 
 
 Double hour, i. 16 
 Double-star method of parallax. 
 
 See Differential method of 
 
 parallax 
 Double stars. See Stars, double 
 
 and multiple 
 
 Draconitic month, n. 40, 43 
 Durchmusterung, xm. 280 
 Dynamics, vi. 133, 134; ix. 179, 
 
 180; xi. 230, 232, 237 
 Dynamique, Iraite de (of D'Alem- 
 
 bert), xi. 232 
 
 Earth, i. i, 15, 17; n. 28, 29, 32, 
 
 39, 41. 43 47, 49, 5 1 ; "i- 66, 
 69; iv. 80, 86; vi. 117, 121, 
 133; vn. 136 . 144, 145, 150; 
 vni. 153, 1 54; ix. 173, 174, 179- 
 
 182, 184, 186, 195; xi. 228, 
 
 245 ; xm. 285, 287, 292, 293, 
 
 297, 320. See also the following 
 
 headings 
 Earth, density, mass of, ix. 180, 
 
 185, 189; x. 219; xi. 235; 
 
 xm. 282, 294 
 Earth, motion of, n. 24, 32, 47 ; 
 
 iv. 73, 76, 77 ; v. 96, 97, 105 ; 
 
 vi. 121, 125-127,129-132; vin. 
 
 161, 162 ; ix. 186, 194 ; xn. 257. 
 
 See also Earth, revolution of and 
 
 rotation of 
 Earth, revolution of, annual motion 
 
 of, n. 23, 24, 28 n, 30, 47 ; iv. 75, 
 
 77, 79-82, 85-88, 89, 90, 92; v. 
 
 in; vi. 119, 126, 129, 131, 133; 
 
 vn. 139, 142, 146 ; vm. 161 ; ix. 
 
 172, 183 ; x. 207-210, 212, 227 ; 
 
 xi. 235, 236, 240 ; xn. 263 ; xm. 
 
 278, 282, 283 
 
 Earth, rigidity of, xm 285, 292 
 Earth, rotation of, daily motion 
 
 of, n. 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 ; xm. 281, 285, 
 
 287, 320 
 Earth, shape of, n. 23, 29, 35, 
 
 45, 47, 54; iv. 76; vm. 161; 
 
 ix. 187, 1 88; x. 196, 213, 215, 
 
 220, 221, 222, 223 ; xi. 229, 231, 
 237, 248 
 
 Earth, size of, 11. 36, 41, 45, 47, 
 49 ; in. 57, 69 ; iv. 85 ; vn. 145 ; 
 vm. 159, 161 ; ix. 173, 174; x. 
 
 221, 222, 223 
 
 Earth, zones of, n. 35, 47 
 Earthshine, in. 69 
 Easter, rule for fixing, 11. 20 
 Eccentric, n. 37, 39, 40, 41 48, 
 51 ; in. 59 ; iv. 85, 89-91 ; vn. 
 
 139, 15 
 
 Eccentricit} 7 , n. 39 ; iv. 85 ; vn. 
 140 n ; xi. 228, 236, 240, 244- 
 246, 250; xm. 294, 318 
 Eccentricity fund, xi. 245 
 Eclipses, i. u, 15, 17; n. 29, 32, 
 
General Index 
 
 429 
 
 \Romanfigures refer to the chapters, Arabic to the articles.] 
 
 40-42, 43, 47-49. 54; "i. 57- 
 
 68 ; iv. 76, 85 ; v. 1 10 ; vi. 127 ; 
 
 vn. 145, 148; vni. 162; x. 201, 
 
 205, 210, 216, 227; xi. 240; 
 
 xiii. 287, 301 
 
 Eclipses, annular, n. 43 ; vn. 145 
 Eclipses, partial, n. 43 
 Eclipses, total, n. 43 ; vn. 145 ; 
 
 x. 205 ; xiii. 301 
 Ecliptic, i. 11, 13, 14; n. 26, 33. 
 
 35. 3 6 38, 40, 42, 5i ; in. 58, 
 
 59, 68 ; iv. 80, 82-84, 87, 89 ; 
 
 v. in; vin. 154; x. 20 -. 2:>>, 
 
 213, 214, 227 ; xi. 235, 236, 24 1. 
 
 246, 250 
 Ecliptic, obliquity of, i. 11 ; n. 
 
 35, 36, 42 ; in. 59, 68 ; iv. 83, 
 , 84 ; xi. 235, 236 
 ^cole Normale, xi. 237, 238 
 Ecole Polytechnique, xi. 237 
 Egyptians, Egyptian astronomy, 
 
 i. 6, II, 12, 16; n. 23, 26, 30, 
 
 45 J iv. 75 
 Elements (of Euclid), in. 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, n. 51 n; in. 66 ; vn. 140, 
 
 141; ix. 175, 176, 190, 194; x. 
 
 200, 209, 214; xi. 228, 236, 242, 
 
 244 ; XIH. 276, 278, 309 
 Ellipticity, x. 221 
 Empty month, n. 19, 2O 
 Empyrean, in. 68 
 Enceladus, xii. 255 
 Encke's comet, xiii. 291 
 Encyclopaedia, the French, xi. 232 
 Energy, xni. 319 
 Ephemerides. See Almanacks 
 Ephemerides (of Regiomontanus), 
 
 in. 68 
 Epicycle, n. 37, 39, 41, 45, 48, 
 
 51, 54; in. 68; iv. 85-87, 89- 
 
 91; vn. 139, 150; vin. 163; ix. 
 
 170, 194 
 
 Epitome (of Kepler), vi. 132 ; vn. 
 
 144, 145 
 
 Epitome (of Purbach), in. 68 
 Equant, n. 51 ; in. 62 ; iv. 85, 89, 
 
 91 ; vii. 139, 150 
 Equation of the centre, n. 39, 48 ; 
 
 in. 60 ; v. ill 
 Equator, i. 9, IO, II ; n. 33, 35, 
 
 39, 42 ; iv. 82, 84 ; v. 98 ; vi. 
 
 129, 133 ; ix. 187 ; x. 207, 220, 
 
 221 ; xiii. 285 
 
 Equator, motion of. See Preces- 
 sion 
 Equinoctial points, i. 11, 13 ; 11. 
 
 42. See als-j Aries, first point of 
 Equinoxes, i. 11 ; n. 39, 42 
 Equinoxes, precession of. See 
 
 Precession 
 Essai philosophiqiie (of Laplace), 
 
 xi. 238 
 
 Ether, xiii. 293, 299 
 Evection, 11. 48, 52; 111. 60; iv. 
 
 85 ; v. in; vii. 145 
 Evening star, i. 14. See also 
 
 Venus 
 Exposition du Systeme du Monde 
 
 (of Laplace), xi. 238, 242 n, 250 
 
 Faculae, vm. 153 ; xni. 300, 301, 
 
 33 
 Figure of the earth. S^ Earth, 
 
 shape of 
 
 Firmament, in. 68 
 First point of Aries, Libia. 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, n. 19, 20 
 Full moon. See Moon, phases of 
 Fundamenta (of Hansen), XIH. 286 
 
430 
 
 General Index 
 
 [Roman figures refer to the chapters, Arabic to the articles.] 
 
 Fundamenta Astronomiae (of 
 Bessel), x. 218; xm. 277 
 
 Funds of eccentricity, inclination, 
 xi. 245 
 
 Galactic circle, xn. 258, 260 
 Galaxy. See Milky Way 
 Gauges, gauging. See Star- 
 gauging 
 
 Georgium Sidus. See Uranus 
 
 Gravitation, gravity, n. 38 n ; vn. 
 
 150; viu. 158, 161 ; IX. passim] 
 
 x. 196, 201, 213, 215, 219, 220, 
 
 223, 226 ; XI. passim ; xn. 264; 
 
 xm. 282, 284, 286-293, 309, 319 
 
 Gravitational astronomy, x. 196; 
 
 xm. 272, 273, 286 
 Gravity, variation of, vm. 161 ; 
 ix. 180; x. 199, 217, 221, 223; 
 xi. 231 
 
 Great Bear, i. 12 ; xn. 266 
 Great circle, i. 11 ; n. 33, 42 ; iv. 
 
 82, 84 
 
 Gregorian Calendar. See Cal- 
 endar, Gregorian 
 Grindstone theory, xn. 258 ; xm. 
 317 
 
 Hakemite Tables, m. 60, 62 
 Halley's comet, vn. 146 ; x. 200, 
 205; xi. 231, 232; xm. 291, 
 
 307 
 
 Harmonics (of Smith), xn. 251 
 Harmony of the World (of Kepler), 
 
 vn. 144 
 
 Helium, xm. 301 
 Herschelian telescope, xn. 255, 
 
 256 
 Histotia Coelestis (of Flam steed), 
 
 x. 198 
 
 Holy Office. See Inquisition 
 Horizon, i. 3, 9; n. 29, 33, 35, 39, 
 
 46; vm. 161 ; xm. 285 
 Horoscopes, v. 99 
 Hour, i. 16 
 
 Hydrostatic balance, vi. 115 n 
 Hyperbola, ix. 190; xi. 236 n 
 Hyperion, xm. 295 
 
 Ilkhanic Tables, in. 62 
 IlSaggiatore (of Galilei), vi. 127 
 Inclination, in. 58; iv. 89; xi. 
 
 228, 244, 245, 246, 250; xm. 
 
 294, 318 
 
 Inclination fund, xi. 245 
 Index of Prohibited Books, vi. 126, 
 
 132; vn. 145 
 Indians, Indian astronomy, i. 6 ; 
 
 in. 56, 64 
 
 Induction, complete, ix. 195 
 Inequalities, long, xi. 243 
 Inequalities, periodic, xi. 242, 
 
 243, 245, 247 
 Inequalities, secular, xi. 242, 
 
 243-247 ; xm. 282. See also 
 
 Perturbations 
 
 Inequality, parallactic, xm. 282 
 Inferior planets, i. 15 ; iv. 87, 88. 
 
 See also Mercurj', Venus 
 Inquisition (Holy Office), vi. 126, 
 
 i3 2 , J 33 
 
 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, n. 23 
 Iris, xm. 281 
 Irradiation, vi. 129 
 Island universe theory, xu. 260 ; 
 
 xm. 317 
 
 Japetus, vm. 160; xn. 267; xm. 
 297 
 
 Julian Calendar. See Calendar, 
 Julian 
 
 Juno, xm. 294 
 
 Jupiter, i. 14-16; ii. 25, 51; iv. 
 81, 87, 88; v. 98, 99; vi. 121, 
 127;- vn. 136 n, 142, 144, 145, 
 150; vm. 154, 156, 162; ix. 
 172, 181, 183, 185-187; x. 204, 
 216; xi. 228, 231, 235, 236, 
 243-245, 248 ; xii. 267 ; xm. 
 281, 288, 294, 297, 305. See 
 also the following headings 
 
 Jupiter, belts of, xn. 267 
 
General Index 
 
 43 i 
 
 [Roman figures refer to the chapters^ Arabic to the articles^ 
 
 Jupiter, mass of, ix. 183, 185 
 Jupiter, rotation of, vm. 160 ; 
 
 ix. 187; xm. 297 
 Jupiter, satellites of, n. 43 ; vi. 
 121, 127, 129, 133; vii. 145, 
 150; vm. 1 60, 162; ix. 170, 
 184, 185 ; x. 210, 216 ; xi. 228, 
 248 ; xii. 267 ; xm. 283, 295, 
 
 297 
 
 Jupiter's satellites, mass of, xi. 
 248 
 
 Kepler's Laws, vn. 141, 144, 145, 
 151; vm. 152; ix. 169, 172, 
 175, 176, 186, 194, 195 ;x. 220; 
 xi. 244 ; xm. 294, 309 
 
 Latitude (celestial), n. 33, 42, 
 
 43; m. 63; iv. 89 
 Latitude (terrestrial), in. 68, 69; 
 
 iv. 73 ; x. 221 ; xm. 285 
 Latitude, variation of, xm. 285 
 Law of gravitation. See Gravi- 
 tation 
 Laws of motion, vi. 130, 133 ; 
 
 vm. 152, 163; ix. 171, 179- 
 
 181, 183, 186, 194, 195 ; xi. 232 
 Leap-year, i. 17; n. 21, 22 
 Least squares, xm. 275, 276 
 Letter to the Grand Duchess (of 
 
 Galilei), vi. 125 
 Level error, x. 225 n 
 Lexell's comet, xi. 248 ; xm. 305 
 Libra, first point of (==), 1. 13 ; n. 
 
 42 
 Librations of the moon, vi. 133 ; 
 
 x. 226 ; xi. 237, 239 
 Libros del Saber, m. 66 
 Light-equation, xm. 283 
 Light, motion of, velocity of, 
 
 vm. 162; x. 208-211, 216, 220; 
 
 xm. 278, 279, 283, 302. See 
 
 also Aberration 
 Logarithms, v. 96, 97 n 
 Long inequalities, xi. 243 
 Longitude (celestial), n. 33, 39, 
 
 42,43; m. 63 ; iv. 87 ; vn. 139 
 
 Longitude (terrestrial), in. 68 ; 
 
 vi. 127, 133; vn. 150; x. 197, 
 
 216, 226 
 Longitudes, Bureau des, xi. 
 
 238 
 
 Lunar distances, in. 68 n 
 Lunar eclipses. See Eclipses 
 Lunar equation, xm. 282 
 Lunar theory, n. 48, 51 ; v. 1 1 1 ; 
 
 vn. 145 ; vm. 156 ; ix. 184, 192 ; 
 
 x. 226; xi. 228, 230, 231, 233, 
 
 234, 240, 241, 242, 248; xm. 
 
 282, 286, 287, 288, 290. See 
 
 also Moon, motion of 
 Lunation, 11. 40. See also Month, 
 
 synodic 
 
 Macchie Solan (of Galilei), vi. 124, 
 
 125 
 
 Magellanic clouds, xm. 307 
 Magnetism, vn. 150; xm. 276, 298 
 Magnitudes and Distances of the 
 
 Sun and Moon (of Aristarchus), 
 
 n. 32 
 Magnitudes of stars, n. 42 ; xn. 
 
 266 ; xn. 280, 316. See also 
 
 Stars, brightness of 
 Mars, i. 14-16 ; n. 25, 26, 30, 51 ; 
 
 m. 68; iv. 81, 87; v. 108; 
 
 vi. 129; vii. 136 n, 139-142, 144, 
 
 145 ; vm. 154, 161 ; ix. 181, 
 
 183, 185 ; x. 223, 227 ; xi. 235, 
 
 245 ; xii. 267 ; xm. 281, 282, 
 
 284, 294, 295, 297. See also 
 
 the following headings 
 Mars, canals of, xm. 297 
 Mars, mass of, xi. 248 
 Mars, opposition of, vm. 161; xm. 
 
 281, 284, 297 
 Mars, rotation of, vm. 160; xm. 
 
 295, 297 
 
 Mars, satellites of, xm. 295 
 
 Mass, ix. 180, 1 8 1, 185 
 
 Mass of the earth, sun, Venus .... 
 See Earth, Sun, Venus .... mass 
 of 
 
 Mecantque Analytique (of La- 
 grange), xi. 237 
 
432 
 
 General Index 
 
 [Roman figures refer to the chapters, Arabic to the articles.] 
 
 Mecanique Celeste (of Laplace), 
 
 xi. 238, 241, 247, 249, 250; 
 
 xin. 292 
 Medicean Planets. See Jupiter, 
 
 satellites of 
 Meraga, in. 62 
 Mercury, I. 14-16 ; n. 25, 26, 45, 
 
 47, 51; m. 66; iv. 73, 75, 81, 
 
 86-89; VI. 121, 124; VII. 136;?, 
 
 139, 142, 144 ; ix. 185 ; xni. 288, 
 
 290, 294, 297. See also the 
 
 following headings 
 Mercury, mass of, xi. 248 
 Mercury, phases of, vi. 129 
 Mercury, rotation of, xin. 297 
 Mercury, transit of, x. 199 
 Meridian, n. 33, 39 ; in. 57 ; vi. 
 
 127 ; vni. 157 ; x. 207, 218, 221 
 Meteorologica (of Aristotle), n. 27 
 Meteors, xin. 305 
 Melon's cycle, n. 20 
 Metric system, xi. 237 
 Micrometer, vni. 155 ; xin. 279, 
 
 281 
 Milky Way, n. 30, 33; vi. 120; 
 
 xii. 258,260-262; xin. 317 
 Mimas, xii. 255 
 Minor planets, xi. 250 n ; xin. 
 
 276, 281, 284, 288, 294, 295, 
 
 297, 318 
 
 Minor planets, mass of, xin. 294 
 Minute (angle), i. 7 
 Mira, xn. 266 
 Mongols, Mongol astronomy, HI. 
 
 62 
 Month, i. 4, 16 ; 11. 19-21, 40, 48 ; 
 
 ix. 173; xi. 240; xm. 293, 320. 
 
 See also the following headings 
 Month, anomalistic, 11. 40 
 Month, draconitic, 11. 40, 43 
 Month, empty, n. 19, 2O 
 Month, full, ii. 19, 20 
 Month, lunar, i. 16; n. 19, 20, 40 
 Month, sidereal, n. 40 
 Month, synodic, n. 40, 43 
 Moon, i. i, 4, 5, n, 13-16; ii. 
 
 19-21, 25, 28, 30, 32, 39, 43 ; 
 
 HI. 68, 69 ; iv. 81, 86 ; v. 104, 
 
 105 w; vi. 119, 121, 123, 129, 
 
 '3. J 33 J vii. 145, 150; vni. 
 
 153; ix. 169, 1 80, 181, 1 88, 189; 
 
 x. 198, 204, 213, 215, 226; xi 
 
 228, 235 ; xn. 256, 257, 271 ; 
 
 xin. 272, 292, 293, 296, 297,, 
 
 301,320. See also the following 
 
 headings 
 Moon, aneular or apparent size of, 
 
 ii. 32,41,43, 46 n, 48; iv. 73, 
 
 85, 90 ; v. 105 n 
 Moon, apparent flattening of, n. 
 
 46 
 
 Moon, atmosphere of, xin. 296 
 Moon, distance of, i. 1 5 ; n. 24, 
 
 25,30,32,41,43,45,48,49,51; 
 
 iv. 85, 90 ; v. 100, 103 ; ix. 173 
 
 185 ; x. 223 ; xin. 293, 320 
 Moon, eclipses of. See Eclipses 
 Moon, librations of, vi. 133; x. 
 
 226; xi. 237, 239 
 Moon, map of, x. 226 ; xin. 
 
 296 
 Moon, mass of, ix. 188, 189 
 
 xi. 235 
 Moon, motion of, i. 4, 8, 13, 15, 
 
 17 ; n. 20, 24-26, 28, 37, 39, 
 
 40, 43, 47, 48, 51 ; in. 60; iv. 
 
 73, 81, 85, 89, 90; v. in ; vi. 
 
 133; vn. 145. 150; vni. 156; 
 
 ix. 169, 173, 174, 179, 184, 185, 
 
 189, 194, 195 ; x. 201, 204,213, 
 
 226; xi. 235, 237, 248 ; xm. 287, 
 
 290, 297, 320. See also Lunar 
 
 theory 
 
 Moon, origin of, xm. 320 
 Moon, parallax of, n. 43, 49 ; iv. 
 
 85. Cf. also Moon, distance of 
 Moon, phases of, i. 4, 16, 17; ii. 
 
 19, 20, 23, 28, 43, 48; m. 68, 
 
 69; vi. 123 
 Moon, rotation of, x. 226 ; xi. 248 ; 
 
 xn. 267 ; xm. 297 
 Moon, shape of, ii. 23, 28, 46 ; 
 
 vi. 119; xi. 237 
 Moon, size of, n. 32, 41 ; iv. 85 
 Moon, tables of. See Tables, 
 
 lunar 
 
General Index 
 
 433 
 
 [Roman figures refer to the chapters^ Arabic to the articles.] 
 
 Moons. See Satellites 
 
 Morning star, i. 14. Sec also 
 
 Venus 
 
 Morocco, in. 6 1 
 Motion, laws of. See Laws of 
 
 motion 
 Multiple stars. See Stars, double 
 
 and multiple 
 
 Mural quadrant, x. 218, 225 n 
 Music of the spheres, n. 23 ; vn. 
 
 144 
 Mysterium Cosmographicum (of 
 
 Kepler), v. 108; vn. 136, 144 
 
 Nadir, in. 64 
 
 Nautical Almanac. See Almanac, 
 Nautical 
 
 Nebula in Argus, xm. 307 
 
 Nebula in Orion, xn. 252, 259, 
 260; xm. 311 
 
 Nebulae, x. 223 ; xi. 250 ; xn. 
 252 256, 259-261 ; xm. SOS- 
 SOS, 310, 311, 317, 318, 319, 320 
 
 Nebulae, spiral, xm. 310 
 
 Nebular hypothesis, xi. 250; xm. 
 318-320 
 
 Nebulous stars, x. 223 ; xn. 260, 
 261 
 
 Neptune, xm. 289, 295, 297 
 
 Neptune, satellite of, xm. 295 
 
 New Almagest (of Kepler), vn 
 148 
 
 New Almagest (of Riccioli), vm. 
 
 *53 
 
 New moon. See Moon, phases of 
 New stars. See Stars, new 
 New Style (N.S.), n. 22. See 
 
 also Calendar, Gregorian 
 Newton's problem, xi. 228, 229, 
 
 249 
 Newtonian telescope, ix. 168 ; 
 
 xn. 252, 253, 256 
 Night-hour, i. 16 
 Node, n. 40, 43; v. in ; ix. 184 ; 
 
 x. 213, 214; xi. 236, 246 
 Nubeculae, xm. 307 
 Nucleus (of a comet), xm. 304 
 Nurnberg school, in. 68 ; iv. 73 
 
 Nutation, x. 206, 207, 213-215, 
 216, 218 ; xi. 232, 248 ; xn. 263 
 , i. i6 
 
 Oberon, xn. 255 
 
 Obliquity of the ecliptic. See 
 
 Ecliptic, obliquity of 
 Observational astronomy, xm. 
 
 272, 273 
 
 Occultations, i. 15 ; 11. 30 
 Octaeteris, n. 19 
 Olbers's comet, xm. 291 
 Old Moore's Almanack, i. 18 n 
 Old Style (O.S.). See Calendar, 
 
 Julian 
 Opposition, n. 43, 48 n ; in. 60 ; 
 
 iv. 87, 88; v. in; vm. 161 ; 
 
 xm. 281, 284, 297 
 Opposition of Mars, vm. 161 ; 
 
 xm. 281, 284, 297 
 Optical double stars, xii. 264 
 Optics (of Gregory), x. 202 
 Optics (of Newton;, ix. 192 
 Optics (of Ptolemy), n. 46 
 Optics (of Smith), xn. 251 
 Opus Majus, Minus, Tertium 
 
 (of Bacon), in. 67 
 Opuscules Mathe'matiques (of 
 
 D'Alembert), xi. 233 
 Orion, nebula in, xn. 252, 259, 
 
 260 ; xm. 311 
 Oscillatorium Horologium (of 
 
 Huygens), vm. 158 ; ix. 171 
 
 Pallas, xm. 294 
 
 Parabola, ix. 190 ; xi. 236 n ; 
 
 xm. 276 
 
 Parallactic inequality, xm. 282 
 Parallax, 11. 43, 49; iv. 85, 92; 
 
 v. 98, 100, no; vi. 129; vii. 
 
 145 ; vm. 161 ; x. 207, 212, 
 
 223, 227; xn. 257, 258, 263, 
 
 264; xm. 272, 278, 279, 281- 
 
 284 
 Parallax, annual, vm. 161. See 
 
 also Parallax, stellar 
 Parallax, horizontal, vm. 161 
 
434 
 
 General Index 
 
 \Romanfigures refer to the chapters, Arabic to the articles .] 
 
 Parallax of the moon. See Moon, 
 
 parallax of 
 Parallax of the sun. See Sun, 
 
 parallax of 
 Parallax, stellar, iv. 92; v. 100 ; 
 
 vi. 129 ; vni. 161 ; x. 207, 212 ; 
 
 xii. 257, 258, 263, 264; xni. 
 
 272, 278, 279 
 
 Parallelogram offerees, ix. i8o;z 
 Parameters, variation of,xi.233 n. 
 
 See also Variation of elements 
 TlapaTrrfY/uLara, n. 2O 
 Partial eclipses, n. 43 
 Pendulum, pendulum clock, v. 
 
 98; vi. 114; vin. 157, 158, 
 
 161; ix. 180, 187; x. 199, 217, 
 
 221, 223; xi. 231. See also 
 
 Gravity, variation of 
 Pendulum Clock (of Huygens), 
 
 vm. 158; ix. 171 
 Penumbra (of a sun-spot), vi. 
 
 124 ; xn. 268 
 Perigee, n. 39, 40, 48; iv. 85. 
 
 See also Apse, apse-line 
 Perihelion, iv. 85 ; xi. 231. See 
 
 also Apse, apse-line 
 Periodic inequalities. See In- 
 equalities, periodic 
 Perturbations, vm. 156; ix. 183, 
 
 184 ; x. 200, 204, 224, 227 ; XI. 
 
 passim ; xm. 282, 293, 294, 297 
 Phases of the moon. See Moon, 
 
 phases of 
 
 Phenomena (of Euclid), n. 33 
 Pbobos, xm. 295 
 Photography, xm. 274, 279-281, 
 
 294, 298, 299, 301, 306 
 Photometry, xm. 316. See also 
 
 Stars, brightness of 
 Photosphere, xii. 268 ; xm. 303 
 Physical double stars, xn. 264. 
 
 See also Stars, double and mul- 
 tiple 
 Planetary tables. See Tables, 
 
 planetary 
 Planetary theory, n. 51, 52, 54; 
 
 in. 68 ; iv. 86-90 ; xi. 228, 230, 
 
 231, 233, 235, 236, 242-247, 
 
 248; xm. 286, 288-290, 293. 
 See also Planets, motion of 
 Planets, i. 13, 14, 15, 16; n. 23- 
 27. 3, 3 2 > 5i J i- 68; iv. 81 ; 
 v. 104, 105, no, 112; vi. 119, 
 121 ; vn. 136, 144; vm. 154, 
 155; x. 200; xi. 228, 250; xn 
 
 253, 255, 257, 267, 271; xm. 
 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, xn. 253, 
 
 254, 255, 267; xm. 289, 294, 
 295, 3i8 
 
 Planets, distances of, i. 15 ; n. 30, 
 51 ; iv. 81, 86, 87 ; vi. 117; 
 vn. 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; xm. 294. See also 
 under the several planets 
 
 Planets, minor. See Minor planets 
 
 Planets, motion of, i. 13, 14, 15; 
 n. 23-25, 26, 27, 30, 41, 45, 47, 
 51, 52 ; m. 62, 68 ; iv. 81, 86- 
 90, 92; v. 100, 104, 105, 112; 
 vi. 119, 121, 129; vn. 139-142, 
 144, 145, 150, 151 ; vm. 152, 
 156; ix. 169,170,172-177, 181, 
 183, 194; x. 199, 204; xi. 228, 
 229, 245, 250; xm. 275, 276, 
 282, 294. See also Planetary 
 theory 
 
 Planets, rotation of, vm. 160 ; ix. 
 187; xi. 228, 250; xn. 267; 
 xm. 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; xn. 260 
 
 Poles (of a great circle), n. 
 33 
 
General Index 
 
 435 
 
 [Roman figures refer to the chapters, Arabic to the articles.'] 
 
 Poles (of the celestial sphere), i. 
 
 8, 9, 10 ; ii. 33, 35; iv. 78; vi. 
 
 129; x. 207, 214; xin. 285 
 Poles (of the earth), iv. 82 ; ix. 
 
 187; x. 220, 221 ; xin. 285 
 Pole-star, i. 8, 9 
 Pollux, xn. 266 
 Pons-Brooks comet, xin. 291 
 Postulates (of Ptolemy), n. 47 
 Postulates (of Coppernicus), iv. 
 
 76 
 
 Praesepe, xn. 260 
 Precession (of the equinoxes), 
 
 ii. 42, 50 ; in. 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; xin. 277, 280 
 Prima Narralio (of Rheticus), 
 
 iv. 74 ; v. 94 
 Primum Mobile, in. 68 
 Principia (of Descartes), vni. 163 
 Principia (of Newton), iv. 75 ; vni. 
 
 152; ix. 164, 177-192, 195 ; x. 
 
 196, 199, 200, 213; xi. 229, 234, 
 
 235. 240 
 Principles of Philosophy (of 
 
 Descartes), vni. 163 
 Probabi life's, The'orie Analytijuc 
 
 des (of Laplace), xi. 238 
 Problem of three bodies. Sec 
 
 Three bodies, problem of 
 Prodromus Cometicus (of Hevel), 
 
 vni. 153 
 
 Prominences, xin. 301, 302, 303 
 Proper motion (of stars), x. 203, 
 
 225 ; xn. 257, 265 ; xin. 278, 280 
 Prosneusis, n. 48 ; in. 60 ; iv. 85 
 Prussian 'tables^ v. 94, 96, 97, 
 
 99; vn. 139 
 Pythagoreans, 11. 24 ; iv. 75 
 
 Quadrant, v. 99 ; x. 218, 225 n 
 Quadrature, n. 48; in. 60; v. in 
 Quadruium, in. 65 
 
 R cherches sur diffe'rens points (of 
 D'Alembert), xi. 233, 235 
 
 Rccherches sur la precession (of 
 
 D'Alembert), xi. 215 
 Red uc ion of observations, x. 
 
 198, 218; xm. 277 
 Reflecting telescopes, ix. 168 ; 
 
 xii. 251-255 
 Refracting telescopes, ix. 168. 
 
 See also Telescopes 
 Refraction, n. 46 ; in. 68 ; v. 98, 
 
 110: vii. 138; vni 159, 160; 
 
 x. 217, 218, 223; xin 277 
 Reiative motion, principle of, 
 
 iv. 77; ix. 1 86 n 
 Renaissance, iv. 70 
 Results of Astronomical Observa- 
 tions (of John HerschelJ, xin. 
 
 308 
 
 Retrograde motion, i. 14 
 Reversing stratum, xin. 303 
 Reviews of the heavens, xn. 252, 
 
 2 53 
 
 Revival of Learning, iv. 70 
 Rri(_a, vni. 160 
 Rigel, in. 64 
 Rigiit ascension, n. 33, 39; x. 
 
 198, 218 ; xin. 276 
 Rills, xin. 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. 1 66, 174, 177, 
 
 191, 192 ; x. 201, 202, 206, 208 ; 
 
 xn. 254, 256, 259, 263; xm. 
 
 292, 308 
 Ritdolphine Tables, v. 94 ; vn 
 
 148, 151 ; vni. 156 
 Ruler, i. 16 
 Running down of the solar 
 
 system, xin. 293, 319 
 
 Sagiatore (of Galilei), vi. 127 
 Sappho, xin. 281 
 
436 
 
 General Index 
 
 [Roman figures refer to the chapters Arabic to the articles.] 
 
 Saros, i. 17 ; n. 43 
 
 Satellites, vi. 121, 127, 129, 133 ; 
 vn. 145, 150; vni. 154, 160, 
 162; ix. 170, 183-185; x. 210, 
 216; xi. 228, 248; xn. 253, 
 255, 267 ; xm. 272, 283, 295, 
 296, 297, 318, 320. See also 
 Jupiter, Saturn, etc., satellites 
 of 
 
 Satellites, direction of revolution 
 of, xi. 250 ; xm. 295, 318 
 
 Satellites, rotation of, xi. 250 ; 
 xn. 267 ; xm. 297 
 
 Saturn, i. 14-16; n. 25, 51; iv. 
 81, 87; v. 99; vi. 123; vn. 
 136 n, 142, 144; vm. 154, 156; 
 ix. 183, 185, 186; x. 204; xi. 
 228, 231, 235, 236, 243-246; 
 xii. 253, 267; xm. 288, 297. 
 See also the following headings 
 
 Saturn, mass of, ix. 185 
 
 Saturn, rings of, vi. 123 ; vni. 
 154, 160 ; xi. 228, 248 ; xn. 267; 
 xm. 295, 297 
 
 Saturn, rotation of, xn. 267 ; xm. 
 297 
 
 Saturn, satellites of, vni. 154, 
 160 ; ix. 184 ; xi. 228 ; xn. 253, 
 255, 267 ; xm. 295, 297, 307 
 
 Scientific method, n. 54 ; vi. 134 ; 
 ix. 195 
 
 Seas (on the moon), vi. 1 19 ; vm. 
 153; xm. 296 
 
 Seasons, i. 3 ; n. 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 ; xm. 287 
 
 Secular inequalities. See In- 
 equalities, secular 
 
 Selenographia (of Hevel), vm. 
 
 153 
 Sele notopograph ische Fragtn ente 
 
 (of Schroeter), xn. 271 
 Sequences, method of, xn. 266 
 Shadow of earth, moon. See 
 
 Eclipses 
 
 " Shining-fluid " theory, xn. 260; 
 
 xm. 310, 311 
 
 Shooting stars. See Meteors 
 Short-period comets, xm. 291 
 Sidereal month, 11. 40 
 Sidereal period, iv. 86, 87 
 Sidereal system, structure of, xii. 
 
 257, 258, 259-262; xm. 317 
 Sidereal year, 11. 42 
 Sidereus Nuncius (of Galilei), vi. 
 
 119-122 
 "Sights," v. no; vm. 155 x. 
 
 198 
 
 Signs of the zodiac, i. 13 
 Sine, ii. 47 n ; m. 59 n, 68 n 
 Sirius, xm. 316 n 
 Solar eclipse. See Eclipse 
 Solar system, stability of, xi. 245 ; 
 
 xm. 288, 293 
 
 Solstices, i. 11 ; 11. 36, 39, 42 
 Solstitial points, i. 1 1 
 Space-penetrating power, xn. 
 
 258 
 
 Spanish astronomy, in. 61, 66 
 Spectroscope, xm. 299. See also 
 
 Spectrum analysis 
 Spectrum, spectrum analysis, ix. 
 
 168; xm. 273, 299-302, 303, 
 
 304,306,309,311-314,317,318 
 Sphaera Mundi (of Sacrobosco), 
 
 m. 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, n. 23; vn. 
 
 144 
 Spherical form of the earth, moon. 
 
 See Earth, Moon, shape of 
 Spherics, 11. 33, 34 
 Spica, ii. 42 
 
 Spiral nebulae, xm. 310 
 Stability of the solar system, xi. 
 
 245 ; xm. 288, 293 
 
General Index 
 
 437 
 
 [Roman figures refer to the chapters, Arabic to the articles] 
 
 Stadium, n. 36, 45, 47 
 
 Star-atlases, star-maps, I. 12 ; 
 x. 198, 223 ; xn. 259, 266; xin. 
 280, 294 
 
 Star-catalogues, n. 32, 42, 50; 
 in. 62, 63 ; iv. 83 ; v. 98. 107, 
 1 10, 112; vin. 153 ; x. 198, 199, 
 205, 218, 223-225 ; xn. 257 ; 
 xm. 277, 280, 316 
 
 Star-clusters, vi. 120; x. 223; 
 xii. 258, 259, 260, 261 ; xni. 
 37, 38, 3*, 3H,3i8 
 
 Star-gauging, xn. 258 ; xin. 307 
 
 Star-groups. See Constellations 
 
 Stars, i. I, 5, 7-10, 12-15, 18; 
 20, 23-26, 29, 30, 32, 33, 39, 40, 
 42, 45-47, 50; m. 56, 57, 62, 
 68 ; iv. 73, 78, 80, 86, 89, 92 ; 
 v. 96-100, 104, 105, no; vi. 120, 
 121, 129; vin. 155, 157, 161 ; 
 ix. 1 86 n ; x. 198, 199, 203, 207- 
 214, 218, 223 ; xi. 228 ; xii. 253, 
 257-266, 267 ; xm. 272, 277- 
 280, 283, 304, 306-318, 320. 
 See also the preceding and fol- 
 lowing headings 
 
 Stars, binary. See Stars, double 
 and multiple 
 
 Stars, brightness of, n. 42 ; xn. 
 258, 266; xm. 278, 280, 316, 
 3 1 7. See also Stars, variable 
 
 Stars, circumpolar, i. 9 ; n. 35 
 
 Stars, colours of, xii. 263 ; xm. 
 
 39 
 Stars, distances of, i. 7 ; n. 30, 32, 
 
 45, 47 ; iv. 80, 92 ; v. 100 ; vi. 
 
 117, 129; xi. 228; xn. 257, 
 
 258, 265, 266 ; xm. 278, 279, 
 
 317. See also Parallax, stellar 
 Stars, distribution of, xn. 257, 
 
 258. See also Sidereal system, 
 
 structure of 
 Stars, double and multiple, xn. 
 
 256, 263, 264; xm. 306-308, 
 
 3u9, 314, 320 
 Stars, magnitudes of, n. 42 ; xn. 
 
 266 ; xm. 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 ; in. 64 
 Stars, nebulous, x. 223 ; xn. 260, 
 
 261 
 Stars, new, n. 42 ; v. 100, 104 ; 
 
 vi. 117, 129; vn. 138 ; xn. 266; 
 
 xm. 312 
 
 Stars, number of, i. 7 n ; xm. 280 
 Stars, parallax of. See Parallax, 
 
 stellar 
 Stars, proper motion of, x. 203, 
 
 225; xn. 257, 265; xm. 278, 
 
 280 
 
 Stars, rotation of, xn. 266 
 Stars, spectra of, xm. 311-314, 
 
 3 ! 7 
 
 Stars, system of. See Sidereal 
 system, structure of 
 
 Stars, variable, xn. 266, 269; 
 xiii. 307, 312, 314, 315 
 
 Stationary points, i. 14; n. 51; 
 iv. 88 
 
 Stjerneborg, v. IOI 
 
 Summer solstice, i. n. See also 
 Solstices 
 
 Sun, i. i, 4, 10, 13, 14, 16; 11. 21, 
 23-26, 28-30, 32, 35, 36, 40, 
 43. 45,48, 51; i". 68,69; iv. 
 73, 75, 77, 79-82, 85-90, 92; 
 v. 98, 103, 105, no, in; vi. 
 119, 121, 123, 124, 126, 127, 
 129, 132; vn. 136, 139-141, 
 144-146, 150; vin. 153, 154, 
 156; ix. 170, 172-175, *8i, 
 183-186, 188-190, 194; x. 198, 
 
 2OO, 2O2, 2O5, 2IO, 213, 223, 
 
 227 ; xi. 228, 235, 236, 240, 243, 
 245, 250; xn. 257, 265, 268, 269; 
 xm. 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, 11. 46 
 
438 
 
 General Index 
 
 [Roman figures refer to the chapters, Arabic to the articles.] 
 
 Sun, distance of, i. 15 ; n. 24, 25, 
 30,32,38,41,43,45,48,49,51; 
 iv. 81, 85, 86, 87, 90, 92; v. 
 in; vii. 144, 145; vni. 156, 
 161; ix. 185, 188; x. 202, 205, 
 223, 227 ; xi. 235 ; xm. 278, 
 281-284 
 
 Sun, eclipses of. See Eclipses 
 
 Sun, heat of, xn. 268, 269 ; xin. 
 303, 307, 319 
 
 Sun, mass of, ix. 183, 184, 185, 
 189; xi. 228; xin. 282 
 
 Sun, motion of, i. 3, 5, 8, 10, 11, 
 J 3> I 5~ I 7J n - 2O ) 2I > 24-26, 35, 
 37, 88,39,40,42,43,47,48,51 ; 
 in. 59 ; iv. 73, 77, 79, 85, 86, 87, 
 
 92; V. 104, 105, III; VI. 121, 
 126, 127, 132; vni. 1 60; ix. 
 186 ; x. 223 ; xi. 235 ; xn. 265 ; 
 
 XIII. 288 
 
 Sun, parallax of, 11. 43 ; v. 98, 
 
 1 10 ; vn. 145 ; vni. 161 ; x. 223, 
 
 227; xin. 281-284. See also 
 
 Sun, distance of 
 Sun, rotation of, vi. 124; vii. 
 
 150 ; xi. 250 ; xm. 297, 298, 302 
 Sun, size of, n. 32 ; iv. 85 ; vn. 
 
 145 ; ix. 173; xin. 319 
 Sun, tables of. See Tables, solar 
 Sun-dials, n. 34 
 Sun-spots, vi. 124, 125; vni. 153; 
 
 xn. 268, 269; xm. 298, 300, 
 
 32, 303 
 Superior planets, i. 15 ; iv. 87, 
 
 88. See also Mars, Jupiter, etc. 
 Svea, xm. 294 
 Synodic month, n. 40, 43 
 Synodic period, iv. 86, 87 
 Synopsis of Cometary Astronomy 
 
 (of Halley), x. 200 
 Sy sterna Saturnium (of Huygens), 
 
 vni. 154 
 Systeme du Monde (of Laplace), 
 
 xi. 238, 242 , 250 
 Systeme du Monde (of Ponte- 
 
 coulant), xm. 286 
 
 Table Talk (of Luther), iv. 73 
 
 Tables, astronomical, in. 58, 60- 
 
 63, 66, 68 ; iv. 70 ; v. 94, 96, 
 
 97, 99, no; vn. 139,148; vni. 
 
 156, 160; x. 216, 217 ; xm. 277. 
 
 See also the following headings 
 Tables, lunar, n. 48; in. 59; x. 
 
 204, 216, 217, 226 ; xi. 233, 234, 
 
 241 ; xin. 286, 290 
 Tables, planetary, in. 63 ; v. 108, 
 
 1 12; vn. 142, 143; x. 204, 216; 
 
 xi. 235, 247; xin. 288, 289, 290 
 Tables, solar, in. 59; iv. 85; 
 
 v. in ; viii. 153 ; x. 224, 225, 
 
 226 ; xi. 235, 247 ; xin. 290 
 Tables, Alfonsine, in. 66, 68; 
 
 v. 94, 96, 99 
 
 Tables, Hakemite, m. 60, 62 
 Tables, Ilkhanic. in. 62 
 Tables, Prussian, v. 94, 96, 97, 
 
 99; vii. 139 
 Tables, Rudolphine, v. 94; vn. 
 
 148, 151 ; vni. 156 
 Tables, Toletan, in. 61, 66 
 Tab:es delaLune (of Damoiseau;, 
 
 xin. 286 
 Tabulae Regiomontanae (of Bes- 
 
 sel), xin. 277 
 Tangent, in. 59 //, 68 n 
 Tartars, Tartar astronomy, in. 63 
 Tebbutt's comet, xin. 305 
 Telescope, HI. 67 ; vi. 118-124, 
 
 134; vn. 138; vni. 152-155; 
 
 ix. 168; x. 207, 213, 218; xii. 
 
 251, 252-258, 260, 262,271; 
 
 xm. 274, 300, 301, 306, 310, 
 
 317 
 Theoria Motus (of Gauss), xin. 
 
 276 
 Theoria MotuutnLunae(of Euler), 
 
 xi. 233 
 The'orie de la Lune (of Clairaut), 
 
 xi. 233 
 Theorie . . . des Probabilites (of 
 
 Laplace), xi. 238 
 Theorie . . . du Systeme du Monde 
 
 (of Pontecculant), xin. 286 
 Theory of the Moon (of Mayer), 
 
 x. 226 
 
General Index 
 
 439 
 
 [Roman figures refer to the chapters^ Arabic to the articles.] 
 
 Theory of the Universe (of Wright), 
 
 xii. 258 
 
 Thetis, vin. 160 
 Three bodies, problem of, xi. 228, 
 
 230-233, 235 
 Tidal friction, xm. 287, 292, 293, 
 
 320 
 Tides, vi. 130; vn. 150; ix. 189; 
 
 xi. 228-230, 235, 248 ; xin. 287, 
 
 292, 293, 297, 320 
 Time, measurement of, i. 4, 5, 16. 
 
 See also Calendar, Day, Hour, 
 
 Month, Week, Year 
 Titan, vin. 154 
 Titania, xn. 255 
 Tolelan Tables, in. 61, 66 
 Torrid zones, n. 35 
 Total eclipse, n. 43 ; vn. 145 ; 
 
 x. 205 ; xm. 301. See also 
 
 Eclipses 
 
 Transit instrument, x. 218, 225 n 
 Transit of Mercury, x. 199 
 Transit of Venus, vin. 156; x. 
 
 202, 205, 224, 227; xm. 281, 
 
 282, 284 
 Translations, HI. 56, 58, 60, 62, 
 
 66, 68 
 
 Transversals, v. lion 
 Trepidation, in. 58, 62, 68 ; iv. 
 
 84; v. 112 
 Trigonometry, 11. 37 , 47 n ; in. 
 
 59 n, 64 n, 68 n ; iv. 74 
 Trivium, in. 65 
 Tropical year, 11. 42 
 Tuttle's comet, xm. 291 
 Twilight, in. 69 
 Twinkling of stars, n. 30 
 Two New Sciences (of Galilei), 
 
 vi. 133, i34; vin. 152 
 Tychonic system, v. 105; vi. 127 
 
 Umbra (of sun-spots), vi. 124 ; 
 
 xii. 268 
 Uniform acceleration, vi. 133. 
 
 See also Acceleration 
 Uraniborg, v. IOI 
 Uranotnetria Nova Oxoniensis, 
 
 xm. 316 
 
 Uranus, xii. 253, 254, 255, 267 ; 
 
 xm. 276, 288, 289, 297 
 Uranus, rotation of, xm. 297 
 Uranus, satellites of, xi. 250 n ; 
 
 xii. 255, 267 ; xin. 272, 295 
 
 Variable stars. See Stars, vari- 
 able 
 
 Variation (of the moon), in. 60 ; 
 v. Ill ; vn. 145 
 
 Variation of elements or para- 
 meters, xi. 233 n, 236, 245 
 
 Variations, calculus of, xi. 237 n 
 
 Vega, in. 64 
 
 Venus, i. 14-16; ii. 25, 26, 45, 
 47, 51; in. 68; iv. 75, 81, 86, 
 87; v. 98, 100, 103; vi. 121, 
 123; vn. 136 n, 139, 142, 144; 
 vin. 154; ix. 181, 185; x. 223, 
 227 ; xi. 235, 245 ; xii. 267, 
 27 1 ; xin. 282, 297. See also 
 the following headings 
 
 Venus, mass of, xi. 235, 248 
 
 Venus, phases of, vi. 123, 129 
 
 Venus, rotation of, vin. 160; xii. 
 267 ; xin. 297 
 
 Venus, transits of. See Transits 
 of Venus 
 
 Vernal equinox, i. n. See also 
 Equinoxes 
 
 Vernier, in. 69 n 
 
 Vertical, n. 33; x. 221 ; xm. 
 285 
 
 Vesta, xin. 294 
 
 Victoria, xin. 281 
 
 Virtual velocities, xi. 237 n 
 
 Vortices, vin. 163; ix. 178, 
 195 
 
 Wave, wave-length (of light) 
 xin. 299, 300, 302 
 
 Weather, prediction of, u. 20 
 vn. 136 
 
 Week, i. 1 6 
 
 Weight, vi. 1 1 6, 130; ix. 180 
 
 Weights and Measures, Com- 
 mission on, xi. 237, 238 
 
440 
 
 General Index 
 
 \Roman figures refer to the chapters, Arabic to the articles .] 
 
 Whetstone of Witte (of Recorde), 
 
 v. 95 
 
 Winter solstice, i. II. See also 
 Solstices 
 
 Year, i. 3, 4, 16; n. 19-22, 42, 
 
 47 ; in. 66 ; v. 1 1 1 
 Year, sidereal, n. 42 
 Year, tropical, n. 42 
 
 ZadkieVs Almanack, i. 18 n 
 Zenith, n. 33, 35, 36, 46 ; in. 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, n. 35, 47 
 
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